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arXiv:physics/0002031v1 [physics.chem-ph] 17 Feb 2000Path-integral Monte Carlo Simulations without the Sign Pro blem: Multilevel Blocking Approach for Effective Actions R. Egger1, L. M¨ uhlbacher1,2, and C.H. Mak2 1Fakult¨ at f¨ ur Physik, Albert-Ludwigs-Universit¨ at, D-7 9104 Freiburg, Germany 2Department of Chemistry, University of Southern Californi a, Los Angeles, CA 90089-0482 (Date: February 2, 2008) The multilevel blocking algorithm recently proposed as a possible solution to the sign problem in path-integral Mont e Carlo simulations has been extended to systems with long- ranged interactions along the Trotter direction. As an appl i- cation, new results for the real-time quantum dynamics of th e spin-boson model are presented. PACS numbers: 02.70.Lq, 05.30.-d, 05.40.+j I. INTRODUCTION Path-integral Monte Carlo (PIMC) simulations are useful for extracting exact results on many-body quan- tum systems [1]. In principle, PIMC methods can be used to study both equilibrium as well as dynamical problems. But in the cases of fermions and real-time dynamics, PIMC suffers from the notorious “sign prob- lem” which renders such simulations unstable. This sign problem manifests itself as an exponential decay of the signal-to-noise ratio for large systems or long real times [2–4]. Its origin is at the heart of quantum mechanics it- self, namely the interference of different quantum paths contributing to the path integral might be destructive due to exchange effects or due to the oscillatory nature of the real-time evolution operator. Besides approximate treatments [2] the sign problem has remained unsolved. Very recently, a new strategy has been proposed as a possible approach to a complete solution of the sign prob- lem. This so-called multi-level blocking (MLB) algorithm [5,6] is a systematic implementation of the simple block- ing idea — by sampling “blocks” instead of single paths, one can always reduce the sign problem [7]. Defining a suitable hierarchy of blocks by grouping them into dif- ferent “levels”, crucial information about the phase can- cellations among different quantum paths can then be recursively transferred from the bottom to the top level. Given sufficient computer memory, such an approach was shown to be able to eliminate the sign problem in a sta- ble and exact manner [5]. But to date, the MLB algo- rithm has only been formulated to solve the sign problem in PIMC simulations with nearest-neighbor interactions along the Trotter direction. This situation is encountered under a direct Trotter-Suzuki breakup of the short-time propagator. In this paper, we report an extension of the MLB ap-proach to the case of effective actions that may include arbitrarily long-ranged interactions. Such effective ac- tions that are non-local in Trotter time may arise from degrees of freedoms having been traced out, e.g., a har- monic heat bath [8], or through a Hubbard-Stratonovich transformation, e.g., in auxiliary-field MC simulations of lattice fermions [3]. Remarkably, because such effective actions capture much of the physics, e.g., symmetries or the dissipative influence of the traced-out degrees of free- dom, the corresponding path integral very often exhibits a significantly reduced “intrinsic” sign problem compared to the original (time-local) formulation. The present gen- eralization of the MLB algorithm was developed to take advantage of this fact. We note that in a PIMC simu- lation with only nearest-neighbor interactions along the Trotter direction, the original MLB approach [5] is more efficient than the method described below, which there- fore should be used only for time-non-local actions. To be specific, we focus on the dynamical sign problem arising in real-time PIMC computations here. The mod- ifications required to implement the method for fermion simulations are then straightforward. The structure of this paper is as follows. In Sec. II the general strategy to deal with long-ranged interactions in a MLB scheme is outlined. A detailed exposition of the computational method can be found in Sec. III. We have studied the real-time dynamics of the celebrated spin-boson system [8] using this approach. Details about this application, performance issues related to the sign problem, and nu- merical results are presented in Sec. IV. Finally, Sec. V offers some conclusions. II. GENERAL CONSIDERATIONS We consider a discretized path integral along a certain contour in the complex-time plane. In a typical real- time calculation, there is a forward branch from t= 0 tot=t∗, where t∗is the maximum time studied in the simulation, followed by a branch going back to the ori- gin, and then by an imaginary-time branch from t= 0 tot=−i¯hβ. We focus on a “factorized” initial prepara- tion where the relevant degrees of freedom, denoted by r(t), are held fixed for t <0 [8,9]. That implies that the imaginary-time dynamics must be frozen at the cor- responding value, and we only need to sample on the two real-time branches. Note that such a nonequilibrium cal- 1culation cannot proceed in a standard way by first doing an imaginary-time QMC simulation followed by analytic continuation of the numerical data [1]. The quantum numbers r(t) at a given time may be discrete or contin- uous variables. Using time slices of length t∗/P, we combine for- ward [ r(tm)] and backward [ r′(tm)] path configurations at time tm=mt∗/Pinto the configuration sm, where m= 1, . . ., P . The configuration at t= 0 is held fixed, and for t=t∗we must be in a diagonal state, r(t∗) =r′(t∗). For an efficient application of the cur- rent method, it is essential to combine several neighbor- ing slices minto new “blocks”. For instance, think of m= 1, . . . ,5 as a new “slice” ℓ= 1, m= 6, . . .,10 as another slice ℓ= 2, and so on. Combining qele- mentary slices into a block sℓ, instead of the original P slices we have L=P/qblocks, where Lis the number of MLB “levels”. In actual applications, there is consid- erable freedom in how these blocks are defined, e.g. if there is hardly any intrinsic sign problem, or if there are only few variables in r, one may choose larger values of q. Additional flexibility can be gained by choosing different qfor different blocks. Say we are interested in sampling the configurations sLon the top level ℓ=Laccording to the appropriate matrix elements of the (reduced) density matrix, ρ(sL) =Z−1/summationdisplay s1,...,sL−1exp{−S[s1, . . . ,sL]},(2.1) where Sis the effective action under study and Zis a normalization constant so that /summationdisplay sLρ(sL) = 1. (2.2) Due to the time-non-locality of this action, there will be interactions among all blocks sℓ. The sum in Eq. (2.1) denotes either an integration over continuous degrees of freedom or a discrete sum. In the case of interest here, the effective action is complex-valued and e−S/|e−S|rep- resents an oscillatory phase factor ( ±1 for the fermion sign problem). The “naive approach” to the sign prob- lem is to sample configurations using the positive definite weight function P ∼ | exp{−S}|, (2.3) and to include the oscillatory phase in the accumulation procedure. Precisely this leads to the exponentially fast decay of the signal-to-noise ratio with t∗. The proposed MLB simulation scheme starts by sam- pling on the finest level ℓ= 1, so only variables in the first block corresponding to m= 1, . . . , q are updated. During this procedure, interference among different paths will take place. Since only relatively few degrees of freedom are sampled, however, the resulting interference informa- tion can be quantified in a controlled way by employingso-called “level- ℓbonds” (here ℓ= 1). As long as qis chosen sufficiently small, the interference cannot lead to numerical instabilities, and the sign cancellations occur - ing while sampling on level ℓ= 1 can thus be synthesized and transferred to the level ℓ= 2, where the sampling is carried out next. Here the procedure is repeated, and by proceeding recursively up to the top level ℓ=L, this strategy can eliminate the sign problem. The main bot- tleneck of the method comes from the immense memory requirements, since one needs to store and update the level-ℓbonds on all levels during the Monte Carlo sam- pling (see below for details). To summarize, the main idea of our approach is to subdivide the allowed inter- ferences among the quantum paths into small subunits (blocks) such that no sign problem occurs when (stochas- tically) summing over the paths within each subunit. The basic observation underlying our method is therefore al- most trivial: The sign problem does not occur in a suffi- ciently small system. The nontrivial computational task then consists of bringing together the interference sig- nals from different blocks, which is done by recursively forming blocks on subsequent higher levels. Instead of the “circular” structure of the time con- tour inherent in the trace operation, it is actually more helpful to view the problem as a linear chain, where the proposed MLB scheme proceeds from left to right. In the case of local actions with only nearest-neighbor inter- actions along Trotter time, a different recursion scheme was implemented in Refs. [5,6] which is close in spirit to the usual block-spin transformations used in renormal- ization group treatments of spin chains. For both MLB implementations, however, the underlying blocking idea is identical, and the non-locality of the effective action studied here only requires one to abandon block-spin-like transformations in favor of the “moving-along-the-chain” picture. Below we assume that one can decompose the effective action according to S[s1, . . . ,sL] =L/summationdisplay ℓ=1Wℓ[sℓ, . . .,sL]. (2.4) All dependence on a configuration sℓis then contained in the “partial actions” Wλwithλ≤ℓ. One could, of course, put all Wℓ>1= 0, but the approach becomes more powerful if a nontrivial decomposition is possible. III. MULTILEVEL BLOCKING APPROACH In the following, we describe in detail how the MLB algorithm for effective actions is implemented in prac- tice. The MC sampling starts on the finest level ℓ= 1, where only the configuration sℓ=1containing the elemen- tary slices m= 1, . . ., q will be updated with all sℓ>1re- 2maining fixed at their initial values s0 ℓ. Using the weight function P0[s1] =|exp{−W1[s1,s0 2, . . . ,s0 L]}|, we generate Ksamples s(i) 1, where i= 1, . . ., K , and store them for later use. To effectively solve the sign problem and to avoid a bias in the algorithm, the sample number Kshould be chosen large enough, see below and Ref. [5]. For K= 1, the algorithm simply reproduces the naive approach. The stored samples are now employed to generate in- formation about the sign cancellations. All knowledge about the interference that occured at this level is en- capsulated in the quantity B1=/angbracketleftbiggexp{−W1[s1, . . . ,sL]} |exp{−W1[s1,s0 2, . . . ,s0 L]}|/angbracketrightbigg P0[s1](3.1) =C−1 0/summationdisplay s1exp{−W1[s1, . . . ,sL]} =K−1K/summationdisplay i=1exp{−W1[s(i) 1,s2, . . .,sL]} |exp{−W1[s(i) 1,s0 2, . . .,s0 L]}| =B1[s2, . . .,sL], which we call “level-1 bond” in analogy to Ref. [5], with the normalization constant C0=/summationtext s1P0[s1]. The third line follows by noting that the s(i) 1were generated ac- cording to the weight P0. This equality requires that K is sufficiently large and that qis sufficiently small in or- der to provide a good statistical estimate of the level-1 bond. Combining the second expression in Eq. (3.1) with Eq. (2.1), we rewrite the density matrix in the follow- ing way: ρ(sL) =Z−1/summationdisplay s2,...,sL−1exp/braceleftBigg −/summationdisplay ℓ>1Wℓ/bracerightBigg C0B1(3.2) =Z−1/summationdisplay s1,...,sL−1P0B1/productdisplay ℓ>1e−Wℓ. When comparing Eq. (3.2) with Eq. (2.1), we see that the entire sign problem has now formally been trans- ferred to levels ℓ >1, since oscillatory phase factors only arise when sampling on these higher levels. Note that B1=B1[s2, . . .,sL] introduces couplings among alllev- elsℓ >1, in addition to the ones already contained in the effective action S. We now proceed to the next level ℓ= 2 and, according to Eq. (3.2), update configurations for m=q+ 1, . . .,2q using the weight P1[s2] =|B1[s2,s0 3, . . . ,s0 L] exp{−W2[s2,s0 3, . . . ,s0 L]}|. (3.3)Under the move s2→s′ 2, we should then resample and update the level-1 bonds, B1→B′ 1. Exploiting the fact that the stored Ksamples s(i) 1are correctly distributed for the original configuration s0 2, the updated bond can be computed according to B′ 1=K−1K/summationdisplay i=1exp{−W1[s(i) 1,s′ 2, . . . ,sL]} |exp{−W1[s(i) 1,s0 2, . . .,s0 L]}|.(3.4) Again, to obtain an accurate estimate for B′ 1, the number Kshould be sufficiently large. In the end, sampling under the weight P1implies that the probability for accepting the move s2→s′ 2under the Metropolis algorithm is p=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationtext iexp{−W1[s(i) 1,s′ 2,s0 3,...]} |exp{−W1[s(i) 1,s0 2,...]}| /summationtext iexp{−W1[s(i) 1,s2,s0 3,...]} |exp{−W1[s(i) 1,s0 2,...]}|/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle×/vextendsingle/vextendsingle/vextendsingle/vextendsingleexp{−W2[s′ 2,s0 3, . . .]} exp{−W2[s2,s0 3, . . .]}/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (3.5) Using this method, we generate Ksamples s(i) 2, store them, and compute the level-2 bonds, B2=/angbracketleftbiggB1[s2,s3, . . .] exp{−W2[s2,s3, . . .]} |B1[s2,s0 3, . . .] exp{−W2[s2,s0 3, . . .]}|/angbracketrightbigg P1[s2](3.6) =C−1 1/summationdisplay s2B1[s2, . . .] exp{−W2[s2, . . .]} =K−1K/summationdisplay i=1B1[s(i) 2,s3, . . .] exp{−W2[s(i) 2,s3, . . .]} |B1[s(i) 2,s0 3, . . .] exp{−W2[s(i) 2,s0 3, . . .]}| =B2[s3, . . . ,sL], withC1=/summationtext s2P1[s2]. Following our above strategy, we then rewrite the reduced density matrix by combining Eq. (3.2) and the second line of Eq. (3.6). This yields ρ(sL) =Z−1/summationdisplay s3,...,sL−1exp/braceleftBigg −/summationdisplay ℓ>2Wℓ/bracerightBigg C0C1B2(3.7) =Z−1/summationdisplay s1,...,sL−1P0P1B2/productdisplay ℓ>2e−Wℓ. Clearly, the sign problem has been transferred one block further to the right along the chain. Note that the nor- malization constants C0, C1, . . .depend only on the initial configuration s0 ℓso that their precise values need not be known. This procedure is now iterated in a recursive manner. Sampling on level ℓusing the weight function Pℓ−1[sℓ] =|Bℓ−1[sℓ,s0 ℓ+1, . . .] exp{−Wℓ[sℓ,s0 ℓ+1, . . .]}| (3.8) requires the recursive update of all bonds Bλwithλ < ℓ. Starting with B1→B′ 1and putting B0= 1, this recursive update is done according to 3B′ λ=K−1(3.9) ×K/summationdisplay i=1B′ λ−1[s(i) λ,sλ+1, . . .] exp{−W′ λ[s(i) λ,sλ+1, . . .]} |Bλ−1[s(i) λ,s0 λ+1, . . .] exp{−Wλ[s(i) λ,s0 λ+1, . . .]}|, where the primed bonds or partial actions depend on s′ ℓ and the unprimed ones on s0 ℓ. Iterating this to get the updated bonds Bℓ−2for all s(i) ℓ−1, the test move sℓ→s′ ℓ is then accepted or rejected according to the probability p=/vextendsingle/vextendsingle/vextendsingle/vextendsingleBℓ−1[s′ ℓ,s0 ℓ+1, . . .] exp{−Wℓ[s′ ℓ,s0 ℓ+1, . . .]} Bℓ−1[sℓ,s0 ℓ+1, . . .] exp{−Wℓ[sℓ,s0 ℓ+1, . . .]}/vextendsingle/vextendsingle/vextendsingle/vextendsingle.(3.10) On this level, we again generate Ksamples s(i) ℓ, store them and compute the level- ℓbonds according to Bℓ[sℓ+1, . . .] =K−1(3.11) ×K/summationdisplay i=1Bℓ−1[s(i) ℓ,sℓ+1, . . .] exp{−Wℓ[s(i) ℓ,sℓ+1, . . .]} |Bℓ−1[s(i) ℓ,s0 ℓ+1, . . .] exp{−Wℓ[s(i) ℓ,s0 ℓ+1, . . .]}|. This process is iterated up to the top level, where the observables of interest may be computed. Since the sampling of Bℓrequires the resampling of all lower-level bonds, the memory and CPU requirements of the algorithm laid out here are quite large. For λ < ℓ−1, one needs to update Bλ→B′ λfor all s(i) ℓ′withλ < ℓ′< ℓ, which implies a tremendous amount of computer mem- ory and CPU time, scaling approximately ∼KLat the top level. Fortunately, an enormous simplification can often be achieved by exploiting the fact that the inter- actions among distant slices are usually weaker than be- tween near-by slices. For instance, when updating level ℓ= 3, the correlations with the configurations s(i) 1may be very weak, and instead of summing over all Ksamples s(i) 1in the update of the bonds Bλ<ℓ, we may select only a small subset. When invoking this argument, one should be careful to also check that the additional interactions coming from the level- λbonds with λ < ℓ are sufficiently short-ranged. From the definition of these bonds, this is to be expected though. Remarkably, this algorithm can significantly relieve the severity of the sign problem. Let us first give a simple qualitative argument supporting this statement for the original MLB method of Ref. [5], where P= 2Lwith Ldenoting the number of levels. If one needs Ksam- ples for each slice on a given level in order to have sat- isfactory statistics despite of the sign problem, the total number of paths needed in the naive approach depends exponentially on P, namely ∼KP. This is precisely the well-known exponential severity of the sign problem un- der the naive approach. However, with MLB the work on the last level [which is the only one affected by a sign problem provided Kwas chosen sufficiently large] is only ∼KL. So in MLB, the work needed to sample theKPpaths with satisfactory statistical accuracy grows∼Klog2P=Plog2K, i.e., only algebraically with P. Pro- vided the interactions along the Trotter time decay suffi- ciently fast, a similar qualitative argument can be given for the generalized MLB algorithm proposed here. For the application described below, we have indeed found only algebraic dependences of the required CPU times and memory resources with the maximum real time t∗, instead of exponential ones as encountered in the naive approach. Further details of the simulation procedure are provided in the next section. IV. APPLICATION: SPIN-BOSON DYNAMICS To demonstrate this MLB algorithm for path integral simulations with long-range interactions in the Trotter direction, we study the real-time dynamics of the spin- boson model, H=−(¯h∆/2)σx+ (¯hǫ/2)σz (4.1) +/summationdisplay α/bracketleftBigg p2 α 2mα+1 2mαω2 α/parenleftbigg xα−cα mαω2ασz/parenrightbigg2/bracketrightBigg . This archetypical model has a number of important ap- plications, e.g., the Kondo problem, interstitial tunnel- ing in solids [8], quantum computing [10], and electron transfer reactions [11], to mention only a few. The bare two-level system (TLS) has a tunneling matrix element ∆ and the asymmetry (bias) ǫbetween the two localized energy levels ( σxandσzare Pauli matrices). Dissipation is introduced via a linear heat bath, i.e., an arbitrary col- lection of harmonic oscillators {xα}bilinearly coupled to σz. Concerning the TLS dynamics, all information about the coupling to the bath is contained in the spectral den- sityJ(ω) = (π/2)/summationtext α(c2 α/mαωα)δ(ω−ωα), which has a quasi-continuous form in typical condensed-phase appli- cations. J(ω) dictates the form of the (twice-integrated) bath correlation function ( β= 1/kBT), Q(t) =/integraldisplay∞ 0dω π¯hJ(ω) ω2cosh[ω¯hβ/2]−cosh[ω(¯hβ/2−it)] sinh[ω¯hβ/2]. (4.2) For the calculations here, we assume an ohmic spectral density of the form J(ω) = 2π¯hαωexp(−ω/ω c), for which Q(t) can be found in closed form [7]. Here ωcis a cutoff frequency, and the damping strength is measured by the dimensionless Kondo parameter α. In the scaling limit ∆≪ωc, and assuming α <1, all dependence on ωc enters via a renormalized tunnel splitting [8] ∆eff= [cos( πα)Γ(1−2α)]1/2(1−α)(∆/ωc)α/(1−α)∆, (4.3) and powerful analytical [8,12] and alternative numerical methods [13,14] are available for computing the nonequi- librium dynamics. 4At this point some remarks are in order. Basically all other published numerical methods except real-time PIMC can deal only with equilibrium quantities, see, e.g., Refs. [15,16], or explicitly introduce approxima- tions [13,14,17,18]. Regarding the latter class, mostly Markovian-type approximations concerning the time- range of the interactions introduced by the influence func- tional have been implemented. Our approach is computa- tionally more expensive than other methods [13–18], but at the same time it is unique in yielding numerically exact results for the nonequilibrium spin-boson dynamics for arbitrary bath spectral densities. It is particularly valu - able away from the scaling regime where important appli- cations, e.g., coherent (nonequilibrium) electron transf er reactions in the adiabatic regime, are found but basically all other methods fail to yield exact results. Finally we briefly compare the present approach to our previously published PIMC method [7]. For not exceedingly small α, it turns out that the latter method is just equivalent to the K= 1 limit of the present method. From Table I and the discussion below, it is thus apparent that MLB is significantly more powerful in allowing for a study of much longer real times than previously. We study the quantity P(t) = /an}bracketle{tσz(t)/an}bracketri}htunder the nonequilibrium initial preparation σz(t < 0) = +1. P(t) gives the time-dependent difference of the quantum- mechanical occupation probabilities of the left and right states, with the particle initially confined to the left stat e. To obtain P(t) numerically, we take the discretized path- integral representation of Ref. [7] and trace out the bath to get a long-ranged effective action, the “influence func- tional”. In discretized form the TLS path is represented by spins σi, σ′ i=±1 on the forward- and backward-paths, respectively. The total action Sconsists of three terms. First, there is the “free” action S0determined by the bare TLS propagator U0, exp(−S0) =P−1/productdisplay i=0U0(σi+1, σi;t∗/P)U0(σ′ i+1, σ′ i;−t∗/P). (4.4) The second is the influence functional, SI=S(1) I+S(2) I, which contains the long-ranged interaction among the spins, S(1) I=/summationdisplay j≥m(σj−σ′ j)/braceleftBig L′ j−m(σm−σ′ m) (4.5) +iL′′ j−m(σm+σ′ m)/bracerightBig , where Lj=L′ j+iL′′ jis given by [7] Lj= [Q((j+ 1)t∗/P) +Q((j−1)t∗/P)−2Q(jt∗/P)]/4 (4.6) forj >0, and L0=Q(t∗/P)/4. In the scaling regime atT= 0, this effective action has interactions ∼α/t2between the spins (“inverse-square Ising model”). The contribution S(2) I=i(t∗/P)/summationdisplay mγ(mt∗/P)(σm−σ′ m) (4.7) gives the interaction with the imaginary-time branch [where σz= +1], where the damping kernel γ(t) =2 π¯h/integraldisplay∞ 0dωJ(ω) ωcos(ωt). (4.8) For clarity, we focus on the most difficult case of an un- biased two-state system at zero temperature, ǫ=T= 0. To ensure that the Trotter error is negligibly small, we have systematically increased Pfor fixed t∗until conver- gence was reached. Typical CPU time requirements per 104MC samples are 4 hours for P= 26, L= 2, K= 1000, or 6 hours for P= 40, L= 3, K= 600, where the simula- tions were carried out on SGI Octane workstations. The memory requirements for these two cases are 60 Mbyte and 160 Mbyte, respectively. Data were collected from several 105samples. Forα= 0, the bare TLS dynamics P(t) = cos(∆ t) is accurately reproduced. As mentioned before, the perfor- mance is slightly inferior to the original MLB approach [6] which is now applicable due to the absence of the in- fluence functional and the associated long-ranged inter- actions. Turning to the situation where a bath is present, we first study the case α= 1/2 and ωc/∆ = 6. The exact α= 1/2 result [8], P(t) = exp( −∆efft), valid in the scal- ing regime ωc/∆≫1, was accurately reproduced, indi- cating that the scaling regime is reached already for mod- erately large ωc/∆. Typical parameters used in the MLB simulations and the respective average sign are listed in Table I. The first line in Table I corresponds to the naive approach. For α= 1/2, it turns out that our previous PIMC scheme [7] yields a comparable performance to the K= 1 version of this MLB method. It is then clear from Table I that the average sign and hence the signal-to- noise ratio can be dramatically improved thus allowing for a study of significantly longer timescales t∗than be- fore. For a fixed number of levels L, the average sign grows by increasing the parameter K. Alternatively, for fixed K, the average sign increases with L. Evidently, the latter procedure is more efficient in curing the sign problem, but at the same time computationally more ex- pensive. In practice, it is then necessary to find a suitable compromise. Figure 1 shows scaling curves for P(t) atα= 1/4 for ωc/∆ = 6 and ωc/∆ = 1. According to the α= 1/2 results, ωc/∆ = 6 is expected to be within the scaling regime. This is confirmed by a comparison to the non- interacting blip approximation (NIBA) [8]. The minor deviations of the NIBA curve from the exact result are in accordance with Refs. [7,12] for α≤1/2. However, forωc/∆ = 1, scaling concepts (and also NIBA) are ex- 5pected to fail even qualitatively. Clearly, the MLB re- sults show that away from the scaling region, quantum coherence is able to persist for much longer, and both frequency and decay rate of the oscillations differ signif- icantly from the predictions of NIBA. In electron trans- fer reactions in the adiabatic-to-nonadiabatic crossover regime, such coherence effects can then strongly influence the low-temperature dynamics. One obvious and impor- tant consequence of these coherence effects is the break- down of a rate description, implying that theories based on an imaginary-time formalism might not be appropri- ate in this regime. A detailed analysis of this crossover regime using MLB is currently in progress. V. CONCLUSIONS In this paper, we have extended the multilevel block- ing (MLB) approach of Refs. [5,6] to path-integral Monte Carlo simulations with long-ranged effective ac- tions along the Trotter direction. For clarity, we have focussed on real-time simulations here, but believe that a similar approach can also be helpful in many-fermion computations, e.g., in auxiliary-field fermion simulation s of lattice fermions. The practical usefulness of the ap- proach was demonstrated by computing the nonequi- librium real-time dynamics of the dissipative two-state system. Here the effective action (influence functional) arises by integrating out the linear heat bath. For a heat bath of the ohmic type, at T= 0 the corresponding in- teractions among different time slices decay only with a slow inverse-square power law. In the present implementation of MLB, the basic block- ing idea operates on multiple time scales by carrying out a subsequent sampling at longer and longer times. Dur- ing this procedure, the interference information collecte d at shorter times is taken fully into account without in- voking any approximation. Under such an approach, at the expense of large memory requirements, the severity of the sign problem can be significantly relieved. The proposed approach allows to study time scales not acces- sible to previous real-time path-integral simulations for the spin-boson system. ACKNOWLEDGMENTS We wish to thank M. Dikovsky and J. Stockburger for useful discussions. This research has been supported by the Volkswagen-Stiftung, by the National Science Foun- dation under Grants No. CHE-9257094 and No. CHE- 9528121, by the Sloan Foundation, and by the Dreyfus Foundation.[1] See, e.g., Quantum Monte Carlo Methods in Condensed Matter Physics, edited by M. Suzuki (World Scientific, Singapore, 1993), and references therein. [2] D.M. Ceperley and B.J. Alder, Science 231, 555 (1986). [3] E.Y. Loh, Jr., J. Gubernatis, R.T. Scalettar, S.R. White , D.J. Scalapino, and R.L. Sugar, Phys. Rev. B 41, 9301 (1990). [4] D. Thirumalai and B.J. Berne, Annu. Rev. Phys. Chem. 37, 401 (1986). [5] C.H. Mak, R. Egger, and H. Weber-Gottschick, Phys. Rev. Lett. 81, 4533 (1998). [6] C.H. Mak and R. Egger, J. Chem. Phys. 110, 12 (1999). [7] R. Egger and C.H. Mak, Phys. Rev. B 50, 15 210 (1994). For a review, see C.H. Mak and R. Egger, Adv. Chem. Phys.93, 39 (1996). [8] A.J. Leggett, S. Chakravarty, A.T. Dorsey, M.P.A. Fisher, A. Garg, and W. Zwerger, Rev. Mod. Phys. 57, 1 (1987); U. Weiss, Quantum Dissipative Systems (World Scientific, Singapore, 1993), and references therein. [9] The calculation of thermal correlation functions is pos si- ble after minor modifications. [10] A. Garg, Phys. Rev. Lett. 77, 964 (1996). [11] D. Chandler, in Liquids, Freezing, and the Glass Tran- sition, Les Houches Lectures, ed. by D. Levesque et al. (Elsevier Science, 1991). [12] F. Lesage and H. Saleur, Phys. Rev. Lett. 80, 4370 (1998) [13] J. Stockburger and C.H. Mak, Phys. Rev. Lett. 80, 2657 (1998). [14] D. Makarov and N. Makri, Chem. Phys. Lett. 221, 482 (1994). [15] T.A. Costi and C. Kieffer, Phys. Rev. Lett. 76, 1683 (1996); T.A. Costi, ibid.80, 1038 (1998). [16] K. V¨ olker, Phys. Rev. B 58, 1862 (1998). [17] M. Winterstetter and W. Domcke, Chem. Phys. Lett. 236, 445 (1995). [18] H. Wang, X. Song, D. Chandler, and W.H. Miller, J. Chem. Phys. 110, 4828 (1999). 0 2 4 6 8 10 efft-1-0.500.51 6FIG. 1. Scaling curves for P(t) forα= 1/4 with ωc/∆ = 6 (closed diamonds) and ωc/∆ = 1 (open circles). The solid curve is the NIBA prediction. The approach of Ref. [7] be- comes unstable for ∆ efft >4 in both cases. Statistical errors are of the order of the symbol sizes. TABLE I. MLB performance for α= 1/2,ωc/∆ = 6, ∆t∗= 10, P= 40, and several L.qℓdenotes the number of slices for ℓ= 1, . . . L. K L q ℓ /angbracketleftsgn/angbracketright 1 1 40 0.03 200 2 30 - 10 0.14 800 2 30 - 10 0.20 200 3 22 - 12 - 6 0.39 600 3 22 - 12 - 6 0.45 7

arXiv:physics/0002032v1 [physics.plasm-ph] 17 Feb 2000Equilibrium orbit analysis in a free-electron laser with a c oaxial wiggler B. Maraghechia,b, B. Farrokhia, J. E. Willettc, and U.-H. Hwangd aInstitute for Studies in Theoretical Physics and Mathemati cs, P.O. Box 19395-5531, Tehran, Iran. bDepartment of Physics, Amir Kabir University, Tehran, Iran . cDepartment of Physics and Astronomy, University of Missouri-Columbia, Columbia, Missouri 65211. dPhysics Department, Korea University of Technology and Edu cation, Chunan, Choongnam 330-860, Korea PACs number(s): 41.60.cr, 52.75.Ms Abstract An analysis of single-electron orbits in combined coaxial w iggler and axial guide magnetic fields is presented. Solutions of the equatio ns of motion are developed in a form convenient for computing orbital veloci ty components and trajectories in the radially dependent wiggler. Simple ana lytical solutions are obtained in the radially-uniform-wiggler approximation a nd a formula for the derivative of the axial velocity v/bardblwith respect to Lorentz factor γis derived. Results of numerical computations are presented and the cha racteristics of the equilibrium orbits are discussed. The third spatial harmon ic of the coaxial wiggler field gives rise to group IIIorbits which are characterized by a strong negative mass regime. 1I. INTRODUCTION Most free-electron lasers employ a wiggler with either a hel ically symmetric magnetic field generated by bifilar current windings or a linearly symm etric magnetic field generated by alternating stacks of permanent magnets. A uniform stati c guide magnetic field is also frequently employed. Single-particle orbits in these heli cal and planar fields combined with an axial guide field have been analyzed in detail and have play ed a role in the development of free-electron lasers [1]. Harmonics of gyroresonance for o ff-axis electrons caused by the radial variation of the magnetic field of a helical wiggler is found b y Chu and Lin [2]. Recently the feasibility of using a coaxial wiggler in a free-electron la ser has been investigated. Freund et al. [3,4] studied the performance of a coaxial hybrid iron wiggl er consisting of a central rod and a coaxial ring of alternating ferrite and dielectric spacers inserted in a uniform static axial magnetic field. McDermott et al.[5] proposed the use of a wiggler consisting of a coaxial periodic permanent magnet and transmission line. Coaxial d evices offer the possibility of generating higher power than conventional free-electron l asers and with a reduction in the beam energy required to generate radiation of a given wavele ngth. In the present paper, single-particle orbits in a coaxial wi ggler are studied. The wiggler magnetic field is radially dependent with the fundamental pl us the third spatial harmonic component and a uniform static axial magnetic field present. In Sec. II the scalar equations of motion are introduced and reduced to a form which is correc t to first order in the wiggler field. In Sec. III solutions of the equations of motion are dev eloped in a form suitable for computing the electron orbital velocity and trajectory in t he radially dependent magnetic field of a coaxial wiggler. The special case of a radially inde pendent wiggler is also analyzed. In Sec. IV the results of numerical computations of the wiggl er field components, velocity components, radial excursions, and the Φ function for locat ing negative mass regimes are presented and discussed. In Sec. V some conclusions are pres ented. 2II. EQUATIONS OF MOTION Electron motions in a static magnetic field Bmay be determined by solution of the vector equation of motion dv dt=−e γmcv×B (1) where v,−e, and mare the velocity, charge, and (rest) mass, respectively, of the electron. Lorentz factor γis a constant given by γ= (1−v2/c2)−1/2(2) where v=|v|is the constant electron speed. The total magnetic field inside a coaxial wiggler will be take n to be of the form B=Brˆr+Bzˆz, (3) Br=BwFr(r, z), (4) Bz=B0+BwFz(r, z), (5) where B0is a uniform static axial guide field, and FrandFzare known functions of cylindrical coordinates randz. Equation (1) may be written in the scalar form dvr dt−v2 θ r=−vθ(Ω0+ Ω wFz), (6) dvθ dt+vθvr r=vr(Ω0+ Ω wFz)−vzΩwFr, (7) dvz dt=vθΩwFr; (8) Ω0and Ω ware relativistic cyclotron frequencies given by Ω0=eB0 γmc, (9) Ωw=eBw γmc. (10) Initial conditions will be chosen such that the transverse m otion of the electron in the B0field vanishes in the limit as Bwapproaches zero. Then, in order to develop a solution to first order in the wiggler field Bw, the scalar equations of motion will be approximated by 3dvr dt=−Ω0vθ, (11) dvθ dt= Ω0vr−v/bardblΩwFr, (12) dvz dt= 0. (13) with the wiggler field approximated by the fundamental plus t he third spatial harmonic component, Fr=Fr1sin(kwz) +Fr3sin(3kwz), (14) where Frn≡G−1 n[SnI1(nkwr) +TnK1(nkwr)], (15) Gn≡I0(nkwRout)K0(nkwRin)−I0(nkwRin)K0(nkwRout), (16) Sn≡2 nπsin(nπ 2)[K0(nkwRin) +K0(nkwRout)], (17) Tn≡2 nπsin(nπ 2)[I0(nkwRin) +I0(nkwRout)], (18) andn= 1,3;RinandRoutare the inner and outer radii of the coaxial waveguide, kw= 2π/λw where λwis the wiggler (spatial) period, and I0,I1,K0, andK1are modified Bessel functions. III. ORBITAL ANALYSIS A. Radially dependent wiggler The scalar equations of motion may be solved to determine the electron orbital velocity and trajectory in a coaxial wiggler. Equation (13) yields vz=v/bardbl (19) where the constant v/bardblis the root-mean-square axial velocity component. With the initial axial position taken to be z0= 0, z=v/bardblt. (20) 4Equations (11), (12), (14), and (20) may be combined to obtai n d2vr dt2+ Ω2 0vr=f(t) (21) where f(t) = Ω 0Ωwv/bardbl[Fr1sin(kwv/bardblt) +Fr3sin(3kwv/bardblt)]. (22) By the method of variation of parameters, a solution of Eq. (2 1) may be obtained in the form vr= [−vθ0+ Ω−1 0/integraldisplayt 0f(τ)cos(Ω0τ)dτ]sin(Ω0t) +[vr0−Ω−1 0/integraldisplayt 0f(τ)sin(Ω0τ)dτ]cos(Ω0t) (23) where vr0andvθ0are initial radial and azimuthal velocity components. Then Eq. (11) yields vθ= [vθ0−Ω−1 0/integraldisplayt 0f(τ)cos(Ω0τ)dτ]cos(Ω0t) +[vr0−Ω−1 0/integraldisplayt 0f(τ)sin(Ω0τ)dτ]sin(Ω0t). (24) The orbital velocity is given to first order in Bwby Eqs. (23), (24), and (19). The trajectory may then be computed using r=r0+/integraldisplayt 0vr(τ)dτ, (25) θ=θ0+/integraldisplayt 0vθ(τ)dτ, (26) and Eq. (20). B. Radially uniform wiggler By neglecting the radial variation of Fr1andFr3, a solution of Eq. (21) may be obtained in the form vr=α1sin(kwv/bardblt) +α3sin(3kwv/bardblt), (27) where 5αn=Ω0Ωwv/bardblFrn Ω2 0−n2k2 wv2 /bardbl(n= 1,3). (28) Equation (11) then yields vθ=−Ω−1 0kwv/bardblα1cos(kwv/bardblt)−Ω−1 0(3kwv/bardblα3)cos(3kwv/bardblt). (29) The corresponding initial conditions are vr0= 0, (30) vθ0=−Ω−1 0kwv/bardblα1−Ω−1 0(3kwv/bardblα3). (31) Root-mean-square values of the velocity components may be d etermined by use of Eqs. (27), (28), and (19). Replacing v2by its root-mean-square value in Eq. (2) then yields v2 /bardbl c2[1 +1 2(α1 v/bardbl)2+1 2Ω−2 0k2 wα2 1+1 2(α3 v/bardbl)2+9 2Ω−2 0k2 wα2 3] = 1−γ−2. (32) The derivative of v/bardblwith respect to γmay be obtained from Eq. (32) and, after some algebra, cast into the form dv/bardbl dγ=c2 γγ2 /bardblv/bardblΦ (33) where Φ = 1 −/summationtext n=1,3(Ω2 0−n2k2 wv2 /bardbl)−3γ2 /bardblΩ2 wF2 rnΩ2 0(Ω2 0+ 3n2k2 wv2 /bardbl) 2 +/summationtext n=1,3(Ω2 0−n2k2wv2 /bardbl)−3Ω2wF2rnΩ2 0(Ω2 0+ 3n2k2wv2 /bardbl). (34) This equation may be used to establish the existence of a nega tive mass regime. IV. NUMERICAL RESULTS A numerical computation is conducted to investigate the pro perties of the equilibrium orbits of electrons inside a coaxial wiggler. Wiggler wavel ength 2 π/kwand lab-frame electron density n0were taken to be 3 cm and 1012cm−3, respectively. The wiggler magnetic field Bw was taken to be 3745 G which corresponds to the relativistic w iggler frequency Ω w/ckw= 60.442. Electron-beam energy ( γ−1)m0c2was taken to be 700 keV, corresponding to a Lorentz factor γ= 2.37. The axial magnetic field B0was varied from 0 to 25.3 kG, corresponding to a variation from 0 to 3 in the normalized relativistic cyclot ron frequency Ω 0/ckwassociated withB0. The inner and outer radii of the coaxial wiggler were assume d to be Rin= 1.5 cm andRout= 3 cm, respectively. Figure 1 shows the variation of the axial velocity of the quas i-steady-state orbits with the axial guide magnetic field for three classes of solutions . Group I orbits for which 0 < Ω0< k wv/bardbl, group II orbits with kwv/bardbl<Ω0<3kwv/bardbl, and group III orbits with Ω 0> 3kwv/bardbl. Existence of group III orbits is due to the presence of the th ird spatial harmonic of the wiggler field, which also produces the second magnetores onance at Ω 0≈3kwv/bardbl. The narrow width of the second resonance at Ω 0/ckw≈2.7 compared to the width of the first magnetoresonance at Ω 0≈kwv/bardblis illustrated in Fig. 1B. This is due to the relatively weak third harmonic compared to the fundamental component of the wiggler field. It should be noted that although the exact resonances Ω 0=kwv/bardbland Ω 0= 3kwv/bardbloccur at the origin where v/bardbl/c= Ω0/ckw= 0, the first ”magnetoresonance” in the literature refers to the group II orbits with the cyclotron frequencies around Ω 0/ckw≈1 in Fig. 1. Similarly we refer to the group III orbits with the cyclotron frequency around Ω 0/ckw≈2.7 as the second magnetoresonance. The rate of change of the electron axial velocity with electr on energy is proportional to Φ and is equal to unity in the absence of the wiggler field. Figur e 2 illustrates the dependence ofΦon the radial wiggler magnetic field and the axial guide magne tic field B0. The curves corresponding to the group I and II orbits are almost unaffect ed by the third harmonic and are almost the same as in Ref. [6] where the third harmonic is n eglected. A negative mass regime (i.e., negative Φ for which a decrease in the axial vel ocity results in an increase in the electron energy) is found for group III orbits which is st ronger than that of the group II orbits. Equations (15)-(18) are used to calculate the radial compon ents of the wiggler field Fr1 andFr3. For the axial component the following expressions are used [4] 7Fz=Fz1cos(kwz) +Fz3cos(3kwz), (35) Fzn=G−1 n[SnI0(nkwr)−TnK0(nkwr)]. (36) Figure 3 shows the variation of the amplitudes of the wiggler magnetic field (divided by Bw= 3745 G) with radius, for the first and third spatial harmonic s. For the first harmonic the radial component, Fr1, has a minimum at r≈2.28 cm and the axial component, Fz1, changes sign around this point. Mc Dermott et al.[5] have demonstrated the stability of a thin annular electron beam when Fr1is minimum at the beam radius. The radial and axial components of the third harmonic of the wiggler Fr3andFz3are also shown in Fig. 3. Magnitudes of Fr3andFz3both are minimum at r≈2.28 cm where Fr1is minimum. This is actually an inflection point for Fz3. Variations of the radial components of the first and third spa tial harmonics of the nor- malized wiggler magnetic field Fr1andFr3with the wiggler wave number kware shown in Fig. 4. Figure 4 also shows the dimensionless transverse vel ocity coefficients ¯ α1=α1/cand ¯α3=α3/cfor the initial orbit radius r0≈2.28 cm where Fr1is minimum. The cyclotron frequency Ω 0/ckw≈2.7 is taken at the second magnetoresonance, and our choice of 3 cm for the wiggler wavelength corresponds to kw≈2.1 cm−1. It can be observed that at this wave number although the radial component of the wiggler fiel d at the first harmonic Fr1 is much larger than the third harmonic Fr3, the transverse velocity coefficients of the third harmonic ¯ α3are larger than ¯ α1. This shows that the third harmonic may have considerable effects around the second magnetoresonance at Ω 0≈3kwv/bardbl. Away from this resonance Eq. (28) shows that α3will be of the order of Fr3. In order to study the transverse motion of electrons in the ra dially dependent wiggler field Eqs. (11), (12), (25), and (26) are solved numerically w ith the initial conditions chosen so that, in the limit of zero wiggler field, there is axial moti on at constant velocity v/bardblbut no Larmor motion. Figure 5 shows the variation of the radial a nd azimuthal components of electron velocity with z(=v/bardblt). The normalized cyclotron frequency Ω 0/ckwis chosen to be 0.5, 1.2, and 3 for group I, II, and III orbits, respectively, which are somewhat away from the 8magnetoresonances. Solid curves correspond to the initial orbit radius r0= 2.28 cm, which is at the point where Fr1is minimum. Broken curves correspond to r0= 1.8 cm, which is away from the Fr1minimum. It can be observed that the spatial periodicity of vrandvθfor the first two groups is equal to one wiggler wavelength, which is the same as that of the first harmonic. Although group III orbits have a clear sinusoidal shape at r0= 2.28 cm (solid curves), the slight deviations from sinusoidal shape are ob vious at r0= 1.8 cm (broken curves). This is because that at r0= 2.28 cm where Fr1is minimum Fr3is very small. Therefore away from the resonance the third harmonic plays a lmost no role, at r0= 2.28 cm. Moving away from Fr1minimum to r0= 1.8 cm, however, increases the magnitude of Fr3slightly making the effect of the third harmonic noticeable o n group III orbits, which are away from the second magnetoresonance, in Fig. 5. Figure 6 shows the variations of vr/candvθ/cwithz(=v/bardblt) for group III orbits when the cyclotron frequency is adjusted at the second magnetoreson ance at Ω 0/ckw≈2.7. At r0= 2.28 cm the periodicity is approximately equal to λw/3, which is the same as that of the third harmonic, and shows the strong influence of the third harmoni c on the transverse velocity components. Going away from the Fr1minimum to r0= 1.8 cm makes the amplitudes of oscillations of vrandvθlarger. Broken curves correspond to the solutions of Eqs. (2 7) and (29) when the radial variation of the wiggler field is neglect ed. These solutions do not differ appreciably from the r-dependent solutions for r0= 2.28 cm because at Fr1minimum the radial excursions are small for group III orbits. Away from t heFr1minimum at r0= 1.8 cm, however, deviations are noticeable due to the larger rad ial excursions. Figure 7 shows vr/cversus zfor group II orbits for r-dependent wiggler (broken curves) and r-independent wiggler (solid curves). The initial orbi t radius is taken at r0= 2.28 cm and the cyclotron frequency is chosen around the first magnet oresonance at Ω 0/ckw= 0.9. Large radial excursions of electrons for group II orbits mak e the transverse velocity strongly affected by the radial dependency of the wiggler field. The amp litude is also modulated in space with the wavelength of around 16 λw= 48 cm. The radial excursion rshown in Fig. 8 corresponds to the cyclotron frequencies awa y 9from the magnetoresonances at Ω 0/ckwequal to 0.5, 1.2, and 3 for the group I, II, and III orbits, respectively. Solid curves correspond to the r0= 2.28 cm and the broken curves correspond to r0= 1.8 cm. It can be noticed that when the electrons are injected in to the wiggler at r0= 2.28 cm, where Fr1is minimum, electron orbits remain well away from the waveguide walls at Rin= 1.5 cm and Rout= 3 cm. Figure 9 compares the radial excursions of group III orbits a t the second magnetores- onance at Ω 0/ckw= 2.7, (solid curve) with those slightly away from the resonance at Ω0/ckw= 3, (broken curve). Influence of the third harmonic can be cle arly seen through the modulation of the third harmonic by the first harmonic whe n the cyclotron frequency is adjusted at the second magnetoresonance. V. CONCLUSIONS The third spatial harmonic of the coaxial wiggler field gives rise to the group III orbits with Ω 0>3kwv/bardbl. This relatively weak third harmonic makes the width of the s econd magnetoresonance narrow compared to the first magnetoreson ance. A strong negative mass regime is found for the group III orbits. By adjusting the cyc lotron frequency at the second magnetoresonance the wiggler induced velocity of the group III orbits was found to increase considerably. When the electrons are injected into the wigg ler where its magnetic field is minimum the electron orbits remain well away from the wavegu ide boundaries. Harmonic gyroresonance of electrons in the combined helica l wiggler and axial guide magnetic field is reported by Chu and Lin [2]. In their analysi s the relativistic single particle equation of motion is used with the axial velocity as well as t he axial magnetic field of the wiggler averaged along the axial direction. By assuming near steady-state orbits for off-axis electrons they found that the radial variation of th e wiggler magnetic field produces a harmonic structure in the transverse force. This force, in turn, comprises oscillations at all harmonics of kwz. It should be noted that there is no harmonic structure in the helical wiggler itself and the higher velocity harmonics vanish for the exact steady-state orbits of 10the on-axis electrons. Moreover, higher harmonics do not ap pear in the one dimensional helical wiggler where the radial variation is neglected. In the present analysis of coaxial wiggler, on the other hand , equation of motion is written to first order in the wiggler amplitude. With this app roximation axial component of the wiggler field has no contribution to the problem leaving t he axial velocity as a constant. Magnetic field of a coaxial wiggler is composed of a fundament al plus a large number of odd spatial harmonics, which directly appear in the magnetic fo rce represented by f(t) in Eq. (22). Third harmonic in f(t) appears in the transverse velocity components as a part of the integrands in Eqs. (23) and (24) and is also demonstrated numerically for the radially dependent coaxial wiggler, but for the radially uniform wig gler the third harmonic is explicit in the solutions Eqs. (27)-(29). 11REFERENCES [1] H. P. Freund and J. M. Antonsen, Jr., Principles of Free-Electron Laser (Chapman and Hall, London, 1996), Chap. 2). [2] K. R. Chu and A. T. Lin, Phys. Rev. E 67, 3235 (1991). [3] H. P. Freund, R. H. Jackson, D. E. Pershing, and J. M. Tacce tti, Phys. Plasmas 1, 1046 (1994). [4] H. P. Freund, M. E. Read, R. H. Jackson, D. E. Pershing, and J. M. Taccetti, Phys. Plasmas 2, 1755 (1995). [5] D. B. McDermott, A. J. Balkcum, R. M. Phillips, and N. C. Lu hmann, Jr, Phys. Plasmas 2, 4332 (1995). [6] J. E. Willett, U. Hwang, Y. Aktas, and H. Mehdian, Phys. Re v. E57, 2262 (1998). 12FIGURES FIG. 1. Normalized axial velocity v/bardbl/cas a function of the normalized axial-guide magnetic field Ω 0/ckwfor group I, II, and III orbits. Narrow width of the second res onance at Ω 0/ckw≈2.7 compared to the width of the first magnetoresonance at Ω 0≈kwv/bardblis illustrated in Fig. 1B. FIG. 2. Factor Φas a function of the normalized axial-guide magnetic field Ω 0/ckwfor group I, II, and III orbits. FIG. 3. Radial dependence of the radial and axial magnetic fie lds (divided by Bw= 3745 G) in the coaxial wiggler for the fundamental and third spatial harmonics. FIG. 4. Wave-number dependence of the radial components of t he normalized wiggler mag- netic field Fr1,Fr3and the dimensionless transverse velocity coefficients ¯ α1, ¯α3. The normalized cyclotron frequency Ω 0/ckw= 2.7 is taken at the second magnetoresonance and r0= 2.28 cm. FIG. 5. Normalized transverse velocity components as a func tion of axial distance zfor the initial orbit radius r0= 2.28 cm (solid curves) and r0= 1.8 cm (dotted curves). The normalized cyclotron frequency Ω 0/ckwis 0.5, 1.2, and 3 for the group I, II, and III orbits, respecti vely. FIG. 6. Normalized transverse velocity components as a func tion of axial distance zfor group III orbits, at the second magnetoresonance Ω 0/ckw= 2.7, solid curves correspond to the radial dependent wiggler and broken curves correspond to the radia l independent wiggler. FIG. 7. Normalized radial velocity as a function of axial dis tance zfor group II orbits at the first magnetoresonance Ω 0/ckw= 0.9 with r0= 2.28 cm. Solid curves correspond to the radial dependent wiggler and broken curves correspond to the radia l independent wiggler. FIG. 8. Radial excursion ras a function of axial distance zforr0= 2.28 cm (solid curves) and r0= 1.8 cm (broken curves) for groups I, II, and III orbits out of res onance at Ω 0/ckw=0.5, 1.2, and 3, respectively. FIG. 9. Radial excursion ras a function of axial distance zfor group III orbits at r0= 2.28 cm. Solid curve corresponds to resonance at Ω 0/ckw= 2.7 and broken curve corresponds to out of resonance at Ω 0/ckw= 3. 130.01.0 0 1 2 3vII / c Ω0 /ckwvII / c 0.910I II III I III IIA B 0.885Fig. 10 1 2 3 Ω0 /ckw-4-20246 ΦI II IIIFig. 2Fr1 Fz1 r (cm)Fr3Fz3 1.5 2 2.5 30.8 0.4 0.0 -0.4 -0.8Fig. 3Fr1 Fr3α1 α3 0.0 0.5 1.0 1.5 2.0−1012 kw (cm-1)Fig. 4Fig. 5 0 1 2 3vr / c vθ / c z (cm)III III III III0.3 0 -0.3 0.35 0 -0.35A Bvθ / c z (cm)0 1 2 300.25 -0.25vr / c0.25 0 -0.25A BIII IIIFig. 6 r0=1.8 cm r0=2.28 cmr0=1.8 cm r0=2.28 cm1 0 -1 0 10 20 30 40 50vr / c z (cm)IIFig. 7III III 0 1 2 32.58 2.48 2.38 2.28 2.18 2.082.1 1.9 1.8 1.72.0 1.6r0=2.28 r0=1.8 r (cm) z (cm)Fig. 80 1 2 32.242.282.30 r (cm) z (cm)IIIFig. 9

arXiv:physics/0002034 18 Feb 2000Feasibility of backward diffraction radiation for non-destructive diagnostics of relativistic charged particle beams. A.P.Potylitsyn, N.A.Potylitsyna Tomsk Polytechnic University, pr. Lenina 30, 634004 Tomsk, Russia Abstract The characteristics of backward diffraction radiation (BDR), i.e. radiation of the charged particle passing through a slit in the tilted screen, has been considered. The technique for non-destructive beam diagnostics based on the measurements of BDR yield for different tilted angles (theta-scan) is suggested. PACS numbers: 41.60.m, 41.75.i Keywords: diffraction radiation, beam diagnostics. Diffraction radiation (DR) is generated when a charged particle rectilinearly moves in the vicinity to a conducting screen (target) in vacuum. As in the case of transition radiation (TR), appearing when a particle crosses a tilted conducting plane, DR characteristics are determined by both the Lorentz – factor of the particle γ and its charge. Also, they do not depend on the mass of the particle. In the paper [1] the exact solution of the Maxwell equations was obtained, describing radiation of charged particle passing above a tilted semi-infinite ideally conducting screen. One of the authors of the present paper showd [2] that for ultrarelativistic particles (γ >>1) the DR characteristics for the geometry considered are quite close to characteristics of TR. Optical transition radiation (OTR) is known to be widely used for charged particle beam diagnostics [3,4]. In the paper [5] M. Castellano considered a possibility of a transverse beam size determination using optical diffraction radiation characteristics when a beam moves through a slit in a perpendicular screen. In the quoted paper it was suggested to measure the DR angular distribution in the vicinity to the initial electron beam direction. In analogy with transition radiation one can talk about “forward diffraction radiation, FDR”. For this case, one should notice that the beam diagnostics is carried out using “backward transition radiation, BTR”, i.e. the radiation from a tilted foil in the vicinity of the specular direction is investigated because in that case the detector can be placed far enough from the beam for soft background conditions. In analogy with BTR one can expect that in case when relativistic particle moves through a slit in a tilted screen, the radiation is generated along the specular direction which can be called as “backward diffraction radiation, BDR” [6]. In the present paper the BDR characteristics of an ultrarelativistic particle when it moves through a slit in an ideally conducting screen has been calculated. Fig. 1 shows the geometry considered and the symbols used. The field strength of DR for the slit /G56/G4F/G4C/G57 /G28→ is considered as a superposition of DR fields from the upper (→ /G58 /G28) and lower (→ /G47 /G28) semi-planes: /G47 /G58 /G4C /G47/G4C /G58 /G56/G4F/G4C/G57 /G48 /G28 /G48 /G28 /G28ϕ ϕ −+ =/G26 /G26 /G26 (1) Here ϕu, ϕd are the phase shifts calculated for the upper and lower slit edges relative to the particle trajectory. The total phase shift ϕ0 can be derived from simple geometrical relations (see Fig.1): ()    −−+== −− =+= βθθθθλπβθθθλπϕϕϕ /G13/G13 /G13/G15 /G14/G13/G13 /G13 /G46/G52/G56/G0C /G46/G52/G56/G0B /G03/G03/G03/G56/G4C/G51/G15/G0C/G46/G52/G56/G0C /G0B/G46/G52/G56/G0B/G15 /G4B /G4B/G44 /G47 /G58 (2)In (2) λ is the DR wavelength and β is the particle velocity. Here and further the system of units /G14=== /G46 /G50 /G21 is used. In the paper [2] it is shown that DR from a relativistic particle is concentrated in the cone /G14/G61−γ near the specular direction. In the coordinate system, where the Z axis coincides with that direction and the X axis is in parallel to the slit edge, from (2) one can derive (up to an accuracy of the terms of /G15−γ): /G5C /G58/G4Bθλπϕ/G14 /G03 /G15−≈ , /G5C /G47/G4Bθλπϕ/G15 /G03 /G15−≈ (3) After substituting in (1) → /G58 /G28 and → /G47 /G28values obtained in the same approximation [2], we have BDR field for the slit:     −+ −+− + ++ ++− −=    −+ −+− + ++ ++− += −− −−−− −− − /G5C /G5B/G5C /G5B /G5C /G5B/G5C /G5B /G56/G4F/G4C/G57 /G5C/G5C /G5B/G5C /G5B /G5C /G5B/G5C /G5B /G5B/G5B/G56/G4F/G4C/G57 /G5B /G4C/G4C/G4B /G4C/G4C/G4B /G3D/G48/G28/G4C/G4C/G4B /G4C/G4C/G4B /G4C/G3D/G48/G28 θθγθθγλπ θθγθθγλπ πθθγθθγλπ θθγθθγλπ θγθ π /G15 /G15/G15 /G15 /G15 /G15 /G15/G15 /G15 /G14 /G15/G0F/G15 /G15/G15 /G15 /G15 /G15 /G15/G15 /G15 /G14 /G15 /G15 /G15/G0F /G15/G48/G5B/G53/G15/G48/G5B/G53 /G17/G0F/G15/G48/G5B/G53/G15/G48/G5B/G53 /G17 (4) In (4) Ze is the particle charge. Expressions (4) coincide with the results of M. L. Ter- Mikaelian [7], obtained for FDR at Z=1. The coincidence of BDR and FDR characteristics is connected with the use of the ideally conducting screen approximation. BDR spectral – angular distribution is simply calculated from (4)         +−   −+ ++− −        + −+   + + +++   +− ==+ =Ω /G5C /G46/G5B /G5C /G5C /G46/G5C /G5B /G5C /G5B/G5B /G46/G5B /G46/G5B /G5C /G5B /G5B/G5B /G46/G5C /G5B /G57 /G57 /G57 /G57 /G57 /G57 /G57 /G57/G57 /G4F /G57 /G4F /G57 /G57 /G57 /G57/G57 /G3D/G28 /G28/G47 /G47/G3A /G47 ωω ωωωω ωω ωω γ παπω /G56/G4C/G51 /G14 /G15 /G46/G52/G56 /G0C /G14 /G0B /G0C /G14 /G0B/G15/G14 /G15 /G48/G5B/G53 /G14 /G15 /G48/G5B/G53 /G0C /G15 /G14 /G0B /G0C /G14 /G0C/G0B /G14 /G0B/G14 /G48/G5B/G53 /G17/G03/G03/G03/G03/G03/G03/G03/G03/G03 /G03/G03/G03/G03/G03/G03/G03/G03/G03/G03 /G03/G03/G03/G03/G03/G03/G03/G03/G03/G03 /G17 /G15 /G15 /G15 /G15 /G15/G15 /G15 /G15 /G15 /G15 /G15/G15 /G15 /G15/G15/G15 /G15 /G15/G15 (5) Here α is the fine structure constant, ()/G15 /G14 /G13 /G56/G4C/G51/G03/G03 /G0F /G03/G03 /G0F/G4B /G4B /G44 /G46 /G5C /G5C /G5B /G5B /G57 /G57+= = = =γ θγϖγθ γθ is the characteristic energy of DR, /G13/G13 /G15 /G14 /G56/G4C/G51/G56/G4C/G51 θθ /G44/G4B/G44 /G4F−= is the relative impact parameter of the particle with respect to the slit center. As in the case of OTR, in the BDR distribution there is a zero minimum at /G13==/G5C /G5B /G57 /G57 when the particle moves through the slit’s center. It is obvious that the optical BDR angular distribution shape is connected with the initial particle beam divergence (the same as in the case of OTR).If the target with a slit is placed on a goniometer and the BDR emitted at the angle θ is detected by means of collimator C, optical filter F and photomultiplier D (see Fig.1), then with changing of the target tilted angle /G13θ one can measure the dependence of BDR yield (theta-scan), which is rather close to BDR angular distribution. Indeed, when target rotates on a value /G13θΔ, specular direction (Z axis) removes on the value /G13 /G15θΔ. Thus, putting /G13 /G15θγΔ→/G5C /G57 in (5) and integrating over the angular variable /G5B /G57 in the range of the collimator aperture /G5BθΔ, we obtain the theta-scan distribution. The energy resolution of the equipment, distribution of the initial beam impact parameters, the beam angular divergence etc. can be taken into account by the fully obvious way. The theta-scan distribution simulation for the beam, which divergence is described by the Gaussian: ()   Δ+Δ − =ΔΔ/G15/G15 /G15 /G15/G14 /G15/G48/G5B/G53 /G0F/G15σ πσ/G5C /G5B /G5C /G5B /G29 (6), has been carried out by the following way. In (6) /G5BΔ, /G5CΔ are the deviation angles of the initial particle from the average direction. For simplification of the calculations the collimator was chosen for /G14 /G14/G03/G03 /G0F− −>>Δ≤Δ γθγθ/G5B /G5C . The convolution of the distributions (5) and (6) with integration over /G5Bθ in the infinite region allows us to obtain a one-dimensional theta-scan distribution for /G46ϖϖ /G18 /G11 /G13= (see Fig.2). As it follows from the figure, for the beam divergence /G14−≥γσ the resulting distribution has a single maximum, instead of the abovementioned minimum at /G13=/G5C /G57. Thus, measuring a similar theta-scan one can obtain the parameter σ describing an isotropic initial beam divergence. For a non-isotropic distribution (i.e. /G5C /G5Bσσ≠) it is necessary either to measure the theta-scan for a narrow aperture (/G14/G61 /G03/G03 /G0F−ΔΔ γθθ/G5C /G5B ) or to measure it during rotation of the target with the slit around a perpendicular axis. When an electron beam the with γ = 2000 moves through a slit with the width a = 0.3mm inclined at the angle /G24/G17/G18/G13=θ, the BDR photon yield in the optical region is only e times less than the OTR photon yield for the same conditions. However, in the latter case beam emittance increases due to multiple scattering in the target, which is absent in the case of DR. Let us consider the DR characteristics for non-relativistic particles ( /G14 /G03/G03 /G0F /G14 /G61<<βγ ). In that case the expression for spectral – angular density can be obtained from the exact result [1]. For non- relativistic particle angular distribution is quite broad and is not determined by particle velocity. Omitting cumbersome derivation, we present the formula for the DR spectrum (neglecting the terms of the order /G15/G61β ):    − =βωαβω/G4B/G3D/G47/G47/G3A /G15/G48/G5B/G53/G15 /G17/G16(7) Here h is the impact parameter. In comparison with the DR from relativistic electrons accelerated nuclear beam emits Z2 times more intensively, however, the factor ()βϖ β /G12 /G15 /G48/G5B/G53 /G4B−⋅ leads to significant suppression of the DR yield in the optical region even for micron-size beams (in that case the impact parameter can also be chosen of a micron size). However, in the infrared region ( /G50/G46/G50 /G03 /G14/G13 /G61λ ) the radiation yield for the beam of protons with () /G13/G11/G1B/G1A /G03/G03/G03 /G15==βγ can be measured if the beam diameter and the impact parameter do not exceed 1mcm. In that case the photon yield in an interval of /G14 /G11 /G13=Δλλ and in the solid angle /G56/G57/G48/G55/G44/G47 /G03 /G14 /G11 /G13=ΔΩ can reach ∼ 10-6 photon/particle,that allows one to discuss the feasibility of using BDR for non-destructive diagnostics of both relativistic electron beams and beams of protons and nuclei. Acknowledgements The authors are thankful to Prof. I.P.Chernov for useful discussions and P.V.Karataev for help in preparing the text of the paper. References 1. A.P. Kazantsev, G.I.Surdutovich, Sov.Phys.Dokl. 7 (1963) 990. 2. Potylitsyn A.P. NIM B 145 (1998) 169-179. 3. L. Wartski, S. Roland, J. Lassale et al., J. Appl. Phys. 46 (1975) 3644. 4. Rule D.W., Fiorito R.B. Beam profiling with optical transition radiation. Proc. 1993 Particle Accelerator Conf., Washington, DC, USA, p.2453 5. M. Castellano, NIM A 394 (1997) 275. 6. Rule D.W., Fiorito R.B., Kimura W.D., Non Interceptive Beam Diagnostics Based on Diffraction Radiation, Proceedings of the Seventh Beam Instrumentation Workshop, Argonne IL, AIP Conference Proceedings 390, NY, 1997. 7. M.L. Ter-Mikaelian, High-Energy Electromagnetic Processes in Condensed Media, Wiley/ Interscience, New York, 1972.θ0e- h1 h 2D C DRF Δθy θ=2θ0+θy Fig.1 Layout of the measurements of BDR yield with changing of the tilted angle /G13θ (theta-scan).Fig.2 Theta-scan distributions for detector placed at the fixed angle /G13 /G15θθ= for beams with different divergence σ.0.6 0.4 0.2 0 2.5 5ω = 0.5 ωc l = 0 σ = 0.5/G14−γ σ = 1/G14−γ σ = 2/G14−γ /G13 /G03θγΔ BDR yield, arb. units

arXiv:physics/0002035v1 [physics.gen-ph] 19 Feb 2000The Emergence of the Planck Scale B.G. Sidharth∗ Centre for Applicable Mathematics & Computer Sciences B.M. Birla Science Centre, Adarsh Nagar, Hyderabad - 500 063 (India) Abstract In this paper we first observe some interesting parallels bet ween Planck scale considerations and elementary particle Compt on wave- length scale considerations, particularly in the context o f Wheeler’s space time foam and a space time arising out of a stochastic ra n- dom heap of elementary particles discussed in previous pape rs. These parallels lead to a semi qualitative picture which shows how the short lived Planck scale arises from the Compton wavelength consi derations. Finally all this is quantified. 1 Introduction About a century ago Max Planck had pointed out that the quanti ty/parenleftBig ¯hG c3/parenrightBig1 2∼ 10−33cmsis a fundamental length. This so called Planck length ties up Quan- tum Mechanics, Gravitation and Special Relativity and lead s to the Planck mass∼10−5gms. It is but natural that the Planck length has played a crucial role in Quantum Gravity as also in String Theory whic h includes a description of Gravitation, unlike Quantum Theory or Quant um Field The- ory. It turns out to be the scale at which we have no longer the smoot h space time of Classical Theory and Quantum Theory, but rather we have th e space time foam of Wheeler[1, 2]. This is inextricably linked with grav itational collapse 0∗Email:birlasc@hd1.vsnl.net.in; birlard@ap.nic.in 1which has been described by Wheeler as ”The greatest crisis o f Physics”. As he puts it, ”These are small scale fluctuations telling one th at something like gravitational collapse is taking place everywhere in space and all the time; that gravitational collapse is in effect perpetually being d one and undone .... at the Planck scale of distances.” In this space time foam, wo rm holes and non local effects abound. On the other hand there is also a stochastic fluctuational pic ture of space time that deals with phenomena at the Compton wavelength sca le and leads to meaningful physics and cosmology including a unified desc ription of grav- itation and electromagnetism consistent with observation [3, 4, 5, 6, 7]. In this picture, space time has been considered to be a random he ap[8] of ele- mentary particles. If we consider a typical elementary part icle to be a pion with Compton wavelength l, then the above picture leads to a dispersion length in the Gaussian distribution ∼√ Nl, N ∼1080being the number of elementary particles in the universe, this being the correc t dimension of the universe itself. We will now show a parallel between the Planck length conside rations and the Compton wavelength considerations referred to above, w hich will then show us how the Planck length considerations emerge. 2 The Emergence of the Planck Scale We first show the parallels between the Compton wavelength pi cture and the Planck length picture. We note that in the former scenari o, particles are fluctuationally created at the Compton wavelength from a background pre space time Zero Point Field (ZPF) of the kind considered i n stochastic electrodynamics[9, ?]. The energy content in terms of the magnetic field of such a particle is given by (Cf.ref.[3]) ∆B∼(¯hc)1/2 L2(1) where Lis the dimension under consideration, which in this case is o f the or- der of the particle’s Compton wavelength. We note that in (1) if ¯hcor equiv- alently 137 e2is replaced by its gravitational counterpart, namely 137 Gm2 2then we get, as in the fluctuation of the metric[1], ∆g∼LP L(2) where LPis the Planck length and Las in (1) is of the order of the dimension under consideration. The space time foam referred to above arises at the Planck sca le because the right hand side in (2) becomes unity, indicating perpetu al collapse and creation. From this point of view, as Wheeler points out our space time i s an approxi- mation, an average swathe at the Planck scale of several prob able spaces and topologies which form the super space (Cf.ref.[2]). There i s an immediately parallel in terms of the Compton wavelength considerations also: As pointed out by Nottale, Abbot-wise, El Naschie, the author and other s[11, 12, 13, 14] the Quantum behaviour below a critical length is fractal and as pointed out by the author[8], our space time is the thick brush stroke of t hickness of the order of the Compton wavelength of a jagged, fractal coastli ne like underpin- ning. In the light of the above considerations the fluctuational cr eation of particles considered by Hayakawa[15] and the author[5] have a paralle l in the non local worm hole related appearance of particles and fields at the Pl anck scale[2]. We will now quantify the above parallels and show the actual e mergence of the Planck scale particles from the Compton wavelength cons iderations. We first observe that in an actual random heap of particles, th e smaller par- ticles (in our case those having smaller Compton wavelength s and therefore higher mass) tend to settle down together due to gravity. In a fluctuationally created random heap of particles, there is no gravity, but as this space time heap is not only non differentiable, but is also not required t o be even a continuum the random motion would have a similar effect: Of th eN′=√ N particles which are less dispersed,√ N′particles would similarly fluctuation- ally, that is non locally be together. This fluctuationally b ound group would have a mass√ N′m∼10−5gmsor the Planck mass, since mis the mass of the pion. (Cf.ref.”Ramification” for another interesting p erspective). One way of looking at this is that in the above scenario, space time no longer has the rigid features of Classical and Quantum Physics - on t he average it is a measure of dispersion of a random distribution of partic les which them- selves have a stochastic underpinning. So the length scale o r dispersion would 3be less, the less dispersed the random collection of particl es is - this leads to the Planck scale from the Compton scale. However it must be borne in mind that a Planck mass has a life time ∼10−42seconds, and can hardly be detected. The Planck scale corresponds to the extreme classical limit of Quantum Me- chanics, as can be immediately seen from the fact that the Pla nck mass mP∼10−5gmscorresponds to a Schwarzchild Black Hole of radius LP∼ 10−33cms, the Planck length. At this stage the spinorial Quantum Mech an- ical feature as brought out by the Kerr-Newman type Black Hol e and the Compton wavelength (Cf.detailed discussion in refs.[3, 4] ) disappears. Infact at the Planck scale we have GmP c2= ¯h/m Pc (3) In (3), the left side gives the Schwarzchild radius while the right side gives the Quantum Mechanical Compton wavelength. Another way of w riting (3) is, Gm2 P e2≈1, (4) Equation (4) expresses the well known fact that at this scale the entire energy is gravitational, rather than electromagnetic, in contras t to equation (1) for a typical elementary particle mass, vi., Gm2≈1√ Ne2∼10−40e2 Interestingly from the background ZPF, Planck particles ca n be produced at the Planck scales given by (3), exactly as in the case of pio ns, as seen earlier. They have been considered to be what may be called a Z ero Point Scale[17, 18, 19]. But these shortlived Planck particles ca n at best describe a space time foam. We will now throw further light on the fact that at the Planck s cale it is gravitation alone that manifests itself. Indeed Rosen[20] has pointed out that one could use a Schrodinger equation with a gravitation al interaction to deduce a mini universe, namely the Planck particle. The Sc hrodinger equation for a self gravitating particle has also been consi dered[21], from a different point of view. We merely quote the main results. 4The energy of such a particle is given by Gm2 L∼2m5G2 ¯h2 (5) where L=¯h2 2m3G(6) (5) and (6) bring out the characteristic of the Planck partic les and also the difference with elementary particles, as we will now see. We first observe that for a Planck mass, (5) gives, self consis tently, Energy = mPc2, while (6) gives, L= 10−33cms, as required. However, the situation for pions is different: They are parts of the universe and do not constitute a mini universe. Indeed, if, as above th ere are Npions in the universe, then the total gravitational energy is give n by, from (5), NGm2 L where now Lstands for the radius of the universe ∼1028cm. As this equals mc2, we get back as can easily be verified, the pion mass! Indeed given the pion mass, one can verify from (6) that L= 1028cmswhich is the radius of the universe, R. Remembering that R≈c H, (6) infact gives back the supposedly mysterious and adhoc Weinberg formula, relating the Hubble constant to the pion mass[22]. This provides a justification for taking a pion as a typical pa rticle of the universe, and not a Planck particle, besides re-emphasizin g the basic unified picture of gravitation and electromagnetism. It must be men tioned that just as the Planck particle constitutes a mini universe or Black H ole, so also the N∼1080pion filled universe can itself considered to be a Black hole[ 23]! To proceed, let us now use the fact that our minimum space time intervals are (lP, τP), the Planck scale, instead of ( l, τ) of the pion, as above. 5With this new limit, it can be easily verified that the total ma ss in the volume ∼l3is given by ρP×l3=M (7) where ρPis the Planck density and Mis the mass of the universe. Moreover the number of Planck masses in the above volume ∼l3can easily be seen to be ∼1060. However, it must be remembered that in the physical time period τ, there are 1020(that isτ τP) Planck life times. In other words the number of Planck particles in the physical interval ( l, τ) isN∼1080, the total particle number, as if all these were the seeds of the fix ed number of Nparticles in the universe. This is symptomatic of the fact th at instead of the elementary particle Compton wavelength scale of the phy sical universe we are using the Planck scale (cf. also considerations befor e equation (3)). That is from the typical physical interval ( l, τ) we recover the entire mass and also the entire number of particles in the universe, as in the Big Bang theory. This also provides the explanation for the above puz zling relations like (7). That is the Big Bang theory is a characterization of the new Co mpton wave- length model in the classical limit at Planck scales, but the n, in this latter case we cannot deduce from theory the relations like the Dira c coincidences or the Weinberg formula. In the spirit of[7], one can now see the semi-classical and Qu antum Mechani- cal divide between Planck particles and elementary particl es in the following way. We will see that Planck particles have a life time given b y the Hawking Radiation Law of Black Hole Thermodynamics, whereas elemen tary particles are characterised by Quantum Mechanical life times. It is well known that[24] the life time due to the Hawking Radi ation Law is given by t=G2m3 ¯hc4(8) which for the Planck particles gives the usual Planck time. However this formulation is not valid for elementary partic les. In this case, we consider the gravitational energy ∆ Eof a pion as given by an equation like (5) and use instead the Quantum Mechanical relation ∆E.∆t∼¯h (9) to get Gm2 π(¯h/m πc)∆t∼¯h (10) 6which is correct if in (9) ∆ t∼1 H, the age of the universe! (cf.also ref.[24])). In this case equation (10) gives the well known and supposedl y mysterious and empirical formula of Weinberg referred to earlier, viz. , m3 π∼H¯h2 Gc(11) One way of looking at this is that it is the emergence of Quantu m Mechanical effects and electromagnetism at the Compton wavelength scal es from classical gravitational considerations at the Planck scale as seen ab ove, which gives stability to the universe as expressed by (9) and (10). All this has been justified from stochastic considerations[ 7]. Another way of looking at all this is the following: The gravi tational constant Gis taken to be a universal constant in most conventional theo ries. However in the above formulation it turns out that, G=G0√ N∝1 T(12) where Nis the number of elementary particles in the universe and Tis the age of the universe. This time varying gravitational constant c an be shown to lead to consistent results including an explanation for the all i mportant precision of the perihelion of the Planet Mercury [6, 25]. The equation (12) also shows a Machian or holistic character. In any case for a single parti cle universe, N= 1 theGabove leads to the Planck length or Planck mass, while for N∼1080the same equation leads to the pion Compton wavelength and the us ual Physics and Cosmology. Infact if the pion Compton time scales ( l, τ) tends to zero or the Planck scale we recover the big bang scenario and the usua l space time of Classical and Quantum Physics or the Prigogine Cosmology [26]. In these cases we cannot explain the large number ”coincidences” and Weinberg’s mysterious formula (11), whereas at the elementary particl e Compton scale these features can be deduced as consequences of the theory. References [1] Misner, C.W., Thorne, K.S., and Wheeler, J.A., (1973), G ravitation, Freeman (San Francisco). 7[2] Wheeler, J.A., (1968) ”Superspace and the Nature of Quan tum Ge- ometrodynamics”, Battelles Rencontres, Lectures, Eds., B .S. De Witt and J.A. Wheeler, Benjamin, New York. [3] Sidharth, B.G., (1998) Int.J. of Mod.Phys.A 13(15), pp2 599ff. [4] Sidharth, B.G., (1998) Gravitation & Cosmology, 4 (2) (1 4), 158ff. [5] Sidharth, B.G., (1998) International Journal of Theore tical Physics, 37 (4), 1307-1312. [6] Sidharth, B.G., ”Effects of Varying G” to appear in Nuovo C imento B. [7] Sidharth, B.G., ”Universe of Chaos and Quanta”, in Chaos , Solitons and Fractals, in press. xxx.lanl.gov.quant-ph: 9902028. [8] Sidharth, B.G., ”Space Time as a Random Heap”, to appear i n Chaos, Solitons and Fractals. [9] De Pena, L., (1983), in ”Stochastic Processes Applied to Physics”, Ed., B. Gomez., World Scientific, Singapore. [10] Haisch, B., Rueda, A., and Puthoff, H.E., (1994), Phys. R ev. A49 (2), pp 678-694. [11] L. Nottale, (1994), Chaos, Solitons & Fractals, 4, 3, 36 1-388 and refer- ences therein. [12] Abbott L.F., and Wise, M.B., (1981), AMJ Phys., 49 , 37-39. [13] El Naschie, M.S., (1993), Vastas Astr. 37 , 249-252. [14] Sidharth, B.G., (1999) ”Dimensionality and Fractals” , Invited contribu- tion to Special Issue of Chaos, Solitons and Fractals on ’Fra ctal Geom- etry in Quantum Physics’. [15] Hayakawa, S., (1965), Suppl of PTP Commemmorative Issu e, 532-541. 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arXiv:physics/0002036v1 [physics.atom-ph] 19 Feb 2000Dielectronic recombination of lithium-like Ni25+ions — high resolution rate coefficients and influence of external crossed E and B fields S. Schippers, T. Bartsch, C. Brandau, A. M¨ uller Institut f¨ ur Kernphysik, Universit¨ at Giessen, 35392 Gie ssen, Germany G. Gwinner, G. Wissler, M. Beutelspacher, M. Grieser, A. Wol f Max-Planck-Institut f¨ ur Kernphysik and Physikalisches I nstitut der Universit¨ at Heidelberg, 69117 Heidelberg, Ge rmany R. A. Phaneuf Department of Physics, University of Nevada, Reno, NV 89557 , USA (December 2, 2013) Absolute dielectronic recombination (DR) rates for lithiu m- like Ni25+(1s22s) ions were measured at high-energy resolu- tion at the Heidelberg heavy-ion storage ring TSR. We stud- ied the center-of-mass energy range 0–130 eV which covers all ∆n=0 core excitations. The influence of external crossed electric (0–300 V/cm) and magnetic (41.8–80.1 mT) fields was investigated. For the measurement at near-zero electric fie ld resonance energies and strengths are given for Rydberg lev- els up to n=32; also Maxwellian plasma rate coefficients for the ∆ n=0 DR at electron temperatures between 0.5 and 200 eV are provided. For increasing electric field strength we fin d that for both the 2 p1/2and the 2 p3/2series of Ni24+(1s22pjnℓ) Rydberg resonances with n >30 the DR rate coefficient in- creases approximately linearly by up to a factor of 1.5. The relative increase due to the applied electric field for Ni25+is remarkably lower than that found in previous measurements with lighter isoelectronic Si11+, Cl14+and also Ti19+ions, [T. Bartsch et al, Phys. Rev. Lett. 79, 2233 (1997); 82, 3779 (1999) and to be published] and in contrast to the results for lighter ions no clear dependence of the electric field enhanc e- ment on the magnetic field strength is found. The Maxwellian plasma rate coefficients for ∆ n=0 DR of Ni25+are enhanced by at most 11% in the presence of the strongest experimen- tally applied fields. 34.80.Lx,32.60.+i,36.20.Kd,52.20.-j I. INTRODUCTION Dielectronic recombination (DR) is an electron-ion col- lision process which is well known to be important in as- trophysical and fusion plasmas [1,2]. In DR the initially free electron is transferred to a bound state of the ion via a doubly excited intermediate state which is formed by an excitation of the core and a simultaneous attachment of the incident electron. This two step process e−+Aq+→[A(q−1)+]∗∗→A(q−1)++hν (1) involves dielectronic capture (time-inverse Auger pro- cess) as the first step with a subsequent stabilization ofthe lowered charge state by radiative decay to a state be- low the ionization limit. This second step competes with autoionization which would transfer the ion back into its initial charge state qwith the net effect being resonant elastic or inelastic electron scattering. Another recombi - nation process, which in contrast to DR is non-resonant, is radiative recombination (RR) e−+Aq+→A(q−1)++hν (2) where the initially free electron is transferred to a bound state of the ion and a photon is emitted simultaneously. The cross section for RR diverges at zero electron en- ergy and decreases rapidly towards higher energies. In the present investigation we regard RR as a continuous background on top of which DR resonances are emerging. In the case of narrow non-overlapping DR resonances the DR cross section due to an intermediate state labelled dcan be well approximated by [3] σd(Ecm) = ¯σdLd(Ecm) (3) with the electron-ion center-of-mass (c. m.) frame en- ergyEcm, the Lorentzian line shape Ld(E) normalized to/integraltext Ld(E)dE= 1 and the resonance strength ¯σd= 4.95×10−30cm2eV2s ×1 Edgd 2giAa(d→i)/summationtext fAr(d→f)/summationtext kAa(d→k) +/summationtext f′Ar(d→f′)(4) where Edis the resonance energy, giandgdare the sta- tistical weights of the initial ionic core iand the doubly excited intermediate state d,Aa(d→i) and Ar(d→f) denote the rate for an autoionizing transition from dto iand the rate for a radiative transition from dto states fbelow the first ionization limit, respectively. The sum- mation indices kandf′run over all states which from d can either be reached by autoionization or by radiative transitions, respectively. Soon after the establishment of DR as an important process governing the charge state balance of ions in the solar corona [4], Burgess and Summers [5] and Jacobs et al. [6] realized that DR cross sections should be sen- sitive to external electric fields present in virtually any 1plasma environment. The electric field enhancement of DR rates was subsequently reproduced in a number of theoretical calculations [7]. Briefly, the effect arises fro m the Stark mixing of ℓstates and the resulting influence on the autoionization rates which, by detailed balancing, de- termine the capture of the free electron. Autoionization rates strongly decrease with increasing ℓand, therefore, only low ℓstates significantly contribute to DR. Electric fields mix low and high ℓstates and thereby increase the autoionization rates of the high ℓstates and consequently also the contribution of high Rydberg states to DR. The control of external fields in experiments using in- tense ion and electron beams is a challenge. Results from early recombination experiments [8,9] could only be brought in agreement with theory under the assumption that external electric fields had been present in the inter- action region. The first experiment where external fields were applied under well controlled conditions was per- formed by M¨ uller and coworkers [10] who investigated DR in the presence of external fields (DRF) of singly charged Mg+ions. They observed an increase of the measured DR cross section by a factor of about 1.5 when increasing the motional /vector v×/vectorBelectric field from 7.2 to 23.5 V/cm. The agreement of these results with theo- retical predictions [11,12] was at the 20% level. Further DRF experiments with multiply charged C3+ions also revealed drastic DR rate enhancements by electric fields [13]; however, the large uncertainties of these measure- ments left ambiguities. The first DRF experiment using highly charged ions at a storage ring was carried out with Si11+ions by Bartsch et al. [14]. It produced results with an unprecedented accuracy, enabling a detailed comparison with theory. Whereas the overall agreement between experiment and theory for the magnitude of the effect (up to a factor of 3 when increasing the field from 0 V/cm to 183 V/cm) was fair, discrepancies remained in the functional de- pendence of the rate enhancement on the electric field strength. This finding stimulated theoretical investiga- tions of the role of the additional magnetic field which is always present in storage ring DR experiments, since it is needed to guide and confine the electron beam within the electron cooler. In a model calculation Robicheaux and Pindzola [15] found that in a configuration of crossed /vectorE and/vectorBfields indeed the magnetic field influences through the mixing of mlevels the rate enhancement generated by the electric field. More detailed calculations [16,17] confirmed these results. It should be noted that in theo- retical calculations by Huber and Bottcher [18] no influ- ence of a pure magnetic field of at least up to 5 T on DR was found. Inspired by these predictions we previously performed storage ring DRF experiments using Li-like Cl14+[19] and Ti19+[20] ions and crossed /vectorEand/vectorBfields where we clearly discovered a distinct effect of the magnetic field strength on the magnitude of the DR rate enhance- ment. The electric field effect decreased monotonicallywith the /vectorBfield increasing from 30 mT to 80 mT. A decrease of the electric field enhancement by a crossed magnetic field is also predicted by the model calculation of Robicheaux and Pindzola [15] for magnetic fields larger than approximately 20 mT where, due to a dominance of the magnetic over the electric interaction energy, the ℓ-mixing weakens and consequently the number of states participating in DR decreases. At lower magnetic fields m-mixing yields an increase of the DR rate with increas- ing/vectorBfield. A corresponding experimental observation has been made recently by Klimenko and coworkers [21] who studied recombination of Ba+ions from a continuum of finite bandwidth which they had prepared by laser ex- citation of neutral Ba atoms. For a given electric field strength of 0.5 V/cm, they find that the recombination rate is increasingly enhanced by crossed magnetic fields up to about 20 mT. However, there is no effect of the magnetic field when it is directed parallel to the electric field vector. For the m-mixing to occur the crossed /vectorE and/vectorBarrangement is essential. In the case of parallel /vectorB and/vectorEfields mremains a good quantum number and no influence of the magnetic field is expected. The aim of the present investigation with Li-like Ni25+ is to extend the previous studies to an ion with even higher nuclear charge Z. Because of the Z4scaling of radiative rates it is expected that with higher Zlessℓ states of a given Rydberg nlevel take part in DR and therefore the sensitivity to ℓ-mixing decreases [10]. Re- sults of Griffin and Pindzola [22] who calculated decreas- ing DR rate enhancements for increasing charge states of iron ions point into the same direction. Another as- pect of going to higher Zis that even high lying 2 pjnℓ Rydberg resonances are more separated in energy and therefore easier to resolve. This has been demonstrated by Brandau et al. who resolved 2 p1/2nℓDR resonances up to n= 41 in the recombination spectrum of Li-like Au76+ions [23]. When choosing Ni25+we hoped to be able to study the field enhancement effect on a single 2pjnℓresonance. This would enable a quantitative com- parison with theory which at present is limited to a single low value of n[16] when explicitly treating all nℓmlevels required for a realistic description of DR in crossed /vectorE and/vectorBfields. In our DRF studies we have chosen Li-like ions as test systems since on the one hand their electronic structure is simple enough to be treated theoretically on a high level of sophistication and on the other hand provides strong DR channels connected to the 2 s→2pjcore excitations. II. EXPERIMENT The measurements have been performed at the heavy ion storage ring TSR [24] of the Max-Planck-Institut f¨ ur Kernphysik in Heidelberg. For a general account of experimental techniques at heavy ion storage rings the reader is referred to an article by M¨ uller and Wolf [25]. 2Recombination measurements at storage rings have been reviewed recently by M¨ uller [26], Schippers [27] and Wolf et al. [28]. Detailed descriptions of the experimental pro- cedure for field free DR measurements have been given by Kilgus et al. [29] and more recently by Lampert et al. [30]. Therefore, we here only describe more explicitly experimental aspects pertaining especially to the present investigation. The58Ni25+ion beam was supplied by the MPI tan- dem booster facility and injected into the TSR with an energy of 343 MeV. Using multiturn injection and e-cool stacking [31] ion currents of up to 3300 µA were stored in the TSR. At these high ion currents, however, intra beam scattering heated the ion beam during DR mea- surements resulting in a considerable loss of energy res- olution. To avoid this and to limit the recombination count rate to below 1 MHz, i. e. to below a count rate where dead time effects are still negligible, we kept ion currents below 1 mA during all DR measurements. In the storage ring the circulating Ni25+ions were merged with the magnetically guided electron beam of the elec- tron cooler. In the present experiment the electron den- sity was 5 .4×106cm−3at cooling energy. Generally the electron density varies with the cathode voltage Uc thereby following a U3/2 cdependence. The distribution of collision velocities in the electron-ion center of mass frame can be described by the anisotropic Maxwellian f(/vector v, vrel) =me 2πkBT⊥exp/parenleftbigg −mev2 ⊥ 2kBT⊥/parenrightbigg ×/bracketleftbiggme 2πkBT||/bracketrightbigg1/2 exp/parenleftbigg −me(v||−vrel)2 2kBT||/parenrightbigg (5) characterized by the longitudinal and transverse temper- atures T||andT⊥. In Eq. (5) meis the electron mass, kBis the Boltzmann constant, and vrelis the detun- ing of the average longitudinal electron velocity from that at cooling, which determines the relative energy Erel≈mev2 rel/2 between the electron and the ion beam. The longitudinal temperature, inferred from the experi- mental resolution for relative energies Erel≫kBT⊥, was kBT||≈0.25 meV. It implies an energy resolution given by ∆E(FWHM)= 4/radicalbig ln(2)kBT||Erel[29]. The longitudi- nal velocity spread of the stored ion beam yields a consid- erable contribution to this temperature, while the veloc- ity spread of the electron beam alone, after acceleration, is estimated to be <0.1 meV. In the transverse direction the electron beam was adiabatically expanded [32] from a diameter dc≈9.5 mm at the cathode to a diameter de= 29.5 mm in the interaction region; the reduction of its transverse velocity spread by this expansion de- termines the low value of the transverse temperature of kBT⊥≈10 meV. Before starting a measurement, the ion beam was cooled for 5 seconds until the beam profiles reached their equilibrium widths. This can be monitored online by employing beam profile monitors based on residual gasionization [33]. The cooled ion beam had a diameter di≈2 mm. During the measurement the electron cooler voltage was stepped through a preset range of values dif- ferent from the cooling voltage, thus introducing non-zero mean relative velocities between ions and electrons. Re- combined Ni24+ions were counted as a function of the cooler voltage with a CsI-scintillation detector [34] lo- cated behind the first dipole magnet downstream of the electron cooler. The dipole magnet bends the circulat- ing Ni25+ion beam onto a closed orbit and separates the recombined Ni24+ions from that orbit. Two different measurement schemes were applied for the measurement of i) a high resolution ‘field-free’ DR spectrum (residual stray electric fields ≤5 V/cm) and ii) DRF spectra with motional electric fields ranging up to 300 V/cm. In view of the result of Huber and Bottcher [18] who calculated that purely magnetic fields below 5 T do not influence DR, the use of the term ‘field-free’ seems justified in case i) even with the magnetic guiding field (up to 80 mT) still present in the electron cooler. A. Procedure for a field free high resolution measurement In between two measurement steps for different values ofErelthe cooler voltage was first set back to the cooling value in order to maintain the ion beam quality and then set to a reference value which is chosen to lead to a rela- tive velocity where the electron-ion recombination signal is very small, favourably being only due to a negligible RR contribution (reference relative energy Eref). Under this condition the recombination rate measured at the reference point monitors the background signal due to electron capture from residual gas molecules. Choosing short time intervals of the order of only 10 ms for dwelling on the measurement, cooling and reference voltages en- sured that the experimental environment did not change significantly between the signal and the background mea- surements. An additional interval of 1.5 ms after each change of the cooler voltage allowed the power supplies to reach the preset values before data taking was started. The electron-ion recombination coefficient α(Erel) =/integraldisplay d3/vector v σ(v)vf(/vector v, vrel) (6) is obtained from the background corrected recombination count rate R(Erel)−R(Eref), the detection efficiency η, the electron density ne, the number of stored ions Ni, the nominal length L= 1.5 m of the interaction zone and the ring circumference C= 55.4 m using the relation α(Erel) =R(Erel)−R(Eref) γ−2 iη ne(Erel)NiL/C+α(Eref)ne(Eref) ne(Erel)(7) where γi= 1 + Ei/(mic2) is the relativistic Lorentz fac- tor for the transformation between the c. m. and the 3laboratory frames where the ions of mass mihave the kinetic energy Ei. The detection efficiency of the CsI- scintillation detector [34] used to detect the recombined ions is very close to unity for count rates up to 2.5 MHz. The second term in Eq. (7) is to be added in case of a non-negligible electron-ion recombination rate at the reference energy. We insert the theoretical RR rate at Eref= 131 .5eVwhich we have calculated to be α(Eref) = 1.39×10−11cm3/s using a semi-classical for- mula for the radiative recombination cross section [35] σRR(Erel) = 2.1×10−22cm2 ×ncut/summationdisplay nminkntnq4R2 nErel(q2R+n2Erel)(8) withRdenoting the Rydberg constant and knbeing cor- rection factors given by Andersen and Bolko [36]. This expression for recombination on bare nuclei is used to approximately describe RR on a lithium-like core by in- troducing the lowest quantum number nmin= 2 and weight factors tnaccounting for partial occupation of n- shells. In our calculation we use t2= 7/8 and tn= 1 for n >2. For the maximum (cut off) quantum number we usencut= 150 as explained below. After the generation of a recombination spectrum from the experimental data via Eq. (7) a correction procedure accounting for non-perfect beam overlap in the merging sections of the cooler is applied [30] which in our case only slightly redistributes the DR resonance strengths, result - ing in DR peaks narrower and taller by small amounts. The systematic uncertainty in the absolute recombina- tion rate coefficient is due to the ion and electron cur- rent determination, the corrections accounting for the merging and demerging sections of the electron and ion beams, and the detection efficiency. It is estimated to be±15% of the measured recombination rate coefficient [30]. The statistical uncertainty of the results presented below amounts to less than 1% of the rate coefficient maximum. B. Procedure for DRF measurements The geometry of the magnetic and electric fields present in the merging section of the electron cooler is sketched in Fig. 1. We choose the z-axis to be defined by the ion beam direction. The magnetic guiding field /vectorB defines the electron beam direction. The field strength B is limited both towards low and high values. Only fields B >25 mT guarantee a reliable operation of the electron cooler. The maximum tolerable current through the gen- erating coils limits Bto at most 80 mT. Correction coils allow the steering of the electron beam in the x-yplane. In the first place these are used to minimize the trans- verse field components BxandBywith respect to the ion beam, such that the two beams are collinear and centered to each other. The collinearity is inferred indirectly frombeam profile measurements of the cooled ion beam [33] with an accuracy of ∼0.2 mrad; i. e. the transverse mag- netic field components caused by imperfections in the beam alignment amounts at most to 2 ×10−4B. Resid- ual fields which may vary in size and direction along the overlap length, are expected to be also of this magni- tude. Since the settings of the various steering magnets result from a rather tedious beam optimization process, they are not exactly reproduced after each optimization procedure that is required e. g. after a change of the magnetic guiding field Bz. This means that the residual transverse magnetic fields for the collinear geometry may also slightly vary from one set of cooler settings to an- other. All uncertainties in the transverse magnetic field translate into an uncertainty in the motional electric field of less than ±10 V/cm in our present experiment. In DRF measurements we offset the current through the correction coils to generate additional magnetic field components BxandBy. Their influence on the stored ion beam is negligible, i. e. the ion beam is still travelling wit h velocity viinz-direction. However, in the frame of the ion beam the magnetic field components BxandBygenerate a motional electric field E⊥=/radicalBig E2y+E2x=vi/radicalBig B2x+B2y in the x-yplane rotated out of the ydirection by the azimuthal angle φ= arctan( By/Bx) = arctan( Ex/Ey), i. e.E⊥=Eyforφ= 0. The electrons (due to their much lower mass) follow the resulting magnetic field vector /vectorBwhich now crosses the ion beam at the an- gleθ= arctan(/radicalBig B2x+B2y/Bz). Two consequences are to be dealt with: i) The ion beam now probes different portions of the space charge well of the electron beam. This reduces the energy resolution. In order to mini- mize this effect we used a rather small electron density of only 5 .4×106cm−3at cooling, i. e. one order of mag- nitude smaller than in the Si11+experiment of Bartsch et al. [14]. ii) The angle θbetween electron beam and ion beam explicitly enters the formula for the transformation from the laboratory system to the c. m. system which is easily derived from the conservation of four-momentum. It reads Erel=mic2(1 +µ)/bracketleftBigg/radicalBigg 1 +2µ (1 +µ)2(Γ−1)−1/bracketrightBigg (9) with the mass ratio µ=me/mi, Γ =γiγe−/radicalBig (γ2 i−1)(γ2e−1)cos θ (10) andγe= 1 + Ee/mec2;Eedenotes the electron labo- ratory energy. It is obvious that the cooling condition Erel= 0 can only be reached for γi=γeandθ= 0, i. e. for Γ = 1. In regular field-free measurements a scheme of intermittent cooling is used during data taking, i. e. after each measured energy a cooling interval ( Erel= 0) is in- serted. For DRF measurements this would require rapid switching from θ∝negationslash= 0 to cooling with θ= 0. It turned out 4that such a procedure heavily distorts the electron beam mainly because of the slow response of the power supplies controlling the steering coils. Under such conditions use- ful measurements could not be performed. Therefore, we omitted the intermittent cooling and reference mea- surement intervals, thereby losing resolution. After each injection into the ring and an appropriate cooling time the correction coils were set to produce a defined E⊥and the cathode voltage was ramped very quickly through a preset range with a dwell-time of only 1 ms per measure- ment point. In such a manner a spectrum for one E⊥ setting was collected within only 4 s. After termination of the voltage ramp the correction coils were set back to θ= 0 and the whole cycle started again with the injec- tion of ions into the ring. In subsequent cycles a range of typically 30 preset E⊥values was scanned. Each spec- trum was measured as many times as needed for reaching a satisfying level of statistical errors. This whole proce- dure was repeated for different settings of the guiding field strength Bz. In order to compare only contributions from DR to the measured spectra we subtracted an empirical background function αBG(Erel) =a0+a1Erel+a2/(1+a3Erel+a4E2 rel) with the coefficients aidetermined by fitting αBG(Erel) to those parts of the spectrum which do not exhibit DR resonances. One should note that a proper calculation of the RR rate coefficient is hampered by the fact that for θ∝negationslash= 0 the electron velocity distribution probed by the ion beam cannot be described by Eq. (5). III. RESULTS AND DISCUSSION A. Recombination at zero electric field 1. DR cross section For the ∆ n= 0 DR channels of Li-like Ni25+, i. e. for DR involving excitations which do not change the main quantum number of any electron in the 1 s22score, Eq. (1) reads more explicitly e−+ Ni25+(1s22s1/2)→Ni24+(1s22pjnℓ) →/braceleftbigg Ni24+(1s22s1/2nℓ) +hν(type I) Ni24+(1s22pjn′ℓ′) +hν′(type II)(11) The lowest Rydberg states which are energetically al- lowed are n= 13 and n= 11 for 2 s1/2→2p1/2and 2s1/2→2p3/2core excitations, respectively. The Ni25+(1s22s1/2) recombination spectrum has been measured for 0 ≤ |Erel| ≤131.5 eV. The result is shown in Fig. 2. At Erel= 0 a sharp rise of the recombination rate due to RR is observed. At higher energies DR reso- nances due to ∆ n= 0 2s1/2→2pjtransitions occur, the lowest resonance appearing at Erel= 0 eV. Individual 2pjnℓresonances are resolved for n≤32. Their reso- nance strengths have been extracted from the measuredspectrum by first subtracting the theoretical recombina- tion rate coefficient due to RR (cf. section II A) where the c. m. velocity spread can be neglected since Erelis very large compared to kT⊥andkT/bardbl. In principle the resulting rate coefficient should be zero at off-resonance energies. However, we find that probably due to our approximate treatment of RR (cf. Eq. (8)) small non-zero rate coeffi- cients remain after subtraction of the calculated RR rate coefficient. These are removed by further subtracting a smooth background before the observed DR resonance structures are fitted by Gaussians. (Details of the ob- served smooth RR rate were not further investigated in the present work.) The resulting values for resonance po- sitions and strengths are listed in Table I. The 2 p1/213ℓ and 2 p3/211ℓresonances at about 2.5 eV and 4.5 eV, respectively, exhibit a splitting due to the interaction of thenℓ-Rydberg electron with the 1 s22pjcore. For higher nresonances this splitting decreases and cannot be ob- served because of the finite experimental energy spread which increases as√Erel. The 2 pjRydberg series limits, E∞, are obtained from a fit of the resonance positions Enwithn≥16 to the Rydberg formula En=E∞− R/parenleftbiggq n−δ/parenrightbigg2 (12) with the quantum defect δas a second fit parameter. The fit results are listed in Table II, where spectroscopic values [37] for the series limits are also given. Our val- ues agree with the spectroscopic values within 0 .6%, i. e. within the experimental uncertainty of the energy scale [29]. The result that the fitted quantum defects are al- most zero reflects the fact that the interaction between the core electrons and the Rydberg electron is weak. The measured DR rate decreases already below the 2p1/2and 2p3/2series limits as obtained from the fit to the peak positions. (cf. Fig. 2). This discrepancy re- sults from field ionization of high Rydberg states with n > n f≈(3.2×108V/cmq3/Edip)1/4in the charge an- alyzing dipole magnet (see Sec. II) with magnetic field strength Bdip= 0.71 T where the moving ion experi- ences the motional electric field Edip=viBdip(nfis the classical field ionization limit). A more realistic value for the cut-off has to account for Stark splitting and tunnelling effects. In Ref. [10] an approximate value ncut≈(7.3×108V/cmq3/Edip)1/4was found. For the calculation of the actual cut-off quantum number ncut relevant in this experiment one has to take into account that on the way from the cooler to the dipole magnet states with n > n fmay radiatively decay to states below nf. An approximate calculation of this effect [38] yields ncut= 150 for the present case. An estimate of the DR line strength escaping detec- tion because of field ionization can be made by extrap- olating the measured DR line strength to n=∞em- ploying the n−3scaling of the autoionization and the type II (cf. Eq. (11)) radiative rates [39]. For autoion- ization rates we make the ansatz Aa(nℓ) =Aa/n3for 50≤ℓ≤ℓmaxandAa(nℓ) = 0 for ℓmax< ℓ < n . Rates for the sum of type I and type II radiative transitions we represent as Ar(nℓ) =A(I) r+A(II) r/n3. The same repre- sentations of the relevant rates have already been used by Kilgus et al. [29] in a recombination study of isoelec- tronic Cu26+ions. After summation over all ℓsubstates Eq. (4) simplifies to ¯σnEn=S0Aa[A(I) r+n−3A(II) r] Aa+n3A(I) r+A(II) r(13) withS0= 2.475(2jc+1)(ℓmax+1)2×10−30cm2eV2s. For the statistical weights in Eq. (4) we have used gi= 2 andgd= 2(2 ℓ+ 1)(2 jc+ 1). Here, jcis the total an- gular momentum quantum number of the core excited state i. e. jc= 1/2 and 3/2 in the present case. The first step of the extrapolation procedure consists of adjusting the model parameters such that Eq. (13) fits the mea- sured DR line strengths. Here the values for A(I) rhave been taken from atomic structure calculations [40] while S0,AaandA(II) rwere allowed to vary during the fit. It turned out that the fit is not very sensitive to large varia- tions of the Auger rates Aa. In this situation we kept also the Auger rates fixed at values which have been inferred from atomic structure calculations and which are meant to be order of magnitude estimates only. In Fig. 3 we have plotted the measured and fitted resonance strengths (multiplied by the resonance energy) as a function of the main quantum number. The actual parameters used for drawing the fit curves are listed in Table II. The fit has been restricted to n≥20 for both the 2 p1/2and the 2 p3/2 series because additional Coster-Kronig decay channels 2p3/2nℓ→2p1/2ǫℓ′open up when Encrosses the 2 p1/2 series limit. A corresponding discontinuous decrease of the 2p3/2nℓDR resonance strength can be clearly dis- cerned in Fig. 3. Since the additional autoionizing chan- nels are included for the 2 p3/2nℓseries of resonances but not for the 2 p1/2nℓseries, Aafor the 2 p3/2nℓseries is more than a factor of 2 higher than Aafor the 2 p1/2nℓ series. A factor of 2 would just correspond to the ratio of statistical weights. Inserting the fit parameters listed in Table II into Eqs. (12) and (13) now allows an extrapolation of the DR resonance positions and strengths, respectively, to be obtained for arbitrary high n. In order to check the quality of the extrapolation we have convoluted the ex- trapolated DR cross section with the experimental elec- tron energy distribution using the electron beam tem- peratures kBT||= 0.25 meV and kBT⊥= 10 meV. Af- ter adding the semiclassically calculated RR rate coef- ficient — as described above — the resulting extrapo- lated DR+RR recombination rate coefficient is plotted in Fig. 4 together with the experimental one. Despite the very simple model assumptions the calculated recombi- nation rate agrees with the measured one also over the energy intervals covered by the 2 pjnℓresonances with n >31 which are not resolved individually and there-fore have not been used for the fits. Deviations for the Ni24+(1s22p3/2nℓ) resonances with n≤19 stem from the fact that for these resonances the Coster-Kronig de- cay channels to Ni25+(1s22p1/2) are closed whereas the fit has been made to resonances where they are open. At energies close to the series limits slight deviations from the model rate occurs even when the expected value of ncut= 150 is inserted into the model (dashed line in Fig. 4). The origin of this discrepancy has probably to be searched for in the approximations made in both the field ionization model and in the model rate descriptions, in particluar regarding the dependence of the angular momentum ℓwhich may be affected by even the small residual electric fields in the interaction region. 2. Maxwellian plasma rate coefficient The comparison between the measured data and the calculated extrapolation to n→ ∞ (full line in Fig. 4) suggests that only a minor part of the total Ni25+∆n= 0 DR resonance strength has not been measured. This enables us to derive from our measurement the Ni25+DR rate coefficient in a plasma. To this end the experimental DR rate coefficient is substituted by the extrapolated one at energies ∼3 eV below the series limits. Compared to using the experimental result without extrapolation this results in a correction of the plasma rate coefficient of at most 5%. The experimental DR rate coefficient including the high n-extrapolation is convoluted by an isotropic Maxwellian electron velocity distribution characterized by the electron temperature Te. The resulting ∆ n= 0 DR plasma rate coefficient is displayed in Fig. 5 (thick full line). Summing experimental and extrapolation errors, the total uncertainty of the DR rate coefficient in plasmas determined in this work amounts to ±20%. A convenient representation of the plasma DR rate co- efficient is provided by the following fit formula α(Te) =T−3/2 e/summationdisplay iciexp (−Ei/kBTe) (14) It has the same functional dependence on the plasma electron temperature as the widely used Burgess formula [41], where the coefficients ciandEiare related to oscil- lator strengths and excitation energies, respectively. Th e results for the fit to the experimental Ni25+∆n= 0 DR rate coefficient in a plasma (thick full line in Fig. 5) are summarized in Table III. The fitted curve cannot be dis- tinguished from the experimental plasma rate coefficient in a plot as presented in Fig. 5. In Fig. 5 we also compare our results with theoreti- cal results for ∆ n= 0 DR by Mewe et al. [42] (dashed line), Romanik [43] (dash-dotted line) and Teng et al. [44] (dashed-dot-dotted line) who interpolated DR cal- culations performed by Chen [45] for selected lithiumlike ions. At temperatures kBTe>1 eV the rate of Mewe et 6al., which is based on the Burgess formula [41], overesti- mates our experimental result by up to a factor of ∼5. At lower temperatures the experimental result is under- estimated by factors up to 10. Above an electron temper- ature of 30 eV Romanik’s theoretical result agrees with our ∆ n= 0 DR rate coefficient to within 15%, which is within the 20% experimental accuracy. In this en- ergy range the interpolation result of Teng et al. under- estimates the experimental rate coefficient by 20–30%. It should be noted that neither the calculation of Ro- manik nor that of Teng et al. covers temperatures below 10 eV. When compared with our RR calculation (thin full line in Fig. 5) our experimental result shows that in the temperature range of 1 to 10 eV, where Ni25+ions may exist in photoionized plasmas, DR is still signifi- cant. The importance of DR in low temperature plasmas has been pointed out recently by Savin et al. [46] who measured DR of fluorine-like Fe17+ions. At higher tem- peratures Romanik’s calculation [43] suggests that above 100 eV ∆ n= 1 DR contributions become significant (up- per dash-dotted line in Fig. 5). B. DRF measurements Fig. 6 shows a series of Ni25+recombination spectra measured in the presence of external electric fields E⊥ ranging from 0 to 270 V/cm. The magnetic field on the axis of the cooler has been Bz= 80 mT. Due to the altered measurement scheme that leaves out the inter- mittent cooling of the ion beam, the energy resolution is reduced compared to Fig. 2. Now individual 2 pjnℓ DR resonances are resolved only up to n= 21. Two features in the series of spectra are to be noted. Firstly, the strength of the DR resonances occurring below 47 eV does not depend on E⊥. Secondly, the strength of the un- resolved high- nDR resonances increases with increasing field strength. This can been seen more clearly from the close-up presented in Fig. 7. At energies of more than 10 eV below the 2 pjseries limits the different DR spec- tra lie perfectly on top of each other whereas at higher energies (i. e. n >30) an increase of the DR intensities by up to a factor of 1.5 at E⊥= 270 V/cm is observed. The degraded resolution of the DR spectrum does not allow us to resolve nlevels in the range of the electric field enhancement. In order to quantify this DR rate enhancement we consider integrated recombination rates with the integration intervals chosen as marked in Fig. 7. The integration intervals 44.0–53.5 eV and 66.2– 76.0 eV include all 2 pjnℓresonances with n≥31 for j= 1/2 and j= 3/2, respectively. In the following we denote the resulting integrals by I1/2andI3/2. The en- ergy range 44.0–53.5 eV also contains 2 p3/2nℓresonances with 16 ≤n≤19. These resonances, however, are not affected by the electric field strengths used in our ex- periment. Consequently, any change in the magnitude ofI1/2as a function of E⊥we attribute to field effectson 2p1/2nℓresonances. As a check of the proper nor- malization of the DR spectra we additionally monitor the integral I0=/integraltext18eV 2eVαDR(Erel)dErelwhich comprises the strengths of the 2 p3/211ℓ, 2p3/212ℓand 2p1/2nℓDR resonances with 13 ≤n≤15. Since these low nreso- nances are not affected by E⊥we expect I0to be con- stant. Any deviation of I0from a constant value would indicate a reduction of beam overlap due to too large a tilting angle θof the electron beam. The maximum an- gleθmaxto which the overlap of the electron beam with the ion beam is ensured over the full interaction length Lis given by tan θmax= (de−di)/L. With the geomet- rical values given above one obtains θmax≈1◦. Apart from the highest E⊥atφ= 180◦and at the lowest mag- netic guiding field, i. e. at Bz= 41.8 mT, the condition θ < θ maxwas always met. This is exemplified in the up- per panel of Fig. 8 where I0,I1/2andI3/2are shown for -270 V/cm ≤E⊥≤220 V/cm. There, positive (nega- tive) field strength indicates φ= 0◦(φ= 180◦). While I1/2andI3/2clearly exhibit a field effect which is nearly symmetric about E⊥= 0 V/cm, I0is independent of E⊥. The fact that the increase of the integrated recombi- nation rate coefficient is independent of the sign of the electric field vector is expected from the cylindrical sym- metry of the merged beams arrangement in the electron cooler. However, for the entire experimental setup this symmetry is broken by the charge analyzing dipole mag- net with an electric field vector lying in the bending plane (thex-yplane of the coordinate frame defined in the in- teraction region). This, in principle, could lead to re- distribution of population between different msubstates in the dipole magnet [47] and a resulting field ionization probability depending on the azimuthal angle φof the motional electric field vector in the cooler. In order to clarify this question we took a series of DRF spectra with φranging from 5◦to 175◦. At the same time the electric and magnetic fields were kept fixed at E⊥= 100 V/cm andBz= 80 mT. A scan around a full circle was prohib- ited by the limited output of the power supplies used for steering the electron beam in the particular arrangement of this experiment. As shown in the lower panel of Fig. 8 no significant dependence of the integrated recombina- tion rates on the azimuthal angle φwas found. As a measure for the magnitude of the field enhance- ment we introduce the field enhancement factor rj(E⊥, Bz) =Cj(Bz)Ij(E⊥, Bz) I0(E⊥, Bz)(15) forj= 1/2 or 3 /2 and the constant Cj(Bz) chosen such thatrj(0, Bz) = 1.0 (see below). Plots of enhancement factors as a function of E⊥are shown in Fig. 9 for differ- ent values of Bz. Since the field effect is independent of the orientation of the electric field in the x-yplane data points for φ= 0◦andφ= 180◦are plotted together. The enhancement factor exhibits a linear dependence on the electric field. Exceptions occur for Bz= 41.8 mT and φ= 180◦atE⊥values where θbecomes maximal, and 7around E⊥= 0 V/cm. In the former case the complete overlap of the ion beam and the electron beam over the full length of the interaction region is lost. This is indi- cated by a reduction of I0, and apparently a consistent normalization cannot be carried out. In the latter case residual electric and magnetic field components resulting e. g. from a non-perfect alignment of the beams prevent us from reaching E⊥= 0 eV. After excluding all data points with E⊥≤10 V/cm and those with φ= 180◦, E⊥≥200 V/cm for Bz= 41.8 mT, we were able to fit straight lines to the measured field enhancement factors as a function of E⊥. The constants Cj(Bz) in Eq. (15) have been chosen such that the fitted straight lines yield r(fit) j(0, Bz) = 1.0. As a measure for the electric-field enhancement of the DR rate enhancement we now consider the slopes sj(Bz) =dr(fit) j(E⊥, Bz) dE⊥(16) of the fitted straight lines, which are displayed as a func- tion of the magnetic field strength in Fig. 10. The er- ror bars correspond to statistical errors only. Systematic uncertainties e. g. due to residual fields are difficult to estimate. Nevertheless, their order of magnitude can be judged from the ≈10% difference between the two data points at Bz= 80 mT which have been measured with different cooler settings. For all magnetic fields used the electric field depen- dence of the rate enhancement factor is steeper for the 2p3/2nℓseries of Rydberg resonances than that for the 2p1/2nℓseries. This can be understood from the fact that the multiplicity of states and consequently the num- ber of states which can be mixed is two times higher for j= 3/2 than for j= 1/2. Following this argument one would expect a ratio of 2 for the respective incremen- tal integrated recombination rates Ij(E⊥, Bz)−Ij(0, Bz). From our measurements we find lower values scatter- ing around 1.5 indicating a somewhat reduced number of states available for field mixing within the series of Ni24+(1s22p3/2nℓ) DR resonances. In calculations for Li- like Si11+and C3+ions ratios, even less than 1 have been found [16,48]. This has been attributed to the electro- static quadrupole-quadrupole interaction between the 2 p and the nℓRydberg electron in the intermediate doubly excited state, which more effectively lifts the degener- acy between the 2 p3/2nℓthan between the 2 1/2nℓlevels. Another reason for the reduced number of 2 p3/2nℓstates participating in DRF might be the existence of additional Coster-Kronig decay channels for these resonances with n≥20 to Ni25+(1s22p1/2) (cf. section III A). Slopes for the relative electric field enhancement ac- cording to Eq. (16) were previously determined in mea- surements by Bartsch and coworkers for the lighter iso- electronic ions Si11+, Cl14+and Ti19+[14,19,20]. The present results are compared with these previous data in Fig. 11. It should be noted that the comparison is only semi-quantitative, because the choice of integrationranges for the calculation of the integrated recombina- tion coefficients is somewhat arbitrary and different cut- off quantum numbers ncutexist for the different ions (cf. Table IV). Clearly, the relative enhancement for a given electric field strength is much lower for Ni25+than for the lighter ions studied so far, with the reduction for the step from lithium-like titanium ( Z= 22) to nickel ( Z= 28) apparently much larger than the step from, e. g., chlorine (Z= 17) to Z= 22. As a general trend with increasing Z, the radiative decay rates Ardecrease ∝Zfor type I transitions (see Eq. (11)) and ∝(Z−3)4for type II transitions, while the autoionization rates are rather in- dependent of Z. This shifts the range where the low- ℓ autoionization rates are larger than the radiative stabi- lization rates (the condition for DR enhancement by ℓ- mixing to occur [15]) down to states of lower principal quantum number nfor increasing Z. For Ni25+the de- gree of mixing reached by the typical experimental field strengths appears to be much reduced as compared to lighter ions. In addition, a clear dependence of the elec- tric field enhancement on the the magnetic field strength is no longer observed for Ni25+. To clarify the reason for the strong reduction of both the electric and the addi- tional magnetic field effect in the heavy system studied here, detailed quantitative calculations are desirable. External electric and magnetic fields are ubiquitous in astrophysical or fusion plasmas. Therefore, it is of in- terest to look into the implications of the field enhance- ment effect for the Ni25+∆n= 0 DR rate coefficient in a plasma. As an example we show in Fig. 12 the ratio of rate coefficients derived from two measurements with and without external electric field. As compared to zero electric field, the recombination coefficient at our highest experimental electric field strength ( E⊥= 270 V/cm at Bz= 80 mT) is enhanced by up to 11%. This value only represents a lower limit for the enhancement at the given field strength, as we observe DR resonances due to Ni24+(1s22pjnℓ) intermediate states only up to ncut≈150. It should also be noted that in our experi- ments we did not reach the electric field strength where the DR rate enhancement saturates. IV. SUMMARY AND CONCLUSIONS The recombination of lithium-like Ni25+ions has been experimentally studied in detail. Spectroscopic infor- mation on individual DR resonances associated with Ni24+(1s22pjnℓ) intermediate states, which have been ex- perimentally resolved up to n= 32, has been extracted and the ∆ n= 0 DR plasma rate coefficient has been de- rived. Our experimental result is underestimated by up to a factor of 2 by semi-empirically calculated rate coef- ficients. At plasma temperatures above 10 eV, results of detailed theoretical calculations are available [43] whic h agree well with the experiment. At lower temperatures, where Ni25+ions may exist in photoionized plasmas, no 8data for the DR rate coefficient have been previously available. In the presence of external electric fields up to 300 V/cm, the measured DR resonance strength is en- hanced by a factor 1.5; a rather weak effect in compari- son with previous measurements at Z= 11, 17 and 22. Due to the overall weakness of the field effect for Ni25+ ions, a marked dependence of the DR rate enhancement on the strength of a crossed magnetic field as observed for Cl14+and Ti19+ions has not been detectable in the present investigation. Experimental limitations prevented us from obtaining an energy resolution in our DRF measurements compara- ble to that achieved in the field-free measurement. Con- sequently, we could not resolve a Ni24+(1s22pjnℓ) DR resonance with nhigh enough to exhibit a field effect. Such an observation would have facilitated a direct com- parison with ab initio calculations, which due to the large number of nℓmstates to be considered, presently can only treat a single nmanifold of Rydberg states in the presence of crossed /vectorEand/vectorBfields [16]. Improvements of this situation can be expected in the near future from the steady increase of computing power on the theoreti- cal side, and on the experimental side from a dedicated electron target which is presently being installed at the TSR. With the electron target and the electron cooler operating at the same time we will be able to perform DRF measurements with continuously cooled ion beams, yielding DRF spectra with increased resolution. 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A 58, 4548 (1998).resonance position resonance strength designation Ed(eV) ¯ σd(10−19eV cm2) 2.079±0.002 2.42 ±0.12 2 p1/213s 2.267±0.003 3.25 ±0.22 2 p1/213p 2.383±0.001 9.53 ±0.26 2 p1/213d 2.625±0.003 94.2 ±5.7 2 p1/213ℓ(ℓ >2) 9.590±0.003 22.7 ±0.71 2 p1/214ℓ 15.21 ±0.001 11.1 ±0.12 2 p1/215ℓ 19.81 ±0.002 6.78 ±0.083 2 p1/216ℓ 23.63 ±0.002 4.85 ±0.077 2 p1/217ℓ 26.83 ±0.003 3.71 ±0.072 2 p1/218ℓ 29.54 ±0.003 2.77 ±0.057 2 p1/219ℓ 33.84 ±0.004 1.98 ±0.046 2 p1/221ℓ 35.57 ±0.004 1.72 ±0.045 2 p1/222ℓ 38.40 ±0.005 1.32 ±0.044 2 p1/224ℓ 39.55 ±0.009 1.13 ±0.065 2 p1/225ℓ 40.63 ±0.006 1.02 ±0.040 2 p1/226ℓ 41.53 ±0.024 1.13 ±0.10 2 p1/227ℓ 42.35 ±0.023 0.935 ±0.14 2 p1/228ℓ 43.09 ±0.008 0.857 ±0.042 2 p1/229ℓ 43.75 ±0.012 0.739 ±0.051 2 p1/230ℓ 44.35 ±0.014 0.744 ±0.063 2 p1/231ℓ 44.91 ±0.013 0.727 ±0.063 2 p1/232ℓ 3.777±0.002 4.71 ±0.13 2 p3/211s 4.190±0.001 17.9 ±0.17 2 p3/211p 4.673±0.001 163.3 ±2.1 2 p3/211ℓ(ℓ >1) 15.96 ±0.001 40.4 ±0.61 2 p3/212ℓ 24.73 ±0.002 20.04 ±0.66 2 p3/213ℓ 31.74 ±0.001 15.2 ±0.14 2 p1/220ℓ,2p3/214ℓ 37.34 ±0.004 10.33 ±0.30 2 p1/223ℓ,2p3/215ℓ 41.95 ±0.003 6.61 ±0.20 2 p3/216ℓ 45.77 ±0.002 5.38 ±0.047 2 p3/217ℓ 48.97 ±0.002 4.45 ±0.045 2 p3/218ℓ 51.70 ±0.003 4.01 ±0.17 2 p3/219ℓ 54.00 ±0.003 2.68 ±0.039 2 p3/220ℓ 55.99 ±0.003 2.32 ±0.038 2 p3/221ℓ 57.73 ±0.004 2.02 ±0.038 2 p3/222ℓ 59.22 ±0.004 1.88 ±0.037 2 p3/223ℓ 60.56 ±0.004 1.75 ±0.035 2 p3/224ℓ 61.72 ±0.005 1.61 ±0.036 2 p3/225ℓ 62.75 ±0.005 1.46 ±0.034 2 p3/226ℓ 63.68 ±0.005 1.40 ±0.035 2 p3/227ℓ 64.49 ±0.005 1.30 ±0.037 2 p3/228ℓ 65.27 ±0.008 1.23 ±0.040 2 p3/229ℓ 65.93 ±0.009 1.16 ±0.040 2 p3/230ℓ 66.52 ±0.01 1.11 ±0.041 2 p3/231ℓ TABLE I. Strengths of the individually resolved 2 pjnℓres- onances as obtained from fits of Gaussians to the experimen- tally observed resonance structures. The errors given are s ta- tistical only (one standard deviation). Systematic uncert ain- ties amount to less than ±15% for the resonance strengths and less than 0 .6% for the resonance positions. 10series 2 p1/2 2p3/2 E∞(eV), spectroscopic 52.95 74.96 E∞(eV), this experiment 53.19 ±0.01 75.34 ±0.01 δ 0.031 ±0.005 0.030 ±0.001 Aa(1015s−1) 0.4 1.2 A(I) r(109s−1) 2.0 5.8 A(II) r(1013s−1) 4.1 ±0.6 6.7 ±0.5 S0(10−27eV2cm2s) 1.19 ±0.09 1.12 ±0.03 ℓmax 14.5 ±0.6 9.5 ±0.2 TABLE II. Parameters obtained from fits of DR resonance positions and strengths. The series limits E∞, their uncer- tainties and quantum defects δresult from a fit of the experi- mental resonance positions to the Rydberg formula Eq. (12). The total experimental uncertainty of E∞is of the order of ±0.5 eV. The spectroscopic values listed for comparison are taken from Ref. [37]. In the fit of Eq. (13) to the measured DR resonance strengths the core radiative rates A(I) rand the Auger rates Aa(as listed) have been taken from atomic struc- ture calculations [40]. S0andA(II) rresult from the fit. ℓmax has been calculated from S0. i c i Ei (10−2cm3s−1K3/2) (eV) 1 0.417 2.91 2 0.683 5.60 3 1.483 19.10 4 3.184 44.54 5 2.928 71.97 TABLE III. Ni25+∆n= 0 plasma DR rate coefficient fit parameters ciandEiaccording to Eq. (14). The fit to the full line in Fig. 5 is accurate to better than 0.5% for 1.0 eV ≤kBTe. The total uncertainty in the rate coefficient is 20%. Ion Reference Ion beam energy nmin nmax [Mev/u] 2 p1/22p3/2 Cl14+[19] 7.1 23 18 79 Ti19+[20] 4.6 27 27 115 Ni25+[this work] 5.9 31 31 150 TABLE IV. Parameters in DRF experiments with Li-like ions. For each ion the field effect has been quantified by considering an integrated DR rate coefficient which com- prises the resonance strengths of 2 pjnℓDR resonances with nmin≤n≤nmaxwhere nmaxis the approximate cut-off quan- tum number due to field ionization in the charge analyzing dipole magnet. It depends on the ion beam energy and the ion’s charge state [38].FIG. 1. Sketch of the electric and magnetic field con- figuration used in DRF measurements. The ion beam is aligned along Bz. The motional electric field is E⊥=viB⊥=vi/radicalbig B2x+B2ywith the ion velocity vi. The azimuthal angle φdenotes the direction of E⊥in the x-y plane. The electron beam is aligned along the resulting /vectorB vector which is inclined by the angle θwith respect to Bz. 0 5 10 15 20 25 30 35020406080100120 2p3/211 l2p1/213 l 20 17 16 1515 14 13 n=12 n=14Recombination rate coefficient (10-10 cm3/s) Relative energy (eV)40 50 60 70012345678 n=26n=16 2p1/22p3/2 FIG. 2. Absolute recombination rate coefficient measured for 340 MeV Ni25+ions. The sharp peak at zero relative energy is due to RR. Energetic positions of the 2 p1/2nland 2p3/2nlresonances according to the Rydberg formula are in- dicated. 1110 12 14 16 18 20 22 24 26 28 30 3210100 σnEn (10-18eV2 cm2) Main quantum number n FIG. 3. Product of strength and resonance energy of the 2pjnℓDR resonances for j= 1/2 (closed symbols) and j= 3/2 (open symbols) as extracted from the experimen- tal recombination spectrum. Statistical error bars are mos tly smaller than the symbol size. The strengths of the 2 p3/214ℓ and 2 p3/215ℓresonances have been obtained by subtract- ing interpolated values for the 2 p1/220ℓand 2 p1/223ℓreso- nance strengths from the measured peak areas at 31.7 eV and 37.3 eV, respectively. The thick full curves represent fits a c- cording to Eq. (13) with the fit parameters listed in Table II. 30 40 50 60 700510∞ ∞ 3025 n=213025 20 n=15 2p3/2nl series 2p1/2nl seriesRecombination rate coefficient (10-10cm3/s) Relative energy (eV) FIG. 4. Comparison between measured (closed symbols) and extrapolated recombination spectra (see text). The ex- trapolations extend to n= 150 (dashed line) and n= 500 (full line) where convergence is achieved.1 10 100110 kBTe (eV)Recombination rate coefficient (10-10cm3/s) FIG. 5. Ni25+∆n= 0 DR plasma rate coefficient as de- rived from our measurement (thick full line, estimated erro r ±20% as indicated). Also shown are theoretical results of Mewe et al. [42] (dashed line), Romanik [43] (dash-dotted line) and Teng et al. [44] (dash-dot-dotted line). At temper - atures kBTe>100 eV two DR rates by Romanik are shown, with the upper one additionally containing ∆ n= 1 DR con- tributions. The RR rate coefficient (thin full line) has been calculated from the RR cross section given by Eq. (8) with q= 25 and nmax= 150. 0 10 20 30 40 50 60 70 8001234 E⊥Recombination rate coefficient (10-9 cm3/s) Relative energy (eV) FIG. 6. Ni25+DR spectra for 28 external electric fields 0≤E⊥≤270 V/cm and |cosφ|= 1. Spectra with φ= 0◦ andφ= 180◦are interleaved. Adjacent spectra differ by ∆E⊥= 10 V/cm. The magnetic field on the axis of the electron cooler has been Bz= 80 mT. The fitted smooth RR contribution has been subtracted. 1230 40 50 60 70012345625222018 16 n=20n=14 2p3/2 2p1/2Recombination rate coefficient (10-10 cm3/s) Relative energy (eV) FIG. 7. Ni25+DR spectra for 4 external electric fields 0, 110, 190 and 270 V/cm clearly showing the DR rate enhance- ment with increasing electric field strength close to the 2 p1/2 and 2 p3/2series limits. Ranges for the determination of in- tegrated rate coefficients are indicated by horizontal arrow s. The magnetic field on the axis of the electron cooler has been Bz= 80 mT. The fitted smooth RR contribution has been subtracted. -200 -100 0 100 2001.61.82.02.22.42.6 ×1/2Integrated recombination rate (10-9eV cm3/s) Electric field (V/cm) 0 30 60 90 120 150 1801.61.82.02.22.42.6 ×1/2 Azimuthal angle (deg)FIG. 8. Integrated recombination rate coefficients I0(full squares), I1/2(full circles) and I3/2(open circles) for the inte- gration ranges defined in the text as a function of electric fie ld (upper panel, φ= 0◦) and of azimuthal angle (lower panel, E⊥= 100 V/cm). The magnetic guiding field is 80 mT in both cases. 0 50 100 150 200 250 3001.01.11.21.31.41.5Bz = 80.1 mT Electric field (V/cm)1.01.11.21.31.41.5Bz = 60.0 mTRecombination rate enhancement factor 1.01.11.21.31.41.5Bz = 41.8 mT FIG. 9. Recombination rate enhancement factors as a func- tion of electric field E⊥for three different magnetic fields Bz. The enhancement is larger for the series of 2 p3/2nℓresonances (open symbols) as compared to the 2 p1/2nℓseries (closed sym- bols). Triangles pointing upwards (downwards) mark data points measured at φ= 0◦(φ= 180◦). The data points which for experimental reasons (see text) have been exclude d from the linear fits (full curves) are marked additionally by crosses. Included and excluded data points for Bz= 41.8 mT andE⊥≥200 V/cm correspond to φ= 0◦andφ= 180◦, respectively. 1340 50 60 70 800.00.20.40.60.81.01.21.41.61.8Slope of enhancement factor (10-3 cm/V) Magnetic guiding field (mT) FIG. 10. Slopes of the enhancement factors as a function of the magnetic field Bzfor the 2 p3/2nℓ(open symbols) and 2p1/2nℓ(closed symbols) series of Rydberg resonances with n≥31. 20 30 40 50 60 70 8002468Slope of enhancement factor (10-3 cm/V) Magnetic guiding field (mT) FIG. 11. Slopes of the enhancement factors as a function of the magnetic field Bzfor the 2 pjnℓresonances of Li-like Si11+(open circle) [14], Cl14+(diamonds) [19], Ti19+(full circles) [20] and Ni25+ions (squares) [this work]. The lines are drawn to guide the eye. For Ti19+and Ni25+where the field effects on the 2 p1/2and 2 p3/2series of DR resonances have been determined separately, the corresponding slopes have been averaged. Clearly the effects of external electric and magnetic fields on DR decrease with increasing nuclear charge.10 1001.001.021.041.061.081.10 α(270 V/cm) / α(0 V/cm) kBTe (eV) FIG. 12. Ratio of Ni25+∆n= 0 DR plasma rate coeffi- cients with ( E⊥= 270 V/cm) and without ( E⊥≈0 V/cm ) external electric field. The magnetic field was Bz= 80 mT. An experimental error bar of ±1% as indicated is estimated for the ratio of the rate coefficient. 14

arXiv:physics/0002037 20 Feb 2000\\ Title :Post acceleraed Frames. Author:Erez M.Yahalomi. Comment:17 pages , 2 figures Report-no:BIU-00-2 Subj-class:General Relativity \\ Special relativistic effects occur only on post accelerated frames, which are frames, which were accelerated in the past to reach a certain uniform velocity before the measurement of these effects. A symmetry breaking in special relativity is then obtained. The non-symmetrical equations for space time, length and charge density are presented. \\/G03 /G03 2 Post Accelerated Frames E. M. Yahalomi Department of Physics, Bar-Ilan University, Ramat-Gan 52900 , ISRAEL Abstract Special relativistic effects occur only on post accelerated frames, which are frames, which were accelerated in the past to reach a certain uniform velocity before the measurement of these effects. A symmetry breaking in special relativity is then obtained. The non-symmetrical equations for space time, length and charge density are presented. *E mail:cherez@techunix.technion.ac.il/G03 /G03 3 I. Introduction As it is known Einstein [1] used the Lorentz transformations in his special theory of relativity by denying the existence of a privileged frame. Lorentz [2], Ives [3], Builder [4], Caruso [5] and Marinov [6] obtained the Lorentz transformations by affirming the existence of a privileged frame. When trying to explain the existence of a privileged inertial reference frame they came with indefinite explanations like motion with respect to distant matter or the existence of ether. In this paper we present a clear definite mathematically formulated explanation for the distinction of a privileged inertial frame and for the cause of special relativistic effects. We find that special relativistic effects occur only in frames which were accelerated to reach the uniform velocity v,and then continue to move with uniform velocity V. The frame which did not accelerate and its velocity is viewed to be v by viewing from the other frame, is the privileged inertial frame. II. Physical Descriptions Consider a rod moving with velocity V relative to an inertial frame. The rod emits two light pulses simultaneously from its edges, perpendicular to the inertial frame below it. Let x'1 and x'2 be the coordinates of the two edges in the reference frame, and x1, x2 the coordinates in the inertial frame, /G03 /G03 4 ()() ()()   − −=−−−− −−=− 1222 12221 222 12 11 10 xxCV xxCVvtx CVvtxxxo (1) Two measuring rods A, B with equal lengths at rest, move with velocity V toward each other. Rods A and B are parallel with infinitesimal vertical distance h between them as illustrated in Fig. 1. FIG. 1. (a) Rod B moves at velocity V, rod A is at rest. (b) Rod A regarded as Moving at velocity -V relative to B which is at rest in its reference frame. /G03 /G03 5 When the front of the two rods overlap two laser pulses are emitted from the two edges of rod B to make marks on rod A. If the two pulses are emitted at the same time, which can be arranged by calculating the overlapping point, they will arrive at rod B at the same time because they both travel distance h and the rods have no vertical velocity. A is at rest and B moves at velocity V, /G22 /G22BAVC =−122/, /G22 /G22BA< (2) then two points would be marked on rod A. When observing from reference frame B, B is at rest and A moves at velocity - V () /G22 /G22ABVC =−122, /G22 /G22BA/G1F (3) one point would be marked on rod A. We see there are two different physical results for the same space time event, which infers a contradiction, which can be solved by looking at the rods later and counting how many points were marked on the rod A. This solves the problem of simultaneously measurement of two moving frames Caruso raised in his paper. The contradiction occurs because we use the special relativistic postulation that states that there is symmetry or the same special relativistic effects, on two frames that move with velocity V relative to each other. Namely, there is symmetry if we define A to be at rest and B to move with velocity V relative to A or if we define B to be at rest and A to move with velocity -V relative to B. This argument leads us to the contradiction. In/G03 /G03 6 order to solve the contradiction we need to find a difference between the two frames A and B. The difference we have found is that only one frame accelerated to reach velocity V. When the velocity is reached, the frame stops accelerating and continues to move at a uniform velocity V. In the first stage both frames A and B are at rest or move at the same velocity. In order to achieve the relative velocity V between A and B one frame has to accelerate until it reaches a relative velocity V compared to the other frame and then stops accelerating. The frame then continue to move in uniform velocity V relative to the other frame. Special relativistic effects occur only in the frame which accelerated before reaching velocity V. This solves the contradiction of different physical results during the same space time event. When frame A moves at a relative velocity compared to frame B we know that A accelerated in an event in the past in order to reach uniform velocity V relative to frame B. On the other hand, in the same situation when assuming frame A is at rest and frame B moves at velocity -V relative to A we know that frame B did not accelerate and its velocity is only a point of view from frame A. Frame A which accelerated in the past is the only frame which has special relativistic effects. The term post acceleration refers to the frame after the acceleration, frame which at the time of the observation moves at uniform velocity V but at some time range in the past before the observation, had accelerated, until it reached the uniform velocity V, then stopped accelerating and continued to move at uniform velocity V./G03 /G03 7 The mathematical solution is the term: Vatdtt =∫() 01 (4) where a(t) is the time dependent acceleration, t1 is the time the body accelerated to reaches velocity V. Now the clock which is considered to move at velocity V is the clock which accelerated from rest to velocity V. The special relativistic effects are derived from the acceleration which brought the system to the uniform velocity V. This phenomenon is different than the acceleration influence on relativistic effects at the moment of acceleration [7,8]. /G03 /G03 8 III.Theoretical Calculation General first order coordinates transformation: xaxayazat yaxayazat zaxayazat taxayazat' ' ' '=+++ =+++ =+++ =+++11 12 13 14 21 22 23 24 31 32 33 34 41 42 43 44 (5) The coordinates set x',y',z',t' move with velocity V in the direction of the positive x axis.Therefore, y'=y,z'=z. xaxayazat taxayazat' '=+++ =+++11 12 13 14 41 42 43 44 (6) The s' move at velocity V compared to frame s,therefore,if x'=0,x=Vt. According to Eq. (4), Vatdtt =∫() 01 , t1 is the time it took an accelerated frame to accelerate from 0 velocity to V. Equation (6) becomes, 0 011 012 13 14 11 14 012 131 1=  +++ = +  ++∫ ∫aatdttayazat aatdtatayazt t() () [7] Since all the variables are independent, the coefficients of the variables should vanish./G03 /G03 9 aaatdt aat 14 11 0 12 131 0=−   ==∫()(8) xaxatdtt yy zz taxatt ' () ' ' '=−     = = =+∫ 11 0 41 441 (9) Consider the light velocity to be C in each direction in any reference frame xyzCt22222++= (10) xyzCt ''''22222++= (11) Substitute the transformation equations [9] into eq. [10] axatdttyzt 112 02 221 −      + ∫() ()( ) = + − ++− +   ∫ CaxataCaxyztatdtaCaaxtt 2 41 442 112 2 412222 112 2 4144 021 () (12) = −      ∫ Catatdtatt 2 442 02 11221 ()/G03 /G03 10 After coefficient comparison between Eq. (10) and Eq. (12) and substituting in Eq. (9), we obtain: xxatdtt atdt C yy zz ttatdtXC atdt Ct t t t'() () ' ' '() ()=−   −         = = =−   −        ∫ ∫ ∫ ∫0 02 02 021 1 1 11 1 (13) x,t equations when observing from frame s' xxatdt Catdtt ttatdt Catdttx Ct t t t=−        +  ⋅ =−        +  ⋅∫ ∫ ∫ ∫'() () '() ()1 102 0 02 021 1 1 1 (14) Eq. 14 is identical to Eq. 13 because the inertial frame is always the frame, which has not, accelerated and the frame which has relativistic effects is the frame which accelerated to velocity V./G03 a /G03 /G03 /G03 b /G03 a /G03 /G03 /G03 b/G03 /G03 11 We obtain a symmetry breaking in special relativity. There is no similarity or symmetry between the description of frame s being at rest as an inertial frame with Frame s' moving at a uniform velocity V as a reference frame, and the description of frame s' being at rest as an inertial frame and s moving at -V as a reference frame. The set of Special relativity equations [1] for frame s' as an inertial and s frame as a reference frame with velocity exchange from v to -v is invalid. There is only one set of equations as obtained in Eq. (13) or Eq. (14), which includes both cases. IV. Rods measurement at different frames In this section we discuss the relativistic effects on rod's measurement at different frames.Consider a case where a rod is at rest along the x' axis of the s' frame.It's edge points are measured to be x'1 and x'2 so that its length in the s' frame is x'2-x'1=Δx'. The rod length as measured by the s frame for which the rod moves at a relative speed V is ΔΔ Δ xxatdtt atdt Ct t'() ()=− ⋅ −        ∫ ∫0 02 11 (15) We can identify Δx as the length of the rod in the s frame if the positions x2 and x1 of the edges' point of the rod are measured in the same time.Thus,with Δt=0./G03 /G03 12 Δ Δ xatdt Cxt =−        ∫ 102 1 () ' (16) Consider now a rod at rest in inertial frame s.Its measured length in the s' frame,which relative to it the rod is moving at velocity -V is from Eq. (15) ΔΔxx atdt Ct' ()= −        ∫ 102 1 since atdt tt () () = = ∫0 0 01 α We get Δx'=Δx. If we measure Δxwith a measuring rod which accelerated to velocity V and now is at rest compared to s',the measurement scales would be according to Eq (16) dxdx atdt Ct' ()= −        ∫ 102 1 and Δx who has not changed because it was not accelerated will be measured on the s frame as/G03 /G03 13 ΔΔxx atdt Ct' ()= −        ∫ 102 1 A rod moving at velocity -V relative to s' will be measured to be longer in the s' frame, opposite of the standard special relativity equations. V.Charge density measurement The charge density ρ0 of a charge at rest in s' frame which moves at velocity V, compared to inertial frame s, is ρρ= −        ∫0 2 101 atdt Ctt () (18) where ρ is the density measured in the inertial frame. If ρ is the density charge at rest in the inertial frame, the ratio between ρ, and ρ0 the density charged ρ frame, is/G03 /G03 14 ρρ02 101 =−          ∫atdt Ctt () [19] Since ρ did not accelerate its value remains ρ, but the measurement instrument which Accelerated to the s' frame velocity,transforms according to Eq. (18). Suggested applicable experiment using Mossbauer effect. Because of the small natural width of high frequency γ /G03 radiation, Mossbauer effect [10] is an excellent tool to verify relativistic effects. The following experiment is suggested. Fig. 2. Two sets up for measurement of /G16 ray frequency shift due to relativist effects by Mossbauer. /G03 /G03 15 (a) A source of /G16 rays moves in a straight line in a constant velocity. The absorber absorbs /G16 rays emitted perpendicular to the source cement. (b) An absorber of /G16 rays moves in a straight line. The absorber absorbs /G16 rays emitted perpendicular from stationary source. On the first measurement, the source of /G16 rays moves in a straight line. The absorber absorbs the /G16 rays emitted perpendicular to the source movements. On the second measurement the absorber of /G16 rays moves in a straight line. The absorber absorbs the /G16 rays emitted perpendicular from stationary source.According to Einstein's special relativity, on both cases it is predicted to measure a frequency red shift, due to transverse Doppler effect. (20) /GFB/G23/G03/G4C/G56/G03/G57/G4B/G48/G03/G49/G55/G48/G54/G58/G48/G51/G46/G5C/G03/G55/G48/G47/G03/G56/G4B/G4C/G49/G57/G0F/G03/G39/G03/G4C/G56/G03/G57/G4B/G48/G03/G59/G48/G4F/G52/G46/G4C/G57/G5C/G03/G45/G48/G57/G5A/G48/G48/G51/G03/G57/G4B/G48/G03/G56/G52/G58/G55/G46/G48/G03/G44/G51/G47/G03/G57/G4B/G48/G03/G44/G45/G56/G52/G55/G45/G48/G55/G0F/G03/G39 /G4C/G56/G03/G57/G4B/G48/G03/G16 ray frequency, C is the speed of light. According to our theory, in the first case the measured frequency is the same as in Einstein's theory because the source accelerated before reaching the velocity V.in the second case, our theory predicted a blue shift. In this case the absorber accelerated before reaching velocity V, which according to Eq. (13a) causes a time dilation, which is interpreted as a red shift in the natural width of the absorber. The source was not accelerated though according to Eq. (13a) there is no time dilation or frequency shift in its natural width. Therefore, /G16 rays from the source measured at the absorber, willv cV 22 2=Δν/G03 /G03 16 appear to be blue shifted. This phenomenon has not been noticed before, since in the former experiment like Pound's and Rebka's [10],the source always moved and the absorber was stationary. In summary, the relativistic effects of special relativity occur only on post accelerated systems, i.e. systems that accelerated to reach the uniform velocity in which they are moving at the time of the measurement. We obtained symmetry breaking in special relativity. There is only one set of relativistic equations for systems with velocity v. There is no second set of relativistic equations for exchanging the frame of reference with the inertial frame and considering it to have velocity -v . A rod at rest in the inertial frame when observed from a reference frame, and compared to the reference frame, as if the inertial frame moves at velocity -v, would not be contracted as stated in the standard special relativity, but it would be measured to expand. The density of charge in inertial frame measured by reference frame was found to be reduced, contrary to standard special relativity equations. This approach to the theory of special relativity is similar to the general theory of relativity where the relativistic effects are determined absolutely by potentials./G03 /G03 17 References [1] A.Einstein, Zur Elektrodinamik bewegter Kroper, "Annalen der Physik",17, 1905, pp. 891 -921. [2]H.A. Lorentz, The Theory of Electrons and its Applications to the Phenomena of Light and Radiant Heat (Columbia U.P., New York, N.Y., 1909). [3]H.E. Ives, Philos. Mag. 36, 392 (1945). [4]G. Builder, Austr. J. Phys., 11, 457 (1958). [5]G. Caruso, Hadronic Journal 20, 85 (1997). [6]S. Mrinov, Foundations of Physics 9, 445 (1979). [7]E.M. Yahalomi, Foundations of Physics Letters 10, 495 (1997). [8]W. Fugmann and M. Kretzschmar, Il Nouvo Cimento B 106, 351 (1991). [9]R.V. Pound and G.A. Rebka, Phys. Rev. Let 4, 337 (1960). [10]R.L. Mossbauer, Z phys. 151, 124 (1958).

arXiv:physics/0002038v1 [physics.atom-ph] 20 Feb 2000Semiclassical description of multiphoton processes Gerd van de Sand1and Jan M. Rost2 1Theoretical Quantum Dynamics, Fakult¨ at f¨ ur Physik, Univ ersit¨ at Freiburg, Hermann-Herder-Str. 3, D-79104 Freiburg, Germany 2Max-Planck-Institute for the Physics of Complex Systems, N ¨ othnitzer Str. 38, D-01187 Dresden, Germany (January 2000) We analyze strong field atomic dynamics semiclassically, ba sed on a full time-dependent description with the Hermann-Kluk propagator. From the properties of th e exact classical trajectories, in particular the accumulation of action in time, the prominen t features of above threshold ionization (ATI) and higher harmonic generation (HHG) are proven to be i nterference phenomena. They are reproduced quantitatively in the semiclassical approxima tion. Moreover, the behavior of the action of the classical trajectories supports the so called strong field approximation which has been devised and postulated for strong field dynamics. PACS numbers: 32.80.Fb, 3.65.Sq, 42.65.Ky I. INTRODUCTION In the last two decades multiphoton processes have been studied intensively, experimentally as well as the- oretically. The inherent time-dependent nature of an atomic or molecular excitation process induced by a short laser pulse renders a theoretical description problem- atic in two respects. Firstly, a full quantum calculation in three dimensions requires a large computational ef- fort. For this reason, quantum calculations have been restricted to one active electron in most cases [1,2]. Sec- ondly, an intuitive understanding for an explicitly time dependent process seems to be notoriously difficult, ex- emplified by pertinent discussions about stabilization in intense laser fields [3–5]. Many studies have been car- ried out to gain an intuitive understanding of the two most prominent strong field phenomena, namely High Harmonic Generation (HHG) and Above Threshold Ion- ization (ATI). In the well established early analytical for - mulation by Keldysh, Faisal and Reiss the atomic poten- tial is treated as a perturbation for the motion of the electron in a strong laser field [6]. This picture is still used in more recent models, where the classical dynamics of the electron in the laser field is explicitly considered, e.g. in Corkum’s rescattering mode l which can explain the cutoff observed in HHG for linearly polarized laser light in one spatial dimension [7]. The corresponding Hamiltonian reads [8] H=H0+E0f(t)xsin(ω0t+δ), (1) where H0=1 2p2+V(x) is the atomic Hamiltonian, f(t) is the time-profile of the laser pulse with maximum am- plitude E0, andω0is the laser frequency. The interaction of the electron with the atom is specified by the potential V. Lewenstein et al. extended Corkum’s rescattering idea to a quasiclassical model which contains one (relevant)bound state not influenced by the laser field on the one hand side and electrons which only feel the laser field on the other side [9]. This simple model explains qualita- tively the features of HHG well. The same is also true for an alternative model, where the electron is bound by a zero-range potential [10] However, the basic ques- tion if and to which extent these multiphoton processes can be understood semiclassically, i.e., by interference o f classical trajectories alone, remains unanswered. It is as - tonishing that no direct semiclassical investigation of th e Hamiltonian Eq. (1) has been performed while a number of classical as well as quantum calculations for Eq. (1) have been published. However, only recently, a semiclas- sical propagation method has been formulated which can be implemented with reasonable numerical effort. This is very important for the seemingly simple Hamiltonian Eq. (1) whose classical dynamics is mixed and in some phase space regions highly chaotic which requires efficient com- putation to achieve convergence. Equipped with these semiclassical tools we have studied multiphoton phenom- ena semiclassically in the frame work of Eq. (1). In com- parison to the exact quantum solution we will work out those features of the intense field dynamics that can be understood in terms of interference of classical trajecto- ries. The plan of the paper is as follows. In section II we provide the tools for the calculation of a semiclassical, time-dependent wavefunction. In section III we discuss Above-Threshold-Ionization (ATI) and work out the clas- sical quantities which structure semiclassically the rele - vant observables. In section IV we use this knowledge for the description of Higher-Harmonic-Generation (HHG). Section V concludes the paper with a comparison of HHG and ATI from a semiclassical perspective and a short summary. 1II. CALCULATION OF THE SEMICLASSICAL WAVE FUNCTION A (multi-dimensional) wave function Ψ β(x, t) can be expressed as Ψ(x, t) =/integraldisplayt 0dx′K(x,x′, t)Ψ(x′). (2) Here, Ψ( x′) is the initial wave function at t= 0 and K(x,x′, t) denotes the propagator. We will not use the well-known semiclassical Van Vleck-Gutzwiller (VVG) propagator which is inconvenient for several reasons. Firstly, one has to deal with caustics, i.e. singularities of the propagator, and secondly, it is originally formu- lated as a boundary value problem. For numerical ap- plications much better suited (and for analytical consid- erations not worse) is the so called Herman-Kluk (HK) propagator which is a uniformized propagator in initial value representation [11,12], formulated in phase space, KHK(x,x′, t) =1 (2π¯h)n/integraldisplay/integraldisplay dpdqCqp(t)eiSqp(t)/¯h gγ(x;q(t),p(t))g∗ γ(x′;q,p) (3) with gγ(x;q,p) =/parenleftigγ π/parenrightign/4 exp/parenleftbigg −γ 2(x−q)2+i ¯hp(x−q)/parenrightbigg (4) and Cqp(t) =/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 2/parenleftbigg Qq+Pp−i¯hγQp−1 i¯hγPq/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 2 .(5) Each phase space point ( q,p) in the integrand of Eq. (3) is the starting point of a classical trajectory with action Sqp(t). The terms Xyin the weight factor Cqp(t) are the four elements of the monodromy matrix, Xy= ∂xt/∂y. The square root in Eq. (5) has to be calculated in such a manner that Cqp(t) is a continuous function of t. The integrand in Eq. (3) is – depending on the system – highly oscillatory. Although we restrict ourselves to one spatial dimension (see Eq. (1)) the number of trajectories necessary for numerical convergence can reach 107. We note in passing that an integration by stationary phase approximation over momentum and coordinate variables reduces the HK-propagator to the VVG-propagator [13]. In all calculations presented here we have used a Gaus- sian wave packet as initial wave function, Ψβ(x′) =/parenleftbiggβ π/parenrightbigg1/4 exp/parenleftbiggβ 2(x′−qβ)2/parenrightbigg . (6) With this choice, the overlap fγβ(q, p)≡/integraldisplay g∗ γ(x′;q, p)Ψβ(x′)dx′(7)can be calculated analytically and Eq. (2) reads together with Eq. (3) ΨH K β(x, t) =/parenleftbigg4γβ α2/parenrightbigg1 41 2π¯h/integraldisplay/integraldisplay dp dq eiSqp(t)/¯h Cqp(t)gγ(x;q(t), p(t))fγβ(q, p) (8) withα=γ+β. For all results presented here we have taken γ=β. For comparison with our semiclassical calculations we determined the quantum mechanical wave function using standard Fast Fourier Transform split operator methods [14]. III. ABOVE THRESHOLD IONIZATION We start from Eq. (1) with δ= 0 and use a rectangular pulse shape f(t) which lasts for 4 .25 optical cycles. This setting is very similar to the one used in [15]. The energy spectrum of the electrons can be expressed by the Fourier transform of the autocorrelation function after the pulse, i.e. for times t > t f, σ(ω) =Re∞/integraldisplay tfeiωt∝angbracketleftΨ(t)|Ψf∝angbracketrightdt , (9) where Ψ f= Ψ(tf) is the wave function after the pulse and correspondingly |Ψ(t)∝angbracketright=eiH0(t−tf)/¯h|Ψf∝angbracketright (10) is calculated by propagating Ψ ffor some time with the atomic Hamiltonian H0only after the laser has been switched off. A. Quantum mechanical and semiclassical spectra for ATI We will present results for two types of potentials to elucidate the dependence of the semiclassical approxima- tion on the form of the potential. 1. Softcore potential First we apply the widely used softcore potential [15,16] V(x) =−1√ x2+a(11) witha= 1 and with an ionization potential Ip= 0.670 a.u.. We have checked that the correlation function differs little if calculated with the exact ground state or with the ground state wave function approximated 2by the Gaussian of Eq. (6) where β= 0.431 a.u. and qβ= 0. However, the semiclassical calculation is consid- erably simplified with a Gaussian as initial state as can be seen from Eqs. (6-8). Therefore we use this initial state and obtain the propagated semiclassical wavefunc- tion in the closed form Eq. (8). In Fig. 1 the quantum and semiclassical results at a frequency ω0= 0.148a.u. and a field strength E0= 0.15a.u. are compared. The Keldysh parameter has the value 1 .14. The quantum mechanical calculation (dotted line) shows a typical ATI spectrum. Intensity maxima with a separation in energy of ¯ hω0are clearly visible. The first maximum has the highest inten- sity while the second maximum is suppressed. 0 1 2 3 4 5 6 7 ω/ω0σ(ω) [arb. units] FIG. 1. Quantum mechanical (dotted line) and semiclas- sical (solid line) ATI spectra for the Hamiltonian of Eq. (1) withE0= 0.15a.u., ω0= 0.148a.u. and the softcore potential Eq. (11). The semiclassical result (solid line) is ambiguous: On the one hand there are clear ATI maxima with a sepa- ration of ¯ hω0. All peaks but the first one have roughly the correct magnitude. Again the second maximum is missing. On the other hand we see a constant shift (about 0 .02a.u.) of the spectrum towards higher energies. Therefore, a quantitative semiclassical description is im - possible, at least with the present parameters and the softcore potential. Next, we will clarify whether the shift in the spectrum is an inherent problem of a semiclassical ATI calculation or if it can be attributed to properties of the softcore potential. 2. Gaussian potential To this end we take a potential which has been used to model the “single bound state” situation mentioned in the introduction [17]. It is of Gaussian form V(x) =−V0exp/parenleftbig −σx2/parenrightbig . (12) With our choice of parameters V0= 0.6 a.u. and σ= 0.025 a.u., the potential contains six bound states and can be approximated, at least in the lower energypart, by a harmonic potential for which semiclassical cal- culations are exact. Hence, the semiclassical ATI spec- trum with this potential should be more accurate ifthe discrepancies in Fig. 1 are due to the potential and not due to the laser interaction. The ground state wave func- tion itself is again well approximated by the Gaussian Eq. (6) with β= 0.154 a.u. and qβ= 0. The laser has a frequency ω0= 0.09 a.u., a field strength E0= 0.049 a.u., and a pulse duration of 4 .25 cycles. The Keldysh parameter has the value 1 .87. 0 1 2 3 4 5 6 7 ω/ω0 σ(ω) [arb. units.] FIG. 2. Quantum mechanical (dotted line) and semiclassi- cal (solid line) ATI spectra for the Hamiltonian of Eq. (1) wi th E0= 0.049a.u., ω0= 0.09a.u. and the Gaussian potential Eq. (12). We obtain a quantum mechanical ATI spectrum (dot- ted line in Fig. 2) with six distinct maxima. The semi- classical spectrum (solid line) is not shifted, the locatio n of the maxima agrees with quantum mechanics. Hence, one can conclude that the softcore potential is respon- sible for the shift. The height of the third maximum is clearly underestimated and the details of the spectrum are exaggerated by the semiclassical calculation. Apart from these deviations the agreement is good enough to use this type of calculation as a basis for a semiclassical understanding of ATI. B. Semiclassical interpretation of the ATI spectrum 1. Classification and coherence of trajectories With the chosen parameters most of the trajectories ionize during the pulse ( ∼92 %). We consider a trajec- tory as ionized if the energy of the atom ε(t) =p(t)2/2 +V(q(t)) (13) becomes positive at some time tnand remains positive, i.e.ε(t)>0 fort > t n. Typically, the trajectories ion- ize around an extremum of the laser field. Tunnelling can not be very important, otherwise the agreement between quantum mechanics and semiclassics would be much worse. The Keldysh parameter of 1 .87 suggests 3that we are in between the tunnelling and the multipho- ton regime. Interestingly, the semiclassical description is successful although we are way below energies of the classically allowed over the barrier regime. 0 T 2T 3T 4T t−0.500.51ε(t) [a.u.] Ι1Ι2Ι3Ι4 FIG. 3. Energy ε(t) from Eq. (13) for trajectories ionized in the intervals I1(solid line), I2(dashed line), I3(dashed-dotted line) and I4(dotted line), respectively. For comparison, the laser field is plotted in arbitrary units (thick dashed line) . An obvious criterion for the classification of the tra- jectories is the time interval of the laser cycle into which their individual ionization time tnfalls, see Fig. 3. Typi- cally ionization of trajectory happens around tn= (2n− 1)T/4 when the force induced by the laser reaches a max- imum. Hence, the ionized trajectories can be attached to time intervals In= [(n−1)T/2, n T/2]. In Fig. 3 we have plotted four trajectories from the intervals I1toI4which end up with an energy E= 0.36 a.u.. After ionization each trajectory shows a quiver motion around a mean momentum pf[18]. One can distinguish two groups of in- tervals, namely those with trajectories ionized with pos- itive momentum pf(the intervals I2k−1) and those with trajectories with negative pf(the intervals I2k). These two groups contribute separately and incoherently to the energy spectrum as one might expect since the electrons are easily distinguishable. One can see this directly from the definition Eq. (9) of the electron energy spectrum. For relative high energies ¯ hωthe (short-range) potential may be neglected in the Hamiltonian H0and we get σ(ω) =Re∞/integraldisplay tfeiωt∝angbracketleftΨf|e−iH0(t−tf)|Ψf∝angbracketrightdt ≈Re∞/integraldisplay 0eiωt∝angbracketleftΨf|e−ip2t/2¯h|Ψf∝angbracketrightdt =∞/integraldisplay −∞δ/parenleftbig ω−p2/2¯h/parenrightbig |Ψf(p)|2dp =/parenleftbigg/vextendsingle/vextendsingle/vextendsingleΨf(−√ 2¯hω)/vextendsingle/vextendsingle/vextendsingle2 +/vextendsingle/vextendsingle/vextendsingleΨf(√ 2¯hω)/vextendsingle/vextendsingle/vextendsingle2/parenrightbigg (¯h/2ω)1/2 ≡σ−(ω) +σ+(ω). (14)Hence, to this approximation, the ATI spectrum is in- deed given by the incoherent sum of two terms belonging to different signs of the momenta of electrons ionized in different time intervals as described above. Figure Fig. 4(a) shows that Eq. (14) is a good approx- imation. σ(ω) [arb. units] 0 1 2 3 4 5 6 ω/ω0 σ(ω) [arb. units](a) (b) FIG. 4. Upper panel (a): Semiclassical spectrum as an in- coherent sum σ+(ω) +σ−(ω) (dashed-dotted line) compared with the full semiclassical spectrum (solid line). Lower pa nel (b): Semiclassical spectrum σ+(ω) , constructed with trajec- tories from the intervals I2,I4,I6andI8(dotted) compared to the incoherent sum ˜ σ+of spectra that belong to the intervals I2toI8(solid line). Only for small ωthe spectra do not agree, where the kinetic energy is comparable with the (neglected) poten- tial energy. Quantum mechanically, all contributions from trajec- tories which lead to the same momentum pfof the elec- tron are indistinguishable and must be summed coher- ently. To double check that the interference from dif- ferent intervals Inis responsible for the ATI peaks, we can artificially create a spectrum by an incoherent super- position ˜ σ+=σ2+σ4+σ6+σ8of contributions from trajectories ionized in the intervals I2j. This artificially incoherent sum (Fig. 4(b)) shows similarity neither with σ+(ω) nor with any kind of ATI spectrum. 2. Classical signature of bound and continuum motion in the laser field The great advantage of an ab initio semiclassical de- scription lies in the possibility to make dynamical be- havior transparent based on classical trajectories, par- ticularly in the case of explicit time dependent problems where our intuition is not as well trained as in the case of conservative Hamiltonian systems. The classical quanti- 4ties enter semiclassically mostly through the phase factor exp (i[Sqp(t)−p(t)q(t)]/¯h)≡exp[iΦ/¯h] (15) which each trajectory contributes to the wave function Eq. (8). Although the prefactor Cqp(t) in Eq. (8) may be complex itself, the major contribution to the phase comes from the effective action Φ in the exponent of Eq. (15). Figure 5 shows the energy εof the atom and the accu- mulated phase Φ. One can recognize a clear distinction between a quasi-free oscillation in the laser field after the ionization and the quasi-bound motion in the potential. The latter is characterized by an almost constant aver- aged bound energy ∝angbracketleftε(t)∝angbracketright(Fig. 5(a)) of the individual trajectory giving rise to an averaged linear increase of the phase (Fig. 5(b)). After ionization the phase decreases linearly with an oscillatory modulation superimposed by the laser field. The almost linear increase of Φ without strong modulation of the laser field during the bound mo- tion of the electron is remarkable, particularly looking at the laser induced modulations of the bound energy seen in Fig. 5(a). The averaged slope of the phase (positive for bound motion, negative for continuum motion) cor- responds via dΦ/dt=−Eto an averaged energy. The behavior can be understood by a closer inspection of the action Φ(t)≡Sqp(t)−p(t)q(t) =/integraldisplayt 0(2T−H−˙p(τ)q(τ)−˙q(τ)p(τ))dτ−qp .(16) Here, T=p2(t)/2 refers to the kinetic energy and H to the entire Hamiltonian of Eq. (1), the dot indicates a derivative with respect to time, and q≡q(t= 0). With the help of Hamilton’s equations and a little algebra Φ from Eq. (16) can be simplified to Φ(t) =−/integraldisplayt 0/parenleftbigg ε(τ)−q(τ)dV dq/parenrightbigg dτ (17) where εis the atomic energy Eq. (13). With Eq. (17) we can quantitatively explain the slope of Φ in Fig. 5(b). For the low energies considered the potential Eq. (12) can be approximated harmonically, V(q)≈ −V0+V0σq2(18) Averaging Φ over some time yields then Φ( t)≈V0t, foranybound energy of a classical trajectory since for an oscillator averaged kinetic and potential energy are equal. Indeed, the numerical value for the positive slope in Fig. 5(b) is 0 .6a.u. in agreement with the value for V0. For the ionized part of the trajectories we may assume that the potential vanishes. The corresponding solutions for electron momentum p(t) follows directly from Hamil- ton’s equation ˙ p=−E0sinω0t, p(t) =E0 ω0cos(ω0t) +p, (19)where pis the mean momentum. Without potential the phase from Eq. (17) reduces to Φ( t) =−/integraltextp2(τ)/2dτ and we obtain with Eq. (19) Φc(t) =−Up 2ω0sin(2ω0t)−E0p ω2 0sinω0t−(Up+p2/2)t(20) with the ponderomotive potential Up=E2 0/4ω2 0. We note in passing that Eq. (20) is identical to the time de- pendent phase in the Volkov state (see the appendix). −0.500.51ε(t) [a.u.] 0 1 2T 3T 4T t−50050100Φ(t) [a.u.](a) (b) FIG. 5. Part (a) shows the atomic energy ε=p2/2 +V(q) as a function of time for three trajectories from the interva ls I2(dashed line), I4(dotted line) and I6(dashed-dotted line), part (b) shows the corresponding phases Φ( t). 3. Semiclassical model for ATI The clear distinction between classical bound and con- tinuum motion in the laser field as demonstrated by Fig. 5 and illuminated in the last section, allows one to derive easily the peak positions of the ATI spectrum. Moreover, this distinction also supports the so called strong field ap- proximation (e.g. [9,19]) where electron dynamics in the laser field is modelled by one bound state and the contin- uum. While this is postulated in [9] as an approximation and justified a posteriori by the results the correspond- ing approximation is suggested in the present context of a semiclassical analysis by the full classical dynamics, i.e ., the behavior of the trajectories, as shown in 5. There, we have seen that each classical bound motion leads to the characteristic linear increase of the phase. If the entire phase space corresponding to the initial (ground state) wave function is probed with many trajectories of differ- ent energy, the dominant contribution will appear at the bound state energy which implies Φb(t)≈Ipt , (21) 5where Ipis the ionization potential. The time for which a trajectory does not fall into one of the two classes, bound or continuum, is very short (Fig. 5). Hence, we can approximately compose the true phase Φ = Φ b+ Φc. However, we don’t know for an electron with mean momentum pwhen it was ionized. Hence, we have to sum over all trajectories with different ionization times τbut equal final momentum p=pfwhich leads to the propagated wavefunction Ψf(t, p)∼/integraldisplayt t0dτexp[i/¯h(Φb(τ) + Φ c(t)−Φc(τ))] ∼/summationdisplay n,mJn/parenleftbiggE0p ω2 0/parenrightbigg Jm/parenleftbiggUp 2ω0/parenrightbigg/integraldisplayt t0dτeiτ∆mn/¯h,(22) where the phase ∆ is given by ∆mn=Ip+Up+p2/2−(n+ 2m)¯hω0.(23) From Eq. (23) and Eq. (22) follows that ATI peaks ap- pear at integer multiples n¯hω0of the laser frequency, when p2 2=n¯hω0−Ip−Up. (24) One can also see from Eq. (22) that the ATI maxima be- come sharper with each optical cycle that supplies ioniz- ing trajectories. Of course, this effect is weakened by the spreading of the wavepacket hidden in the prefactor of each trajectory contribution (see Eq. (8)) not considered here. Trajectories that are ionized during different laser cy- cles accumulate a specific mean phase difference. The phase difference depends on the number kof laser cycles passed between the two ionization processes: ∆Φ(p) =k T/parenleftbigg Ip+p2 2+Up/parenrightbigg . (25) The trajectories interfere constructively if ∆Φ(p) = 2 πl⇒1 2p2=l kω0−Ip−Up.(26) If an energy spectrum is calculated exclusively with tra- jectories from two intervals separated by kcycles there should be additional maxima in the ATI spectrum with a distance ¯ hω0/k. As a test for this semiclassical interpretation of the ATI mechanism we have calculated three spectra with trajectories where the mean time delay between ionizing events is given by ∆ t=T, ∆t= 2Tand ∆ t= 3T. For the spectrum Fig. 6 (a) we have used exclusively trajec- tories from the intervals I2andI4(∆t=T). One can see broad maxima separated by ¯ hω0in energy. Trajec- tories from the intervals I2andI6(see Fig. 6 (b)) form a spectrum where the maxima are separated by ¯ hω0/2 – as predicted for ∆ t= 2T. In analogy the separation forthe ATI maxima in a spectrum with trajectories from the intervals I2andI8is given by ¯ hω0/3 (Abb. 6 (c)). The interference of trajectories ionized in many subsequent cycles suppresses the non-integer maxima according to Eq. (23). If the field strength is high enough the atom is completely ionized during the first cycle. The oppor- tunity for interference gets lost and we end up with an unstructured energy spectrum. σ(ω) [arb. units] σ(ω) [arb. units] 0 1 2 3 4 5 6 ω/ω0σ(ω) [arb. units](a) (b) (c) FIG. 6. Semiclassical spectra calculated with trajectorie s from the intervals I2andI4(a),I2andI6(b), and I2andI8 (c). In an extreme semiclassical approximation we would have evaluated the integral in Eq. (22) by stationary phase. The condition d/dτ[Φb(τ)−Φc(τ)]≡Ip+p2(τ)/2 = 0 (27) leads to complex ionization times tnwhose real part is periodic and allows for two ionizing events per laser cycle, close to the extrema of the laser amplitude. The deriva- tion is simple but technical, therefore we don’t carry it out explicitely here. However, it explains the observa- tion that ionization occurs close to the extrema of the laser field and it also makes contact with the tunnelling process often referred to in the literature since the com- plex time can be interpreted as tunnelling at a complex ”transition” energy. Clearly, our semiclassical analysis as described here supports the picture which has been sketched in [20] interpreting a quantum calculation. The authors as- sume that wave packets are emitted every time the laser reaches an extremum. The interference of the different wave packets gives rise to the ATI peaks. In the following we will discuss the process of higher harmonic generation (HHG) which is closely related to ATI. In fact, the separation into a bound and continuum 6part of the electron description is constitutive for HHG as well, the prominent features, such as cutoff and peak locations, can be derived from the same phase properties Eq. (22) as for ATI. However, there is a characteristic difference, howthese phases enter. IV. HIGH HARMONIC GENERATION First, we briefly recapitulate the findings of [21], where we have calculated the harmonic spectrum with the soft- core potential Eq. (11). With our choice of a= 2 the ion- ization potential is given by Ip= 0.5a.u.. The laser field has a strength E0= 0.1 a.u., a frequency ω0= 0.0378 a.u. and a phase δ=π/2. The initial wave packet with a width of β= 0.05a.u. is located at qβ=E0/ω2 0= 70a.u.. Note, that the cutoff energy ECin such a symmetric laser scattering experiment is given by EC=Ip+ 2Up. (28) From the dipole acceleration (see Fig. 7) d(t) =−/angbracketleftbigg Ψ(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingledV(x) dx/vextendsingle/vextendsingle/vextendsingle/vextendsingleΨ(t)/angbracketrightbigg , (29) follows by Fourier transform σ(ω) =/integraldisplay d(t) exp( iωt)dt (30) the harmonic power spectrum (see Fig. 8). 0 T/2 T 3T/2 2T 5T/2 3T t−0.006−0.00300.003d(t) [a.u.]−0.00300.003d(t) [a.u.](a) (b) FIG. 7. Quantum (a) and semiclassical (b) dipole acceler- ation of higher harmonics according to Eq. (29).0 40 80 120 Harmonic Order10−710−610−510−4σ(ω) [a.u.]10−710−610−510−4σ(ω) [a.u.](a) (b) FIG. 8. Quantum (a) and semiclassical (b) spectrum of higher harmonics according to Eq. (30). Clearly, our semiclassical approach represents a good approximation. The dipole acceleration shows the char- acteristic feature that fast oscillations (which are respo n- sible for the high harmonics in Fourier space) show only up after some time, here after t=T. This is the first time where trajectories are trapped. Trapping can only occur if (i) tn=nT/2, (ii) the trajectories reach a turning point (i.e. p(tn) = 0), and (iii) if at this time the electron is close to the nucleus ( q(tn)≈0). The trapped trajecto- ries constitute a partially bound state which can interfere with the main part of the wavepacket (trajectories) still bouncing back and forward over the nucleus driven by the laser. The group of briefly bound (i.e. trapped or stranded) trajectories can be clearly identified, either by their small excursion in space (Fig. 9a) or by the positive slope of their action (Fig. 9b) as it was the case for ATI (compare with Fig. 5). By artificially discarding the ini- tial conditions in the semiclassical propagator which lead to trapped trajectories one can convincingly demonstrate that the plateau in HHG generation is a simple interfer- ence effect [21]. Here, we are interested firstly in linking ATI to HHG by using the same separation in bound and continuum parts of the dynamics already worked out for ATI. Secondly, we want to go one step further and con- struct a wavefunction based on this principle. Semiclassically, we have to look first at the phases of the observable. Therefore, we define a linear combination for the wavefunction from the respective phase factors for bound and continuum motion. Considering only terms in the exponent the harmonic spectrum Eq. (30) reads simply 7−1000100q(t) [a.u.] 0 T/2 T 3T/2 2T 5T/2 3T t−600−400−2000Φ(t) [a.u.](a) (b) FIG. 9. Examples for direct (solid line), trapped (dotted line), and stranded (dashed line) trajectories, see text. σ(ω)∼/integraldisplay dtexp(iωt) |exp(iΦc(t)/¯h) +cexp(iΦb(t)/¯h)|2, (31) where c∝negationslash= 0 is a (so far) arbitrary constant. In princi- ple,c=c(t), however its change in time is much slower than that of the optical oscillations of the phases Φ( t), hence we may approximate cby a constant. The bound and continuum phases, Φ band Φ c, are defined in Eq. (21) and Eq. (20), respectively. For Φ cwe have p= 0, since this is the dominant contribution from the center of the wavepacket which was initially at rest. The re- sult is shown in Fig. 10. Indeed, the plateau with the harmonics is generated, however, the initial exponential decrease is missing since we have neglected all prefac- tors of the semiclassical wavefunction which describe the dispersion of the wavepacket. 0 40 80 120 Harmonic Orderσ(ω) [arb. units] FIG. 10. Harmonic spectrum according to Eq. (31). Consequently, one can evaluate Eq. (31) in stationary phase approximation. The integrand of Eq. (31) becomesstationary if d dt[¯hωt±(Φb(t)−Φc(t))] = 0 (32) which happens at ¯hω= 2Upsin2(ωt) +Ip. (33) From Eq. (33) we conclude the cut-off law ωmax= 2Up+Ip, (34) as expected for laser assisted electron ion scattering [21] . Using the same expansion into Bessel functions as in Eq. (22) we obtain for the spectrum Eq. (31): /integraldisplay dtexp/parenleftbiggi ¯h/bracketleftbigg (¯hω−Up−Ip)t+Up 2ω0sin (2ω0t)/bracketrightbigg/parenrightbigg =∞/summationdisplay k=−∞/integraldisplay dt eit(¯hω−Up−Ip+2k¯hω0)/¯hJk/parenleftbiggUp 2¯hω0/parenrightbigg .(35) Therefore, we see maxima in the harmonic spectrum for ¯hωk=Up+Ip−2kω0. (36) We can go one step further and construct a full time- dependent wavefunction from this semiclassical approxi- mation, namely Ψ(x, t) = Ψsc β(x, t) +cΨ0(x)exp(itIp/¯h). (37) Here, Ψ 0(x)exp(iIpt/¯h) is the time dependent ground state wave function (without the laser field) and Ψsc β(x, t) is a (semiclassical) wavepacket in the laser field but with- out potential. Calculating the dipole acceleration and the resulting harmonic spectrum with this wavefunction leads to a remarkably good approximation of the true quantum spectrum (compare Fig. 8 with Fig. 11). The dispersion of the wavepacket leads to the lower plateau compared to Fig. 10. 0 40 80 120 Harmonic Order10−710−610−510−4σ(ω) [a.u.] FIG. 11. Harmonic spectrum, generated from the wave- function Eq. (37) with c= 0.025 and β= 0.05 a.u.. V. CONCLUSIONS 8A. Semiclassical comparison between ATI and HHG Clearly, the main structure such as the plateau, cutoff (HHG) and the occurrence of peaks and their separation in energy (ATI and HHG) is a property of the differ- ence of the classical time-dependent actions Φ b(t)−Φc(t) alone. However, the HHG power spectrum Eq. (30) is an inte- gral over all the time for which the electron wavepacket is exposed to the laser field. In contrast, the ATI spectrum is obtained in the long-time limit t→ ∞ after the laser has been switched off. This difference may explain why the HHG results tend to be better than the ATI results semiclassically: Any semiclassical approximation (which is not exact) become worse for large times. A second point refers to the fact that the character- istic phase difference Φ b(t)−Φc(t) appears already in the wavefunction Eq. (22) for ATI, while for HHG it oc- curs only in the expectation value Eq. (29). However, this difference is artificial, since the expectation value, or better its Fourier transform the power spectrum, is not the observable of higher harmonic radiation. The correct expression is the dipole-dipole correlation func- tionRwhich can be approximated as R∝ |σ(ω)|2under single atom conditions or in the case of an ensemble of independent atoms which radiate [10,22]. Hence, in both cases, ATI and HHG, the peak structure appears already on the level of the quantum amplitude (or wavefunction) and is amplified in the true observable. B. Summary We have given a time-dependent fully semiclassical description of multiphoton processes. The prominent ATI and HHG features emerge naturally from proper- ties of the classical trajectories whose contributions to the semiclassical wavefunction interfere semiclassicall y. Any effect of this semiclassical interference can be double- checked by disregarding the phases. This leads (with the same trajectories) to a classical observable. As we have seen, to a good approximation the classical action for an individual trajectory can be composed of one part Φ b for the time the electron is bound (disregarding the laser field) and of another part Φ cfor the time the electron is in the continuum (disregarding the atomic potential). The relevant phase difference Φ b−Φcleads in both cases, ATI and HHG, to the prominent harmonic structures in terms of the laser energy ¯ hω0. Finally, we have been able to construct a simple wavefunction for higher har- monics generated in laser assisted scattering. Its key ele- ment is an explicitely time-dependent wavepacket of the electron under the influence of the laser field. Starting from an initial Gaussian distribution localized in space the wavepacket disperses in time providing the correct decrease of the intensity of the lower harmonics and in turn the correct height of the plateau.Financial support from the DFG under the Gerhard Hess-Programm and the SFB 276 is gratefully acknowl- edged. APPENDIX: We want to calculate the semiclassical wave function of a free particle in a laser field according to Eq. (8). A particle in a laser field VL(x, t) =E0sin(ωt) moves with p(t) =p+E0 ωcos(ωt)≡p+ ˜p(t) (A1) q(t) =q+p t+E0 ω2sin(ωt)≡q+p t+ ˜q(t) (A2) The weight factor Cqp(t) is given by Cqp(t) =/parenleftbigg 1−i¯hγ 2t/parenrightbigg1 2 . (A3) For the phase factor Sqp(t)−p(t)q(t) we get: Sqp(t)−p(t)q(t) =−Up 2ωsin(2ωt)−Upt −p2 2t−˜q(t)p−q p (A4) Evaluating Eq. (8) with the stationary phase approxima- tion, which is exact for quadratic potentials, leads to the condition that f(q, p) =i ¯h/parenleftbigg xp(t)−p2 2t−˜q(t)p−γ αq p−β αqβp/parenrightbigg −γ 2(x−q(t))2−γβ 2α(q−qβ)2−1 2¯h2αp2 (A5) must have an extremum. With ∂f ∂q= 0 = γ[x−q(t)]−γβ α(q−qβ)−i ¯hγ αp(A6) ∂f ∂p= 0 = γ[x−q(t)]t−1 ¯h2αp +i ¯h/parenleftbigg x−p t−˜q(t)−γ αq−β αqβ/parenrightbigg (A7) we find qs=x−˜q(t) +i¯hβtq β 1 +i¯hβt(A8) ps=i¯hβ 1 +i¯hβt(x−˜q(t)−qβ). (A9) 9After some algebra we arrive at the stationary exponent f(qs, ps) =i ¯hx˜p(t)−β 2 (1 + i¯hβt)(x−˜q(t)−qβ)2 =i ¯hx˜p(t)−i ¯h¯h2β2t 2σ(t)(x−˜q(t)−qβ)2 −β 2σ(t)(x−˜q(t)−qβ)2, (A10) where σ(t) is given by σ(t) = 1 + β2¯h2t2. (A11) The determinant of the second derivatives of fstill has to be calculated. With ∂2f ∂q2=−γ4+ 2γβ α∂2f ∂p2=−i ¯ht−γt2−1 ¯h2α ∂2f ∂q∂p=−i ¯hγ α−γt (A12) we get det ∂2f ∂q2∂2f ∂q ∂p ∂2f ∂p ∂q∂2f ∂p2 =2γ ¯h2α/parenleftig [1−iγ¯ht/2][1 + iβ¯ht]/parenrightig . (A13) The factor γcancels as it should be and we are left with Ψsc β(x, t) =/parenleftbiggβ π/parenrightbigg1/4/radicalbigg1 1 +i¯hβt exp/parenleftbiggi ¯h/bracketleftbigg ˜p(t)x−Up 2ωsin(2ωt)−Upt/bracketrightbigg/parenrightbigg exp/parenleftbiggi ¯h¯h2β2 2σ(t)(x−˜q(t)−qβ)2t/parenrightbigg exp/parenleftbigg −β 2σ(t)(x−˜q(t)−qβ)2/parenrightbigg .(A14) This semiclassical time dependent wavepacket is quan- tum mechanically exact and corresponds to a superpo- sition of Volkov solutions according to a Gaussian dis- tribution at time t= 0 [23]. The fact that the semi- classical wavefunction is exact is a direct consequence of the Ehrenfest theorem which implies that interactions V∝xn,n= 0,1,2 have quantum mechanically exact semiclassical solutions. [1] K. C. Kulander, K. J. Schafer, and J. L. Krause, Adv. At. Mol. Phys. Suppl. 1 (Atoms in intense Laser Fields), 247 (1992). [2] M. Protopapas, C. H. Keitel, and P. L. Knight, Rep. Prog. Phys. 60, 389 (1997).[3] K. C. Kulander, K. J. Schafer, and J. L. Krause, Phys. Rev. Lett. 66, 2601 (1991). [4] B. Sundaram and R. V. Jensen, Phys. Rev. A 47, 1415 (1993). [5] M. Yu. Ivanov, O. V. Tikhonova, and M. V. Feodorov, Phys. Rev. A. 58, R793 (1998); O. V. Tikhonova, E. A. Volkova, A. M. Popov, and M. V. Feodorov, Phys. Rev. A.60, R749 (1999). [6] F. H. M. Faisal, J. Phys. B 6, L89 (1973); H. R. Reiss, Phys. Rev. A 22, 1786 (1980). [7] P. B. Corkum, Phys. Rev. Lett. 71, 1994 (1993). [8] M. Protopapas, D. G. Lappas, C. H. Keitel, and P. L. Knight, Phys. Rev. A. 53, R2933 (1995). [9] M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huiller, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994). [10] W. Becker, S. Long, and J. K. McIver, Phys. Rev. A 50, 1540 (1994). [11] M. F. Herman and E. Kluk, Chem. Phys. 91, 27 (1984). [12] K. G. Kay, J. Chem. Phys. 100, 4377 (1994). [13] F. Großmann, Comments At. Mol. Phys. 34, 141 (1999). [14] J. A. Fleck, J. R. Morris, and M. D. Feit, App. Phys. 10, 129 (1976). [15] J. Javanainen, J. H. Eberly, and Q. Su, Phys. Rev. A 38, 3430 (1988). [16] D. G. Lappas, A. Sanpera, J. B. Watson, K. Burnett, P. L. Knight, R. Grobe, and J. H. Eberly, J. Phys. B 29, L619 (1996). [17] C. Figueira de Morisson Faria, M. D¨ orr, and W. Sandner, Phys. Rev. A 55, 3961 (1997). [18] As already discussed in [15] it is therfore important th at the calculatuions are stopped at times tm= (2m−1)T/4 so that the ionized trajectories terminate at a time where they have their mean momentum. [19] M. Lewenstein, K. C. Kulander, K. J. Schafer, P. H. Bucksbaum, Phys. Rev. A 51, 1495 (1995). [20] D. G. Lappas and P. L. Knight, Comments At. Mol. Phys. 33, 237 (1997). [21] G. van de Sand and J. M. Rost, Phys. Rev. Lett. 83, 524 (1999). [22] B. Sundaram, P. W. Milonni, Phys. Rev. A 41, 6571 (1990). [23] S. Virito, K. T. Taylor, and J. S. Parker, J. Phys. B 32, 3015 (1999). 10

1Structural information from multilamellar liposomes at full hydration: full q-range fitting with high quality X-ray data Georg Pabst, Michael Rappolt, Heinz Amenitsch and Peter Laggner Institute of Biophysics and X-ray Structure Research, Austrian Academy of Sciences, Steyrergasse 17, A-8010 Graz, Austria. (E-mail: Peter.Laggner@oeaw.ac.at). Receipt date: Nov. 1999 PACS number(s): 61.30.Cz, 87.64.Bx, 61.10.Eq, 61.30.Eb2Abstract We present a novel method for analyzing Small Angle X-ray Scattering data on multilamellar phospholipid bilayer systems at full hydration. The method utilizes a modified Caillé theory structure factor in combination with a Gaussian model representation of the electron density profile such that it accounts also for the diffuse scattering between Bragg peaks. Thus, the method can retrieve structural information even if only a few orders of diffraction are observed. We further introduce a new procedure to derive fundamental parameters, such as area per lipid, membrane thickness, and number of water molecules per lipid, directly from the electron density profile without the need of additional volumetric measurements. The theoretical apparatus is applied to experimental data on 1-palmitoyl-2-oleoyl- sn-glycero-3- phosphocholine and 1,2-dipalmitoyl- sn-glycero-3-phosphoethanolamine liposome preparations.3I. INTRODUCTION Phospholipids are the main constituents of biological membranes by forming the structural matrix into which functional membrane units such as proteins are imbedded. Among the various structures that are formed by phospholipid membranes, the lamellar liquid crystalline phase is the biologically most relevant one. The interest in the structure and physical properties of this particular phase has therefore always been an important subject in biophysical and biochemical research, since the structure is directly related to the function of the molecular aggregates. But not only the efforts to understand the function of biological membranes drive the progress in phospholipid structure research, also phospholipid-based rational drug design and bio-mimetic material development rely on physical interaction predictions. The structural characterization of phospholipid model membranes was initiated by the pioneering work of Luzzati and coworkers [1, 2] on unoriented multilayers of diacyl- phosphocholines and was followed by a large number of X-ray and neutron scattering experiments on different phospholipid bilayer structures [3, 4]. However, the major difficulties in obtaining accurate structural data arise, apart from thermal disorder ("disorder of first kind"), from disorder in the crystal lattice ("disorder of second kind"), which is mostly dominant in the liquid crystalline phases due to their liquid properties. Two theories have been developed to model the lattice structure factor of model membranes, both accounting for the deficiencies in long range order: the paracrystalline theory (PT), a general theory for disorder of first and the second kind originated by Hosemann and Bagchi [5] and Guinier [6], and the Caillé theory (CT) [7], which was invoked for smectic liquid crystals only. The main difference between the two models is that the paracrystalline theory describes the stochastic4fluctuations of the single, ideally flat layers, whereas the Caillé theory considers also bilayer undulations by applying a Hamiltonian description derived from the free energy density of a lipid bilayer, originally derived by De Gennes [8]. In 1994 the Caillé theory was modified by Zhang et al. [9] (MCT), in order to take the finite size of the lamellar stack into account; a similar expression was obtained by Nallet et al. [10]. Both theories (PT & MCT) have been applied to experimental data [10-17], but with the help of the high resolution capabilities of modern synchrotron radiation sources the superiority of the Caillé theory could clearly be demonstrated [15]. The facts therefore encourage to use MCT for smectic A liquid crystals, and moreover tests on our own data gave better fits for MCT than for a PT model (results not shown). But having an employed theory describing well the crystal lattice and thus the position and shapes of the diffraction peaks does not overcome a principle problem of liquid crystallography: As a consequence of the lattice disorder multilamellar liposomal suspensions give hardly rise to a sufficient number of diffraction orders to derive structural information. Among the zwitterionic phospholipids the situation is somewhat better for phosphatidylethanolamine (PE) membrane stacks, exhibiting 4 Bragg peaks throughout the whole L α-phase, whereas the higher water content in phosphatidylcholine (PC) bilayers leads to a higher lattice disorder and thus to even less diffraction peaks observed. As a consequence, the electron density profiles are very poor in detail and likely to be affected by Fourier truncation errors. There are two ways to circumvent this problem, both applying osmotic pressure techniques. (1) One is to incubate multilamellar liposomes in aqueous solutions containing various concentrations of large, neutral polymers such as dextran or polyvinylpyrrolidone (PVP) [14-21]. With such "swelling experiments" the system is partly dehydrated, and consequently the number of observed diffraction orders increases. Structural5parameters for the fully hydrated bilayer are then obtained by extrapolating the areas per lipid, derived from the partly dehydrated systems to full hydration [14-17]. (2) Even more structural information can be obtained by exposing oriented multilayers to constant relative humidity atmospheres [21-25] and depending on the degree of hydration, up to 10 diffraction orders have been recorded [23, 24]. The electron density profile from such experiments is much richer in information and even allows for a quasimolecular modeling, first applied by Wiener and White [23, 26, 27]. The phospholipid molecule is partitioned into quasimolecular fragments, and the contribution of each fragment to the bilayer profile is modeled by a Gaussian distribution. In this manner structural details have been obtained by a joint refinement of neutron and X-ray data sets [23]. Still, the major drawback of measuring oriented sample in humidity chambers is that the bilayer repeat does not swell to the value reached in the unoriented case under excess water conditions, even at 100% relative humidity. Consequently, the fully hydrated L α-phase cannot be exploited with this technique. The so- called “vapor pressure paradox” has been for a long time a disputed topic in the lipid community. Recently, Katsaras installed a new cell for oriented bilayers [28] and could demonstrate that the vapor pressure paradox originates simply from experimental inadequacy and has no theoretical background [29]. Hence the ghost of the vapor pressure paradox ceased haunting through the brains of lipid scientists and diffraction experiments on oriented membrane stacks will be of prime importance in future phospholipid structure research. However, unoriented multilamellar liposomes at full hydration are still a frequent measurement situation. Not least simulations of biological systems and development of new drugs, e.g., carrier systems, will always demand the work with liposomal dispersions in the excess water situation. Here the information content is very low, if only Bragg-peaks are considered in the data analysis. We invoke a model that accounts also for the diffuse6scattering of the bilayer between diffraction peaks, and thus, exploits the complete data recorded in a continuous q-range. In this way our method is capable to retrieve fundamental structural parameters, such as membrane thickness, area per lipid, and number of waters, even under above conditions, when only a few orders of diffraction are observed. We further introduce a procedure, based on simple geometric relationships, to calculate the above named parameters directly from a electron density model of the bilayer, without the need of extra volumetric measurements. II. THEORY The intensity scattered from a finite stack of unoriented bilayers is described by ()()() 22 qqsqf qI∝ , (1) where q is the absolute value of the scattering vector ( q = 4 p sinq /l), f(q) the form factor and s(q) the structure factor. The form factor characterizes the electron density distribution and is given in the case of a layered structure by the Fourier transform () () ( )dzzqi z qf exp∫=r (2) of the electron density profile r along the z-axis. The structure factor accounts for the crystalline or quasi-crystalline nature of the lattice of the bilayer stack in the liquid crystalline phase. Both, structure and form factor, are averaged over the bilayer fluctuations. By assuming that the fluctuations within the bilayer are independent of the fluctuations of the lattice points, the structure factor and the form factor can be treated separately according to Debye [6]7() ()() () () ( ) [ ]2 2 2 21qf qfN qs qfqqI − + ∝ . (3) The last term in Eq. (3) gives rise to a diffuse scattering and is usually neglected, when structural information is derived from Bragg peaks only. The standard data analysis procedure is then to fit the Bragg reflections with the appropriate structure factor multiplied by a constant form factor for each single peak, which is a reasonable assumption in the vicinity of the diffraction peaks only. The electron density profile relative to the constant electron density of the buffer (water) is calculated by the Fourier synthesis ()∑=±=max 12cos *h hh dzhF zpr , (4) wherein h is the order of reflection and d the size of the unit cell. We invoke a model that tries to solve the problem in the backward direction by means of an inverse Fourier transform. Since we record data in a continuous q-range, we should rather model the scattering function I(q) in the whole range studied. The electron density profile - at a given resolution of 4 diffraction orders – can be modeled according to Wiener et al. [30] by a summation of 2 Gaussians, each representing the polar headgroup and the methyl terminus1, respectively ()() ()   − +        +−+   −− += 22 22 22 2exp2exp2exp2 CC HH HH H CHz zz zzzsrs sr r r , (5) where the electron densities of the headgroup Hr and hydrocarbon tails Cr are defined relative to the methylene electron density2CHr 1 Wiener and White were able to model the bilayer profile with a summation of 8 Gaussians [23] representing quasi-molecular phospholipid fragments for oriented dioleoylphosphatidylcholine bilayers at 66% RH. However, this model is not applicable for the present case of the resolution limit of 4 and less diffraction orders.82CH H Hr r r −≡ 2CH C Cr r r −≡ , (6) (Fig. 1). The position of the Gaussian peak is at zi (i = H, C; zC = 0), with a standard deviation of si. The form factor of this electron density model can be calculated analytically by applying Eq. (2) () () () ()qFqF qF qf C H+ = = 2 , (7) where the individual components denote the form factor of the headgroup () ()HH H H HzqqqF cos2exp 222    − =srsp (8) and the form factor of the hydrocarbon chains ()   − =2exp 222qqFC C C Csrsp . (9) Equation (7) gives the time averaged form factor of the bilayer as a continuous function of the scattering vector q. Since the structure factor retained from the Caillé theory considers the lattice disorder, a full q-range description will also account for the diffuse scattering term in Eq. (3). We choose the discrete formula of the MCT structure factor [9] in the equivalent form of ()() ()() () ∑− =−  − − +==1 122 122 122 cos 2N kqd qd k e kqd kN N qS qsh pghpp , (10) given in a manuscript of Lemmich et al. [31]. The mean number of coherent scattering bilayers in the stack is denoted as N, g is Eulers constant. The Caillé parameter h1 involves both, the bending modulus K of lipid bilayers and the bulk modulus B for compression [7, 9] KBkTq h ph 82 = (11)9with 2 1h hh h=. (12) However, we discovered during our data analysis an additional diffuse scattering contribution, which is not described by the MCT. Its origin is attributed to bilayers with strong lattice defects or unilamellar vesicles, which display neither short-range nor (quasi) long-range order. The total scattered intensity is therefore given by the diffraction of the phospholipid multilayers within the quasi long-range order lattice, plus the additional diffuse scattering of single, uncorrelated bilayers () ()() () ( )2 2 21qF NqSqFqqI diff+ ∝ . (13) We will refer in the further context of this article to the above described model as MCG, since it is a combination of MCT and a Gaussian electron density representation of the headgroup [30]. A further benefit of this method is that one can derive structural parameters from simple geometric relationships, without the need of volumetric data as, e.g., in the approach of McIntosh and Simon [32], or Nagle et al. [14]. For determining the area per lipid, we follow the formalism given by Lemmich et al. [33] by calculating the ratio C H rr r r /~≡ (see Eq. (5)), which yields ()    −−= He H Ce Cr r CHdn dnAr r r~ 1~1 2, (14) where e Cn is the number of hydrocarbon electrons and e Hn the number of headgroup electrons, respectively. The headgroup size dH can be estimated from the FWHM of the Gaussian, representing the headgroup and the hydrocarbon chain length dC can be derived from 2H H CFWHMz d −= . (15)10Further parameters of interest are the bilayer thickness + =22H H BFWHMz d , (16) the thickness of the water layer B Wdd d −= , (17) and the number of interbilayer free water per lipid molecule WW W VdAn 2*= (18) (see, e.g., [1, 14, 32]), where VW is the volume of one water molecule (approx. 30 Å3). The total number of water, including the molecules intercalated into the bilayer, can be estimated from the distance of the headgroup to the bilayer center zH ( ) WH W Vz dAn−=2/. (19) Finally, the electron density profile can be set on an absolute scale. Here we follow the procedure introduced by Nagle and Wiener [34] by calculating the integral ()( )()∫ ∫       − +   −− = −2/ 022 222/ 02exp2exp2d CC HH Hd CHdzz zzA dz z Asrsr a r r , (20) wherein α is the instrumental scaling constant. The evaluation of the left integral gives ()( )22 22/ 0dAn n dz z ACH e We Ld CHrr r −+= −∫ , (21) with e Ln being the number of electrons of the phospholipid molecule and e Wn the number of water electrons, i.e., the total number of waters per lipid molecule times the number of electrons in one water molecule. The Integral on the right is given by ()∫       − +   −− =Γ2/ 022 22 2exp2expd CC HH Hdzz zz srsr11        −   −−   −= HH HH HH H Hz za z d s s ssrp 2erf 2erf2 22/erf 2        −   + C CC Ca d s ssrp 2erf2 22/erf 2, (22) the parameter a is the root of the function ()2CHzr r −. By combining both results, Eq. (21) and (22), one arrives at Γ−+ = AdAn nCH e We L 22r a (23) for the instrumental scaling constant. The electron density on an absolute scale is then given by ()() ()        − +        +−+   −− + =22 22 22 2exp 2exp 2exp2 CC HH HH H CH absz zz zzz sr s sra r r (24) (cf. Eq. (5)). III. EXPERIMENTAL METHODS A. Sample preparation 1-palmitoyl-2-oleoyl- sn-glycero-3-phosphocholine (POPC) and 1,2-dipalmitoyl- sn-glycero-3- phosphoethanolamine (DPPE) were purchased from Avanti Polar Lipids, Birmingham Alabama, and used without further purification. Multilamellar liposomes were prepared by dispersing weighted amounts of dry lipids, typically 20-30% w/w, in bidistilled water. To ensure complete hydration, the lipid dispersions were incubated for about 4 hours at least 10 °C above the main transition temperature. During this period the lipid dispersions were12vigorously vortexed. Aqueous dispersions of this lipids display narrow, cooperative melting transitions within the limits of published values, thus proving that the lipid purity corresponds to the claimed one of 99%. The POPC dispersions were further subjected to a centrifugation (centrifuge: 3K18, Sigma, Germany / rotor: 12 x 1.5 (max. 2.2 ml) / time: 10 min / 12000 rpm) to determine the content of unilamellar vesicles [35]. The phospholipid content in the supernatant was assayed by an enzymatic kit test ( Phospholipides enzymatiques PAP 150, bioMérieux, France). A proportion of 0.1-0.2% of the total phospholipids was found as unilamellar vesicles in the supernatant. Thus, diffuse scattering from unilamellar vesicles can be neglected. B. Experimental protocol Small angle X-ray scattering (SAXS) experiments were carried out at the SAXS beam-line, ELETTRA [36, 37]. The diffraction patterns were recorded with a one-dimensional position sensitive detector [38] monitoring the q-range between 2 π/90 and 2 π/10 Å-1 at a photon energy of 8 keV. The lipid dispersions were kept in a thin-walled 1 mm diameter Mark capillary held in a steel cuvette, which provides good thermal contact to the Peltier heating unit. Exposure times were typically in the range of 5 minutes. Random thin layer chromatography tests for radiation damage resulted normal, i.e., they showed no decomposition products. The position calibration of the detector was performed by using the diffraction pattern of silver behenate powder (CH 3(CH2)20COOAg) (repeat unit = 58.38 Å) [39]. C. Data analysis13The X-ray data was analyzed in terms of the model developed in section II. After substracting the background scattering from water and the sample cell, we applied the following procedure. First, the Bragg reflections were fitted by Lorentzians taking the square root of the peak area as an estimate for the constant form factor of each peak. Utilizing Eq. (4) a raw electron density profile was calculated with the appropriate phases (- - + - -) [24, 32]. The profile was then fitted with the electron density model Eq. (5), taking the results as input parameters for the further calculations. Thereafter, the diffraction pattern was fitted in the complete q-range by operating Eq. (7) and (10), where the finite instrumental resolution has to be accounted for by the convolution () ()()∫∞+ ∞−′−′ = 'dqqqrqI qIobsb , (25) b is the instrumental scaling constant. We chose an instrumental resolution function r with a Gaussian profile ()   −=22 2exp rqqrs, (26) where the standard deviation sr is typically in the range of 1.2 10-3 Å-1 for the given experimental set-up. The number of fit parameters is 9 compared to 8 for the MCT model at 4 orders of diffraction [9]. Least square fitting was performed with self-written IDL (Interactive Data Language) procedures, utilizing MPFIT [40], which is based on the MINPACK library [41]. Structural parameters have been calculated according to Eqs. (14) – (19). IV. EXPERIMENTAL RESULTS We measured X-ray diffraction profiles from unoriented liposomal suspensions of POPC and DPPE at 20 and 30% w/w lipid concentration, respectively. Both phospholipid samples were14measured in the lamellar liquid crystalline phase (smectic A); POPC was equilibrated at 2°C and 50°C, DPPE at 75°C, respectively. Figure 2 shows the diffraction pattern of POPC. Diffraction orders number 1, 2, 3, and 5 are observed, the 4th order is ruled out by the form factor. The background between the Bragg reflections is clearly modulated by the bilayer form factor, most dominantly between the first and third order. The solid line gives the best fit of the MCG model, developed in the theory section (Eqs. (1), (7), (10), and (25)). The results for the fit parameters are given in the second column of Table I. Note, that no diffuse background is fitted. The system has been equilibrated at 2°C, only, and hence lattice defects are much more suppressed than at higher temperatures, where molecular motions are more destructive to the lattice order. Figure 2 depicts further the MCT fit (dashed line) within a q-range of ± 0.01 Å-1 around each Bragg peak (cf. [14]); a close view of the first-order peak is drawn in the insert to Fig. 2. The comparison demonstrates two facts: First, standard MCT uses only a small fraction of the available diffraction data. Second, MCT gives a better fit for the peak tops, but a poorer fit for the peak tails, as it applies a constant form factor within the fitted peak region. Neither of the model functions perfectly describes the experimental data points. With the MCT method it is apparently easier to model the scattered intensity in a limited regime around the Bragg peaks, while MCG proved to be better suited to model the asymmetric tails. A quantitative comparison of the two models in terms of the respective, reduced χ2 sums is not expedient, as different numbers of data points are being considered. It is more important to state that MCG gives a qualitatively good fit for the full q-range, i.e., the diffraction peaks including the diffuse scattering, whereas MCT works in the vicinity of Bragg peaks only.15Figure 3 shows the differences between MCT and MCG in terms of the electron density profiles. The Fourier synthesis for the MCT fit shows an anomalous, small hump at the center of the water layer, due to truncation errors. The MCG model, on the other hand, gives a smoother representation of the bilayer profile, since it excludes by definition Fourier truncation errors (Eq. (5)). However, with 4 diffraction orders given, both profiles yield similar structure results. Thus full advantage of MCG can be taken only on data with less Bragg peaks. At 50°C the scattered intensity of POPC exhibits different features (Fig. 4). Evidently, the number of clearly recognizable diffraction orders has decreased from 4 to 2, an effect which is attributed to stronger thermal induced fluctuations of the bilayers, but not only. The position of the 3rd order Bragg peak is close to a minimum of the bilayer form factor, therefore the 3rd order is also attenuated because of the bilayer structure. Applying Fourier methods, such as MCT, gives in this case only very rough structural information, as only 2 diffraction orders can be used to construct the electron density profile (cf. insert to Fig. 4, dashed line). The MCG model (solid line), on the other hand, gives a clearly refined picture of the bilayer, which affects especially the headgroup region, whereas the terminating methylene group remains strongly smeared. Further, one should expect a diffuse scattering from lattice defects, as the temperature has increased from 2°C to 50°C. Indeed, we find a diffuse contribution of the bilayer form factor (cf. Table I). An additional fingerprint for enhanced fluctuations at higher temperatures is the Caillé parameter h1, which is almost 2 times greater than at 2°C. Compared to POPC, the diffraction pattern of DPPE (Fig. 5) exhibits a completely different characteristic, regarding both, the number of observed Bragg peaks - here we detect the first 4 orders - as well as the diffuse background between the reflections. The solid line gives again16the best fit of the MCG model. The fit is in good agreement with the experimental data, the fit results are given in Table I. The model fits also here a contribution of diffuse scattering, which is again attributed to the enhanced molecular motions at 75°C. The insert to Fig. 5 illustrates the effect of the MCG on Fourier artifacts. The unreal Fourier ripples of the Lorentzian model (dashed line), a consequence of the Fourier synthesis with 4 terms only, are suppressed resulting in a smooth bilayer profile (solid line) that corresponds to the resolution of the experiment. Further structural parameters have been calculated according to the geometric considerations expressed in Eqs. (14)-(19). The number of headgroup electrons is 164 and the number of hydrocarbon chain electrons is 256 for POPC, whereas 140=e Hn and 242=e Cn for DPPE, respectively. The methylene electron density is 0.317 ± 0.003 e/Å3 according to Wiener et al. [30]. The results for the two measured samples are listed in Tab. II. The structural parameters of POPC at 2°C are compared to the values obtained by the volumetric method, which was introduced by McIntosh and Simon [32, 42] for phospatidylethanolamines and further adopted for lecithins by Nagle et al. [14]. A brief description of the formalism is given in the Appendix. For the lipid volume, which is an input parameter of the method, we refer to the measurement of Hianik et al. [43] and extrapolate to 2°C, so that we get l LV = 1223 Å3. Within measurement errors, which are larger for the volumetric method, mostly due to uncertainties in the headgroup thickness [12, 13] both methods result in the same values for the structural parameters (cf. column 1 & 2 of Tab. II). At 50°C, the repeat distance is reduced by 2 Å, and the bilayer thickness by approx. 8 Å. On the other hand, the interbilayer water thickness is increased by roughly 6 Å, a sign for water uptake from the excess phase as observed in the increase of parameter nW or * Wn, respectively, due to reduced van der Walls interactions between opposing bilayers [44] at stronger undulations [45]. A further parameter,17which increases with temperature is the area per lipid. The structural results for DPPE, give a very thin water layer of 10 water molecules per lipid molecule out of which approx. 6 are intercalated into the bilayer. These values are in good agreement with the data published by McIntosh and Simon for dilauroylphosphatidylethanolamine (DLPE) [32]. The small fluid space in PE bilayers could arise from interbilayer hydrogen bond formation through the water molecules or electrostatic interactions between the amine and phosphate groups of opposing bilayers [32]. Finally, the electron density profiles were put on an absolute scale by applying Eqs. (20)-(24). An input parameter is the total number of electrons per lipid molecule, which is 420 for POPC and 382 for DPPE, respectively. The results are plotted in Fig. 6; Fig. 6 (a) and 6 (b) give the absolute electron density of POPC at 2°C and 50°C, respectively, whereas Fig. 6 (c) depicts the absolute electron density of DPPE at 75°C. V. DISCUSSION A new model has been introduced to analyze small angle diffraction data of unoriented phospholipid membrane stacks at high instrumental resolution. The formalism combines a form factor, related to a Gaussian representation of the electron density profile (Fig.1), with a MCT structure factor. The proposed electron density model gives the mean structure of a phospholipid bilayer time averaged over all fluctuations and is well suited to represent the X- ray picture one sees from not more than 5 orders of diffraction. Higher orders - which can be obtained by aligning the layers only - would result in a more detailed electron density profile for which other electron density model, like ,e.g., hybrid types of Gaussians and strip-models18[34] would give a better representation. Such models have also been tried out on our data, but were found to fail because of too many correlating fit parameters for the given instrumental resolution. It is reasonable to model the electron density profile by means of analytic functions, as the features of its structure are well known since the pioneering work of Luzzati and Tardieu [1, 2]. The difference in the distinct phospholipid bilayer structures are then accounted for by adjusting the parameters, i.e., headgroup position, headgroup width, etc., of the analytical function. The inverse Fourier method, which takes the form factor of the bilayer model and fits it together with a structure factor to the scattered intensity has further the advantage of excluding Fourier truncation errors. The MCG model has been tested experimentally on POPC and DPPE multilayers giving good fit results [see results section & Fig. 2, Fig. 4, Fig. 5, Tab. I]. Several other models have already been published [5, 6, 9, 10, 33], in order to perform the same task. We shall briefly discuss the most prominent ones. In 1994, Nagle and coworkers introduced the modified Caillé theory and gave an experimental proof of its superiority to the classical paracrystalline theory [9, 15]. The group usually records high-resolution data at a synchrotron beam-line by means of a diffractometer, but in the vicinity of the Bragg reflections only. Electron density profiles are computed by applying the standard Fourier synthesis (Eq. (4)). In contrary, we use an equivalently brilliant source, but a detecting system, which is able to monitor the diffraction pattern in a continuous range of scattering angles. In this case, applying the standard MCT-data analysis, which works only in the regime close to diffraction peaks, means to reject all the information hidden in the diffuse background scattering between the Bragg peaks (Fig. 2). This information becomes even more valuable if less than 4 orders are observed. Nagle and coworkers report only two diffraction orders for unoriented dipalmitoylphosphatidylcholine (DPPC), egg19phosphatidylcholine (EPC), dimyristolphosphatidylcholine (DMPC), and dioleoyl- phosphatidylcholine (DOPC) bilayers in excess water [14 - 17], which is insufficient to obtain satisfactory structural information, if only the Bragg peaks are considered. The common circumvention of this problem are osmotic stress experiments [14-21, 23, 24], where the system is partly dehydrated, and thus more diffraction orders are detected as bilayers are consequently hindered in undulation. Structural information of the fully hydrated phase is accessible then only through a numerical extrapolation to zero osmotic pressure. It is well known that extrapolations are always inherent to large uncertainties and should be avoided if possible. The MCG model, on the other hand, describes also the diffuse scattering and is thus capable of obtaining structural information even at low Bragg reflection information content, e.g., POPC at 50°C (Fig. 4). Moreover, the assumption of a constant form factor for each Bragg peak is not very accurate for higher diffracting orders, as peaks broaden strongly and more and more scattered intensity is smeared to the peak tails. For instance, the third order peak of the 2°C-POPC diffraction pattern displays an asymmetric shape (Fig. 2), which is obviously due to the modulation by a non-constant bilayer form factor. Such effects are not seen in the X-ray data published by Nagle and coworkers, because the observation of asymmetric peak shapes is likely to depend on the lipid type and on its specific form factor, e.g., the diffraction pattern of DPPE does not exhibit any asymmetric peaks (Fig. 4). Further, data treated with MCT only, has not always been presented in a uniform fashion, i.e., with increasing order (h = 1 to 3) decreases the data point density [16, 17] or the selected q-range [15]. Thus, peak asymmetries, even if present are difficult to be seen. Nallet et al. [10] suggested a model similar to MCT [9] to analyze small angle scattering data on bis 2-ethylhexyl sodium sulphosuccinate (AOT) and didodecyl dimethyl ammonium bromide (DDAB) / water systems. They combined the structure factor with the form factor of20a strip model for a continuous q-range fit function. Although the strip model for the AOT/water and DDAB/water systems differs somewhat from a reasonable strip model for phospholipid bilayers, this method could in principle easily be adopted with the advantage of less fit parameters. Still we refer to the common criticism on strip models, which is that discontinuous boundaries between the different regions of the bilayers are an unrealistic picture of a fluctuating bilayer. A quite different approach was introduced by Lemmich et al. [33] for neutron scattering experiments. He proposed a strip model for the bilayer, but averaged its form factor together with a paracrystalline structure factor without decoupling the two entities as the two other theories do (Eqs. (1) and (3)). Lemmich analyzed his data in terms of both, his model and MCT, but the fits gave equally good results for phospholipids in the lamellar liquid crystalline phase. The most convincing explanation is that the strong instrumental smearing, inherent to neutron scattering experiments, does not allow for any decision. Since not even Lemmich could show better fit results for phospholipids in the L α-phase, we see no argument to apply his model which would imply a recalculation of the whole formalism, since X-rays "see" a different contrast than neutrons do. Concluding the last paragraph, we should state that the models that have been discussed are without any doubts appropriate for the measurement methods applied by the individual groups. This is clearly demonstrated by the good fits to their experimental data. However, for the given reasons our method is best tailored to extract as much information as possible from high resolution X-ray data recorded in a continuous range.21A further benefit of MCG is that structural parameters like bilayer thickness, area per lipid, water distribution, etc., can be estimated from simple geometric considerations. Despite the gravimetric method of Luzzati [1], the commonly used method, initiated by McIntosh and Simon [32, 42] and applied by Nagle et al. [14], relies on additional information of the lipid volume, which is supplied by specific volumetric measurements. The algorithm is build up upon a comparison with a known gel phase structure, assuming that the volume of the headgroup is the same for both phases (cf. Appendix, Eq. (A1), (A2)). For phospholipids with a PC headgroup one usually employs the structural data of DPPC in the Lβ‘-phase, published by Sun et. al [46, 47]. A further structural input, i.e., the headgroup thickness, is needed to calculate the bilayer thickness according to the steric definition [42] (Eq. (A4)). McIntosh and Simon suggested a value of 10 Å for PC headgroups and 8 Å for PE’s, derived from space filling molecular models. The headgroup conformation of DPPC has been measured by Büldt et al. [48, 49], by means of neutron diffraction and deuteron labels, but at very low water content (10 & 25 % w/w). From the published data the heagroup thickness can be extracted as dH = 9 ± 1.2 Å, a value which is employed by Nagle and coworkers, without considering the measurement error within which the values given by McIntosh and Büldt are equal. However, the headgroup conformation is likely to depend on temperature, pressure, chain tilt [30] or hydration [24], which directly affects the headgroup dimensions, so that the volume of the PC headgroup in the Lβ’-phase is not evidently the same as in the L α-phase. Hence, a method which utilizes the assumption of constant headgroup volume and size, respectively, and even relies on measurements on systems different from the situation of fully hydrated bilayers, can be justifiable but certainly leads to a rough estimate. A way out of this dilemma should be structural data from highly aligned multilayers at full hydration according to the method of Katsaras et al. [28]. However, it is possible to obtain also reasonable estimates for unoriented systems without the need of extra data input by the simple geometric relationships of the22Gaussian electron density model (Eqs. (14)-(19)). The results compare well to those obtained by the volumetric method (cf. Table II) and even display smaller errors. The Gaussian electron density profile can be set on an absolute scale, which is often desirable. The scaling factor is computed by integrating the profile from the center to the border of the unit cell (Eq. (20)). This can be easily done, since the electron density profile is given as an analytic function. However, we argue to take absolute electron densities with great care, since the relative error of the scaling factor is large (0.2 for POPC at 50°C), a consequence of the large number of error contributors in the calculation procedure. This implies also to absolute electron densities published by other groups [14-17, 30], but has not been discussed there. In conclusion, we remark that the MCG model gives considerable more structural information than standard MCT, provided that the number of recorded diffraction orders is less than 4. At 4 orders of diffraction one obtains equally good results (Fig. 3). The advantages of the model are due to a cancellation of Fourier artifacts, and a simple method to derive structural parameters. Since the model can retrieve structural information from the diffuse scattering its potential increases in importance, when less than four orders of diffraction are recorded (Fig. 4). This is a common situation for fully hydrated phosphaditylcholine bilayers, which include about 3 times more interbilayer water than phosphaditylethanolamine bilayer systems. ACKNOWLEDGEMENTS The authors are grateful to J. F. Nagle and H. I. Petrache, for helpful discussions and for providing us the source code of the program MCT. We also express our thanks to F. Nallet23and J. Lemmich for sending additional manuscripts and to H. Sormann for helpful discussions on statistics. This work has been supported by the “Elettra-Project” of the Austrian Academy of Sciences. M. Rappolt is the recipient of a long-term grant from the European Commission under the program “Training and Mobility of Researchers” [Contract no. SMT4-CT97- 9024(DG12-CZJU)].24APPENDIX Structural parameters for bilayers in the lamellar liquid crystalline phase can be derived upon the assumption that the volume of the phospholipid headgroup is equal to the volume in the gel phase [14] g Hl HV V= , (A1) where the superscript l denotes the liquid phase and g the gel phase. By calculating the difference in the total lipid volume g Ll LV V- one arrives at 2g HHl HH g Cg Hl L l d ddV VA -+-= (A2) for the area of the fluid bilayer, where dC is the hydrocarbon chain length and dHH the head-to head-group distance over the bilayer. For phospholipids with a PC headgroup one usually employs the structural data of Lβ’-DPPC as published by Sun et al. [46]: Å6 319± =g HV , 0.2Å 3.17 ± =g Cd , and the corrected value of the head-to-head-group distance [47] Å 2.08.42 ±=g HHd . The hydrocarbon chainlength is given by lg Hl L l C AV Vd-= (A3) and the bilayer thickness, according to the steric definition of McIntosh and Simon [42], by ( )Hl Cl Bd d d + =2 . (A4) The headgroup thickness dH has been estimated from space filling models to be 10 Å for PC’s and 8 Å for PE’s, whereas Büldt et al. found a value of 9 ± 1.2 Å with neutron diffraction experiments at a hydration of 10% w/w [48, 49]. The interbilayer water thickness and the number of free water is given according to Eqs. (17) and (18).25Sometimes it is desirable to compare the structural results with already published data derived by applying the gravimetric method of Luzzati [1]. The Luzzati bilayer thickness is calculated as AVdL Luzzati B2= , (A5) with the corresponding interbilayer water thickness the total number of water molecules per lipid are obtained according to Eq. (19).26References [1] V. Luzzati, in Biological Membranes , edited by D. Chapman (Academic Press, London, 1967), p. 71. [2] A. 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Cevc (Marcel Dekker, New York, 1993), p. 553. [26] M. C. Wiener and S. H. White, Biophys. J. 59, 162 (1991). [27] M. C. Wiener and S. H. White, Biophys. J. 59, 174 (1991). [28] J. Katsaras, Biophys. J. 75, 2157 (1998). [29] J. F. Nagle and J. Katsaras. Phys. Rev. E. 59, 7018 (1999). [30] M. C. Wiener, R. M. Suter and J. F. Nagle, Biophys. J. 55, 315 (1989). [31] J. Lemmich et al. (submitted to Biophys.J.). [32] T. J. McIntosh and S. A. Simon, Biochemistry 25, 4948 (1986). [33] J. Lemmich et al. , Phys. Rev. E 53, 5169 (1996). [34] J. F. Nagle and M. C. Wiener, Biophys. J. 55, 309 (1989). [35] K. Lohner et al., Biochemistry 38, 16514 (1999). [36] H. Amenitsch, S. Bernstorff, and P. Laggner, Rev. Sci. Instrum. 66, 1624 (1995). [37] H. Amenitsch et al., J. Appl. Cryst. 30, 872 (1997). [38] A. Gabriel, Rev. Sci. Instrum. 48, 1303 (1977). [39] T. C. Huang, H. Toraya, T. N. Blanton, and Y. Wu, J. Appl. Cryst. 26, 180 (1993). [40] C. Markwardt, documentation for MPFIT, internet site: http://cow.physics.wisc.edu/~craigm/idl/idl.html.28[41] B. S. Garbow, K. E. Hillstrom, and J. J. Moore, documentation for MINPACK, internet site: http://www.zrz.tu-berlin.de/~zrz/software/numerik/minpack. [42] T. J. McIntosh and S. A. Simon, Biochemistry 25, 4058 (1986). [43] T. Hianik et al., Colloids Surfaces A: Physicochem. Eng. Aspects 139, 189 (1998). [44] J. Israelachvili, Intermolecular & Surface Forces (Academic Press, 1991), p. 395. [45] W. Helfrich, Z. Naturforsch. 33a, 305 (1978). [46] W.-J. Sun et al. , Phys. Rev. E 49, 4665 (1994). [47] W.-J. Sun et al. , Biophys. J. 71, 885 (1996). [48] G. Büldt, H. U. Gally, J. Seelig, and G. Zaccai, J. Mol. Biol. 134, 673 (1979). [49] G. Zaccai, G. Büldt, A. Seelig and J. Seelig, J. Mol. Biol. 134, 693 (1979).29TABLE I. Fit results for the diffraction patterns of POPC at 2°C and 50°C, and DPPE at 75°C (cf. Fig. 1). The parameters Hrand Cr are given in absolute units according to Eq. (24) (see also Fig. 6). Fit parameter POPC DPPE T = 2°C T = 50°C T = 75°C zH (Å) 20.2 ± 0.1 17.0 ± 0.3 19.2 ± 0.1 sH (Å) 3.6 ± 0.1 3.6 ± 0.2 3.3 ± 0.1 Hr(e/Å3) 0.11 ± 0.01 0.11 ± 0.01 0.15 ± 0.01 sC (Å) 4.8 ± 0.2 6.8 ± 0.7 2.5 ± 0.2 Cr (e/Å3) -0.08 ± 0.01 -0.10 ± 0.02 -0.06 ± 0.01 d (Å) 66.2 ± 0.1 64.3 ± 0.1 51.4 ± 0.1 h1 (Å) 0.0504 ± 0.0005 0.092 ± 0.001 0.016 ± 0.001 N 28.0 ± 1.0 23.0 ± 1.0 52 ± 1 Ndiff 0.0 0.17 ± 0.09 1.08 ± 0.0430TABLE II. Derived structural parameters calculated by using Eqs. (14)-(19). The results for POPC at 2°C are compared to the values obtained by using the volumetric method [16, 17, 32] (cf. Appendix). parameter POPC DPPE T = 2°C T = 50°C T = 75°C volumetric geometric geometric geometric d (Å) 66.2 ± 0.1 66.2 ± 0.1 64.3 ± 0.1 51.4 ± 0.1 dB (Å) 50.2 ± 3.6 48.9 ± 0.3 42.5 ± 1.1 46.2 ± 0.4 dW (Å) 16.0 ± 3.7 17.3 ± 0.4 21.7 ± 1.2 5.3 ± 0.5 dC (Å) 16.1 ± 0.6 16.0 ± 0.2 12.8 ± 0.6 15.4 ± 0.2 A (Å2) 56 ± 2 54 ± 1 62 ± 1 52 ± 1 nW 22 ± 2 24 ± 1 31 ±1 11.3 ± 0.3 * Wn 15 ± 4 16 ± 1 23 ± 2 4.6 ± 0.431Figure Captions FIG. 1. The Gaussian electron density profile representation of a phospholipid bilayer corresponding to a X-ray resolution of 4 Bragg peaks. FIG. 2. The best fit of the MCG model (solid line) and MCT (dashed line within marked peak region) to the diffraction pattern of POPC at 2°C. The insert gives a zoom of the first order Bragg peak. FIG. 3. Comparison of the electron density profile for POPC bilayers at 2°C obtained by a Fourier synthesis (dashed line), using MCT and the MCG refined profile (solid line). FIG. 4. The best fit of the MCG model (solid line) to the diffraction pattern of POPC at 50°C. The insert gives the electron density profile obtained by a Fourier synthesis (dashed line), using Lorentzians to fit the Bragg peaks, and the profile refined with MCG (solid line). FIG. 5. The best fit of the MCG model (solid line) to the diffraction pattern of DPPE at 75°C. The insert gives the electron density profile obtained by a Fourier synthesis (dashed line), using Lorentzians to fit the Bragg peaks, and the profile refined with MCG (solid line). FIG. 6. Absolute electron density profiles of POPC at 2°C (a), POPC at 50°C (b), and DPPE at 75°C (c). Deviations due to the error of the instrumental scaling factor a are depicted as a gray area enveloped by the maximal positive (dashed line) and negative (dot-dashed line) divergence.FIG. 1 / Georg PabstCH2ρρH ρC σHσCzH z d/2 - d/2 ρ0.1 0.2 0.3 0.4 0.5101102103104105I (a. u.) q (Å-1)0.085 0.090 0.095 0.100 0.105h = 1I (a. u.) q (Å-1) FIG. 2 / Georg Pabst-30 -20 -10 0 10 20 30ρ (a. u.) z (Å) FIG. 3 / Georg Pabst0.1 0.2 0.3 0.4 0.5100101102103104I (a. u.) q (Å-1)-30 -20 -10 0 10 20 30ρ (a. u.) z (Å) FIG. 4 / Georg PabstFIG. 5 / Georg Pabst0.1 0.2 0.3 0.4 0.5101102103104105106107108109I(q) q (Å-1)-20 -10 0 10 20ρ (a. u.) z (Å)ρ (e/Å3) z (Å)ρ (e/Å3) z (Å)ρ (e/Å3) z (Å)a b c-30 -20 30 -10 0 20 100.20.30.4 -30 -20 30 -10 0 20 100.20.30.40.5 0 -10 -20 10 200.30.40.5 FIG. 6 / Georg Pabst

1X-RAY KINEMATOGRAPHY OF TEMPERATURE-JUMP RELAXATION PROBES THE ELASTIC PROPERTIES OF FLUID BILAYERS Georg Pabst*, Michael Rappolt*, Heinz Amenitsch*, Sigrid Bernstorff #, and Peter Laggner* *Institute of Biophysics and X-ray Structure Research, Austrian Academy of Sciences, Steyrergasse 17, A-8010 Graz, Austria. #Synchrotron Trieste (ELETTRA), SS 14, Km 163.5, I-34012 Basovizza (TS), Italy. Running Title: Bilayer Elasticity from T-jump Relaxation Keywords: phosphatidylcholine, phase transition, x-ray diffraction, synchrotron, intermediates Corresponding Author: Peter Laggner Institute of Biophysics and X-ray Structure Research, Austrian Academy of Sciences, Steyrergasse 17, A-8010 Graz / Austria Tel: ** 43 316 812003 Fax: ** 43 316 812367 Email: Peter.Laggner@oeaw.ac.at23ABSTRACT The response kinetics of liquid crystalline phosphatidylcholine bilayer stacks to rapid, IR- laser induced temperature jumps has been studied by millisecond time-resolved x-ray diffraction. The system reacts on the fast temperature change by a discrete bilayer compression normal to its surface and a lateral bilayer expansion. Since water cannot diffuse from the excess phase into the interbilayer water region within the 2 ms duration of the laser pulse, the water layer has to follow the bilayer expansion, by an anomalous thinning. Structural analysis of a 20 ms diffraction pattern from the intermediate phase indicates that the bilayer thickness remains within the limits of isothermal equilibrium values. Both, the intermediate structure and its relaxation into the original equilibrium L α-phase, depend on the visco-elastic properties of the bilayer/water system. We present an analysis of the relaxation process by an overdamped one-dimensional oscillation model revealing the concepts of Hooke's law for phospholipid bilayers on a supramolecular basis. The results yield a constant bilayer repulsion and viscosity within Hooke's regime suggesting that the hydrocarbon chains act as a buffer for the supplied thermal energy. The bilayer compression is a function of the initial temperature and the temperature amplitude, but is independent of the chain length.4INTRODUCTION Fluid membranes are characterized by only two types of possible elastic deformations: stretching and bending, as the shear modulus within a fluid membrane is zero (for review see Lipowsky, 1995). Experimental methods for resolving the elastic properties of liquid crystalline phospholipid bilayers include a mode analysis of bending fluctuations, pipette aspiration or electric fields induced deformations on unilamellar vesicles (for reviews see, Helfrich, 1995; Evans, 1995). For multilamellar vesicles additional forces arise from interactions between opposing bilayer surfaces. In the case of electrically neutral phospholipids the equilibrium structure is determined by attractive van der Waals forces and repulsive hydration and steric forces (for review see Parsegian and Rand, 1995). The experimental techniques for exploring the interaction potentials are the surface force apparatus, the osmotic stress method, atomic force microscopy and pipette aspiration (Parsegian and Rand, 1995). However, a unifying theoretical description of the liquid crystalline layered structures is still lacking, and presently membrane biophysics relies on a rather empirical exponential model-function of the repulsive forces ( Parsegian and Rand, 1995). Since the elastic properties of phospholipid bilayers determine the equilibrium structure, the bilayer elasticity appears to be even more important for biological non-equilibrium "switching" processes, such as cell fusion or pore formation: only a conservation of the membrane structure will preserve the vital function of the biological membrane in cell compartimentation and communication. Rapid pressure and temperature jump experiments on phospholipid phase transitions ( Pressl et al. 1997; Rapp and Goody, 1991) in combination with time resolved x-ray diffraction ( Gruner, 1987; Caffrey, 1989; Rapp, 1992; Cunningham5et al. 1994) have revealed that the fast phase transitions proceed without any detectable disruption of the multilamellar stack order (Laggner and Kriechbaum 1991; Rapp et al., 1993; Laggner et al., 1999). The most intriguing observation with such experiments is an anomalous, thin lamellar intermediate structure, induced by a T-jump through the pretransition of dipalmitoylphosphatidylcholine (Rappolt M., G. Pabst, G. Rapp, M. Kriechbaum, H. Amenitsch, C. Krenn, S. Bernstorff, and P. Laggner. New evidence for gel- liquid crystalline phase co-existence in the ripple phase of phosphatidylcholines. Eur. Biophys. J. , submitted). T-jumps in the L α-phase region have revealed a similar, non- equilibrium intermediate structure – designated as L α* - with a lifetime in the sub-second range (Laggner et al., 1999). In this article we present an explanation for the observed phenomena of the L α-Lα*-Lα transition based on a structural calculation of the intermediate structure through a recently developed x-ray analysis method (Pabst, G., M. Rappolt, H. Amenitsch, and P. Laggner. Structural information from multilamellar liposomes at full hydration: X-ray data analysis method. Phys. Rev. E , submitted.). We further present a relaxation model which gives qualitative insight into mechanical membrane properties. The results demonstrate that the T- jump technique provides a sensitive way to determine the interacting forces between opposing bilayers and within the bilayer itself, i.e., between the lipid molecules. Thus, the method is an additional tool to gain valuable information on the physical interactions of fluid membrane stacks.6MATERIALS AND METHODS I. Sample preparation POPC (1-palmitoyl-2-oleoyl- sn-glycero-3-phosphocholine), DPPC (1,2-palmitoyl-2-oleoyl- sn-glycero-3-phosphocholine) and DSPC (1,2-stearoyl-2-oleoyl- sn-glycero-3- phosphocholine) were purchased from Avanti Polar Lipids (Alabaster, AL, > 99% purity) and used without further purification. Multilamellar liposomes were prepared by dispersing weighted amounts of lipids (20 - 30% w/w) in pure water ( Fluka, Neu Ulm, Germany: double quartz distilled water, with a specific resistance 18 M Ω cm) and incubating the dispersions for 4 hours at least 10°C above the main transition temperature to guarantee full hydration. During this period the lipid dispersions were vigorously vortexed. To avoid radiation damage the total exposure-time was kept as low as possible, i.e., each single experiment was carried out with fresh sample but of the same stock. Random thin layer chromatography tests on silica gel-plates 60 (Merck, Darmstadt, Germany) resulted normal. The solvent used was chloro- form/methanol/water (65/25/4). II. Instrumentation and X-ray Diffraction Diffraction patterns were recorded at the Austrian Small Angle X-ray Scattering (SAXS) beamline at ELETTRA (Amenitsch et al., 1998; Bernstorff et al., 1998), using a one- dimensional position sensitive detector (Gabriel, 1977) which covered the q-range (q = 4 π sinθ/λ) of interest from about 0.03 Å-1 to 0.52 Å-1. The angular calibration of the detector was determined by using the SAXS-diffraction patterns of silver- behenate (CH 3(CH2)20COOAg: d-spacing = 58.38 Å) (Huang et al., 1993). The lipid dispersions were kept in a thin-walled 17mm diameter Mark capillary held in a steel cuvette, which provides good thermal contact to the Peltier heating unit (Anton Paar, Graz, Austria). T-jumps were generated by an erbium- glass IR-laser (Kriechbaum et al., 1990; Rapp and Goody, 1991). The experimental setup is schematically depicted in Fig. 1. The laser-pulse energy varied from 0.78 to 2.40 J resulting in an average T-jump amplitude ΔT of 5 to 16 °C (Tab. 1). The variation given for the T-jump amplitude derives mainly from the difficulty to measure precisely the energy deposited onto the sample, which depends on the accuracy of measurement of the laser-pulse energy (± 5%), the estimation of the geometric properties given in the experimental set-up (10% error), and the estimation of the absorption in the sample (10% error); however, the reproducibility of the laser-pulse energy lies within 2%. A time-resolved experiment (one cycle) consisted of a series of 256 time-frames of pattern detection, with the laser (pulse-length 2 ms), triggered at the beginning of the 11th frame. The maximum time resolution of the X-ray experiment, i.e., right after the laser shot, was set to 5 ms. The total exposure time of one cycle was 16 s. Each experiment was repeated 3 times, i.e., every diffraction pattern was averaged over 3 cycles. The raw data of the time-resolved experiments were normalized for the integration time of each time-frame, and the background (capillary with water) was subtracted. III. Structure Calculations The X-ray diffraction patterns of POPC have been analyzed in terms of the MCG method (Pabst, G., M. Rappolt, H. Amenitsch, and P. Laggner. Structural information from multilamellar liposomes at full hydration: X-ray data analysis method. Phys. Rev. E , submitted.). MCG is a diffraction model that combines a modified Caillé theory structure factor (Caillé, 1967; Zhang et al., 1994) with the form factor of a Gaussian representation of8the electron density profile such that it accounts for both, Bragg diffraction and diffuse scattering. The method is thus capable of retrieving structural information even if only a few orders of diffraction are observed. The basic concepts of the model are summarized as follows. The total scattered intensity of stack of lamellae is described by () ()() () ( )2 2 21qF NqSqFqqI diff+ ∝ , (1) where q is the absolute value of the scattering vector, F(q) the bilayer form factor given by the Fourier transform of a Gaussian representation of the electron density profile, S(q) the Caillé structure factor and Ndiff a multiplicator that controls the term for additional diffuse scattering of single, uncorrelated bilayers. Structural parameters are calculated by applying a geometric model, where the bilayer thickness is given by ( )2/ 2 H H BFWHM z d + = , (2) with zH being the position of the headgroup with respect to the bilayer center at the methyl terminus and the FWHM of the Gaussian representing the headgroup in the electron density profile. The interbilayer water layer is the difference of the length of the unit cell minus the bilayer thickness B Wdd d −= . (3) For further details on the geometric model and MCG we refer to (Pabst, G., M. Rappolt, H. Amenitsch, and P. Laggner. Structural information from multilamellar liposomes at full hydration: X-ray data analysis method. Phys. Rev. E , submitted.).9IV. Relaxation Kinetics The relaxation kinetics of the lamellar repeat distance d is given by a double-exponential decay (Laggner et al., 1999) () ( ) ( )b b a a eqt d t d dtd t t / exp / exp − − − −= , (4) where deq is the equilibrium d-spacing at the given system temperature before the T-jump, da and db are relaxation components and ta, tb the respective relaxation time constants. The relaxation process can be compared to a damped oscillation. The differential equation of a one-dimensional damped harmonic oscillator is 0= + + xcx xm &&&r , (5) with the oscillating mass m, the friction coefficient r, and c, the restoring force constant. By solving Eq. 5 one has to distinguish three cases, namely a quasi-periodic motion for slight damping, an aperiodic motion for strong (over-) damping and the aperiodic limit, giving the fastest relaxation to equilibrium without any oscillation. For details on harmonic oscillators see any mathematics texts on differential equations (e.g. G. Joos and E.W. Richter, 1978; C. Schaefer and M. Päsler, 1970). The case of an aperiodic motion gives the solution () ( ) ( )Ct Bt Atx +− +− = 2 1exp exp b b 2 02 1w d d b −+= 2 02 2w d d b −−= , (6) where the frequency w0 is related to the "spring constant" c by mc/ 0=w (7) and the damping factor d to the friction coefficient by m2rd=. (8)10The similar analytic form of Eq. 4 and 6 suggest that the relaxation kinetics of the d-spacing can be seen as a harmonic one dimensional overdamped oscillation. The relaxation time constants ta and tb are related to these quantities by ( )2/1 0− =batt w (9) and bab a ttt td2+= . (10) The relaxation components da, db can be expressed by the relaxation velocity b b a ad d v t t / / 0+ = (11) and the relaxation acceleration 2 2 0/ / b b a ad d a t t− −= (12) at t = 0. RESULTS AND DISCUSSION I. Equilibrium Structure We took static X-ray diffraction patterns of POPC liposomal dispersions in a range of 10°C to 70°C within the L α-phase of POPC in excess of water. The sample was held at each temperature for 5 minutes before the measurement was started, so that the system can be regarded as being in thermal equilibrium. Fig. 2 shows the changes in the d-spacing, bilayer thickness, and water layer thickness, as the temperature is increased. The lamellar repeat decreases as the temperature is raised up to 30°C down to a value of d = 63.6 ± 0.1 Å. Above 30°C, the bilayer-water system swells again and finally exhibits a larger lattice parameter at1170°C than at 10°C. The decomposition of the d-spacings into bilayer and interbilayer water thickness reveals that this is caused by an uptake of water, as the membrane thickness continuously decreases with increasing temperature, but the bilayer separates more and more, such that the sum of both gives the observed re-increase in d-spacing. With respect to the results which will be presented in subsection III, we draw the attention to the membrane thickness, which first decreases linearly with temperature, but shows an asymptotic behavior above 50°C. The characteristics of the L α-phase are “molten” hydrocarbon chains due to a transition from a all- trans to a trans -gauche state. An increase in temperature progressively induces trans -to-gauche conformation changes – the minimal energy difference between trans and gauche states is 500 cal mol-1 (Flory, 1969) –, but since the hydrocarbon tails are finite in length, only a limited number of gauche isomers can be generated. This corresponds to a saturation effect in hydrocarbon chain melting as has been found before, e.g., for DPPC (Seelig and Seelig, 1974) that causes the asymptotic behavior of the membrane thickness, which is also observed for POPC above 50°C (Fig. 2). The reduction of the bilayer thickness has been first described by Luzzati and co-workers, and explained in terms of a rubber-like model for the hydrocarbon chain elasticity (Luzzati, 1968). Concomitantly with the decrease in thickness, the bilayer responds to the temperature increase by a lateral expansion, as each single phospholipid molecule requires more space due to the increasing degrees of motional freedom at higher temperatures (for review see Cevc and Marsh 1987; Hianik et al., 1998). The linear thermal expansion coefficient a, defined as T dd BB ΔΔ=1a (13) is given as a function of temperature in Fig. 3. The mean value of α = -2.2·10-3 K-1 below 50°C is close to the linear thermal expansion coefficient of –2.7·10-3 K-1 for K-soaps (Luzzati, 1968) and –2.5·10-3 K-1 for DPPC ( Seelig and Seelig, 1974). The thermal expansion12coefficient decreases linearly from 50°C to70°C to a value of a = -0.5·10-3 K-1, indicating a reduced bilayer elasticity. At first sight, this appears to be surprising as one would expect a more fluid bilayer at higher temperatures with increased disorder. However, fluid DPPC bilayers have been found to have less order, but a ten times larger viscosity than soaps, e.g., sodium decanoate-decanol bilayers ( Seelig and Seelig, 1974). II. Intermediate Structure Fig. 4 shows the kinematographic diffraction patterns of a typical jump/relaxation experiment. The structure of the L α-phase under equilibrium conditions at the given system temperature T 0 is the starting point and reference for the T-jump experiment (Fig. 4). With the laser flash (duration: 2 ms) the original structure converts quasi discontinuously, i.e., within the time resolution of the experiment*, into a lamellar phase, whose d-spacing is clearly thinner than the corresponding lattice parameter under equilibrium conditions at T = T 0 + ΔT (Laggner et al., 1999). The excited phase, denoted as L α*, relaxes within the time scale of seconds back to the equilibrium d-spacing. The FWHM of the L α*-phase is smaller than the corresponding values of the original L α-phase. In the following, the intermediate structure is to be compared with the equilibrium structure obtained in the previous subsection. As a reference we choose the structure of POPC bilayers at 20°C, with a d-spacing of 63.9 ± 0.1 Å, a bilayer thickness of 46.7 ± 0.3 Å, and a bilayer * The diffraction pattern between the well defined L α and L α*-phase is recorded during the heating of the laser pulse. The broad diffuse peak of this pattern is due to the fast changes in the lattice parameter and not to a less ordered phase.13separation of 17.2 ± 0.3 Å (Fig. 2). The laser voltage was set to 600 V, which corresponds to a temperature jump of 10 ± 2°C (Tab. 1). In this particular experiment, the T-jump/relaxation cycles were repeated 12 times, and the single diffraction patterns were added up to reduce statistic noise. We further summed up the first four 5 ms-diffraction patterns after the laser pulse, during which the d-spacing relaxes by approximately 0.3 Å only, again to improve the statistics of the L α* diffraction pattern. The structural analysis on L α* has been performed by applying the MCG model (Pabst, G., M. Rappolt, H. Amenitsch, and P. Laggner. Structural information from multilamellar liposomes at full hydration: X-ray data analysis method. Phys. Rev. E , submitted.), yielding 62.1 ± 0.5 Å for the lamellar repeat distance, i.e., by 1.5 Å thinner than the minimum repeat distance found under equilibrium conditions (Fig. 2), 45 ± 1.7 Å for the bilayer thickness, and a water layer of 17.1 ± 1.7 Å thickness. The corresponding bilayer thickness at 30°C under equilibrium conditions is 45.6 ± 0.3 Å, and the value of dw is 18 ± 0.5 Å (Fig. 2). Within the limits of measurement error it can be assumed that the bilayer shrinks proportionally to the heat deposited, according to ( )T Td d B BΔ+ =0 *, (14) wherein dB* denotes the bilayer thickness of L α*. We can further calculate the change in interbilayer water volume by W W WdA dA V − =Δ**, (15) where A and A* are the respective areas per phospholipid molecule, which can be estimated according to the formalism of Nagle and co-workers (Nagle et al., 1996). For the given temperature of T 0 = 20°C, we calculate a volume change of DVW = -17 Å3. For comparison: the volume of one water molecule is approx. 30 Å3. Thus, the change in interbilayer water volume can be regarded as zero. However, we estimate the error to be larger than 50 Å3, since the employed estimates of the T-jump amplitude and of the area per lipid include uncertainties, which strongly influence the result.14The result predicts a anomalous thin water layer of the intermediate structure: the temperature-induced decrease in membrane thickness is accompanied by a lateral expansion of the bilayer ( Cevc and Marsh, 1987). As there is no water exchange with the excess aqueous phase directly after the laser shot ( DVW = 0), the interbilayer water will follow the lateral bilayer expansion which gives the observed thin layer. The transient water deficit can also explain the observed sharp Bragg peaks of the L α*-phase, as more layers can contribute coherently to Bragg diffraction, similarly to phosphatidylethanolamine multilayers, which incorporate less than half the amount of water compared to phosphatidylcholine multilayers, and which exhibit sharper Bragg peaks than phosphatidylcholines in the L α-phase (McIntosh and Simon, 1986). III. Relaxation Kinetics The relaxation kinetics of the IR-laser T-jump induced L α-Lα* transitions have been analyzed in terms of the relaxation model presented in the theory section (Eqs. 4, 9-12). Fig. 5a shows the changes in the lattice parameter as a function of time for a 16°C T-jump from T 0 = 20°C. Time equals zero at the first frame after the laser shot. The individual data points were obtained by fitting the first order Bragg peak of each single diffraction pattern by a Lorentzian function. The solid line in Fig. 5a gives the best fit of the d-spacing relaxation to the double exponential decay model (Eq. 6) yielding a fast time of constant 0.45 ± 0.02 s and a slower component of t = 3.0 ± 0.1 s. The lower part of Fig. 5a further illustrates the time course of the maximum intensity and FWHM of the first order Bragg peak. The intermediate phase, at t = 0, has a thinner FWHM and an increased peak intensity as compared to the original L α- phase. Both, maximum intensity and FWHM, decay rapidly to values below the equilibrium15Lα reference values and start to recover after 2 s. Even after 16 s, when the d-spacing has retained its equilibrium value, the peak intensity and FWHM indicate that the relaxation process is still going on. The total relaxation time is about 30 to 40 s. Fig. 5b depicts the first and the second derivative of the d-spacing relaxation, i.e., the relaxation velocity and the relaxation acceleration, for the first 5 seconds of the relaxation process. The relaxation is initially driven by a strong acceleration which diminishes after 2-3 seconds, whereupon the relaxation proceeds with constant velocity. We shall try to obtain further insight from energetic aspects. Fig. 6 depicts the interaction potential between two bilayer surfaces. For neutral phospholipids the interacting forces are attractive van der Waals, repulsive hydration, and steric forces ( Marra and Isrealachvili, 1985; Lipowsky, 1995; Parsegian and Rand, 1995). The bilayer separation is given by the minimum of the potential curve. At higher temperatures, the potential well is shallower, and the minimum moves to larger bilayer separations. Through the T-jump, the bilayer/water system is quasi discontinuously forced into the intermediate state (Fig. 6 Ä), sensing the strong repulsive part of the interaction potential that drives the water layer thickness towards the equilibrium separation (Fig. 6 Å). As the temperature of the systems does not remain at T 0 + ΔT, but slowly decreases to T 0, the interaction potential changes continuously, and the equilibrium separation moves again to smaller values. The bilayer separation has to follow the changes via a series of potential wells. So far, we have considered the bilayer/water system as a closed system that does not interact with the excess water phase. In reality, also diffusion of bulk water through the phospholipid bilayers will have an impact on the relaxation process. Diffusion proceeds either directly through the membrane or through local membrane defects, i.e., several times faster, and is16driven by the hydration forces due to the water deficit. Thus, the rate of water transport into the interbilayer space will affect the relaxation velocity by retarding (damping) the relaxation process. Since the temperature of the heated sample volume decreases in the later stages of the experiment by heat diffusion into the unheated sample regions, we have to consider two processes, at the same time: a relaxation – due to the anomalously thin water layer – and a bilayer swelling – due to the decrease in temperature. Both processes are governed by the elastic properties of the bilayer/water system. However, we expect the first process, i.e., the structural relaxation, to be initially dominant over the temperature dependent bilayer swelling. The rapid drop in intensity (Fig. 5a) and the relaxation acceleration (Fig. 5b) within the first 2-3 seconds after the laser pulse can be therefore interpreted as the fingerprint of this relaxation process. The constant relaxation velocity (Fig. 5b) is then attributed to the second – swelling – process. The swelling velocity is governed by the interplay of temperature dependent molecular rearrangements within the phospholipid bilayer and water diffusion in and out of the water layers, as the system will generally face both non-equilibrium positions on the interaction potential well (Fig. 6), attracting and repelling. A possible relaxation pathway of the bilayer separation is sketched in Fig. 6. A further interesting aspect of T-jump experiments in the L α-phase is the relaxation behavior of the lattice parameter as a function of starting temperature (T 0), and temperature jump amplitude ( ΔT), respectively. We first present the results upon varying the starting temperature. The liposomal dispersion of POPC was equilibrated at temperatures between 10 and 70°C. At each temperature we performed a T-jump experiment with a jump amplitude of ΔT = 10 ± 2°C. The relaxation kinetics of the d-spacings were analyzed in terms of the double17exponential decay model (Eqs. 4, 9 – 12). Fig. 7 shows the results for the most important parameters, i.e., the change in repeat distance Dd, the square of oscillation frequency w02 , the damping factor d, the zero velocity v0 and the zero acceleration a0. As T 0 is increased, the absolute value of Dd decreases linear up to a temperature of 50°C. This agrees well with the findings under equilibrium conditions, where the membrane thickness also exhibited a linear decrease in the range of 10°C to 50°C (Fig. 2). For the present purposes, water can be regarded as an incompressible medium, and therefore, Dd is directly related to the compressibility of the phospholipid membrane stacks. The results therefore demonstrate that below 50°C, the transmitted thermal energy can be directed into a membrane thinning. Mechanically, this regime corresponds to a linear relationship between applied tension and deformation, i.e., Hooke’s law. This is also expressed in the constancy of the other parameters w02 , d , v0 and a0. However, these parameters cannot be attributed to membrane properties only. Here, the bilayer/water system has to be considered as an entity, since it is the interplay of repulsive and attractive forces of bilayer/water interactions which governs the relaxation process and thus influences these parameters. As described in the theory section, the square of the oscillation frequency w02 is directly related to the proportionality constant of the repelling forces, which are dominated by hydration potentials (Fig. 6). The results depicted in Fig. 8 clearly show that, as long as the linearity of Hooke’s law is given, the relaxation proceeds with the same "spring constant". This further emphasizes that the transmitted energy can be transformed into hydrocarbon trans -gauche transitions within the regime of Hooke’s law, so that the bilayer separation and hence also the bilayer repulsion after the laser pulse is practically the same. In this sense, the hydrocarbon chains act as a buffer for the thermal energy.18A similar behavior is found for the damping factor, which is related to the system inertia or viscosity. As the viscosity of water can be neglected, the d parameter is a measure of viscosity of the phospholipid bilayers. The viscosity also includes the permeability of the membrane for water, since we concluded before, that water will diffuse from the excess phase into the water layer of the membrane stacks. Above 50°C, the total change in d-spacing exhibits saturation, no further compression of the bilayer/water system can be achieved with the supplied laser energy. As found under equilibrium conditions, the bilayer is less elastic above 50°C due to the maximum number of gauche isomers reached. The transmitted energy cannot be completely buffered by the bilayer, so that the reaction, i.e., membrane thinning, to the sudden temperature increase is not strictly linear, resulting in a minimal constant Dd above 50°C. The excess of thermal energy results in a stronger repulsion of the membrane surfaces as observed in the increase of w02 , d , v0 and a0. Fig. 8, depicts the changes in the relaxation parameters as a function of the temperature jump amplitude. The sample was equilibrated at 40°C and the laser voltage was varied from 500 to 750 V in steps of 50 V, yielding the different T-jump amplitudes (Tab. 1). With increasing T- jump amplitude, the absolute value of Dd increases linear. The square of the oscillation frequency and the damping factor show a statistic variation, but can be regarded as constant within in the measurement error. On the other hand, both zero velocity and zero acceleration show a strong statistic variation, but a tendency to increase with increasing T-jump amplitude. At 40°C, the system is very close to the end of validity of Hooke’s law. This proximity to non-linearity may be the reason for the strong statistic fluctuations of the relaxation parameters w02 , d , v0 and a0. Nevertheless, the results clearly state that the discrete jump in the repeat distance Dd is a linear function of the transmitted laser energy and that the other19relaxation parameters are independent of the T-jump amplitude, as long as the linearity of Hooke’s law holds. If T-jumps are sensible to the hydrocarbon chain elasticity, then one would expect different results for phospholipids with different chain lengths. Table 2 shows the fitted relaxation parameters for fluid DPPC, POPC and DSPC multilayers. The laser energy was adjusted to a T-jump amplitude of 16°C. The results reveal that the discrete change in d-spacing Dd is within the measurement error for all three lipids equal to ~ 2.6 Å. Since this shrinkage can be attributed to hydrocarbon chain melting, we conclude that an equal amount of energy deposit causes an equal amount of trans -gauche transformations. However, the other parameters exhibit a dependence on the lipid type. Most interestingly, the square of the oscillation frequency is highest for POPC, such that the increasing order with respect to w02 is DPPC < DSPC < POPC. This might be an effect of the unsaturated 18:1c9 chain of POPC, but as w02 is related to the repelling forces, the reason might also be a better packing of the phospholipids at lower temperatures (Tab. 2), because a more compact bilayer will provide a better surface for the affecting repulsive forces. Nevertheless, the damping coefficient d increases with hydrocarbon chain length, which is reasonable as one would expect a higher viscosity for thicker membranes. The zero velocity v0 and zero acceleration a0, respectively, exhibit the same dependence on hydrocarbon chain length.20CONCLUSION The reaction of the liquid-crystalline multilayer system to the T-jump proceeds via an excited state. This intermediate structure L α* is characterized by a membrane thickness which compares to its equilibrium value at T 0 + ΔT and an anomalously thin water layer, as the result of a lateral expansion of the membrane and the impossibility of a sufficiently fast water diffusion from the excess phase into the water layers of the membrane stacks. Strong repulsive hydration forces then initiate the relaxation process (Fig. 5), where the membrane viscosity and its permeability for water controls the relaxation velocity. As the heat within the sample volume dissipates into the surrounding unheated sample, the relaxation proceeds over a series of potential wells and the finial state is equal to the original L α-phase (Fig. 6). Fig. 9 shows a cartoon of the T-jump induced structural transformation; the POPC bilayer has been generated by a molecular dynamics simulation ( Heller et al., 1993). Since the only two types of elastic deformations for fluid membranes are stretching and bending ( Lipowsky, 1995), the possible forms of the deformed intermediate liposomes are elongated vesicles or liposomes with a rippled "harmonica-like" topology (left inset of Fig. 9). In addition to a structural model for the L α-Lα* transition, the T-jump/relaxation experiments were found to provide a qualitative, fundamental insight into the elastic properties of the phospholipid bilayers. The elastic properties of model membranes have been discussed ever since Luzzati and co-workers described the hydrocarbon chain properties as rubber-like (Luzzati, 1968), where the impact of the elasto-mechanic membrane features is of high interest for membrane protein interactions (e.g. Mouritsen and Bloom, 1984; Nielsen et al., 1998, Lundbæk and Andersen, 1999). The standard measurement techniques for mechanical bilayer properties are pipette aspiration, the surface force apparatus, the osmotic stress method21and atomic force microscopy (Evans, 1995, Parsegian and Rand, 1995). However, the generic interactions of stacks of fluctuating membranes are still missing an accepted theoretical formulation, i.e., the physical basis of the hydration forces is not yet understood. Here T-jump experiments can give additional information, since they directly probe the membrane stacks mechanic characteristics as has been demonstrated in the present work for three different phospholipids (Tab. 2). So far the concept relies on a qualitative analysis given by the relaxation parameters Dd, w0 and d. Future research will be directed towards a relaxation theory that allows for a quantitative evaluation of membrane properties. As physical interaction prediction is of prime importance also for liposomal based rational drug design or nano-materials, the T-jump/relaxation method shall open the door to a wider field of applications.22REFERENCES Amenitsch, H., M. Rappolt, M. Kriechbaum, H. Mio, P. Laggner, and S. Bernstorff. 1998. First Performance Assessment of the Small-Angle X-ray Scattering Beamline at ELETTRA. J. Synchrotron Rad. 5:506-508. Bernstorff, S., H. Amenitsch, and P. Laggner. 1998. High-Throughput Asymmetric Double- Crystal Monochromator for the SAXS Beamline at ELETTRA. J. Synchrotron Rad. 5:1215- 1221. Caillé, A. 1972. Remarques sur la diffusion des rayon x dans les smectiques A, C.R. Acad. Sc. Paris, Ser. B 274:891-893. Caffrey, M. 1989. 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Polymer Sci. 93:25-29. Schaefer, C., and M. Päsler. 1970. Einführung in die theoretische Physik. DeGruyer, Berlin. Seelig, A. and J. Seelig. 1974. Structure of fatty acyl chains in a phospholipid bilayer measured by deuterium magnetic resonance, Biochemistry 13:4839-4845. Tardieu, A., V. Luzzati, and F.C. Reman. 1973. Structure and polymorphism of the hydrocarbon chains of lipids: a study of phosphatidylcholine-water phases. J. Mol. Biol. 75:711-733. Zhang, R., R. M. Suter, and J. F. Nagle. 1994. Theory of the structure factor of lipid bilayers, Phys. Rev. E 50:5047-5060.2728TABLE 1 Summary of the laser-pulse energy chosen and the resulting temperature amplitudes ΔT. Applied Laser Voltage (V) Energy of the laser-pulse (J) Temperature-jump amplitude ΔT (°C) 500 0.79 ± 0.04 5 ± 1 550 1.15 ± 0.06 8 ± 1 600 1.42 ± 0.07 10 ± 2 650 1.76 ± 0.09 12 ± 2 700 2.04 ± 0.10 14 ± 2 750 2.41 ± 0.12 16 ± 229TABLE 2 Relaxation parameters for three phosphatidylcholines (all 20% w/w) at a T-jump amplitude of 16°C. DPPC POPC DSPC chains 16:0/16:0 16:0/18:1c9 18:0/18:0 T0 (°C) 50 30 70 Δd (Å) 2.5 ± 0.3 2.8 ± 0.3 2.5 ± 0.3 ω02 (s-2) 0.044 ± 0.002 1.1 ± 0.1 0.34 ± 0.02 δ (s-1) 0.72 ± 0.03 1.6 ± 0.1 2.6 ± 0.2 v0 (Å/s) 1.5 ± 0.1 5.0 ± 0.7 7.7 ± 0.7 a0 (Å/s2) - 2 ± 0.4 - 13 ± 2 - 40 ± 230Figure Captions FIGURE 1 Experimental set-up of a T-jump experiment at the SAXS-beamline (Elettra). The IR-beam of a Erbium glass laser is directed via a prism onto the sample capillary which is thermostated by a Peltier heating unit. The high flux x-ray beam transverses the sample normal to the IR-beam. The structural changes are recorded by a position sensitive detector with a maximum time resolution of 5 ms. FIGURE 2 The equilibrium structure of POPC bilayers in the L α-phase at different temperatures. The changes on d-spacing, membrane thickness and interbilayer water thickness are depicted. The observed re-increase in lamellar repeat distance is due to a uptake of water which increases the bilayer separation, whereas the membrane itself gets thinner with temperature and exhibits an asymptotic behavior above 50°C. FIGURE 3 The elastic properties of POPC bilayers at equilibrium. The linear expansion coefficient α exhibits a constant value up to 40°C, and reduces linear as the temperature is increased further. FIGURE 4 The L α-Lα*-Lα transition in a liposomal dispersion of POPC induced by a 16°C temperature jump. The series of time-sliced diffraction shows the time course of the first order Bragg reflections. The laser is triggered at t = 0. FIGURE 5 The relaxation kinetics of the L α-Lα*-Lα transition. (a) The temporal evolution of the lamellar repeat distances, obtained by a Lorentzian fit to the first order Bragg peaks. The straight line gives the best fit of the relaxation model Eq. 4 to the data points. The lower part31depicts the changes in the Bragg peak intensity ( s) and FWHM ( m) during a T-jump relaxation experiment. (b) The relaxation velocity and the relaxation acceleration during the first 5 seconds after the laser shot. FIGURE 6 Illustration of the interaction potential as a function of bilayer separation at different temperatures. The IR-laser shoots the system into a situation ( ⊗) far away from equilibrium, where it faces the repulsion of hydration forces. As the temperature does not remain at T 0 + ΔT, but decreases with time to T 0 the equilibrium position ( ⊕) at the reached temperature will in general not be reached. FIGURE 7 The relaxation parameters of POPC liposomes as a function of initial temperature T0. The T-jump amplitude was 16°C at each temperature. FIGURE 8 The relaxation parameters of POPC liposomes as a function of the T-jump amplitude. The samples were equilibrated at T 0 = 40°C before each T-jump experiment. FIGURE 9 Cartoon of the T-jump induced structural chances on fluid bilayers (using PDB- files by Heller et al., 1993). An increase in temperature induces trans -gauche transitions in the fatty acid tails and an increase in lateral area per each single phospholipid molecule (bottom right insert). This leads in the compound of the membrane to a compression normal to the bilayer surface and a lateral bilayer expansion. Since water cannot diffuse fast enough from the excess phase into the interbilayer water region the water layer thickness reduces to an anomalous thin value. The bottom left insert depicts the possible intermediate forms of the liposomes.Detector Toroidal Mirror Double Crystal MonochromatorThermoelectric Sample Cuvette WigglerErbium IR-Laser He-Ne Alignment LaserIR-Beam Synchrotron Beam Pabst / Fig. 110 20 30 40 50 60 7016182022dW (Å) T (°C)444648dB (Å)646566d (Å) Pabst / Fig. 2Pabst / Fig. 320 30 40 50 60 70-2.5x10 -3-2.0x10 -3-1.5x10 -3-1.0x10 -3-5.0x10 -4 T (°C)α (K-1)Pabst / Fig. 4Pabst / Fig. 50 1 2 3 4 5-6-4-20a (Å/s 2) t (s)024v (Å/s)a b61626364d (Å) 0 2 4 6 8 10 12 14 160.020.040.060.08FWHM (Å) t (s)3456789I (a. u.)Pabst / Fig. 6ÅÄ Possible Recovery PathSystem after the T-jump @ T0 + ΔT System before the Perturbation @ T0Interaction Potential Bilayer Separation10 20 30 40 50 60 70-80-60-40-200 T0 (°C)a0 (Å/s 2)0246810v0 (Å/s)02468δ (s-1)-101234ω02 (s-2)-2.5-2.0-1.5-1.0Δd (Å) Pabst / Fig. 7Pabst / Fig. 85 10 15 20-30-20-10010 ΔT0 (°C)a0 (Å/s 2)0510v0 (Å/s)02δ (s-1)01ω02 (s-2)-4-3-2-10Δd (Å)lateral expansion compressionT-jump: 2 ms ΔT = 5 - 15 °C Relaxation: ~ 30 - 40 s Diffusion of water + decrease in temperature equilibrium form intermediate forms T0T0 + ΔTLαLα* Pabst / Fig. 9

arXiv:physics/0002041v1 [physics.atom-ph] 21 Feb 2000Josephson effects in dilute Bose-Einstein condensates S. Giovanazzi∗, A. Smerzi and S. Fantoni Istituto Nazionale di Fisica della Materia and Internation al School for Advanced Studies, via Beirut 2/4, I-34014, Trieste, Italy, (February 2, 2008) We propose an experiment that would demonstrate the “dc” and “ac” Josephson effects in two weakly linked Bose- Einstein condensates. We consider a time-dependent barrie r, moving adiabatically across the trapping potential. The ph ase dynamics are governed by a “driven-pendulum” equation, as in current-driven superconducting Josephson junctions. A t a critical velocity of the barrier (proportional to the crit ical tunneling current), there is a sharp transition between the “dc” and “ac” regimes. The signature is a sudden jump of a large fraction of the relative condensate population. Anal yti- cal predictions are compared with a full numerical solution of the time dependent Gross-Pitaevskii equation, in an experi - mentally realistic situation. PACS: 03.75.Fi,74.50.+r,05.30.Jp,32.80.Pj The Josephson effects (JE’s) are a paradigm of the phase coherence manifestation in a macroscopic quan- tum system [1–3]. Observed early on in superconduc- tors [2], JE’s have been demonstrated in two weakly linked superfluid3He-B reservoirs [4]. Weakly interacting Bose-Einstein condensate (BEC) gases [5] provide a fur- ther (and different) context for JE’s. Indeed, magnetic and optical traps can be tailored and biased (by time- dependent external probes) with high accuracy [6–8], al- lowing the investigation of dynamical regimes that might not be accessible with other superconducting/superfluid systems. The macroscopic BEC’s coherence has been demonstrated by interference experiments [6,7], and the first evidence of coherent tunneling in an atomic array, related to the “ac” JE, has been recently reported [8]. A superconducting Josephson junction (SJJ) is usu- ally biased by an external circuit that typically includes a current drive Iext. The striking signatures of the Josephson effects in SJJ are contained in the voltage- current characteristic ( V-Iext), where usually one can distinguish between the superconductive branch or “dc”- branch (V= 0,Iext/negationslash= 0), and the resistive branch or “ac”-branch ( V≈RIext) (see for example [2]). Exter- nal circuits and current sources are absent in two weakly linked Bose condensates and the Josephson effects have been related, so far, with coherent density oscillations be - tween condensates in two traps or between condensates in two different hyperfine levels [9–14]. This collective dynamical behavior is described by a non-rigid pendu- lum equation [9], predicting a new class of phenomena not observable with SJJ’s. Now the following question arises: can two weaklylinked condensates exhibit the analog of the voltage- current characteristic in SJJ? Although BECs are obvi- ously neutral, the answer is positive. A dc current-biased SJJ can be simulated by considering a tunneling barrier moving with constant velocity across the trap. At a crit- ical velocity of the barrier a sharp transition between the “dc” and “ac” (boson) Josephson regimes occurs. This transition is associated with a macroscopic jump in the population difference, that can be easily moni- tored experimentally by destructive or non-destructive techniques. In the following we will briefly introduce the phe- nomenological equations of the resistively shunted junc- tion (RSJ) model for the SJJ. We will describe the cor- responding experiment for two weakly linked BECs and show that the relevant equations are formally equivalent to the RSJ equations. Then we compare the analytical re- sults with a numerical integration of the Gross-Pitaevskii equation in a realistic 3D setup. In the RSJ model, SJJ is described by an equivalent circuit [2] in which the current balance equation is Icsin(θ) +GV+C˙V=Iext (1) whereIcis the upper bound of the Josephson supercur- rentI(which is represented, in the ideal case, by the sinusoidal current-phase relation I=Icsin(θ));Gis an effective conductance (offered by the quasiparticles and the circuit shunt resistor), and Cis the junction capac- itance. The voltage difference Vacross the junction is related to the relative phase θby ˙θ= 2eV/¯h. (2) In the low conductance limit G≪ωpCwhereωp=/radicalbig 2eIc/¯hCis the Josephson plasma frequency, combining equations (1) and (2) leads to the “driven pendulum” equation ¨θ=−ω2 p∂ ∂θU(θ) (3) whereUis the tilted “washboard” potential: U(θ) = 1−cos(θ) +iθ (4) withi=Iext/Ic. This equation describes the tran- sient behavior before the stationary dissipative behavior is reached (resistive branch). If we start from equilib- rium, with i= 0, and increase adiabatically the current, 1no voltage drop develops until the critical value i= 1 is reached (neglecting secondary quantum effects). At this pointVcontinuously develops until a stationary asymp- totic dissipative behavior is reached in a time scale ap- proximately of order C/G. Similar phenomenology may occur in BECs and we will derive equations formally iden- tical to Equations (3) and (4). A weak link between two condensates can be created by focusing a blue-detuned far-off-resonant laser sheet into the center of the magnetic trap [6]. The weak link can be tailored by tuning the width and/or the height of the laser sheet. Raman transitions between two conden- sates in different hyperfine levels provide a different weak link [7], in analogy with the “internal Josephson effect” observed in 70s with3He−A[15]. Here we consider a double well potential in which the laser sheet slowly moves across the magnetic trap with velocityv(but our framework can be easily adapted to investigate the internal Josephson effect). In the limit of very lowv, the two condensates remain in equilibrium, i.e. in their instantaneous ground state, because of the non-zero tunneling current that can be supported by the barrier. In fact, an average net current, proportional to the velocity of the laser sheet, flows through the barrier, sustained by a constant relative phase between the two condensates. This keeps the chemical potential difference between the two subsystems locked to zero, as in the SJJ dc-branch. However, the superfluid component of the current flowing through the barrier is bounded by a critical value Ic. As a consequence there exists a critical velocityvc, above which a non-zero chemical potential difference develops across the junction. This regime is characterized by a running-phase mode, and provides the analog of the ac-branch in SJJ’s. The ”dc” and ”ac” BEC regimes are governed by a phase-equation similar to the current-driven pendulum equations (3) and (4). Such equations together with the sinusoidal current-phase relation I=Icsin(θ) describe the phase difference and current dynamics. The dimen- sionless current iis related to the barrier velocity by i=v/v c (5) with the critical velocity vcgiven by vc=¯hω2 p F(6) whereFis to a good approximation represented by dou- ble the average force exerted by the magnetic trap on single atoms in one well. Equations (3)-(6) can be derived by a time-dependent variational approximation and have also been verified, as we discuss below, by the full numerical integration [18] of the Gross-Pitaevskii equation (GPE) [16,17]. The GPE describes the collective dynamics of a dilute Bose gas at zero temperature:i¯h∂ ∂tΨ =/bracketleftbig H0(t) +g|Ψ|2/bracketrightbig Ψ (7) whereH0(t) =−¯h2 2m∇2+Vext(r,t) is the non interacting Hamiltonian and where g= 4π¯h2a/m, withathe scat- tering length and mthe atomic mass. The order param- eter Ψ = Ψ ( r,t) is normalized as/integraltext dr|Ψ (r,t)|2=N, withNthe total number of atoms. The external po- tential is given by the magnetic trap and the laser bar- rierVext(r,t) =Vtrap(r) +Vlaser(z,t). We consider a harmonic, cylindrically symmetric trap Vtrap(r) = 1 2mω2 r/parenleftbig x2+y2/parenrightbig +1 2mω2 0z2whereωrandω0are the radial and longitudinal frequency, respectively. The barrier is provided by a Gaussian shaped laser sheet, focused near the center of the trap Vlaser(z) =V0exp/parenleftbig −(z−lz)2/λ2/parenrightbig with the coordinate lz(t) describing the laser motion and v=dlz/dtits velocity. The equations (3) to (6) can be derived by solv- ing variationally the GPE using the ansatz: Ψ ( r,t) = c1(t)ψ1(r)+c2(t)ψ2(r), wherecn=/radicalbig Nn(t) exp(iθn(t)) are complex time-dependent amplitudes of the left n= 1 and rightn= 2 condensates (see also [9]). The trial wave functionsψ1,2(r) are orthonormal and can be interpreted as approximate ground state solutions of the GPE of the left and right wells. The equations of motion for the rel- ative population η= (N2−N1)/Nand phaseθ=θ2−θ1 between the two symmetric traps are ¯h˙η= (2EJ/N)/radicalbig 1−η2sin (θ), (8) ¯h˙θ=Flz(t)−2EJ Nη/radicalbig 1−η2cos(θ)−NEc 2η, (9) whereEc= 2g/integraltextdrψ1(r)4is the variational ana- log of the capacitive energy in SJJ, while EJ= −N/integraltext drψ1(r)/bracketleftbig H0+gNψ2 1(r)/bracketrightbig ψ2(r) is the Josephson coupling energy. The current-phase relation I= Ic/radicalbig 1−η2sin(θ) is directly related to Eq. (8) where the critical current is given by Ic=EJ/¯h. Flz(t) represents the contribution to the chemi- cal potential difference in the two wells due to the laser displacement lz(after linearizing in lz), and where F=/integraltext dr/parenleftbig ψ1(r)2−ψ2(r)2/parenrightbig∂ ∂lzVlaser≃ mω2 0/integraltext drz/parenleftbig ψ1(r)2−ψ2(r)2/parenrightbig . The above variational method provides a simple and useful interpolating scheme between the low interacting limit N2Ec≪EJ and the opposite limit N2Ec≫EJ. In the last case, and withη≪1, we recover the driven-pendulum phase equa- tion (3) and the critical velocity relations (5) and (6) with ¯hωp=√EJEc. In particular, it is legitimate to consider the Josephson coupling as a perturbation, with the the phase dynamics entirely determined by the difference in the chemical potentials µ1(N1,lz) andµ2(N2,lz) in the two wells. In this case Eccorresponds to 2 ( ∂µ1/∂N1)lz and ¯h2ω2 p=EJ(∂µ1/∂N1)lz. The critical velocity is pro- portional to the critical current: vc=/parenleftBig d N1 dlz/parenrightBig−1 Ic, with 2/parenleftbiggdN1 dlz/parenrightbigg−1 =/parenleftbigg∂µ1 ∂lz/parenrightbigg−1 N1/parenleftbigg∂µ1 ∂N1/parenrightbigg lz(10) and (∂µ1/∂lz)N1beingF/2 in Eq.(6). These deriva- tives can be computed numerically. In the Thomas-Fermi (TF) limit they reduce to /parenleftbigg∂µ1 ∂N1/parenrightbigg lz=g VTF(11) and /parenleftbigg∂µ1 ∂lz/parenrightbigg N1=1 VTF/integraldisplay VT Fdr∂ ∂lzVlaser (12) whereVTFis the volume of the region in which Ψ 1is different from zero (in the TF approximation). We make the comparison of Eqs. (8) and (9) with a full numerical integration of the GPE in an experimentally realistic geometry relative to the limit N2Ec≫EJ. In particular, we show that Eq. (6), derived in the limit ofη≪1, still remains a good approximation even for η≈0.4. The details of the numerical calculation are given elsewhere [18]. We have considered the JILA setup, with N= 5×104 Rb atoms in a cylindrically symmetric harmonic trap, having the longitudinal frequency ω0= 50 s−1and the radial frequency ωr= 17.68 s−1. The value of the scatter- ing length considered is a= 58.19˙A. A Gaussian shaped laser sheet is focused in the center of the trap, cutting it into two parts. We assume that the (longitudinal) 1 /e2 half-width of the laser barrier is 3 .5µm and the barrier heightV0/¯h= 650 s−1. Although the lifetime of a trapped condensate can be as long as minutes, we have made a quite conservative choice, by considering a time scale on the order of one sec- ond. The possibility to perform experiments on a longer time-scale will improve the observability of the phenom- ena we are discussing. With this choice of time scale, that corresponds only to few plasma oscillations, an adi- abatic increase of the velocity is not possible, therefore we proceed as follows. For t<0 the laser is at rest in the middle of the trap, lz= 0, and the two condensates are in equilibrium. For t>0 the laser moves across the trap, with constant velocity, and the relative atomic popula- tion is observed at tf= 1s. With this initial condition, which introduces small plasma oscillations in the rela- tive population, it is expected, in absence of dissipation, to slightly reduce the critical current by the numerical factor≈0.725 (see the general properties of the driven pendulum equation [2]). In Fig.1 we show the relative condensate population η= (N2−N1)/N, calculated after 1 second, for differ- ent values of the laser velocity v. The crosses are the results obtained with the full numerical integration of the time-dependent GPE (7). The dot-dashed line showsthe equilibrium values ηeqof the relative population cal- culated with the stationary GPE and with the laser at rest in the ”final” position lz=v tf. The displacement ofη(tf) fromηeqis a measure of the chemical potential difference, being ∆ µ=µ2−µ1≈NEc(η(tf)−ηeq)/2. Forv<0.42µm/s, the atoms tunnel through the bar- rier in order to keep the chemical potential difference ∆µlocked around zero. The dc component of the tun- neling current is accounted for by an averaged constant phase difference between the two condensates. This is the close analog of the dc Josephson effect in superconduct- ing Josephson junctions. The small deviations between the dashed line and the crosses are due to the presence of plasma oscillations (induced by our initial condition). Atv≈0.42µm/s there is a sharp transition, connected with the crossover from the dc-branch to the ac-branch in SJJ. For v >0.42µm/s, the phase difference starts running and the population difference, after a transient time, remains on average fixed. A macroscopic chemical potential difference is established across the junction. In this regime ac oscillations in the population difference are observed. The frequency of such oscillations are approx- imatively given by ∆ µ(t)/¯h(not visible in the figure). 0.0000 0.0002 0.0004 0.0006 laser velocity [ mm/s]0.00.20.4(N1-N2)/N FIG. 1. Fractional population imbalance versus the veloc- ity of the laser creating the weak link. A sharp transition between the ”dc” and the ”ac” branches occurs at a bar- rier critical velocity. The solid line and the crosses are th e analytical and the numerical calculations, respectively. The dashed-dot line represents the static equilibrium value ηeqcal- culated with the center of the laser at v tf. The solid line of Fig.1 corresponds to the solutions of Eqs. (8) and (9) in which the value of the energy integrals EcN/¯h= 2.46ms−1andEJ/N¯h= 2.41×10−4ms−1 are chosen in order to give the correct value of ωp= 2.44×10−2ms−1andIc= 12.1ms−1. The val- uesωp,Icare calculated numerically studying the fre- quency of small oscillations around equilibrium and the current-phase relation, respectively. The force integral is 3F/¯h= 1.060ms−1µm−1. The parameters ωp,IcandF are calculated with the laser at rest ( v= 0) inlz= 0. Using these values in Eq. (6) and taking into account the reducing factor 0 .725 we obtain the value 0 .407µm s−1 for the critical velocity, in agreement with the value ob- served in the simulation. Small deviations between the variational solutions (full line in Fig.1) and the numerical results (crosses in Fig.1), above the critical velocity, are due to “level-crossing” ef - fects. Numerical results [18] show that when the con- densate ground state of the “upper” well is aligned with the excited collective dipole state in the “lower” well, a finite number of atoms go from the “upper” well to the “lower” well. Close to this tunneling resonance it is possible to control, by manipulating the barrier veloc- ity below a fraction of vc, the dc flux of atoms from the ground state condensate in the “upper” well to the longi- tudinal intrawell collective dipole mode of the condensate in the “lower” well. This effect is directly observable in the macroscopic longitudinal oscillations of the two con- densates (at frequencies ≈ω0). Concerning a possible realization of the phenomenon described in this work, we note that for small barrier velocitiesv, the motion of the laser sheet with respect to the magnetic trap with velocity vor,viceversa , the motion of the magnetic trap with velocity −v, are equiv- alent, there being negligible corrections due to different initial accelerations. Thus far we have discussed the zero temperature limit. At finite temperature dissipation can arise due to inco- herent exchange of thermal atoms between the two wells. This can be described phenomenologically by including a term−EcG˙θ/ω2 pin Eq. (3) where Gis the conductance. Dissipation will be negligible as long as the characteristi c time scale ( EcG)−1≈(20G/¯h)sis bigger than the time scale of the experiment ( ≈1s). To conclude we note that while it could be difficult to measure directly the plasma oscillations, since their amplitude is limited by ∆ η <4 N/radicalBig EJ Ec, the macroscopic change in the population difference may be easily de- tected with standard techniques. Moreover the frame- work that we have discussed can be easily adapted to investigate the internal Josephson effect. Our phenomenological equations are similar to the driven pendulum equation governing the Josephson ef- fects in SJJs. As a consequence, within this framework we can study the “secondary quantum phenomena”, such as the Macroscopic Quantum Tunneling between differ- ent local minima of the washboard potential (see for in- stance [19]). It is a pleasure to thank L. P. Pitaevskii, S. Raghavan and S. R. Shenoy for many fruitful discussions.∗Present Address: Department of Chemical Physics, Weizmann Institute of Science, 76100 Rehovot, Israel. [1] P. W. Anderson, Basic Notions of Condensed Matter Physics (Benjamin-Cummings, Menlo Park, 1984). [2] A. Barone and G. Paterno, Physics and Applications of the Josephson Effect (Wiley, New York, 1982). [3] A. Barone, NATO ASI Series Quantum Mesoscopic Phe- nomena and Mesoscopic Devices in Microelectronics , Ankara June 1999, (I.O. Kulik and R. Ellialtioglu Eds.) Kluwer (in press). [4] O. Avenel, and E. Varoquaux, Phys. Rev. Lett. 55, 2704 (1985); S. V. Pereverzev et al., Nature 388, 449 (1997); S. Backhaus, et al., Science 278, 1435 (1998); S. Backhaus, et al., Nature 392, 687 (1998). [5] M. H. Anderson et al., Science 269, 198 (1995); K. B. Davis, et al., Phys. Rev. Lett. 75, 3969 (1995); C. C. Bradley, et al., Phys. Rev. Lett. 75, 1687 (1995); D. G. Fried, et al., Phys. Rev. Lett. 81, 3811 (1998). [6] M. R. Andrews et al., Science 275, 637 (1997). [7] D. S. Hall et al., 81, 1539, 1543 (1998). [8] B. P. Anderson and M. A. Kasevich, Science 282, 1686 (1998). [9] A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy, Phys. Rev. Lett., 79, 4950 (1997); S. Raghavan, A. Smerzi, S. Fantoni, and S. R. Shenoy, Phys. Rev. A, 59, 620 (1999). [10] C. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls, Phys. Rev. A 55, 4318 (1997). [11] J. Ruostekoski and D.J. Walls, Phys. Rev. A 58R50 (1998) [12] P. Villain and M. Lewenstein, Phys. Rev. A 59, 2250 (1999). [13] I. Zapata, F. Sols, and A. Leggett, Phys. Rev. A 57, R28 (1998). [14] J. Williams, R. Walser, J. Cooper, E. Cornell, and M. Holland, Phys. Rev. A 59, R31 (1999). [15] R. A. Webb et al., Phys. Lett 48A, 421 (1974); Phys. Rev. Lett33, 145 (1974); A. J. Leggett, Rev. Mod. Phys. 47, 331 (1975); K. Maki and T. Tsuneto, Prog. Theor. Phys. 52, 773 (1974). [16] L. P. Pitaevskii, Sov. Phys. JETP, 13, 451 (1961); E. P. Gross, Nuovo Cimento 20, 454 (1961); J. Math. Phys. 4, 195 (1963). [17] F. Dalfovo, S. Giorgini, L. P. Pitaevskii and S. Stringa ri, Rev. Mod. Phys. 71, 463 (1999). [18] S. Giovanazzi, Ph.D. Thesis, SISSA, Trieste, Italy, (1998), unpublished. [19] G. Schon, and A. D. Zaikin, Phys. Rep. 198, 237 (1999); P. Silvestrini, B. Ruggiero and A. Esposito, Low Temp. Phys.22, 195 (1996). 4

arXiv:physics/0002042v1 [physics.soc-ph] 22 Feb 2000Heat and Kinetic Theory in 19th-Century Physics Textbooks: The Case of Spain JOS´E M. VAQUERO and ANDR ´ES SANTOS Departamento de F´ ısica, Universidad de Extremadura, E-06 071 Badajoz, Spain 1. INTRODUCTION The period of Enlightenment was a time when Spanish physics, which had been lagging behind the level of the rest of Europe, was ab le to recover. However, the political events of the end of the 18th -century, the War of Independence, and then the reign of Fernando VII ru ined the panorama of Spanish science. History was sadly to repeat itself with the Spanish Civil War of 1936, which brought to nothing t he efforts of physicists and mathematicians of the end of the 19t h and the beginning of the 20th centuries to regain the time lost durin g the reign of Fernando VII until the “generation of ’98”. Part of the disastrous situation during the 19th-century ca n be traced back to the legislation and curricular plans of Spain ’s univer- sities. The Faculty of Exact, Physical, and Natural Science s was not created until 1857, with the Ley (Law) Moyano. Physics had be en relegated to a minor faculty, that of Arts, as preparation fo r the major faculties, and formed part of the “Philosophical Instituti ons” which were studied in the following order: I. History of philosoph y and ele- ments of mathematics; II. Logic and metaphysics; III. Gener al physics; IV. Special physics. All was in a Latin which had become progr essively less suited to teaching, so that enlightened reformers alwa ys attempted to publish textbooks in Spanish. Physics texts in Latin were still being imposed on students even up to the time of the absolutist peri od of Fernando VII (Moreno Gonz´ alez, 1988). After the death of Fernando VII, an ordinance regulating pri nting was promulgated in 1834 which allowed a certain freedom in pu bli- cation, in particular in scientific publishing. One of the co nsequences of the centralization of curricular plans, such as the Plan P idal, was the impulse given to the production of textbooks associated with the new programs of study. They were original productions as wel l as translations of foreign authors, and formed the beginning o f a national scientific output which was to be of greater or lesser quality according to each particular case. The study plans usually counselled the use of certain textbooks which most closely approached the spir it of the SCED524.tex; 2/02/2008; 1:08; p.12 J. M. VAQUERO AND A. SANTOS courses. The procedure consisted in appointing committees to decide on the most suitable texts which then put forward a number of t hem to be chosen from. In the meantime, Europe was seeing a major change in vision to - wards atomism thanks to such personalities as Herapath, Wat erston, Clausius, van der Waals, Maxwell, and Boltzmann (Brush, 198 6). Im- ponderable fluids had been abandoned as the explanation of ph ysical phenomena, and Mechanics, Heat, Electromagnetism, and Opt ics had been unified by the concept of energy. Teaching was based on a p rogram of mechanical explanations (Harman, 1982). The goal of the present communication is to analyze the impac t of modern ideas concerning energy and the constitution of ma tter on the textbooks of 19th-century Spain. We have examined 45 wor ks on general physics at secondary and university education leve ls, with pub- lication dates from the mid 19th-century to the early 20th-c entury. One of the first things that struck us was the great similarity bet ween the books of different authors. With centralism, teachers were e ncouraged to write their own textbooks and, indeed, were rewarded for d oing so. At the same time, the administration’s goal was for teaching to be uni- form nationwide. The result was that textbooks were written essentially with the Ministerio de Fomento’s (Ministry of Development) official program as the Table of Contents, and that the texts served so lely to expound known science and in no way to serve as the basis for fu rther research. There were those who, being interested in science teaching, protested about the policy concerning textbooks. Represen tative of them was Eduardo Lozano y Ponce de Le´ on. Under the pseudonym L. Opando y Uceda, Lozano published “Programas y Libros de Te xto” (Programs and Textbooks) in Revista de la Sociedad de Profesores de Ciencias (Opando y Uceda, 1875), and under the pseudonym “Un Ex- treme˜ no” the same article in the journal El Magisterio Extreme˜ no (The Extreme˜ nan Schoolmaster ) (Extreme˜ no, 1875). In this same journal and on the same theme, Ildefonso Fern´ andez S´ anchez published another critical article (Fern´ andez S´ anchez, 1876). In his writi ngs Lozano asked for the programs to have reference strictly to the subject ma tter to be dealt with and not to the methods to be used, and that the progr ams should include suggestions of the books that were best suite d to the subject, but without the obligation of following them. The immediate consequence of that educational policy was th at the different editions of textbooks seemed rather to be reprinti ngs: textbook “immutability” from one edition to another was astonishing . Let us take two cases as example. The Trait´ e ´El´ ementaire de Physique (Elemen- tary Treatise on Physics ) by A. Ganot was used in many Spanish and European Institutes (secondary education) and Universiti es. The first SCED524.tex; 2/02/2008; 1:08; p.2Heat and Kinetic Theory in 19th-Century Textbooks 3 Spanish language edition dates from 1853. The 18th Spanish e dition was printed in 1923, and even as late as 1945 an edition was pub lished in San Sebasti´ an (Moreno Gonz´ alez, 1988). In all this time there were only small modifications made with respect to the first French edition. An even more extreme example, since it was a work with even mor e antiquated ways of putting the material than Ganot, was the t extbook of Gonz´ alez Valledor & Ch´ avarri. The 2nd edition dates fro m 1851 (Gonz´ alez Valledor & Ch´ avarri, 1851) and the 10th edition from 1870 (Gonz´ alez Valledor & Ch´ avarri, 1870). In these twenty yea rs and eight new editions the work remained without a single significant c hange, notwithstanding the archaic ways of setting forth the subje ct matter of the first editions. A third example is less disappointing: Elementos de F´ ısica (Elements of Physics ) by Enrique Iglesias Ejarque. The first edition of 1897 (Iglesias Ejarque, 1897) has quite a modern m anner of exposition. Nevertheless, while the following editions in troduce small amounts of additional material into the text, there are neve r any sig- nificant changes. The 8th edition dates from 1924 (Iglesias E jarque, 1924) and the last that we can find a reference to is the 10th edi tion in 1933. Clearly one may conclude that physics textbooks had quite a long effective lifetime. While there was a certain degree of difficulty in introducing n ew material into the textbooks, the great problem was to elimin ate con- tent which was included by tradition even though it was antiq uated. The case of Eduardo S´ anchez Pardo, the translator of Ganot’ s work, is significant. In the prologue to that popularly used book: The editor D. C´ arlos Bailly-Bailli` ere being for his part d esirous that our public in general, and the pupils of our Institutes and Facul ties in partic- ular, follow science in her latest advances, charged us with the translation of the latest edition of the cited Treatise on Physics. On tak ing on this commitment, with the object that the Spanish edition be more complete, we judged it convenient, corresponding thereby also to the w ishes of the editor, to conserve certain theories, the exposition of var ious experiments, and the description of some of the instruments or apparatus t hat had figured in earlier editions of the forementioned work, and th at in the latest edition had been totally or partially suppressed. (Ganot 18 76, p. v)1 2 Another of the characteristics of Spanish physics textbook s was their orientation to student success in tests and examinations. W orks that are halfway between textbook and simple program of the curri culum abound. Secondary education teachers often encouraged the students to think little about the physical phenomena themselves but to learn definitions, laws, and descriptions of apparatus. As clear e xamples we could cite, amongst several works, the Resumen de F´ ısica y Nociones de SCED524.tex; 2/02/2008; 1:08; p.34 J. M. VAQUERO AND A. SANTOS Qu´ ımica (A Summary of Physics and Notions of Chemistry ) (Santos de Castro, 1865), and Definiciones, Principios y Leyes de la F´ ısica (Defini- tions, Principles, and Laws of Physics ) (Paz Sabugo, 1892). Hence, for most students learning was not meaningful but purely memori zation. Together with the lack of practical classes, this may have be en one of the causes of Spanish physics’ slow rate of development du ring the 19th-century. 2. ANALYSIS OF THE TEXTBOOKS In order to analyze what we understand to have been the introd uc- tion of modern physics into Spanish textbooks, we asked ours elves the following questions relative to the works that we consulted : 1. Are imponderable fluids studied in the textbook? 2. Is the term caloric used to refer to heat? 3. Does the concept of energy appear in a general form? 4. Does the mechanical theory of heat appear? 5. Does the kinetic theory of gases appear? Affirmative responses to the first two questions would indicat e tra- ditionalism, and to the last three, modernity. We would clas sify a textbook as modern if questions 3 and 4 are answered affirmativ ely. If the kinetic theory of gases appears as well (question 5), t hen we would consider the textbook to be quite complete. Table I lis ts the responses to these questions for the 19th-century textbook s to which we had access. The works are ordered by publication date. Whe n the edition is not given, reference is to the first edition. Most of the works consulted belong to the library of the Real S o- ciedad Econ´ omica de Amigos del Pa´ ıs in Badajoz (Spain). Wh ile not exhaustive, the sample is significant, since students of the city of Bada- joz in the 19th-century were prepared in this centre for the e ntrance examinations of the Schools of Engineering (high-level tec hnical degree courses, rather than purely Engineering in the English lang uage sense) or Military Academies. Some comments are necessary concerning Table I. The books th at were published in Spain in the 1840s studied imponderable flu ids, and dealt neither with energy, nor with the mechanical theory of heat, nor with the kinetic theory of gases. This situation was natural since these concepts were only being developed in Europe at this time. Ho wever SCED524.tex; 2/02/2008; 1:08; p.4Heat and Kinetic Theory in 19th-Century Textbooks 5 Table I. Responses to the five questions posed in the text, acc ording to the different works consulted. Textbook 1 2 3 4 5 (Ribero Serrano, 1844) Yes Yes No No No (Deguin, 1845), 2nd ed. No No No No No (Morquencho Palma, 1845) Yes Yes No No No (Santos de Castro, 1846) Yes Yes No No No (Pinaud, 1847) Yes Yes No No No (Gonz´ alez Valledor & Ch´ avarri, 1851), 2nd ed. Yes No No No N o (Gonz´ alez Valledor & Ch´ avarri, 1856), 4th ed. Yes No No No N o (Gonz´ alez Valledor & Ch´ avarri, 1857), 5th ed. Yes No No No N o (Rodr´ ıguez, 1858) Yes Yes No Yes No (Fern´ andez de Figares, 1861), 2nd ed. Yes Yes No No No (Santos de Castro, 1865) Yes Yes No Yes No (Boutet de Monvel, 1866) No No No Yes No (Gonz´ alez Valledor & Ch´ avarri, 1868), 9th ed. Yes No No No N o (Rico Sinobas & Santisteban, 1869), 7th ed. Yes Yes No No No (Gonz´ alez Valledor & Ch´ avarri, 1870), 10th ed. Yes No No No No (Feli´ u P´ erez, 1874), 2nd ed. Yes No No No No (Rico Sinobas & Santisteban, 1875), 8th ed. Yes Yes No No No (Ganot, 1876), 7th ed. No No No Yes Yes (Fuertes Acevedo, 1879) No No No Yes No (Ramos Lafuente, 1880), 6th ed. Yes Yes No Yes No (Fuertes Acevedo, 1882), 2nd ed. No No No Yes No (Rico Sinobas & Santisteban, 1882), 10th ed. Yes Yes No No No (M´ arquez Chaparro, 1886) No No Yes Yes No (Pina Vidal, 1887) No No No Yes No (Amig´ o Carruana, 1889) No No Yes Yes Yes (Picatoste, 1889) No No No No No (Feli´ u P´ erez, 1890), 7th ed. No No Yes Yes No (Escriche Mieg, 1891) No No Yes Yes No (Paz Sabugo, 1892) No No Yes Yes Yes (Garagarza Dujiols, 1892) No Yes Yes No No (Lozano, 1893), 3rd ed. No No Yes Yes No (Mart´ ın de Argenta & Mart´ ınez Pacheco, 1893) No No Yes Yes N o (Rodr´ ıguez Largo, 1895), 2nd ed. Yes Yes Yes Yes No (Ribera et al., 1895) No Yes Yes Yes No (Feli´ u P´ erez, 1896), 8th ed. No No Yes Yes No (Iglesias Ejarque, 1897) No No Yes Yes Yes (Soler S´ anchez, 1900), 2nd ed. No No Yes Yes Yes (Lozano, 1900) No No Yes Yes No SCED524.tex; 2/02/2008; 1:08; p.56 J. M. VAQUERO AND A. SANTOS some foreign authors already did not study imponderable flui ds (neither did they use the term caloric to refer to heat), and their Span ish transla- tions set an example for such fluids to be abandoned. This was t he case with the textbook of Deguin, translated by Venancio Gonz´ al ez (Deguin, 1845). The translator would write his own textbook later (Go nz´ alez Valledor & Ch´ avarri, 1851), and despite going through nume rous edi- tions, would never abandon imponderable fluids. Another tex tbook of a French author is Pinaud (1847), translated by Florencio Mart´ ın Castro, although here imponderable fluids were still being s tudied in the original. The handbook of Morquencho Palma (1845) presents physics as a science of Nature in general, including subjects such as geo logy and geography. This was the concept of physics that existed in Eu rope at the beginning of the century. On the positive side, the mecha nical ideal which was at the base of the development of physics in the 19th -century did indeed have a reflection in Spanish textbooks. One reads i n the prologue of the work of Ribero Serrano (1844): (. . . ) Effectively, all the phenomena attributed to caloric, to light, and to the electric fluid are mensurable and calculable effects; all are attributed to forces, all consist of movements, and constitute, in sum, mechanics. (Ribero Serrano 1844, p. i )3 The texts that we were able to consult from the 1850s are mainl y editions of the work of Gonz´ alez Valledor & Ch´ avarri (Gonz ´ alez Valle- dor & Ch´ avarri 1851, 1856, 1857). The principal characteri stic of this text was its immutability, edition after edition. Although it studied imponderable fluids, the term caloric was not used, and neith er energy, nor the mechanical theory of heat, nor the kinetics of gases a ppear in the text. The other textbook that we consulted is that of Rodr ´ ıguez (1858). Although imponderable fluids are studied and the ter m caloric is used, the author briefly explains the theory of ondulation s4(as the mechanical theory of heat was first known), and says of that th eory that “it is the one which today seems more correct” (p. 173). T his work was awarded a prize in a public competition under the aus pices of the Real Academia de Ciencias (Gaceta de Madrid, 9 Septemb er 1854). Neither the concept of energy in a general form nor the kinetic theory of gases appear in Rodr´ ıguez’s text. It is interesti ng too that in this decade there appeared a physics textbook written in Lat in which followed the scholastic tradition (Sant, 1857). The textbooks of Spanish authors of the 1860s were still stud ying imponderable fluids. They were also still using the term calo ric to refer to heat, except for the editions of Gonz´ alez Valledor & Ch´ a varri (1868, 1870), as we noted before. One French-authored textbook, Bo utet de SCED524.tex; 2/02/2008; 1:08; p.6Heat and Kinetic Theory in 19th-Century Textbooks 7 Monvel (1866), did not study imponderable fluids or use the te rm caloric, but then neither did it use the energy concept or the theories we are looking for. The same was the case with the textbook of D eguin (1845), commented on above. A possible reason for these to be missing is that the translator, Ram´ on de la Sagra, used the 7th Frenc h edition. The note of modernity is found in the textbook of Santos de Cas tro (1865), in which there appears an idea concerning the mechan ical the- ory of heat, namely the hypothesis of ondulations, although for didactic purposes the theory of emissions (caloric) was preferred. T hus, one reads: The system of ondulations is the most scientific, and the most admitted in modern physics; but that of emission lends itself more to d emonstra- tions, for which reason it is generally preferred for the exp lanation of the phenomena of the caloric. (Santos de Castro 1865, p. 204)5 The textbooks of the 1870s begin to abandon the traditional t heses, and show signs of modernity. The most traditional are Rico Si nobas & Santisteban (1875), an 8th edition, and Feli´ u P´ erez (187 4). The latter already does not use the term caloric, and later editi ons were progressively modernized. The textbook of Ramos Lafuente (1880) studies imponderable fluids, and uses the term caloric. In its treatment of radiant heat, h owever, the work seems very modern. As noted above for Santos de Castro (1 865), the author prefers the caloric hypothesis for its simplicit y in teaching: The admissible hypothesis is at present that of ondulations , in the light of the advances in modern physics; but as it simplifies the dem onstra- tions, many physicists prefer the hypothesis of emission to explain the phenomena of heat. (Ramos Lafuente 1880, p. 148)6 The textbooks with a more modern spirit are those of Fuertes Acevedo (1879) and Ganot (1876). M´ aximo Fuertes Acevedo’s work neither studies imponderable fluids nor uses the term calori c. But en- ergy does not appear as a general concept either, despite the mechanical theory of heat being explained. Adolphe Ganot’s textbook wa s much used in Europe. The first Spanish edition is of the year 1853. W e con- sulted the 7th Spanish edition (Ganot, 1876). It contains a p aragraph explaining the dynamic theory of gases: gases are described as formed by elastic molecules in motion, and the elasticity of a gas at a given volume is proportional to the vis viva (total mass of the molecules multiplied by the square of their speed). While making use of such concepts as this (today, of course, replaced by kinetic ener gy), the text does not deal with energy in a general form. As is to be expecte d with this perspective, imponderable fluids and the term caloric h ave been forgotten, and the mechanical theory of heat is studied. SCED524.tex; 2/02/2008; 1:08; p.78 J. M. VAQUERO AND A. SANTOS All the textbooks of the 1880s that we consulted have abandon ed im- ponderable fluids, with the exception of the handbook of Rico Sinobas & Santisteban (1882) which is already in its 10th edition. Of the other works, the most traditional is that of Picatoste (1889), sin ce it neither deals with energy in a general way nor introduces the theorie s we are looking for. We then have the 2nd edition of the work of Fue rtes Acevedo (1882), with no substantial changes from the 1st edi tion. We also consulted a Portuguese secondary education textbook, Pina Vidal (1887), which has a similar perspective to those of Fuertes A cevedo (1879, 1882). The textbook of M´ arquez Chaparro (1886) is th e first of the series of books that we consulted in which energy is dea lt with in a general fashion. The mechanical theory of heat is studie d, but the kinetic theory of gases has still not appeared. The 7th ed ition of the work of Feli´ u P´ erez (1890) has almost nothing to do with the 2nd edition that we commented on above (Feli´ u P´ erez, 1874). No w, energy and the mechanical theory of heat are presented. In the prolo gue to the 6th edition, also included in the 7th, one reads: With great insistence I have attempted in the treatment of he at to relate together all the phenomena of thermo-dynamic theory. (Feli ´ u P´ erez 1890, p. v)7 The outstanding textbook of this decade is that written by Am ig´ o Carruana (1889), published in Tarragona. The author held th e chair of physics and chemistry in the Instituto Provincial of Tarrag ona. This is a modern text which includes the kinetic theory of gases, and in general explains physical phenomena mechanistically. In the work’ s prologue, the author speaks about a book of his on mechanics published i n 1885, and gives great importance to this branch of physics: The criterion that has inspired this treatise responds to th e necessity already recognized by all to explain the subject of Physics i n a single course, always preceded by a short course of Mechanics as fou ndation and basis of the former (. . . ). (Amig´ o Carruana 1889, p. 3)8 As a continuation of thermodynamics, Amig´ o Carruana descr ibes the kinetic theory of gases by following the ideas of Clausiu s and introducing the definition of free path of a molecule as the di stance travelled between two consecutive collisions. He deduces M ariotte’s Law, obtaining the formula pV=1 3nmv2, where pis the pressure of the gas, Vits volume, nthe number of molecules, vthe mean velocity, andmthe mass of a molecule. He also deduces Avogadro’s hypothesi s from the kinetic theory of gases (Vaquero, 1998). As an indic ation of the little that this personality and his work have been studi ed, there only appears one book of Amig´ o Carruana in the Collected Catalogue of Spain’s Bibliographic Heritage: 19th Century (Biblioteca Nacional, SCED524.tex; 2/02/2008; 1:08; p.8Heat and Kinetic Theory in 19th-Century Textbooks 9 1889). This is a textbook on elementary chemistry (Amig´ o Ca rruana, 1892). The main characteristic of the books of the 1890s is that they all now treat energy in a general form, as well as the mechanical t heory of heat. The textbook of Rodr´ ıguez Largo (1895) simultaneo usly uses imponderable fluids and the term caloric, but only as additio nal in- formation. A more curious case is that of Ribera et al. (1895) whose textbook uses the term caloric without studying imponderab le fluids. The work of Escriche Mieg (1891) has the interest of studying heat and light side by side9, to bring out their relationship as vibratory phenomena: (. . . ) The molecular vibrations of the bodies, transmitted b y the aether, produce the feelings of HEAT in the touch and of LIGHT in the si ght. (Escriche Mieg 1891, p. 496)10 Other modern textbooks are those of Lozano (1893), the pharm a- cist Garagarza Dujiols (1892), Mart´ ın de Argenta & Mart´ ın ez Pacheco (1893), and Feli´ u P´ erez (1896). The small book of Paz Sabugo (1892) contains only definitions and principles. At no time is any idea developed, since the book’ s object is to serve as a collection of phrases for students to learn in pr eparation for their examinations. As part of these examination aids, t he author prepared two plates which accompany the text, one on units an d abbre- viations of the decimal metric system, and another on physic al units. The kinetic theory of gases does not appear explicitly, but s ome of its results do. For instance, one can read Maxwell’s Law: The viscosity of a gas measured by the coefficient of friction i s independent of the density. (Paz Sabugo 1892, p. 60)11 Another modern work is that of Iglesias Ejarque (1897). The a uthor indicates on the first page in a footnote that the works that ha d been consulted were “Spanish: Escriche, Feli´ u, Mu˜ noz, Rojas, and Rodr´ ıguez Largo. Foreign: Ganot, Jamin, Joubert, Maxwell, and Tyndal l”. The text contains a short paragraph on the Theory of Gases, in whi ch Bernouilli, Clausius, and Maxwell are cited. A kinetic inte rpretation of pressure is also given, but neither is the concept of mean f ree path introduced nor estimates of molecular speeds given. The textbook of Soler S´ anchez (1900), another modern work, has in the book dedicated to heat an article on the Thermal Constitution of Gases , in which the kinetic theory is described. A kinetic interpr etation of pressure is given, together with an explanation of high mo lecular velocities, including numerical estimates. A short sectio n is dedicated to the Height of the Atmosphere , and another to the Mean Free Path , in which one reads that “Crookes calls mean free path the space t ravelled SCED524.tex; 2/02/2008; 1:08; p.910 J. M. VAQUERO AND A. SANTOS Figure 1. The evolution of textbook contents with respect to the aband oning of imponderable fluids and the introduction of the concept of en ergy, the mechanical theory of heat, and the kinetic theory of gases. The figure has been constructed by calculating for each decade the percentage of affirmative res ponses to the questions formulated in the text. The intention of these results is not to provide full statistical certainty, but to present qualitatively the findings of this investigation. One clearly appreciates the decline in the use of imponderable fluids and the rise of content related to an atomic view of matter. by the molecule between two of those collisions”. On the othe r hand, the kinetic theory of gases is not mentioned by Lozano (1900) . We also consulted some 20th-century books. They all had a mod ern perspective. The textbooks of Carrasco Garrorena (1925) an d Monz´ on Gonz´ alez (1928) did not include the kinetic theory of gases . The 6th and 8th editions of Iglesias Ejarque (1915, 1924) present im provements over the 1st edition. With respect to the kinetic theory of ga ses, a new paragraph is included to explain molecular velocities. The textbook of Gonz´ alez Mart´ ı (1904) explains van der Waals’s equation o f state. The last textbook that we wish to comment on is a translation o f an Italian book. It is the work of Castelfranchi (1932), a unive rsity text- book of modern physics where the kinetic theory of gases is de veloped completely. The topics in the book are surprising in their br eadth and modernity. As a qualitative summary, Figure 1 shows the evolution of the re- sponses to the questions that we formulated, according to th e data listed in Table I. SCED524.tex; 2/02/2008; 1:08; p.10Heat and Kinetic Theory in 19th-Century Textbooks 11 3. CONCLUSIONS Spain’s educational policies in the 19th-century encourag ed a great similarity between the general physics textbooks of differe nt authors, although there were voices such as that of Eduardo Lozano y Po nce de Le´ on, in discord with this policy. This situation led to the differences between the successive editions of a textbook being minimal , and as a result, the effective lifetime of these textbooks was extrem ely long. A noteworthy example is the Spanish translation of the work of A. Ganot — the 1st edition was published in 1853 and the last we know of w as 1945. Likewise, there were difficulties in rooting out the antiquat ed content of physics textbooks. In many of them, modern and old theorie s shared the pages. A significant example was the case of Eduardo S´ anc hez Pardo, who amplified his translation (Ganot, 1876) with elem ents of old editions that had been discarded from consideration in n ew foreign editions. The result was to help make the recovery of Spanish physics was a very slow process. With respect to the term caloric, it was used in the Spain of th is period with different acceptations: as an imponderable fluid responsi- ble for thermal phenomena (its original meaning), as a synon ym for heat (the latter being understood according to the mechanic al theory of heat), and as the cause of the phenomena of heat, whatever t heir nature. Also, during the analysis of the sources that were available , we found the orientation towards examination preparation to be exce ssive in the textbooks that we consulted. Some are a simple cookbook of la ws and physical phenomena that the students would have to learn if t hey were to pass their examination. Hence, no interest was aroused in reflecting on content or in carrying out experiments in practical class es. Certain textbooks shamelessly encouraged totally memoristic lear ning, so that content was not applied to new situations and was readily for gotten. Another fact revealed by the analysis of the textbooks was th e speed with which the mechanical theory of heat gained acceptance. This the- ory had become quite usual in many of the Spanish physics text books well before the concept of energy had begun to be treated in a g eneral form. While this fact seems disconcerting from today’s view point, it must be pointed out that both the mechanical theory of heat an d the second law of thermodynamics had been proposed before the la w of conservation of energy was generally accepted. The introduction of the kinetic theory into 19th-century Sp ain oc- curred from approximately the 1870s onwards. In the 1880s, t he theory already appears more developed in one of the textbooks that w e con- SCED524.tex; 2/02/2008; 1:08; p.1112 J. M. VAQUERO AND A. SANTOS sulted. This was the work of Amig´ o Carruana (1889), which st ands out for its mechanistic deductions of Mariotte’s Law and Avogad ro’s Law. The topics dealt with concerning the kinetic theory in the te xtbooks that we consulted were: generalities of the theory, kinetic interpretation of pressure, molecular velocities, mean free path, height o f the atmo- sphere, and the van der Waals equation. Lastly it should be re marked that we found no original contribution to kinetic theory in t hese Spanish works of the end of the 19th-century. 4. Acknowledgements This work has been partially supported by the DGES (Spain) th rough Grant No. PB97-1501 and by the Junta de Extremadura (Fondo So cial Europeo) through Grant No. IPR98C019. Notes 1The original text is: “Deseoso por su parte el editor D. C´ arl os Bailly-Bailli` ere de que nuestro p´ ublico en general, y particularmente los al umnos de nuestros In- stitutos y Facultades sigan ´ a la ciencia en sus ´ ultimos ade lantos, nos encomend´ o la traduccion de la ´ ultima edicion del citado Tratado de F´ ısi ca. Mas al encargarnos de este cometido, con objeto de que la edicion espa˜ nola fuera m as completa, juzgamos conveniente, correspondiendo por otra parte as´ ı ´ a los des eos del editor, conservar algunas teor´ ıas, la exposicion de varios experimentos y la descripcion de algunos instrumentos ´ o aparatos que en ediciones anteriores de la m encionada obra figuraban, y en la ´ ultima h´ anse total ´ o parcialmente suprimido.” 2Among the old material that was revived one can find, for insta nce, sections on Animal Electricity, Perreaux’s dynamometer, Alvergnat’s barometer, Carr´ e’s device for making ice or Sturm’s vision theory. 3The original text is: “(...) Efectivamente, todos los fen´ o menos atribuidos al cal´ orico, ´ a la luz y al fluido el´ ectrico, son efectos mensu rables y calculables; todos se atribuyen ´ a fuerzas, consisten todos en movimientos, y c onstituyen en fin la mec´ anica.” 4According to the hypothesis of ondulations, as defined in the 19th-century Span- ish books we have consulted, heat is caused by the rapid motio n of the molecules and is transmitted through the aether by ondulations. Thus, all the heat phenomena are referred to a unique cause, motion, in contrast to the heat un derstood as a substance (caloric). The hypothesis of ondulations is not exactly the same as the wave theory of heat (Brush, 1986), according to which heat is the vibrati ons of aether itself. 5The original text is: “El sistema de las ondulaciones es el ma s cient´ ıfico, y el mas admitido en la f´ ısica moderna; pero el de la emisi´ on se pres ta m´ as a las demostra- ciones, por lo que se prefiere generalmente para la esplicaci on de los fen´ omenos del cal´ orico.” 6The original text is: “La hip´ otesis de las ondulaciones es l a admisible en la actualidad, atendidos los progresos de la f´ ısica moderna; pero simplific´ andose las SCED524.tex; 2/02/2008; 1:08; p.12Heat and Kinetic Theory in 19th-Century Textbooks 13 demostraciones por la hip´ otesis de la emisi´ on, muchos f´ ı sicos la prefieren para esplicar los fen´ omenos del calor.” 7The original text is: “Con insistencia grande he procurado e n el tratado del calor hacer relacionar todos los fen´ omenos con la teor´ ıa t ermo-din´ amica.” 8The original text is: “El criterio que ha inspirado este trat ado, responde ´ a la necesidad ya reconocida por todos, de explicar en un s´ olo cu rso la asignatura de F´ ısica, precedida siempre de un cursillo de Mec´ anica como fundamento y base de aquella (...)” 9On the other hand, Escriche Mieg does not distinguish radian t heat from heat as energy of molecular motion 10The original text is: “(. ..) Las vibraciones moleculares de los cuerpos, trans- mitidas por el ´ eter, producen en el tacto la sensaci´ on de CA LOR y en la vista de LUZ.” 11The original text is: “La viscosidad de un gas medida por el co eficiente de frotamiento es independiente de la densidad.” References Amig´ o Carruana, J. M.: 1889, Tratado de F´ ısica Elemental , Establecimiento tipogr´ afico de Adolfo Alegret, Tarragona. Amig´ o Carruana, J. M.: 1892, Tratado de Qu´ ımica Elemental , Imprenta de Hip´ olito Garc´ ıa, Cartagena. Biblioteca Nacional: 1989, Cat´ alogo Colectivo del Patrimonio Bibliogr´ afico Espa˜ no l. Siglo XIX , (letter A and indices), Arco: Direcci´ on General del Libro y Bibliotecas, Madrid. Boutet de Monvel, B.: 1866, Nociones de F´ ısica Elemental y M´ edica (translated by Ram´ on de la Sagra), L. Hachette y C´ ıa, Paris. Brush, S. G.: 1986, The Kind of Motion We Call Heat , North-Holland, Amsterdam. Carrasco Garrorena, P.: 1925, Elementos de F´ ısica General , Librer´ ıa de Victoriano Su´ arez, Madrid. Castelfranchi, C.: 1932, F´ ısica Moderna , Gustavo Gili, Barcelona. Deguin, M.: 1845, Curso Elemental de F´ ısica (translated by Venancio Gonz´ alez Valledor), Imprenta de D. I. Boix, Madrid. Escriche Mieg, T.: 1891, Elementos de F´ ısica y Nociones de Qu´ ımica , Imprenta de Pedro Ortega, Barcelona. Extreme˜ no, Un: 1875, ‘Programas y Libros de Texto’, El Magisterio Extreme˜ no , 265–267. Feli´ u P´ erez, B.: 1874, Curso Elemental de F´ ısica Experimental y Aplicada y Nocion es de Qu´ ımica Inorg´ anica (2nd edition), Imprenta de Jos´ e Rius, Valencia. Feli´ u P´ erez, B.: 1890, Curso Elemental de F´ ısica Experimental y Aplicada y Nocion es de Qu´ ımica Inorg´ anica (7th edition), Imprenta de Pedro Ortega, Barcelona. Feli´ u P´ erez, B.: 1896, Curso Elemental de F´ ısica Experimental y Aplicada (8th edition), Tipograf´ ıa de Comas Hermanos, Zaragoza. Fern´ andez de Figares, M.: 1861, Manual de F´ ısica y Nociones de Qu´ ımica (2nd edition), Imprenta y librer´ ıa de Don Jos´ e Mar´ ıa Zamora, G ranada. Fern´ andez S´ anchez, I.: 1876, ‘Libros de Texto’, El Magisterio Extreme˜ no , 210–212. Fuertes Acevedo, M.: 1879, Curso de F´ ısica Elemental y Nociones de Qu´ ımica , Imprenta y litograf´ ıa de V. Brid, Oviedo. SCED524.tex; 2/02/2008; 1:08; p.1314 J. M. VAQUERO AND A. SANTOS Fuertes Acevedo, M.: 1882, Elementos de F´ ısica y Nociones de Qu´ ımica (2nd edition), La Minerva Extreme˜ na, Badajoz. Ganot , A.: 1876, Tratado Elemental de F´ ısica Experimental y Aplicada y de Me - teorolog´ ıa (7th edition, translated by Eduardo S´ anchez Pardo & Eduard o Le´ on Ortiz), C´ arlos Bailly-Bailli` ere, Madrid. Garagarza Dujiols, F., 1892, Instrumentos y Aparatos de F´ ısica de Aplicaci´ on a la Farmacia , librer´ ıa de la viuda de Hernando y C´ ıa, Madrid. Gonz´ alez Mart´ ı: 1904, Tratado de F´ ısica . With no place of publication or printer given in the copy consulted. Gonz´ alez Valledor, V. & Ch´ avarri, J.: 1851, Programa de un Curso Elemen- tal de F´ ısica y Nociones de Qu´ ımica (2nd edition), Imprenta del colegio de sordo-mudos, Madrid. Gonz´ alez Valledor, V. & Ch´ avarri, J.: 1856, Programa de un Curso Elemental de F´ ısica y Nociones de Qu´ ımica (4th edition), Imprenta del colegio de sordo-mudos, Madrid. Gonz´ alez Valledor, V. & Ch´ avarri, J.: 1857, Programa de un Curso Elemental de F´ ısica y Nociones de Qu´ ımica (5th edition), Imprenta del colegio de sordo-mudos, Madrid. Gonz´ alez Valledor, V. & Ch´ avarri, J.: 1868, Programa de un Curso Elemental de F´ ısica y Nociones de Qu´ ımica (9th edition), Imprenta del colegio de sordo-mudos, Madrid. Gonz´ alez Valledor, V. & Ch´ avarri, J.: 1870, Programa de un Curso Elemental de F´ ısica y Nociones de Qu´ ımica (10th edition), Imprenta del colegio de sordo- mudos, Madrid. Harman, P. M.: 1982, Energy, Force, and Matter. The Conceptual Development of Nineteenth Century Physics , Cambridge University Press, Cambridge. Iglesias Ejarque, E.: 1897, Elementos de F´ ısica , Imprenta de Crespo Hermanos, Jerez. Iglesias Ejarque, E.: 1915, Elementos de F´ ısica (6th edition), Establecimiento tipogr´ afico de Domingo Sar, Vitoria. Iglesias Ejarque, E.: 1924, Elementos de F´ ısica (8th edition), Imprenta y Librer´ ıa de Andr´ es Mart´ ın S´ anchez, Valladolid. Lozano, E.: 1893, Elementos de F´ ısica , Imprenta de Jaime Jep´ us y Roviralta, Barcelona. Lozano, E.: 1900, F´ ısica , Manuales Soler, Barcelona. M´ arquez Chaparro, B.: 1886, Resumen de un Curso de F´ ısica Experimental y Nociones de Qu´ ımica , Imprenta y litograf´ ıa de Jos´ e Mar´ ıa Ariza, Sevilla. Mart´ ın de Argenta, V. & Mart´ ınez Pacheco, J.: 1893, Nuevo Tratado de F´ ısica y Qu´ ımica , Librer´ ıa de Victoriano Su´ arez, Madrid. Monz´ on Gonz´ alez, J.: 1928, Nociones de F´ ısica y Qu´ ımica , Sucesores de Rivadeneyra, Madrid. Moreno Gonz´ alez, A.: 1988, Una Ciencia en Cuarentena. Evoluci´ on de la F´ ısica en la Universidad y Otras Instituciones Espa˜ nolas de Mediado s del Siglo XVIII a la Crisis Finisecular del XIX , CSIC, Madrid. Morquencho Palma, G.: 1845, Manual o Resumen de un Curso de F´ ısica Ex- perimental, y Nociones de Qu´ ımica , Imprenta y librer´ ıa de D. Ignacio Boix, Madrid. Opando y Uceda, L.: 1875, ‘Programas y Libros de Texto’, Revista de la Sociedad de Profesores de Ciencias ,Vol. VI , 232–235. Paz Sabugo, M.: 1892, Definiciones, Principios y Leyes de la F´ ısica , Tipograf´ ıa La Econ´ omica, Badajoz. SCED524.tex; 2/02/2008; 1:08; p.14Heat and Kinetic Theory in 19th-Century Textbooks 15 Picatoste, F.: 1889, Elementos de F´ ısica y Qu´ ımica , Librer´ ıa de la Vda. de Hernando y C´ ıa, Madrid. Pina Vidal, A.A.: 1887, Principios de Physica , Typographia da Academia Real das Ciencias, Lisbon. Pinaud: 1847, Programa de un Curso Elemental de F´ ısica (translated by Florencio Mart´ ın Castro), Imprenta de Concha y C´ ıa, C´ aceres. Ramos Lafuente, M.: 1880, Elementos de F´ ısica (6th edition), Imprenta y librer´ ıa de la Vda. e hijo de Aguado, Madrid. Ribera, J., Nacente, F. & Soler, P.: 1895, F´ ısica Industrial o F´ ısica Aplicada a la Industria, la Agricultura, Artes y Oficios , J. Rom´ a, Barcelona. Ribero Serrano, A.: 1844, Tratado Elemental de F´ ısica Elemental y M´ edica (7th edition), Joaqu´ ın del R´ ıo, Madrid. Rico Sinobas, M. & Santisteban, M.: 1869, Manual de F´ ısica y Qu´ ımica (7th edition), Imprenta de Manuel Minuesa, Madrid. Rico Sinobas, M. & Santisteban, M.: 1875, Manual de F´ ısica y Qu´ ımica (8th edition), Imprenta de Manuel Minuesa, Madrid. Rico Sinobas, M. & Santisteban, M.: 1882, Manual de F´ ısica y Qu´ ımica (10th edition), Moya y Plaza, Madrid. Rodr´ ıguez, E.: 1858, Manual de F´ ısica General y Aplicada a la Agricultura y a la Industria , Eduardo Aguado, Madrid. Rodr´ ıguez Largo: 1895, Elementos de F´ ısica , Madrid. Sant, A.: 1857, Physica Theoretica et Experimentalis Ejusque Propagines e t Appen- dices Chimie, Astronomie et Historia Naturalis , Ex. Tip. Petra Sant, Caelsonae [Solsona]. Santos de Castro, F.: 1846, Curso Elemental Completo de F´ ısica Esperimental , Imprenta de D. F. ´Alvarez y C´ ıa, Sevilla. Santos de Castro, F.: 1865, Resumen de F´ ısica y Nociones de Qu´ ımica , Francisco ´Alvarez, Sevilla. Soler S´ anchez: 1900, Elementos de F´ ısica (2nd edition), Alicante. Vaquero Mart´ ınez, J.M.: 1998, ‘Dos Deduciones Cin´ eticas de la Ley de los Gases Perfectos en Libros Espa˜ noles de F´ ısica del Siglo XIX’, Revista Espa˜ nola de F´ ısica ,12(4), 43–46. SCED524.tex; 2/02/2008; 1:08; p.15This figure "s_e.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0002042v1

arXiv:physics/0002043v1 [physics.flu-dyn] 22 Feb 2000The decay of multiscale signals – deterministic model of the Burgers turbulence S.N. Gurbatov1,3, A.V.Troussov2,3 February 2, 2008 1Radiophysics Dept., University of Nizhny Novgorod 23, Gagarin Ave., Nizhny Novgorod 603600, Russia. E-mail: gurb@rf.unn.runnet.ru (Permanent address) 2Joint Institute of Physics of the Earth RAS, Molodezhnaya Str., 3, GC RAS, Moscow, 117296 Russia. E-mail: shtrssv@cityline.ru, troussov@hotmail.com (Permanent address) 3Observatoire de la Cote d’Azur, Laboratorie G.D. Cassini, B.P. 229, 06304, Nice Cedex 4, France. Corresponding author: 3731007. Sergey N. Gurbatov, Radiophysics Dept., University of Nizhny Novgorod, 23, Gagarin Ave., Nizhny Nov gorod 603600, Russia. E-mail: gurb@rf.unn.runnet.ru. Tel.: (8 312) 656002, Fax: (8 312) 656416, Tel.(Home): (8 312) 340969. Abstract This work is devoted to the study of the decay of multiscale de terministic solutions of the unforced Burgers’ equation in the limit of vanishing v iscosity. It is well known that Burgers turbulence with a power law ener gy spectrum E0(k)∼ |k|nhas a self-similar regime of evolution. For n <1 this regime is characterised by an integral scale L(t)∼t2/(3+n), which increases with the time due to the multiple mergings of the shocks, and therefore, the energy of a random wave decays more slowly than the energy of a periodic signal. In this paper a deterministic model of turbulence-like evol ution is considered. We construct the initial perturbation as a piecewise linear an alog of the Weierstrass func- tion. The wavenumbers of this function form a ”Weierstrass s pectrum”, which ac- cumulates at the origin in geometric progression. ”Reverse ” sawtooth functions with negative initial slope are used in this series as basic funct ions, while their amplitudes are chosen by the condition that the distribution of energy o ver exponential intervals of wavenumbers is the same as for the continuous spectrum in B urgers turbulence. Combining these two ideas allows us to obtain an exact analyt ical solution for the 1velocity field. We also notice that such multiscale waves may be constructed for mul- tidimensional Burgers’ equation. This solution has scaling exponent h=−(1+n)/2 and its evolution in time is self- similar with logarithmic periodicity and with the same aver age law L(t) as for Burgers turbulence. Shocklines form self-similar regular tree-li ke structures. This model also describes important properties of the Burgers turbulence s uch as the self-preservation of the evolution of large scale structures in the presence of small scales perturbations. PACS 43.25.Cb,47.27.Eq. Keywords: Burgers’ equation; Burgers turbulence 1 Introduction The nonlinear diffusion equation ∂v ∂t+v∂v ∂x=ν∂2v ∂x2;v(x, t= 0) = v0(x). (1.1) was originally introduced by J.M.Burgers in [6] (1939) as a m odel for hydrodynamical tur- bulence. Burgers’ equation (1.1) describes two fundamenta l effects characteristic of any turbulence [10]: the nonlinear redistribution of energy ov er the spectrum and the action of viscosity in small scales. Burgers’ equation was used lat er to describe a large class of physical systems in which the nonlinearity is fairly weak (q uadratic) and the dispersion is negligible compared to the linear damping [29]. The most imp ortant example of such waves are acoustical waves with finite amplitude [24]. Another cla ss of problems, arising, e.g., in surface growth, also leads to Burgers’ equation [5],[8],[3 0]. The three dimensional form of (1.1) has been used in cosmology to describe the formation of large scale structures of the Universe at a nonlinear stage of gravitational instability ( see e.g. [15], [25], [13], [28] ). In the physically important case of large Reynolds number, t he action of viscosity is significant only in the small regions with high gradient of th e velocity field. In the limit ν→0, the solution of Burgers’ equation has the following form ( see [18], [7], [13]): v(x, t) =x−y(x, t) t, (1.2) where y(x, t) is the coordinate of the maximum of the function G(x, y, t) = Ψ 0(y)−(x−y)2 2t, v0(x) =−∂Ψ0(x) ∂x. (1.3) Strong interaction between coherent harmonics leads to the appearance of local self- similar structures in Burgers’ equation. A periodic initia l perturbation with zero mean velocity is transformed asymptotically into a sawtooth wav e with gradient ∂xv= 1/tand with the same period l0. It is important that at this stage the amplitude a(t) =l0/tand the energy density σ2(t)≃l2 0/12t2do not depend on the initial amplitude. 2An initial one-signed pulse with the area m >0, localised at t= 0 in the neighbourhood of the point x= 0, also has asymptotically a universal form: it transforms into a triangular pulse with the gradient ∂xv= 1/tand increasing coordinate of the shock xs≈(2mt)1/2. Due to the increase of the integral scale the amplitude of such a p ulsea(t) =xs(t)/t∼m1/2t−1/2 and its energy will decrease more slowly than for a periodic s ignal, like t−1/2. Continuous random initial fields are also transformed into s equences of regions with the same gradient ∂xv= 1/t, but with random locations of the shocks separating them. Du e to the multiple merging of the shocks the statistical proper ties of such random fields are also self-similar and may be characterised by the integral s cale of the turbulence L(t). The merging of the shocks leads to an increase of the integral sca leL(t), and because of this the energy σ2(t)∼L2(t)/t2(1.4) of a random wave decreases more slowly than the energy of peri odic signals. The type of turbulence evolution is determined by the behavi our of the large scale part of the initial energy spectrum E0(k) =α2knb0(k); (1.5) E0(k) =1 2π/integraldisplay /angbracketleftv0(x), v0(x+z)/angbracketrighteikzdz. (1.6) Here b0(k) is a function which falls off rapidly for k > k 0∼l0, and b0(0) = 1. For n >1 the law of enrgy decay strongly depends on the statistical p roperties of the initial field (see e.g. [30] and references therein). For the initial Gaussian perturbation the integral scaleL(t)∼t1/2times logarithmic correction obtains and is determined by t wo integral characteristics of the initial spectrum: the variances of t he initial potential Ψ 0and the velocity v0(x) [20],[9],[13],[16]. Forn <1 the structure function of the initial potential increases as a power law in space. Then the initial potential field is Brownian, or fractional B rownian motion, and some scaling may be used [7],[20], [13], [26], [3],[22], [23]. In this cas e the turbulence is also self-similar and the integral scale L(t) increases as L(t) = (αt)2/(3+n). (1.7) The energy of the turbulence is derived from (1.4): σ2(t)∼t−p, p=2(n+ 1) n+ 3. (1.8) The difference between these two cases ( n <1 and n >1 ) is connected to the process of parametric generation of low frequency component of the spe ctrum. For the case n <1 the newly generated low frequency components are relatively sm all and we have the conservation of large scale part of the spectrum: E(k, t) =E0(k) =α2kn, for k << 1/L(t). (1.9) Thus, the laws of turbulent decay are more complex than for si mple signals, which can be attributed to multiple merging of the shocks. In [12] a mod el of a regular fractal signal 3with decay lower than for single one-signed pulse was introd uced. The initial signal v0(x) was constructed as a sequence of one-signed pulses whose pos itions form a Cantor set with capacity (fractal dimension) D= lnN/lnβ, where Npis the number of pulses in the scale Lp≃L1Nβp−1, 0< D < 1. Multiple merging makes the decay of the wave slower and the general behaviour of the energy decay may be approximate d by the power law with the exponent in (1.8): p=1−D 2−D,0< p < 1. (1.10) The evolution proves to be self-similar in successive time p eriods ( ti, ti−1) and ( ti+1, ti+2), where ti+1/ti=β2/N. This shows log-periodical self-similarity of the field evo lution. Linear and non-linear decay of fractal and spiral fields given by the sequences of regular pulses was also investigated in [1]. It was shown that the power law (1.8 ) with the exponent given by the formula (1.10) holds also true for homogeneous fractal p ulse signal with capacity D. Another model of a multiscale signal, which has the same gene ral behaviour on the external scale L(t) (1.7), and energy of the Burgers turbulence, was also discu ssed in [12]. It was assumed therein that the initial signal is a discrete set of modes - the spatial harmonics v0(x) =∞/summationdisplay p=0apsin(kpx+ϕp), (1.11) with wavenumbers kpand amplitudes apgiven in terms of a parameter ǫby kp=k0ǫp, a p=a0ǫ−hp, h=−(n+ 1 2), a0=αk(n+1)/2 0 . (1.12) Amplitudes apand the scaling exponent hare chosen from the condition that the mean energy of harmonics in the interval ∆ p=kp−kp+1be identical to that corresponding to the spectral density (1.5): a2 p=E(kp)∆p. For ǫ << 1 and n >−1 the harmonics are spread over the spatial spectrum and accumulate at the point k= 0 with decreasing amplitude. The main approach in this model was that the energy of the wave is the sum of energies of independent modes. The approach is nontrivial, but neverth eless leads to the same laws of the integral scale L(t) (1.7) and the energy decay (1.8) as in the case of continuous spectrum (1.5). Let us point out that representation of the field given by the formula (1.11) is similar to shell models, which were introduced as useful models addr essing the problem of analogous scaling in fully developed turbulence (see, e.g., [11], [19 ] and references in there). In present paper we consider the evolution of a regular signa l whose behaviour in general is similar to the evolution of the Burgers turbulence with co ntinuous spectrum (1.5). The main difference,as compared to the model discussed above, is that we construct the exact solution of Burgers’ equation using as an initial mode the ”r everse” sawtooth wave. The frequency ratio in our model is ǫ= 1/N, where Nis an integer and N≥2. These perturba- tion are similar to the well known Weierstrass and Weierstra ss-Mandelbrot fractal functions (see [21], [4]). For the analysis of Burgers’ equation it is convenient to use a mechanical interpretation. There is a one-to-one correspondence between the solution o f Burgers’ equation and the 4dynamic of a gas of inelastically interacting particles [13 ], [14]. Let us take a one-dimensional particle flux with a contact interaction: as long as the parti cles do not run into each other they move with constant velocity. In the collision they stic k together, forming a delta- function singularity in the matter density. This leads to th e appearance of gas of two species: a hydrodynamical flux of the ”light” initial partic les, and a gas of ”heavy” particles arising in the adhesion process of light particles. The evol ution of the particle velocity field will be described by the solution of Burgers’ equation if we a ssume that the initial density of the light particles is ρ0=const, the velocity of particles is equal to the initial velocity in (1.1), and that the collision of the particles conserve th eir mass and momentum. This analogy permits construction of a very fast (linear time) al gorithm of solution of Burgers’ equation [27]. In our case, for the initial reverse sawtooth wave, all the ma tter turns into heavy particles at the same moment of time. Thus, after this time the evolutio n of the Burgers turbulence is fully determined by the motion of heavy particles, whose p ositions are positions of shocks, and masses are equal to ∆ v·t, where ∆ vare the amplitudes of the shocks. The paper is organized as follows. In Section 2 we consider th e evolution and interac- tion of “reverse” sawtooth modes. In section 3 we consider th e interaction of small scale mode with large scale structures. In Section 4 we investigat e the properties of the sawtooth Weierstrass-Mandelbrot fractal function. In section 5 we s how that deterministic model has logarithmic periodic self-similarity. We also discuss her e the multi-dimensional generaliza- tion of this model. Section 5 presents concluding remarks. 2 Evolution and interaction of ”reverse” sawtooth modes in Burgers’ equation Let us introduce the pth ”reverse” saw tooth mode as v(p)(x,0) =apA(kpx+ϕp) (2.1) Hereap, kpare amplitude and wavenumber of the mode, −ϕpis its phase. The function A(x) is 2πperiodic function, given on its first period by the following expression A(x) =π−x, x ∈[0,2π[ (2.2) The set of ”reverse sawtooth” functions is not orthogonal, b ut nevertheless we will introduce a set of modes satisfying equation (1.12), whose wavenumber s and amplitudes satisfy the same relations as the sinusoidal modes of [12], i.e. relatio n (1.11). We introduce the term ”reverse sawtooth” because this signal is a sawtooth with te eth facing to the right, but the term ”sawtooth” by itself is widely used in Burgers turbulen ce literature to refer to the late stage of the evolution of the wave profile a sequence of sawtee th with positive slope 1 /t. The solution of Burgers’ equation with a linear velocity pro filev0=−γ(x−x+) is well-known (see, e.g., [13]): v(x, t) =−γ(x−x+) 1−γt(2.3) 5The value γ−1has the dimension of time, and for γ >0, at the finite time t=γ−1the gradient ∂xvbecames infinite. For γ <0 the gradient becomes equal to ∂xv=t−1, independent of the value of γat times t≫ |γ|−1. Thus, we have from (1.12), (2.1),(2.2), that the evolution of the pth mode is characterised by the nonlinear time tp=γ−1 p=apkp−1=t0/(ǫ(n+3)/2)p, t0= 1/αkn+3/2 0 (2.4) Based on solution (2.3 ) it is easy to see that for the ”first” pe riod (if ϕp= 0) the evolution of the pth reverse mode at the initial stage ( t < t p) may be described as v(p)(x, t) =  x/t, if 0< x <t tpπ kp; 1 tp(π kp−x)/1 1−t/tp,if|π kp−x|<t tpπ kp; (x−π kp)/t, ift tpπ kp< x−π kp<π kp.(2.5) On the other hand, at time t=tp, the mode transforms into a ”direct” sawtooth wave with slope∂xv= 1/t, independent of the amplitude and wavenumber of the mode: v(p)(x, t) =/braceleftBiggx/t, if 0< x <π kp; (x−π kp)/t,ifπ kp< x <2π kp.(2.6) The density of energy σ2(t) =/angbracketleftv2(x, t)/angbracketrightL, where /angbracketleft/angbracketrightLdenotes averaging over the period, is conserved before t < t p, and decreases like ( kpt)−2aftert > t p. Consider now the evolution of a gas of sticky particles in the case of independent evolution of the pth mode. In the general case, the density of the gas is calcula ted by using the Jacobian of the transformation from Lagrangian to Eulerian coordina tes and may be written in the form ( see, e.g., [13] ) ρ(x, t) =ρ0(1−t∂xv(x, t)) (2.7) Then it is obvious that at the initial stage ρ(x, t) =ρ01 1−t/tp,|π kp−x|<t tpπ kp(2.8) while ρis zero outside this interval in each period. At time t=tpall the light particles in each period collide into a single heavy particle with mass mp=ρ0Lp=ρ02π kp, (2.9) and the heavy particles have positions xp,l=π kp−ϕp kp+2π kpl;l= 0,±1,±2, ... (2.10) equal to the zero positions of the initial pth mode. The process of light merging particles and the evolution of the velocity is shown in Fig.1. 6Consider now the joint evolution of two successive modes: pth and ( p+1)th. From (2.4) one can see that the ratio of nonlinear times of the successiv e modes is tp+1 tp=ǫ−(n+3)/2≡ǫ1−h, (2.11) which does not depend on pand increases if the exponent nis greater than −3. The gradient of the initial field v(p)(x) +v(p+1)(x) is −(γp+γp+1), so the effective nonlinear time for such a sum is tp,eff=tp,p+1=1 γp+γp+1=tp 1 +tp/tp+1, (2.12) Because all parts of the initial perturbation have the same s lope, all light particles will collide at the same time t=tp,p+1. The mass of heavy particles after merging are mp,i=ρ0∆i,i+1, where ∆ i,i+1is the distance between adjacent shocks in the initial perturbation. The ra tio of periods of two adjacent modes is Lp+1/Lp=kp/kp+1=ǫ−1. IfN=ǫ−1is an integer larger than 1, there will be N+ 1 heavy particles on the period of the larger scale mode ( p+ 1)th: ( N−1) with the massmp=ρ0Lp(2.9), and two particles with total mass equal to mp. These two particles only exist when the shock of the ( p+ 1)th mode is located in the interval between shocks of thepth mode. For simplicity, we will consider the case where the s patial relations kp+1ϕp+1=kpϕp+ 2πr/N, (2.13) between the phases of successive modes, hold. In this case th e discontinuities of the ( p+1)th mode do not produce new shocks in the total perturbation v(p)(x)+v(p+1)(x). Thus, at times tlarger than t > t r,eff, the masses of all heavy particles will be the same, as would b e the case without the large scale modes (2.9). The positions of these heavy particles at time t=tp,effare X(p,l)(tp,eff) =xp,l+v(p+1)(xp,l)tp,eff, (2.14) where xp,lare the zero positions of the pth mode (2.10). The velocity of this particle is equal tov(p+1)(xp,l). Equation (2.14) is obvious if we use the trivial equality v(p)(xp,l)+v(p+1)(xp,l) = v(p+1)(xp,l), and also note that the position of the heavy particle xp,l(tp,eff) is equal to the position at the same time of all light particles with initial coordinate x=xp,l. From (2.14) we immediately have that after time tp,effthe positions of the particles are X(p,l)(t) =x(p,l)+v(p+1)(xp,l)t. (2.15) The difference between the coordinates of the adjacent parti clesXp,l(t) and Xp,l+1(t) de- creases with time, proportionally to the gradient of v(p+1)(x): X(p,l+1)(t)−Xp,l(t) = (xp,l+1−xp,l)−t∂v(p+1)(x) ∂x(xp,l+1−xp,l)≡ (xp,l+1−xp,l)(1−t/tp+1). (2.16) 7These particles collide at time t=tp+1(2.4) and the newly created heavy particles will have masses m(p+1)=ρ0Lp+1 (2.17) and positions xp+1,l=π kp+1−ϕp+1 kp+2πl kp+1;l= 0,±1,±2, ... (2.18) The velocity of these particles is zero. Thus, at times tlarger then tp+1, the evolution of the initial perturbation v0(x) =v(p)(x)+ v(p+1)(x) will be the same as the evolution of only the large scale mode v(p+1)(x). The process of particles merging and the evolution of the velocity for th e sum of two successive modes with the periods ratio N= 2 are shown in Fig. 2. By recurrence, it is evident that for finite number of modes v(x) =v(p)(x) +v(p+1)(x) + ...+v(M)(x) the evolution of the field after tMwill be the same as the evolution of only the largest mode v(M)(x). The reason for this, is of course, the special relation bet ween the phases ϕpand the wavenumbers kpof all interacting modes: kp=k0/Np( see equation (1.12) ). For integer Nthe minimal value of any combination of these wavenumbers is equal to the largest mode wavenumber kM=k0/NM. So the nonlinear interaction does not produce new components at frequencies less than kM. 3 Interaction of small scale ”reverse” sawtooth mode with large scale structures Let us now consider the interaction of the pth mode with an infinite series of larger scale modes Wp(x) =∞/summationdisplay r=p+1vr(x)≡∞/summationdisplay r=p+1arA(krx+ϕr), (3.1) assuming that the phases of the modes satisfy the relations ( 2.13 ) and that kr=k0ǫr, ar= a0ǫ−hr, h=−(n+ 1)/2. From (3.1) and (2.4) we have for the gradient of the initial perturbation v0(x) =vp(x) +Wp(x) ∂xv0(x) = ∂xvp(x) +∂xWp(x) =/summationtext∞ r=pγp= =γ0/summationtext∞ r=p(ǫ(n+3) 2)r=γ0ǫ(n+3) 2p 1 1−ǫ(n+3) 2, (3.2) the condition n >−3 (h <1) being necessary for the series to converge. From (3.2), we have for the effective time of nonlinearity of pth mode ˜tp= 1/∂xv0(x) =tp(1−ǫ(n+3) 2), (3.3) with the original tpdetermined by the equation (2.4). 8Thus, after the time of collision ˜tp, heavy particles with mass mp=ρ0Lp(3.2) appear. The coordinates of these particles will be determined by an e quation similar to (2.15) xp,l(t) =xp,l(t) +Wp(xp,l)t, (3.4) with the velocity of particles determined by the function Wp(x) (see 3.1), which is a sum of all larger modes, and xp,lare the coordinates of the zeros of the pth modes. The difference between the coordinates of adjacent particles xp,landxp,lwill then decrease with time like (1−t/˜tp+1), where ˜tp+1=tp+1(1−ǫ(n+3)/2) is the inverse of the gradient of the function Wp(x), see (3.1) and (3.2). Thus, the time of particle collision f or this generation will be described by equation (3.3) with p=p+ 1, and the new masses will be determined by the period of the ( p+ 1)th mode - see equation (2.17). The extrapolation of this particle merging process to the ne xt generations is evident by recurrence. The qth collision of heavy particles takes place at time ˜tp+q=tp+q(1−ǫn+3 2), the masses of these particles at this time are determined by t he period of the ( p+q)th mode mp+q=ρ0Lp+q= 2πρ0/kp+q(2.9), (2.17). In the time interval t∈[˜tp+q,˜tp+q+1], ˜tp+q+1 ˜tp+q=ǫ−(n+3) 2=Nn+3 2; (3.5) the coordinates of particles will be determined by the equat ion (3.4) with p=p+q. Here Wp+q(x) is the sum of the velocities of all larger modes with r > p +q, and xp+q,lare the zeros of ( p+q)-th mode. It is important to note that at time t >˜tp+qthe evolution of the particles is solely determined by the modes with r≥p+q. It means, that at times t >˜tp+q the position of the particles does not depend on the presence in the initial condition of the small scale modes with r < p +q. Thus, two processes with different initial velocities: ˜ v0(x), the field with small scales, andv0(x), the field without small scales modes: v0(x) =Wp+q−1(x); ˜v0(x) =Wp−1(x) (3.6) will have the same evolution after t >˜tp+q−1. Even if p→ −∞ (when modes with very small scales Lp∼ǫ−p=Npand very large amplitudes ap∼a0(ǫ(n+1)/2)p=a0(N(n+1)/2)−p) are present in the initial perturbation) the multiple mergi ng of the particles will lead to the independence of the evolution of large scale modes with resp ect to the small scale modes. This effect is similar to the self–preservation of large scal e structures in Burgers turbu- lence [2],[17]. When the initial field v0(x) is noise, the highly nonlinear structures contin- uously interact and due to the merging of shocks, their chara cteristic scale L(t) constantly increases. The presence of small scale noise perturbation vh(x) results in additional fluc- tuations in the shock coordinates ∆ xk(t), and these fluctuations increase in strength with the passage of time. Thus, the final result of the evolution of the field is determined by the competition of two factors, the increase in the external sca leL(t) of the structures and the increase in the strength ∆ xk(t) of shock coordinates fluctuations, the later being related to the perturbation vh(x). In a turbulence, having power index n <1 (1.5), multiple merging of shocks leads to self-preservation of the large scale stru ctures independently of the presence 9of small scale components. For the model signal this effect ap pears for arbitrary ndue to the special choice of wavenumbers and phases of interacting modes. It was stressed in the introduction that the solution of Burg ers’ equation has a one-to-one correspondence with the dynamics of the gas of inelasticall y interacting particles ([13]). The stage when all light particles collide, forming heavy parti cles, corresponds to the solution of Burgers’ equation with a well-defined slope ∂xv= 1/t. In this case the profile of the fieldv(x, t) is fully determined by the coordinates and amplitudes of th e shocks. Their coordinates Xs(t) are equal to the coordinates of heavy particles, their velo city vs(t) =dXs(t) dt= (vs(xs−0, t) +vs(xs+ 0, t))/2 (3.7) is equal to the velocity of the particles, and the amplitude o f the shock ∆vs(x) = (v(xs−0, t)−v(xs+ 0, t)) =m/t (3.8) is determined by the mass of the particle ( ρ0≡1) (see, e.g., [13]). Thus, the investigation of the motion of heavy particles per mits to fully reconstruct the properties of the velocity field v(x, t) of Burgers’ equation. 4 The sawtooth Weierstrass-Mandelbrot fractal func- tion It was shown in the previous section that the evolution of the particles (shocks) is deter- mined by the function Wp(x) (3.1). The basis functions of Wp(x) are the reverse sawtooth periodic functions with wavenumbers kr=k0ǫrand amplitudes ar=a0ǫ−hr, h=−(n+1)/2 satisfy relations (1.12). Wavenumbers form a geometrical p rogression like in the Weier- strass function (see [4]) and accumulate at the origin k= 0. In the original Weierstrass function, the situation was the opposite with increasing fr equencies, but nevertheless the function Wp(x) has many properties of Weierstrass function and of its gene ralisation – the Weierstrass-Mandelbrot function (see [21], [4]). We consider here a deterministic function Wp(x) with the special phase relation ϕp= (2πk/N )p(k= 1,2, ..., N ;N= 1/ǫ), thus, the discontinuities in the largest modes r > p + 1 coincide with some of the discontinuities of the smaller m oder=p+ 1. The function Wp(x) is continuous in the intervals 2 π/kp+1= 2π/(k0ǫp+1) with the same slope in each interval. The inverse value of this slope ˜tp+1 ˜tp+1=tp+1(1−ǫn+3 2);tp+1=t0(ǫn+3 2)p+1(4.1) is proportional to the nonlinear time tp+1of the smallest mode. Of course, we need n >−3, so that the convergence of (3.2) is assured and the inequalit iestp+1> tphold. The amplitudes of the modes are proportional to ǫ(n+1)/2and for n >−1 the function Wp(x) is bounded Wp(x)≤∞/summationdisplay r=p+1ar=a0(ǫn+1 2)p+1 1 1−ǫn+1 2. (4.2) 10Thus, for finite pthe energy of Wp(x) is also finite. For the case of the phase relation introduced above, the functions Wp(x) also have scaling properties, so that for instance for k= 0, we have Wp(x) =ǫ−hpW0(ǫpx);Wp(ǫmx) =ǫhmWm+p(x). (4.3) The case −1< n < 1 is similar to the initial conditions with generalized whit e noise in Burgers turbulence. The energy of the initial signal in su ch turbulence is determined by the largest cutoff wavenumber, so in our model by the smallest scalep. Ifp→ −∞ the energy of the model signal (as the energy of white noise) will tend to infinity. But from the considerations in the previous section we have that at th e finite time tall the modes withtp< thave finite energy ∼L2 p/t2due to the nonlinear dissipation, so that the whole energy of the turbulence is also finite. Thus, even in the case of ”divergent” initial conditions (p→ −∞ ), we will have a ”convergent” solution for any time t >0. The case n <−1 is similar to having fractional Brownian motion initial co ndition in Burgers turbulence. In this case, the series (4.2) diverges and the initial signal Wp(x) is unbounded. But for Burgers turbulence ( for the process of pa rticles motion and colli- sions ) only relative velocity of the particles matters. So w e can use the same regularisation procedure with Wp(x). Such a procedure was done with the Weierstrass function in [21]. In our case, taking ϕp≡0, we can introduce the function W∞ p(x) =Wp(x)−Wp(0), according to [21], which is finite in all finite spatial interv als. The other way to get a bounded function is to use special phase relations for the modes. 5 Self-similarity properties of deterministic model in one and two dimension Here we summarise the properties of the evolution of the mult iscale deterministic signal using some additional information about scaling characteristic s ofWp(x), and compare them with the properties of the Burgers turbulence. Let us consider the evolution of the multiscale signal v0(x) =vp(x) +Wp(x). (5.1) It was shown that at times tfor which t >˜tp, heavy particles with mass Mp=ρ0Lpappear and their coordinates are determined by the relation (3.5). These particles collide at time ˜tp+1, (˜tp+1/˜tp=N(n+3) 2, N=ǫ−1), and new particles with masses mp+1=ρ0Lp+1=mpN appear. Their motion will be determined by the same law (3.5) with substitution p→p+1. Using the scaling properties of Wp(x) (4.3) we have, that the motion of the particles in this interval will be similar to the motion of the particles in the interval [ ˜tp,˜tp+1] if we rescale the time t/˜tp⇒t/˜tp+1. Since the ratio tp+1/tpdoes not depend on p, one can speak about the logarithmic periodic self-similarity of the motion of t he particles. This means that at arbitrary interval [ ˜tq,˜tq+1] the motion of the particles will be similar to the motion of t he particles in the interval ˜tp,˜tp+1, by the scaling factor xp/xq=ǫp−qin space, and the scaling function tp/tq= (ǫ−n+3 2)p−qin time. The coordinates and masses of the particles fully 11determine the velocity field, and so the solution of Burgers’ equation is also logarithmic periodic self–similar. With each collision, the mass M(t) of the particles increases N= 1/ǫtimes. The time interval between the two successive collisions increases a stp+1/tp=Nn+3 2. Thus, by the approximation of piecewise constant function m(t) by the power law m(t)≃m0(t/t0)(n+3)/2. (5.2) we obtain the same result as for the Burgers turbulence. In ou r case, mis proportional to the period of the smallest mode at time t, and is analogous to the integral scale in Burgers turbulence. In the case n >−1 we can also estimate the energy decay of the model signal. Fo r n >−1 and ǫ≪1 the main energy of the signal at time tis in the smallest mode and is proportional to L2(t)/t2. Thus, we have here again the same law for the energy decay as f or Burgers turbulence. The numerical simulation based on the algorithm [27] was don e to illustrate the process of particles merging and velocity field evolution. The traject ories of the particles and profile of the field at different times are plotted for the initial ”white noise” signal ( n= 0, h=−1/2) in Fig.3, and for the initial ”Brownian” motion ( n=−2, h= 1/2) in Fig.4. Ten modes with the ratio of successive wavenumbers ǫ= 1/N= 1/2 were used. The plots show the initial stage of the evolution in some relatively small region where the finiteness of the number of modes is not significant. In Fig. 3 one can see that for n= 0 the initial ”sawtooth” multiscale function oscillates nearv= 0 like a ”white noise” with finite variance. After the collis ion of light particles, when the reverse sawtooth function transforms into a sawtoo th wave with positive gradient ∂xv= 1/t, the structure of the signal is relatively simple, and even f orN= 2 the main energy remains in the mode with smallest wavenumber. In the case n=−2 the initial profile has a large deviation behaviour which is typical for Brownian motion functions. After merging of light particle s, the sawtooth profile has a set of small shocks with different amplitudes, which is also simi lar to the properties of Brownian signal in the Burgers turbulence [28]. In Figs. 3(c) and 4(c) the velocity field at three successive m erger times t∗/t∗∗=N(n+3)/2 are plotted. These figures show the logarithmic periodic sel f-similarity of the evolution of multiscale signals. We notice now that such multiscale waves may be constructed f or multidimensional Burgers’ equation. Let us assume that the initial vector fiel dVp(x) is an infinite series of ”reverse” modes vr(x): Vp(x) =∞/summationdisplay r=pvr(x), (5.3) In the two dimensional case, the rth ”reverse” mode may be composed of piecewise linear functions defined on a system of regular triangles of size Lrcovering the plane. We consider here the special case when the ratio of the scales of two adjac ent modes is Lr+1/Lr=ǫ−1= N= 2. We assume also the special symmetry and phase relation be tween the different modes. In our case one big triangle is divided into four small er triangles with vertices 12located at midpoints of its sides ( see Fig.5 ). We assume that inside each triangle in the rth mode the velocity has a linear profile vr(x) =−γr(x−x+), where x+is the coordinate of the center of the triangle. The solution of the multidimensi onal Burgers equation for such initial perturbation v(x, t) =−γ(x−x+) 1−γt(5.4) is now valid inside the triangle of size Lr(t) =Lr(1−tγr) . The value γ−1 rhas the dimension of time, and at the finite time tr=γ−1 rthe velocity gradient becomes infinite. On the other hand, at time t=tr, the mode transforms into a ”direct” sawtooth wave with the universal behaviour inside the new set of triangles and with the gradient 1 /tinde- pendent of the amplitude and wavenumber of the mode: v(x, t) =x−xc t, (5.5) wherexcis now the center of the triangle, coinciding with the top of t he initial triangular set. Consider now the evolution of a gas of sticky particles in the case of independent evolution of the rth mode. Then, it is obvious that at the initial stage, inside the ”collapsing” triangle of size Lr(t) =Lr(1−tγr) the density increases as ρ(x, t) =ρ01 (1−t/tp)2, (5.6) while ρis zero outside ”collapsing” triangular in each cell. At tim et=tr, all the light particles in each cell collide into a single heavy particle w ith mass mr=ρ0L2 r√ 3/4, (5.7) and the heavy particles’ positions are equal to the center of the initial triangle x+. We assume also that the evolution of the rth mode is characterised by a nonlinear time trthe same as in the one dimension case (2.4): tr=γ−1 r=t0/(2−(n+3)/2)r. (5.8) Let us now consider the evolution of the vector field Vp(x) (5.3) which is an infinite series of ”reverse” modes. The evolution of the vector field i s very similar to the evolution of the scalar field (3.1). For the gradient of the initial pert urbation Vp(x) we have the same relation (3.2) as in 1 D. The effective time of nonlinearity of the smallest pth mode in the vector field (5.3), in presence of all large scale modes, is de termined by the equation (3.3). Thus, after the time of collision ˜tp, heavy particles with mass mp(5.7) appear. Velocities of these particles will be determined by the function Vp+1(x) (5.3), which is a sum of all larger modes, but the number of particle collisions is determined b y the next p+ 1 mode. At time ˜tp+1we have a collision of four heavy particles. The extrapolation of this particle merger process to next ge nerations is evident by recur- rence. The qth collision of heavy particles takes place at time ˜tp+q), (5.8) the masses of these particles at this time are determined by the scale of the ( p+q)th mode mp+q=ρ0L2 p+q√ 3/4 13(5.7). Here also one can speak about the logarithmic periodi c self-similarity of the motion of the particles. This means that at arbitrary interval [ ˜tq,˜tq+1] the motion of the particles will be similar to the motion of the particles in the interval ˜tp,˜tp+1, by the scaling factor xp/xq= 2(q−p)in space, and the scaling function tp/tq= (2n+3 2)(p−q)in time. We used computer simulation for studying two dimensional ca se; results of the simulation were generated in so called VRML (Virtual Reality Modeling L anguage), which enables to handle three dimensional figure in different projections. Fi g. 6 presents snapshots of this modeling. On the Fig. 6(a) one can see that particles formed b y small triangles move towards the center of an embracing triangle; this center in its turn, moves towards the center of the next bigger triangle in the hierarchy; e.t.c. Fig.6(b) give s the side-view of this process. 6 Conclusion In conclusion, we would like to point out that the evolution o f the multiscale signal with the Weierstrass spectrum simulates properties of Burgers t urbulence such as self-similarity, conservation of large scale structures and has the same laws of the energy decay and integral scale. The difference between the deterministic model and Bu rgers turbulence is that here we have the exact solution for the evolution of multiscale si gnals and these properties are not stochastic but deterministic. The evolution of the multisc ale signal is exactly self-similar in logarithmically spaced time intervals. The evolution of th e large scale modes is completely independent of the small scales modes, even if these have ver y large amplitudes. These properties take place for Burgers turbulence in the st ochastical sense and, more- over, for a signal with cutoff frequencies of small scales, on ly asymptotically. Of course, these properties of the multiscale signal are determined by the sp ecial form of modes (reverse saw- tooth function), the special relations between wavenumber s of modes ( kr+1=kr/N, where Nis an integer) and their phase relations. On the other hand, these model signals do not reflect such prop erties of Burgers turbu- lence as qualitative difference in the behaviour of the turbu lence for n <1 and n >1 in the power spectrum (1.9) due to the process of generation of l arge scale components in the spectrum. For the deterministic model this process is not pr esent due to the special relation of wavenumbers. Let as now move to the mechanical interpretation of solution of the Burgers equation. For the initial reverse sawtooth wave, all the matter turns i nto heavy particles at the same moment of time. Thus, after this time, the evolution of Burge rs turbulence is fully deter- mined by the motion of heavy particles. The trajectories of h eavy particles form regular tree-like structure on the plane ( X, t), see Figs. 3 and 4. The properties of this structure depend on the parameters of our model. The integer ǫ−1=Nis the number of trajectories which intersect at one point and form a new branch of our struc ture. For ǫ−1=N= 2 we, thus, obtain binary tree structure. Changing of the paramet erhstretches or contracts the structure in the tdirection. One can say that our structure is the plane repres entation of theN−tree; the root of our tree is located at t= +∞. This tree is similar to the flattened fractal model of botanical umbrella tree (see [21]). If we ta ke some node ( X, T) of this 14structure as a root, and consider the trajectories of all hea vy particles, which will merge at the moment of time Tat this point ( X, T) we shall also obtain an n-tree. The whole tree seems self-similar, because every branch plus the bran ches it carries is a reduced scale version of the whole. We also notice that such multiscale waves may be constructed for multidimensional Burgers’ equation. 7 Acknowledgements The authors are grateful to U. Frisch for useful discussions and for his hospitality at the Observatoire de la Cˆ ote d’Azur, to G.M. Molchan, A.I. Saich ev, A. Noullez and W.A. Woy- czynski for useful discussions. This work was partially sup ported by the French Ministry of Higher Education, by INTAS through grant No 97–11134, by RFB R through grant 99-02- 18354. References [1] J.R. Angilella, J.C. Vassilicos, Spectral, diffusive an d convective proprties of fractal and spiral fields, Physica D, 124 (1998) 23–57. [2] E. Aurell, S. Gurbatov, I. Wertgeim, Self-preservation of large-scale structures in Burg- ers turbulence, Physics Lett. A 182 (1993) 104–113. [3] M. Avellaneda, W. E, Statistical properties of shocks in Burgers turbulence, Comm. Math. Phys. 172 (1995) 13-38. [4] M.V. Berry, Z.V. Lewis, On the Weierstrass-Mandelbrot f ractal function, Proc. Roy. Soc. A340 (1980) 459–484. [5] J.P. Bouchaud, M. M´ ezard, G. Parisi, Scaling and interm ittency in Burgers turbulence, Phys. Rev. E 52 (1995) 3656–3674. [6] J.M. Burgers, Mathematical examples illustrating rela tions occurring in the theory of turbulent fluid motion, Kon. Ned. Akad. Wet. Verh. 17 (1939 ) 1–53, (also in: F.T.M. Nieuwstadt, J.A. Steketee (Eds.), Selected Papers o f J.M. Burgers, Kluwer Academic Publishers, Nederlands (1995) 281–334). [7] J.M. Burgers, The Nonlinear Diffusion Equation, D. Reide l, Dordrecht, 1974. [8] A.-L. Barabasi, H.E. Stanley, Fractal Concepts in Surfa ce growth, Cambridge University Press, Cambridge, 1995. [9] J.D. Fournier, U. Frisch, L’´ equation de Burgers d´ eter ministe et statistique, J. M´ ec. Th´ eor. Appl. 2, Paris (1983b) 699–750. 15[10] U. Frisch, Turbulence: the Legacy of A.N. Kolmogorov, C ambridge University Press, 1995. [11] J.L. Gilson, I. Daumond, T. Dombre, A two-fluid picture o f intermittency in shell- models of turbulence, in: U. Frisch (Ed.), Advances in Turbu lence VII, Kluwer Aca- demic Publishers, Nederlands (1998) 219-222. [12] S.N. Gurbatov, D.B. Crighton, The nonlinear decay of co mplex signals in dissipative media, Chaos 5(3) (1995) 524–530. [13] S.N. Gurbatov, A.N. Malakhov, A.I. Saichev, Nonlinear Random Waves and Turbu- lence in Nondispersive media: Waves, Rays, Particles, Manc hester University Press, Manchester, 1991. [14] S.N. Gurbatov, A.I. Saichev, Inertial nonlinearity an d chaotic motion of particle flux, Chaos 3 (1993) 333–358. [15] S.N. Gurbatov, A.I. Saichev, S.F. Shandarin, The large -scale structure of the Universe in the frame of the model equation of non-linear diffusion, Mo nth. Not. Roy. Astr. Soc. 236 (1989) 385–402. [16] S.N. Gurbatov, S.I. Simdyankin, E. Aurell, U. Frisch, G . Toth, On the decay of Burgers turbulence, J. Fluid Mech. 344 (1997) 339-374. [17] S.N. Gurbatov, G.V. Pasmanik, Self-preservation of la rge-scale structures in nonlinear viscous medium described by the Burgers equation, J.Exp.Th eor.Phys. 88, N.2 (1999) 309–319. [18] E. Hopf, The partial differential equation ut+uux=uxx, Comm. Pure Appl. Mech. 3 (1950) 201–230. [19] L. Kadanoff, A cascade model of turbulence, in: U. Frisch (Ed.), Advances in Turbulence VII, Kluwer Academic Publishers, Nederlands (1998) 201-20 2. [20] S. Kida, Asymptotic properties of Burgers turbulence, J. Fluid Mech. 93 (1979) 337– 377. [21] Benoit B. Mandelbrot, The fractal geometry of nature, F reeman, New York, 1982. [22] G.M. Molchan, Burgers equation with self-similar Gaus sian initial data: tail probabili- ties, J. Stat. Phys. 88 (1997) 1139-1150. [23] R. Ryen, Large-deviation analysis of Burgers turbulen ce with white-noise initial data, Comm. Pure Appl. Math. L1 (1998) 47–75. [24] O. Rudenko, S. Soluyan, Theoretical foundation of Nonl inear Acoustics, Plenum, New- York, 1997. 16[25] S.F. Shandarin, Ya.B. Zeldovich, The large-scale stru cture of the Universe: turbulence, intermittency, structures in a self-gravitating medium, R ev. Mod. Phys. 61 (1989) 185– 220. [26] Ya.G. Sinai, Statistics of shocks in solutions of invis cid Burgers equation, Com- mun. Math. Phys. 148 (1992)601–622 . [27] A. Troussov, Linear time algorithms for discrete Legen dre transform and for the nu- merical simulation of Burgers’ equation, Comp. Seism. 29 (1 996) 45-53 (In Russian). [28] M. Vergassola, B. Dubrulle, U. Frisch, A. Noullez, Burg ers’ equation, Devil’s staircases and the mass distribution for large-scale structures, Astr on. Astrophys. 289 (1994) 325– 356. [29] G.B. Whitham, Linear and Nonlinear Waves, Wiley, New Yo rk, 1974. [30] W.A. Woyczynski, Burgers–KPZ Turbulence, Springer-V erlag, Berlin, 1998. 17Figure captions Figure 1: Evolution of one-mode pulse. (a) particle traject ories; (b) evolution of the initial velocity field given in the same spatio-temporal scale. Bold lines on the time axis denote moments at which the profiles of the velocity are plotted. Figure 2: Evolution and interaction of two modes. (a) partic le trajectories; (b) evolution of the initial velocity field given in the same spatio-temporal scale. Bold points on the time axis denote moments at which the profiles of the velocity are p lotted. Figure 3: Evolution of the multiscale fractal signal with n= 0 (h=−1/2), corresponding to ”white noise” signal. (a) particle trajectories; (b) evolu tion of the initial velocity field; (b) velocity field taken at the initial moment of time and then at t hree successive time moments of self-similarity. Figure 4: Evolution of the multiscale fractal signal with n=−2 (h= 1/2), corresponding to ”Brownian motion” signal. (a) particle trajectories; (b ) evolution of the initial velocity field; (b) velocity field taken at the initial moment of time an d then at three successive time moments of self-similarity. Figure 5: Plane construction for two dimensional case. The h ierarchy of triangles, used for the construction of the multiscale signal, with four lay ers shown. The initial signal is constructed as a series of signals piece-wise linear on tria ngles. Figure 6: Particle trajectories for the multiscale fractal signal in two dimensional case pre- sented in spatio-time three dimensional space; the width of particle trajectory reflects its mass: (a) top view; (b) side-view. 18This figure "Fig5.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0002043v1This figure "Fig6a.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0002043v1This figure "Fig6b.jpg" is available in "jpg" format from: http://arXiv.org/ps/physics/0002043v1

arXiv:physics/0002044v1 [physics.atom-ph] 23 Feb 2000Coherent backscattering of light by atoms: a factor two smal ler than 2 Thibaut Jonckheere†, Cord A. M¨ uller∗, Robin Kaiser∗,Christian Miniatura∗and Dominique Delande† †Laboratoire Kastler-Brossel, Universit´ e Pierre et Marie Curie Tour 12, Etage 1, 4 Place Jussieu, F-75252 Paris Cedex 05, Fra nce ∗Institut Non Lin´ eaire de Nice, UMR 6618 du CNRS, 1361 route d es Lucioles, F-06560 Valbonne, France (September 21, 2013) Coherent backscattering is a multiple scattering interfer - ence effect which enhances the diffuse reflection off a disor- dered sample in the backward direction. For classical scatt er- ers, the enhanced intensity is twice the average background under well chosen experimental conditions. In this paper, w e show how the quantum internal structure of atomic scatter- ers leads to a significantly smaller enhancement. Theoretic al results for double scattering are presented here which confi rm recent experimental observations. When monochromatic light is elastically scattered off a disordered medium, the interference between all par- tial waves produces strong angular fluctuations of the intensity distribution known as a speckle pattern [1]. Its detailed shape depends very sensitively on the positions of the scatterers. In a dilute medium such that kℓ≫1 (k= 2π/λis the light wave number and ℓthe elastic mean free path) – the only situation we will consider in this letter – the phases associated with different scatter- ing paths may thus be expected to be completely uncor- related. Averaging over the scatterers’ positions would then wash out all interferences and produce a smooth re- flected intensity. There is, however, an exception: the ensemble average cannot scramble the interference be- tween a wave travelling along any scattering path and the wave travelling along the same path in reverse order [2]. These two-wave interferences enhance the average reflected intensity in the backscattering direction. This phenomenon, known as coherent backscattering (CBS), has been extensively studied in various media and is now a hallmark of interference effects in multiple scattering [3]. In this letter, we study CBS by scatterers having an internal structure so that their internal states can be affected by the scattering events. This is the case for atoms exposed to a near resonant light, since their in- ternal state may change depending on the polarization of the incident and scattered photons. This dramatically affects the properties of the interfering paths and in gen- eral induces a reduction of the enhancement factor in the backscattered direction, although reciprocity is perfect ly preserved. The average intensity Iscattered at angle θcan be written as a sum of three terms, I(θ) =IS(θ) +IL(θ) + IC(θ) [4]. Here, ILis the raw contribution of multi-ple scattering paths (the so-called ladder terms), ICis the CBS contribution (the so-called maximally crossed terms), and ISreduces to the single scattering contribu- tion in the limit kℓ≫1.ILandISdo not contain any interference term and thus vary smoothly with θ. The contribution to ILof a pair of direct and reverse paths is essentially |Tdir|2+|Trev|2whereas the contribution to IC is 2|Tdir||Trev|cosφ,withφ= (k+k′)·(r′−r),wherek,k′ are the incident and scattered wave vectors and r,r′the positions of the first and last scatterer along the path. From these expressions, it follows that ICis always smaller or equal than IL. For a small scattering angle θ, the phase difference φis essentially of the order of θkℓ. Thus, IC(θ) is peaked around the backscattering direction θ= 0 and rapidly decreases to zero over an angular width ∆ θ∼1/kℓ≪1 [5]. We define the enhancement factor as the ratio of the average intensity in the backward direction I(θ= 0) to the average background intensity I(1/kℓ≪θ≪1) = IS+IL. It is thus α= 1 +IC(0) IS+IL. (1) Since IC≤IL, its largest possible value is 2, reached if and only if ISvanishes and IC=ILatθ= 0. In usual experiments, the incident light is polarized either linear ly or circularly (with a given helicity h) and one studies the scattered light with the same or orthogonal polarization. This defines four polarization channels: lin/bardbllin,lin⊥ lin,h/bardblh(helicity is preserved) and h⊥h(helicity is reversed). For classical scatterers, the essential result s are: ( i)IS= 0 for spherical scatterers (Mie spheres, Rayleigh scatterers) in the lin⊥linandh/bardblhchannels; (ii)IC=ILin the lin/bardbllinandh/bardblhchannels provided reciprocity holds [6]. As a consequence, the maximum enhancement α= 2 is expected for spherical scatterers in the h/bardblhchannel, a prediction confirmed by experiment [7]. Reciprocity is a symmetry property which holds when- ever the fundamental microscopic description of a system is time reversal invariant [8]. For classical scatterers, i t imposes equal transition amplitudes Tfor direct and re- versed scattering processes, even in absorbing media but provided no magnetic field is present [9]: T(kǫ→k′ǫ′) =T(−k′ǫ′∗→ −kǫ∗) (2) 1where ( k,ǫ) and ( k′,ǫ′) are the incident and scattered wave vectors and polarizations (the star indicates com- plex conjugation). At exact backscattering ( k′=−k) and in the parallel polarization channels ( ǫ′=ǫ∗), the scattering amplitudes of any couple of direct and reverse paths are identical, leading to complete constructive in- terference and to IC=IL. In the perpendicular chan- nels, the interfering amplitudes are not reciprocal which results in a decreased contrast in the interference pat- tern and – after configuration averaging – in IC< IL andα <2. Let us now turn to the case of atomic scatterers. Re- cently, coherent backscattering of light was observed with cold atoms [10]. The reported enhancement factors are much smaller than 2, with the lowest value in the h/bardblh channel. The internal structure of the atoms can account for a major part of this astonishing observation. In order to understand the reduction of the enhancement factor, two different reasons must be distinguished: ( i) Due to Raman transitions (see below), the single scattering con- tribution is in general non zero, even in the lin⊥linand h/bardblhchannels. As IS/negationslash= 0 also occurs for classical non- spherical scatterers [6], we will not focus on this point any further; ( ii) The existence of a quantum internal struc- ture leads to IC< IL. The rest of the paper is devoted to elucidating this point. We consider a collection of atoms at rest exposed to monochromatic light, quasi-resonant with an electric dipole transition between some ground state with angular momentum Fand some excited state with angular mo- mentum F′(in the experiment F= 3 and F′= 4). For sufficiently weak light intensity, a perturbative approach is in order: an atom with initial state |FmF/angbracketright=|m/angbracketright undergoes a transition viaa virtual intermediate state |F′mF′/angbracketrightinto a final state |Fm′ F/angbracketright=|m′/angbracketrightwhile scattering an incoming photon ( k,ǫ) into an outgoing mode ( k′,ǫ′). When no magnetic field is present, this scattering process is purely elastic, both for Rayleigh ( m′=m) and Raman (m′/negationslash=m) scattering. As a general rule in quantum mechanics, only transi- tion amplitudes which connect the same initial state to thesame final state can interfere. Here the states of the complete system are the photon modes andthe inter- nal states |m/angbracketright=|m1, m2, . . ./angbracketrightof all atoms. Here again, CBS originates from the interference between amplitudes associated with direct and reversed scattering sequences with identical transitions m→m′. But in this case, reci- procity fails to predict the enhancement factor. Indeed, the reciprocity relation now reads [11] T(kǫ,m→k′ǫ′,m′) =(−1)/summationtext i(m′ i−mi) ×T(−k′ǫ′∗,−m′→ −kǫ∗,−m). (3) It shows that, except for the very special case m′=−m, reciprocal amplitudes will notcontribute to CBS. Theyare associated to different initial and final states of the system and cannot interfere. This is true for all polar- ization channels, and stands out in sharp contrast to the classical case. No fundamental reason is left for IC=IL to hold. We point out, however, that eq. (3) reduces to eq. (2) in the case of an F= 0→F′= 1 transition since the ground state then has no internal structure. The en- hancement factor for this transition will thus be the same as in the classical case. Before turning to the complete calculation, let us sim- ply illustrate why IC< ILin general. Consider dou- ble Rayleigh scattering ( m1=m′ 1;m2=m′ 2) on an F= 1/2→F′= 1/2 transition in the h/bardblhchannel with positive incident helicity [12]. The atoms are sup- posed to be initially prepared in the |m1=−1/2/angbracketrightand |m2= +1/2/angbracketrightsubstates (quantization axis parallel to the incoming light wavevector). In this configuration (see fig. 1), atom 1 can scatter the incident photon. The emitted photon can be scattered back by atom 2 with the required helicity. But atom 2 cannot scatter the incident photon. This simple example shows a situation where the reverse amplitude is strictly zero while the direct one is not. Consequently, this interference does not contribute at all to the CBS enhancement factor. More generally, a path and its reverse partner will have non-zero but dif- ferent amplitudes, resulting in an overall enhancement factor less than 2. Non zero amplitudeAtom 2Atom 1 Atom 2Atom 1 Zero amplitude FIG. 1. Example of a direct and reverse scattering path having different amplitudes: double Rayleigh scattering on aF= 1/2→F′= 1/2 transition in the helicity preserv- ing channel with positive incident helicity. The arrows sho w atomic transitions corresponding to absorption and emissi on of photons; the dashed lines show the process which has a vanishing amplitude. We now sketch the general lines of the complete cal- 2culation. Details will be given elsewhere [19]. We con- sider the double scattering (higher orders can be calcu- lated similarly) of an incident photon ( k,ǫ) into the mode (k′,ǫ′) by two atoms i= 1,2 at fixed positions riwith initial and final states |mi/angbracketrightand|m′ i/angbracketright. The transition am- plitudes associated with the direct and reverse scatter- ing paths can be calculated using standard diagrammatic techniques [13]. The dependence of the direct amplitude on the internal atomic structure can be completely fac- torized into t=/angbracketleftm′|(ǫ′∗·d2)[(d1·d2)−(ˆ n·d1)(ˆ n·d2)](ǫ·d1)|m/angbracketright (4) where ˆ n=r12/r12is the unit vector joining the two atoms and dithe dipole operator of atom ibetween the FandF′subspaces. The internal part of the amplitude for the reverse path is obtained by exchanging d1andd2 (butnotthe magnetic quantum numbers). To calculate the reflected average intensity, one has to sum over all possible final atomic states and to average over their ini- tial distribution. In the following, we assume this to be a complete statistical mixture, which is likely to be the case under usual experimental conditions. Standard tech- niques for irreducible tensor operators [14] then permit to obtain the following expression (valid for any transition F→F′): /angbracketleftIL,C/angbracketright=I(0) L,C2/summationdisplay K,K′=0/braceleftbigg 1 1 K F F F′/bracerightbigg2/braceleftbigg 1 1 K′ F F F′/bracerightbigg2 MKK′ L,C (5) The prefactor I(0) L,C(k,k′,r12) is proportional to the inten- sity corresponding to resonant Rayleigh scatterers with- out any internal structure. All information about the atomic transition is contained in the “6 j” Wigner coeffi- cients whereas the 3 ×3 matrices ML,C(ˆ n,ǫ,ǫ′) depend only on the geometry ( i.e.on scalar products between ˆ n,ǫandǫ′), see [19] for the complete expressions. In general, ML/negationslash=MC. Thus IC/negationslash=ILand the enhance- ment factor is reduced. Let us check the limit F= 0: the Wigner coefficients essentially reduce to δK0factors which let only survive the classical (dipole scattering) termM00 L=M00 C=|(ǫ∗·ǫ′)−(ǫ∗·ˆ n)(ǫ′·ˆ n)|2, so that IL=ICin the backscattering direction as expected in this case. In order to calculate the enhancement factor and the angular intensity profile, one further has to average eq. (5) over all possible pair configurations. To facilitate comparison with existing results on classical Rayleigh scatterers [15], we assume the atoms to be uniformly dis- tributed in half-space and compute the backscattered in- tensity as a function of the reduced scattering angle kℓθ. The angular intensity profiles for the double scatter- ing contribution are shown in fig. 2 for the four usualpolarization channels. Table I lists the corresponding en- hancement factors in the backscattering direction. For F= 3→F′= 4, they qualitatively agree with the ex- perimental observation [10]: the highest value is reached in the h⊥hchannel, followed by the lin/bardbllin,lin⊥lin andh/bardblhchannels. As far as their shape is concerned, the CBS curves in fig. 2 are essentially identical with those for Rayleigh scatterers, with a width ∆ θof the order of 1/kℓ. −40 −20 0 20 4011.21.41.611.21.41.611.21.41.611.21.41.6 klθEnhancement Factor(a) (b) (c) (d) FIG. 2. Enhancement factor for double scattering of light by atoms ( F= 3→F′= 4, uniformly distributed in a semi-infinite medium), as a function of kℓθ(kis the wave vector, ℓthe mean free path and θthe angle with the exact backscattered direction) in the different polarization cha nnels: a)h⊥h(helicity non preserving) b) lin/bardbllinc)lin⊥lind) h/bardblh(helicity preserving). In the linear channels, θis in the plane of incident polarization. The dotted curve in d) shows the curve obtained by artificially removing the Raman scat- tering contributions from the interference terms. F→F′0→1 (Rayleigh scatterers) 1 /2→1/2 3→4 h⊥h 2.00 1.67 1.71 lin/bardbllin 2.00 1.73 1.60 lin⊥lin 2.00 1.43 1.45 h/bardblh 2.00 1.42 1.22 TABLE I. Enhancement factor of the average intensity scattered in the backward direction for double scattering on atoms ( F→F′transition, uniformly distributed in a semi-infinite medium) in the four polarization channels. Th e case 0 →1 is equivalent to classical Rayleigh scatterers. 3Let us emphasize that the enhancement factors smaller than 2 here are notdue to single scattering (we consider only double scattering) and notdue to any geometrical effect. In the same conditions, for classical Rayleigh scat- terers, the enhancement factor is 2 in the four polariza- tion channels, see Table I. It is also very important to stress that the enhancement reduction is notdue to any coherence property of light emitted in Raman transitions. If it is true that light emitted in a Raman transition can- not interfere with the incident light (simply because the final atomic states are different) [18], it can very well interfere with the Raman light scattered along the re- verse path and contribute to CBS. Even more, Raman transitions yield the dominant CBS contributions in the h/bardblhchannel as can be seen in fig. 2: the dashed curve shows the profile obtained if one excludes Raman scatter- ing terms from the crossed intensity. The enhancement factor in this case is only 1 .034, clearly below the exper- imentally observed factor of 1 .06. The enhancement factors reported in ref. [10] are sig- nificantly smaller than the ones presented here. We think that the main reason for this is single scattering. Even in the lin/bardbllinandh/bardblhchannels, single (Raman) scat- tering contributes to the backscattered intensity. This effect can be treated with the methods exposed above but the result depends on the geometry of the scattering medium. Two experimental caracteristics make the sin- gle scattering contribution particularly important here: firstly, the medium has a small optical depth ( ≈4), in- creasing the relative weight of single scattering as com- pared to the case of a semi-infinite medium. Secondly, the atomic cloud has a spherical geometry, which also tends to increase the importance of single scattering. The importance of these two effects can be observed consid- ering a model with classical scatterers of scalar waves contained in a sphere of small optical depth. Numer- ical calculations show that although IC=IL, the en- hancement factor drops to α≈1.4 for an optical depth of 4, to be contrasted with α= 1.88 for a semi-infinite slab [4]. For a quantitative comparison with the experi- ment, also higher orders of scattering have to be included. According to preliminary calculations, we predict that increasing scattering orders lead to decreasing enhance- ment factors. This causes a further reduction of the total enhancement factor. In conclusion, we have shown that reciprocity is irrele- vant in coherent backscattering of light by atoms because the internal structure implies that the interfering ampli- tudes are not reciprocal. The observed reduction of the enhancement factor is confirmed by a perturbative cal- culation and not attributed to different coherence prop- erties of light emitted in Raman or Rayleigh transitions. Further studies will include the effect of the sample ge- ometry on the one side, higher scattering orders as well as effects of saturation of the atomic transition on the other side.We thank Serge Reynaud, Jakub Zakrzewski and Bart van Tiggelen for numerous fruitful discussions. Labora- toire Kastler Brossel de l’Universit´ e Pierre et Marie Curi e et de l’Ecole Normale Sup´ erieure is UMR 8552 du CNRS. [1] Laser Speckle and related phenomena, ed. J.C. Dainty, Topics in Applied Physics, vol. 9, Springer-Verlag (Berlin, 1975). [2] S. Chakravarty and A. Schmid, Phys. Rep. 140, 193 (1986) [3]New Aspects of Electromagnetic and Acoustic Wave Dif- fusion , Springer Tracts in Modern Physics, Springer, Berlin (1998). [4] M. B. van der Mark, M. P. van Albada and A. Lagendijk, Phys. Rev. B, 37, 3575 (1988). [5] E. Akkermans, P. E. Wolf and R. Maynard, Phys. Rev. Lett.56, 1471 (1986); E. Akkermans, P. E. Wolf, R. Maynard and G. Maret, J. Phys. France 49, 77 (1988). For simplicity, we identify here the transport mean free pathℓ∗and the elastic mean free path ℓ, which is valid for Rayleigh scatterers, much smaller than the wavelength of the light. [6] M. I. Mishchenko, J. Opt. Soc. Am. A, 9, 978 (1992). [7] D. S. Wiersma, M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, Phys. Rev. Lett. 74, 4193 (1995). [8] L. Onsager, Phys. Rev. 38, 2265 (1931); H. B. G. Casimir, Rev. Mod. Phys. 17, 343 (1945). [9] D. S. Saxon, Phys. Rev. 100, 1771 (1955); B. A. van Tiggelen and R. Maynard, in : Wave Propagation in Complex Media (IMA Volume 96), ed. G. Papanicolaou, p. 252 (Springer, 1997). [10] G. Labeyrie, F. de Tomasi, J.-C. Bernard, C.A. M¨ uller, C. Miniatura and R. Kaiser, Phys. Rev. Lett. 83, 5266, (1999). [11] L.D. Landau and E.M. Lifshitz, Quantum Mechanics non-relativistic Theory , Butterworth-Heinemann (1981). [12] In this configuration there is no single scattering con- tribution since the h/bardblhchannel would require ∆ m= m′−m= 2 for one atom. The enhancement reduction is entirely due to IC< IL. [13] V.B. Berestetskii, E.M. Lifchitz and L.P. Pitaevskii, Quantum Electrodynamics , Butterworth-Heinemann (1982). [14] A.R. Edmonds, Angular Momentum in Quantum Me- chanics , Princeton University Press (1960). [15] B.A. van Tiggelen, A. Lagendijk, & A. Tip, J. Phys.: Condens. Matter 2, 7653 (1990). [16] M. P. van Albada and A. Lagendijk, Phys. Rev. Lett. 55, 2692 (1985). [17] P. E. Wolf and G. Maret, Phys. Rev. Lett. 55, 2696 (1985). [18] K. M. Watson, Phys. Rev. 105, 1388 (1957). [19] C. A. M¨ uller et al., to be published. 4

arXiv:physics/0002045v1 [physics.gen-ph] 24 Feb 2000Can Science ‘explain’ Consciousness ? M. K. Samal Non-Accelerator Particle Physics (NAPP) Group, Indian Institute of Astrophysics, Bangalore-560 034, Indi a. (e-mail: mks@iiap.ernet.in ) Consciousness is the process by which one attributes ‘meani ng’ to the world. Considering Fφllesdal’s definition of ‘meaning’ as the joint product of all ‘evidence’ that is available to those who ‘communicate’, we conclude that science can, not only reduc e all the evidence to a Basic Entity (we call BE), but also can ‘explain’ consciousness once a suitab le definition for communication is found that exploits the quantum superposition principle to incor porate the fuzzyness of our experience. Consciousness may be beyond ‘computability’, but it is not b eyond ‘communicability’. I. INTRODUCTION Among all the human endeavours, science can be considered to be the most powerful for the maximum power it endowes us to manipulate the nature through an understanding of our p osition in it. This understanding is gained when a set of careful observations based on tangible perceptions, acqui red by sensory organs and/or their extensions, is submitted to the logical analysis of human intellect as well as to the intuiti ve power of imagination to yield the abstract fundamental la ws of nature that are not self-evident at the gross level of phen omenal existence. There exists a unity in nature at the level of laws that corresponds to the manifest diversity at the level of phenomena. Can consciousness be understood in this sense by an appropri ate use of the methodology of science ? The most difficult problem related to consciousness is perhaps, ‘how to define i t ?’. Consciousness has remained a unitary subjective exper ience, its various ‘components’ being reflective (the recognition by the thinking subject of its own actions and mental states) , perceptual (the state or faculty of being mentally aware of e xternal environment) and a free will (volition). But how these components are integrated to provide the unique exper ience called ‘consciousness’, familiar to all of us, remain s a mystery. Does it lie at the level of ‘perceptions’ or at the le vel of ‘laws’ ? Can it be reduced to some basic ‘substance’ or ‘phenomenon’ ? Can it be manipulated in a controlled way ? Is t here a need for a change of either the methodology or the paradigm of science to answer the above questions ? In this article, I make a modest attempt to answer these questions, albeit in a speculative manner. II. CAN CONSCIOUSNESS BE REDUCED FURTHER ? Most of the successes of science over the past five hundred yea rs or so can be attributed to the great emphasis it lays on the ‘reductionist paradigm’. Following this approach, can consciousness be reduced either to ‘substance’ or ‘phenome na’ in the sense that by understanding which one can understand c onsciousness ? A. Physical Substratum The attempts to reduce consciousness to a physical basis hav e been made in the following ways by trying to understand the mechanism and functioning of the human brain in various d ifferent contexts. •Physics The basic substratum of physical reality is the ‘state’ of th e system and the whole job of physics can be put into a single question : ‘given the initial state, how to predict it s evolution at a later time ?’. In classical world, the state and its evolution can be reduced to events and their spatio-t emporal correlations. Consciousness has no direct role to play in this process of reduction, although it is responsi ble to find an ‘objective meaning’ in such a reduction. But the situation is quite different in the quantum world as al l relevant physical information about a system is contained in its wavefunction (or equivalently in its state vector), which is not physical in the sense of being directly measurable. Consciousness plays no role in the determinist ic and unitary Schr¨ odinger evolution (i.e. the U-process of Penrose [1]) that the ‘unphysical’ wavefunction undergo es. To extract any physical information from the wavefuction on e has to use the Born-Dirac rule and thus probability enters in a new way into the quantum mechanical description d espite the strictly deterministic nature of evolution 1of the wavefunction. The measurement process forces the sys tem to choose an ‘actuality’ from all ‘possibilities’ and thus leads to a non-unitary collapse of the general wavefunc tion to an eigenstate (i.e. the R-process of Penrose [1]) of the concerned observable. The dynamics of this R-process is not known and it is here some authors like Wigner have brought in the consciousness of the observer to cause th e collapse of the wavefunction. But instead of explaining the consciousness, this approach uses consciousness for th e sake of Quantum Mechanics which needs the R-process along with the U-process to yield all its spectacular successes. TheR-process is necessarily non-local and is governed by an irreducible element of chance, which me ans that the theory is not naturalistic: the dynamics is controlled in pa rt by something that is not a part of the physical universe. Stapp [2] has given a quantum mechanical model of the brain dy namics in which this quantum selection process is a causal process governed not by pure chance but rather by a ma thematically specified non-local physical process identifiable as the conscious process . It was reported [3] that attempts have been made to explain c onsciousness by relating it to the ‘quantum events’, but any such attempt is b ound to be futile as the concept of ‘quantum event’ in itself is ill-defined ! Keeping in view the fundamental role that the quantum vacuum plays in formulating the quantum field theories of all four known basic interactions of nature spreading over a period from the big-bang to the present, it has been suggested [4] that if at all consciousness be reduced to anyt hing ‘fundamental’ that should be the ‘quantum vacuum’ in itself. But in such an approach the following questions ar ise: 1) If consciousness has its origin in the quantum vacuum that gives rise to all fundamental particles as well a s the force fields, then why is it that only living things possess consciousness ?, 2) What is the relation between the quantum vacuum that gives rise to consciousness and the space-time continuum that confines all our perceptions t hrough which consciousness manifests itself ?, 3) Should one attribute consciousness only to systems consisting of ‘ real’ particles or also to systems containing ‘virtual’ par ticles ? Despite these questions, the idea of tracing the origin of ‘ consciousness’ to ‘substantial nothingness’ appears quit e promising because the properties of ‘quantum vacuum’ may ul timately lead us to an understanding of the dynamics of theR-process and thus to a physical comprehension of consciousn ess. One of the properties that distinguishes living systems fro m the non-living systems is their ability of self-organisat ion and complexity. Since life is a necessary condition for poss essing consciousness, can one attribute consciousness to a ‘degree of complexity’ in the sense that various degrees of consciousness can be caused by different levels of complexity? Can one give a suitable quantitative definition of consciousness in terms of ‘entropy’ that describes the ‘degree of self-organisation or complexity’ of a system ? What is the role of non-linearity and non-equilibrium thermodynamics in such a definition of consciousness ? In thi s holistic view of consciousness what is the role played by the phenomenon of quantum non-locality , first envisaged in EPR paper [5] and subsequently confirmed e xperimentally [6] by Aspect et. al ? What is the role of irreversibility and d issipation in this holistic view ? •Neuro-biology On the basis of the vast amount of information available on th e structure and the modes of communication (neuro- transmitters, neuro-modulators, neuro-hormones) of the n euron, neuroscience has empirically found [7] the neural ba sis of several attributes of consciousness. With the help of mod ern scanning techniques and by direct manipulations of the brain, neuro-biologists have found out that various human a ctivities (both physical and mental) and perceptions can be mapped into almost unique regions of the brain. Awareness , being intrinsic to neural activity, arises in higher level processing centers and requires integration of activity ov er time at the neuronal level. But there exists no particular region that can be attributed to have given rise to conscious ness. Consciousness appears to be a collective phenomena where the ‘whole’ is much more than the sum of parts ! Is each neuron having the ‘whole of consciousness’ within it, although it does work towards a particular attribute of c onsciousness at a time ? Can this paradigm of finding neural correlates of the attribu tes of consciousness be fruitful in demystifying con- sciousness ? Certainly not ! As it was aptly concluded [8] the currently prevalent reductionist approaches are unlikely to reveal the basis of such holistic phenomenon as c onsciousness. There have been holistic attempts [9,1] to understand consciousness in terms of collective quantum effects arising in cytoskeletons and microtubles; minute substructures lying deep within the brain’s neurons. The eff ect of general anaesthetics like chloroform (CHCl 3), isofluorane (CHF 2OCHClCF 3) etc. in swiching off the consciousness, not only in higher an imals such as mammals or birds but also in paramecium, amoeba, or even green slime mou ld has been advocated [10] to be providing a direct evidence that the phenomenon of consciousness is related to the action of the cytoskeleton and to microtubles. But all the implications of ‘quantum coherence’ regarding cons ciousness in such approach can only be unfolded after we achieve a better understanding of ‘quantum reality’, which still lies ahead of the present-day physics. •Artificial Intelligence Can machines be intelligent ? Within the restricted definiti on of ‘artificial intelligence’, the neural network approac h has been the most promising one. But the possibility of reali sing a machine capable of artificial intelligence based 2on this approach is constrained at present [11] by the limita tions of ‘silicon technology’ for integrating the desired astronomical number of ‘neuron-equivalents’ into a reason able compact space. Even though we might achieve such a feat in the foreseeable future by using chemical memories, it is not quite clear whether such artificially intelligent machines can be capable of ‘artificial consciousness’. Beca use one lacks at present a suitable working definition of ‘consciousness’ within the frame-work of studies involvin g artificial intelligence. Invoking G¨ odel’s incompleteness theorem, Penrose has arg ued [1] that the technology of electronic computer-control led robots will not provide a way to the artificial construction o f an actually intelligent machine–in the sense of a machine that ‘understands’ what it is doing and can act upon that unde rstanding. He maintains that human understanding (hence consciousness) lies beyond formal arguments and bey ond computability i.e. in the Turing-machine-accessible sense. Assuming the inherent ability of quantum mechanics to incor porate consciousness, can one expect any improvement in the above situation by considering ‘computation’ to be a p hysical process that is governed by the rules of quantum mechanics rather than that of classical physics ? In ‘Quantu m computation’ [12] the classical notion of a Turing machine is extended to a corresponding quantum one that take s into account the quantum superposition principle . In ‘standard’ quantum computation, the usual rules of quant um theory are adopted, in which the system evolves according to the U-process for essentially the entire operation, but the R-process becomes relevant mainly only at the end of the operation, when the system is ‘measured’ in ord er to ascertain either the termination or the result of the computation. Although the superiority of the quantum computation over cl assical computation in the sense of complexity theory have been shown [13], Penrose insists that it is still a ‘computat ional’ process since U-process is a computable operation andR-process is purely probabilistic procedure. What can be ach ieved in principle by a quantum computer could also be achieved, in principle, by a suitable Turing-machin e-with-randomiser. Thus he concludes that even a quantum computer would not be able to perform the operations require d for human conscious understanding. But we think that such a view is limited because ‘computation’ as a proces s need not be confined to a Turing-machine-accessible sense and in such situations one has to explore the power of qu antum computation in understanding consciousness. We conclude from the above discussions that the basic physic al substrata to which consciousness may be reduced are ‘neuron’, ‘event’ and ‘bit’ at the classical level, whereas at the quantum level they are ‘microtuble’, ‘wavefunction’ and ‘qubit’; depending on whether the studies are done in neuro- biology, physics and computer science respectively. Can th ere be a common platform for these trio of substrata ? We believe the answer to be in affirmative and the first hint rega rding this comes from Wheeler’s [14] remarkable idea: “it from bit i.e.everyit– every particle, every field of force, even the spacetime con tinuum itself – derives its function, its meaning, its very existence entirely – even if in some con texts indirectly – from the apparatus-elicited answers to y es or no questions, binary choices, bits”. This view of the world refers not to an object, but to a visio n of a world derived from pure logic and mathematics in the sense that an immaterial source and explanation lies at the bottom of every item of the physical world. In a recent report [15] the remarkable extent of embodiment of this vision in modern physics has beed discussed alongwith the possible difficulties faced by s uch a scheme. But can this scheme explain consciousness by reducing it to bits? Perhaps not unless it undergoes some modification. Why ? Because consciousness involves an awareness of an endless m osaic of qualitatively different things –such as the colour o f a rose, the fragrance of a perfume, the music of a piano, the ta ctile sense of objects, the power of abstraction, the intuit ive feeling for time and space, emotional states like love and ha te, the ability to put oneself in other’s position, the abili tiy to wonder, the power to wonder at one’s wondering etc. It is almo st impossible to reduce them all to the 0-or-1 sharpness of the definition of ‘bits’. A major part of human experience and consciousness is fuzzyand hence can not be reduced to yes or no type situations. Hence we believe that ‘bit’ has to be mo dified to incorporate this fuzzyness of the world. Perhaps the quantum superposition inherent to a ‘qubit’ can help. Ca n one then reduce the consciousness to a consistent theory of ‘quantum information’ based on qubits ? Quite unlikely, t ill our knowledge of ‘quantum reality’ and the ‘emergence of classicality from it’ becomes more clear. The major hurdles to be cleared are (1) Observer or Participator ? (In such equipment-evoked, quantum- information-theoretic approach, the inseparability of th e observer from the observed will bring in the quantum measur ement problem either in the form of dynamics of the R-process or in the emergence of classicality of the world fro m a quantum substratum. We first need the solutions to these long-standi ng problems before attempting to reduce the ‘fuzzy’ world of consciousness to ‘qubits’! ); (2) Communication ? (Even if we get the solutions to the above problems that enabl e us to reduce the ‘attributes of consciousness’ to ‘qubits’, stil l then the ‘dynamics of the process that gives rise to conscio usness’ will be beyond ‘quantum information’ as it will require a sui table definition of ‘communication’ in the sense expressed b y Fφllesdal [16] “ Meaning is the joint product of all evidence th at is available to those who communicate”. Consciousness helps us to find a ‘meaning’ or ‘understanding’ and will depen d upon ‘communication’. Although all ‘evidence’ can be reduced to qubits, ‘communication’ as an exchange of qubits has to be well-defined. Why do we say that a stone or a tree is 3unconscious ? Is it because we do not know how to ‘communicate ’ with them ? Can one define ‘communication’ in physical terms beyond any verbal or non-verbal language ? Where does o ne look for a suitable definition of ‘communication’ ? Maybe one has to define ‘communication’ at the ‘substantial n othingness’ level of quantum vacuum.); (3) Time’s Arrow ?(How important is the role of memory in ‘possessing consciou sness’ ? Would our consciousness be altered if the world we experience were reversible with respect to time ? Can our c onsciousness ever find out why it is not possible to influence the past ?). Hence we conclude that although consciousness may be beyond ‘computability’, it is not beyond ‘quantum communicabil- ity’ once a suitable definition for ‘communication’ is found that exploits the quantum superposition principle to incor porate the fuzzyness of our experience. Few questions arise: 1) how to modify the qubit ?, 2) can a suitable definition of ‘com- munication’, based on immaterial entity like ‘qubit’ or ‘mo dified qubit’, take care of non-physical experience like dre am or thoughts ? We assume, being optimistic, that a suitable modi fication of ‘qubit’ is possible that will surpass the hurdles of communicability, dynamics of R-process and irreversibility. For the lack of a better word w e will henceforth call such a modified qubit as ‘Basic Entity’ (BE). B. Non-Physical Substratum Unlike our sensory perceptions related to physical ‘substa nce’ and ‘phenomena’ there exists a plethora of human expe- riences like dreams, thoughts and lack of any experience dur ing sleep which are believed to be non-physical in the sense that they cannot be reduced to anything basic within the confi nement of space-time and causality. For example one cannot ascribe either spatiality or causality to human thoughts, d reams etc. Does one need a frame-work that transcends spatio - temporality to incorporate such non-physical ‘events’ ? Or can one explain them by using BE ? The following views can be taken depending on one’s belief: •Modified BE [ M(BE) ] What could be the basic substratum of these non-physical ent ities ? Could they be understood in terms of any suitably modified physical substratum ? At the classical lev el one might think of reducing them to ‘events’ which, unlike the physical events, do not have any reference to spat iality. Attempts [17] have been made to understand the non-physical entities like thoughts and dreams in terms of t emporal events and correlation between them. Although such an approach may yield the kinematics of these non-physi cal entities, it is not clear how their dynamics i.e. evolution etc. can be understood in terms of temporal compon ent alone without any external spatial input, when in the first place they have arose from perceptions that are me aningful only in the context of spatio-temporality ?! Secondly, it is not clear why the ‘mental events’ constructe d after dropping the spatiality should require new set of laws that are different from the usual physical laws. At the quantum level one might try to have a suitable modificat ion of the wavefunction to incorporate these non- physical entities. One may make the wavefunction depend on e xtra parameters [18], either physical or non-physical, to give it the extra degrees of freedom to mathematically inclu de more information. But such a wavefunction bound to have severe problems at the level of interpretation. For exa mple, if one includes an extra parameter called ‘meditation ’ as a new degree of freedom apart from the usual ones, then how w ill one interpret squared modulus of the wavefunction ? It will be certainly too crude to extend the Born rule to conc lude that the squared modulus in this case will give the probability of finding a particle having certain meditation value ! Hence this kind of modification will not be of much help except for the apparent satisfaction of being able to wr ite an eigenvalue equation for dreams or emotions ! This approach is certainly not capable of telling how the wavefun ction is related to consciousness, let alone a mathematical equation for the evolution of consciousness ! If one accepts consciousness as a phenomenon that arises out of execution of processes then any suggested [19] new physical basis can be shown to be redundant. As we have con cluded earlier, all such possible processes and their execution can be reduced to BE and spatio-temporal cor relations among BE using a suitable definition of communication. Hence to incorporate non-physical entities as some kind of i nformation one has to modify the BE in a subtle way. Schematically M(BE)= BE ⊗X, where ⊗stands for a yet unknown operation and X stands for fundament al substratum of non-physical information. X has to be different from BE; ot herwise it could be reducible to BE and then there will be no spatio-temporal distinction between physical and non -physical information. But, how to find out what is X ? Is it evident that the laws for M(BE) will be different from tha t for BE ? •Give up BE One could believe that it is the ‘Qualia’ that constitutes co nsciousness and hence consciousness has to be understood at a phenomenological level without disecting it into BE or M (BE). One would note that consciousness mainly consists 4of three phenomenological processes that can be roughly put as retentive, reflective and creative. But keeping the tremendous progress of our physical sciences and their util ity to neuro-sciences in view, it is not unreasonable to expe ct that all these three phenomenological processes, involvin g both human as well as animal [20] can be understood oneday in terms of M(BE). •Platonic BE It has been suggested [21] that consciousness could be like m athematics in the sense that although it is needed to comprehend the physical reality, in itself it is not ‘real’. The ‘reality’ of mathematics is a controversial issue that b rings in the old debate between the realists and the constructivists whether a mathematical truth is ‘a discove ry’ or ‘an invention’ of the human mind ? Should one consider the physical laws based on mathematical truth as re al or not ?! The realist’s stand of attributing a Platonic existence to the mathematical truth is a matter of p ure faith unless one tries to get the guidance from the knowledge of the physical world. It is doubtful whether our k nowledge of physical sciences provides support for the realist’s view if one considers the challenge to ‘realism’ i n physical sciences by the quantum world-view, which has been substantiated in recent past by experiments [6] that vi olate Bell’s inequalities. Even if one accepts the Platonic world of mathematical forms , this no way makes consciousness non-existent or unreal. Rather the very fact that truth of such a platonic wor ld of mathematics yields to the human understanding as much as that of a physical world makes consciousness all th e more profound in its existence . III. CAN CONSCIOUSNESS BE MANIPULATED ? Can consciousness be manipulated in a controlled manner ? Ex perience tells us how difficult it is to control the thoughts and how improbable it is to control the dreams. We discuss bel ow few methods prescribed by western psycho-analysis and oriental philosophies regarding the manipulation of consc iousness. Is there a lesson for modern science to learn from t hese methods ? A. Self The subject of ‘self’ is usually considered to belong to an ‘i nternal space’ in contrast to the external space where we dea l with others. We will consider the following two cases here: •Auto-suggestions There have been evidences that by auto-suggestions one can c ontrol one’s feelings like pain and pleasure. Can one cure oneself of diseases of physical origin by auto-suggest ions ? This requires further investigations. •Yoga and other oriental methods The eight-fold ( asthanga ) Yoga of Patanjali is perhaps the most ancient method prescr ibed [22] to control one’s thought and to direct it in a controlled manner. But it requir es certain control over body and emotions before one aspires to gain control over mind. In particular it lays grea t stress on ‘breath control’ ( pranayama ) as a means to relax the body and to still the mind. In its later stages it pro vides systematic methods to acquire concentration (dhyan ) and to prolong concentration on an object or a thought ( dharna ). After this attainment one can reach a stage where one’s aware ness of self and the surrounding is at its best. Then in its last stage, Yoga prescribes one’s acute awareness to b e decontextualized [23] from all perceptions limited by spatio-temporality and thus to reach a pinnacle called ( samadhi ) where one attains an understanding of everything and has no doubts. In this sense the Yogic philosophy believes that pure consci ousness transcends all perceptions and awareness . It is difficult to understand this on the basis of day to day exp erience. Why does one need to sharpen one’s awareness to its extreme if one is finally going to aband on its use ? How does abandonning one’s sharpened awareness help in attaining a realisation that transcends s patio-temporality? Can any one realise anything that is beyond the space, time and causality ? What is the purpose of s uch a consciousness that lies beyond the confinement of space and time ? 5B. Non-Self The Non-Self belongs to an external world consisting of othe rs, both living and non-living. In the following we discuss whether one can direct one’s consciousness towards others s uch that one can affect their behaviour. •Hypnosis, ESP etc... It is a well-known fact that it is possible to hypnotise a pers on and then to make contact with his/her sub-conscious mind. Where does this sub-conscious lie ? What is its relatio n to the conscious mind ? The efficacy of the method of hypnosis in curing people of deep-rooted psychological pro blems tells us that we are yet to understand the dynamics of the human brain fully. The field of Para-Psychology deals with ‘phenomena’ like Ext ra Sensory Perception (ESP) and telepathy etc. where one can direct one’s consciousness to gain insight into futu re or to influence others mind. It is not possible to explain [23] these on the basis of the known laws of the world. It has be en claimed that under hypnosis a subject could vividly recollect incidents from the previous lives includ ing near-death and death experiences which is independent o f spatio-temporality. Then, it is not clear, why most of these experiences are related to past ? If these phenomena are truely independent of space and time, then studies shoul d be made to find out if anybody under hypnosis can predict his/her own death, an event that can be easily verifia ble in due course of time, unlike the recollections of past-life ! •PK, FieldREG etc. Can mind influence matter belonging to outside of the body ? Th e studies dubbed as Psycho-Kinesis (PK) have been conducted to investigate the ‘suspect’ interaction of the h uman mind with various material objects such as cards, dice, simple pendulum etc. An excellent historical overvie w of such studies leading upto the modern era is available as a review paper, titled “ The Persistent Paradox of Psychic Phenomena: An Engineering Perspective”, by Robert Jahn of Princeton University published in Proc. IEEE (Feb. 1 982). The Princeton Engineering Anomalies Research (PEAR) progr amme of the Department of Applied Sciences and Engineering, Princeton University, has recently develope d and patented a ‘Field REG’ (Field Random Event Generator) device which is basically a portable notebook computer with a built-in truely random number generator (based on a microelectronic device such as a shot noise resistor or a so lid-state diode) and requisite software for on-line data processing and display, specifically tailored for conducti ng ‘mind-machine interaction’ studies. After performing large number of systematic experiments ov er the last two decades, the PEAR group has reported [24] the existence of such a consciousness related mind-machine interaction in the case of ‘truely random devices’. They attribute it to a ‘Consciousness Field Effect’. They have als o reported that deterministic random number sequences such as those generated by mathematical algorithm or pseudo -random generators do not show any consciousness related anomalous behaviour. Another curious finding is tha t ‘intense emotional resonance’ generates the effect whereas ‘intense intellectual resonance’ does not ! It is also not clear what is the strength of the ‘consciousnes s field’ in comparison to all the four known basic force fields of nature . One should not reject outright any phenomenon that cannot be explained by the known basic laws of nature. Because each such phenomenon holds the key to extend the boundary of o ur knowledge further. But before accepting these effects one should filter them through the rigours of scientific metho dology. In particular, the following questions can be asked : •Why are these events rare and not repeatable ? •How does one make sure that these effects are not manifestatio ns of yet unknown facets of the known forces ? •Why is it necessary to have truely random processes ? How does one make sure that these are not merely st atistical artifacts ? If the above effects survive the scrutiny of the above questio ns (or similar ones) then they will open up the doors to a new world not yet known to science. In such a case how does one acco modate them within the existing framework of scientific methods ? If these effects are confirmed beyond doubt, then one has to explore the possibility that at the fundamental level of nature, the laws are either different from the known physic al laws or there is a need to complement the known physical laws with a set of non-physical laws ! In such a situation, the se ‘suspect’ phenomena might provide us with the valuable clue for modifying BE to get M(BE) that is the basis of everyth ing including both physical and mental ! 6IV. IS THERE A NEED FOR A CHANGE OF PARADIGM ? Although reductionist approach can provide us with valuabl e clues regarding the attributes of consciousness, it is the holistic approach that can only explain consciousness. But the dualism of Descarte [25] that treats physical and mental processes in a mutually exclusive manner will not suffice for u nderstanding consciousness unless it makes an appropriate use of complementarity for mental and physical events which is analogous to the complementarity evident in the quantum world. V. CONCLUSION Where does the brain end and the mind begin ? Brain is the physi cal means to acquire and to retain the information for the mind to process them to find a ‘meaning’ or a ‘structure’ which we call ‘understanding’ that is attributed to consciousnes s. Whereas attributes of consciousness can be reduced to BE [or to M(BE)], the holistic process of consciousness can only be understood in terms of ‘quantum communication’, where ‘com munication’ has an appropriate meaning. Maybe one has to look for such a suitable definition of communication at the le vel of ‘quantum vacuum’. VI. ACKNOWLEDGEMENTS It is a pleasure to thank the organisers, in particular to Pro f. B. V. Sreekantan and Dr. Sangeetha Menon; for the hospitality and encouragement as well as for providing the c onducive atmosphere that made this article possible. [1] Penrose, R., Emperor’s New Mind andShadows of the Mind , Vintage Editions, (1998). [2] Stapp, H. P., Chance, Choice, and Consciousness: The Role of Mind in the Qu antum Brain , quant-ph/9511029 (electronic archive at LANL). [3] Nair, Ranjit, Consciousness and the Quantum in this conference. [4] Sreekantan, B. V., Scientific Explanations and Consciousness in this conference. [5] Einsten, A., Podolsky, P., and Rosen, N., (1935), Can quantum-mechanical description of physical reality be considered complete ? , Phys. Rev. 47, 777-80. [6] Aspect, A., Grangier, P., and Roger, G. (1982), Experimental realization of Einstein-Podolsky-Rosen-Bo hm Gedankenex- periment: a new violation of Bell’s inequalities , Phys. Rev. Lett., 48, 91-4. [7] Rao, Shobhini, Neural Correlates and Consciousness in this conference. [8] Tondon, P. N., Exploring Consciousness–Neurobiological Approaches in this conference. [9] Hameroff, S. R., (1987) Ultimate computing, Biomolecular consciousness and nano- technology , North-Holland, Amsterdam. [10] Hameroff, S. R. and Watt, R. C. (1983), Do anesthetics act by altering electron mobility ? , Anesth. Analg., 62, 936-40. [11] Vidyasagar, M., Artificial Intelligence in this conference. [12] Deutsch, D. (1985), Quantum theory, the Church-Turing principle and the univer sal quantum computer in Proc. Roy. Soc. (London), A 400 , 97-117; Feynman R. P., (1986), Quantum mechanical Computers , Found. of Phys., 16(6) , 507-31. [13] Deutsch, D. and Jozsa, R. (1992), Rapid solution of problems by quantum computation , Proc. Roy. Soc. (London), A 439 , 553-8. [14] Wheeler, J. A., (1989), Information, Physics, Quantum: the Search for the Links , Proc. 3rd Int. symp. Foundations of Quantum Mechanics, Tokyo, pp. 354-368. [15] Wilczek, F., (1999), Getting its from bits , Nature 397, 303-6. [16] D. F φllesdal, (1975), Meaning and Experience in ‘Mind and Language’, ed. S. Guttenplan (Clarendon, Oxfor d), pp. 25-44. [17] Singh, N., Fundamental Laws of mental events in this conference. [18] Kaushal, R. S., Plurality of Consciousness in Vedantic Philosophy and its r ole in Scientific Observations in this conference. [19] Singh, R. K., A Physical Basis of Consciousness in this conference. [20] Sinha, A., Almost Minds? The Search for Animal Consciousness in this conference. [21] Sarukkai, S., Reality and Consciousness in this conference. [22] Iyengar, B. K. S., Light on Yoga , Unwin Publishers, London. [23] Krishna Rao P. V., Yoga and Transformation of Consciousness in this conference. [24] Srinivasan, M., Experimental studies on interaction of human consciousnes s with physical systems in this conference. [25] Narasimhan, M. G., The Emergence of Cartesian Paradigm and Its Impacts on Later Developments in this conference. 7

arXiv:physics/0002046v1 [physics.gen-ph] 24 Feb 2000Cosmological Models with “Some” Variable Constants Jos´ e Antonio Belinch´ on Grupo Inter-Universitario de An´ alisis Dimensional Dept. F ´isica ETS Arquitectura UPM Av. Juan de Herrera 4 Madrid 28040 Espaa E-mail: jabelinchon@usa.net Various models are under consideration with metric type flat FRW i.e. with k=0 whose energy- momentum tensor is described by a perfect fluid whose generic equation state is p=ωρand taking into account the conservation principle div(Tij) = 0, but considering some of the “constants” as variable. A set of solutions through dimensional analysis i s trivially found. The numeric calcula- tions carried out show that the results obtained are not disc ordant with those presently observed for cosmological parameters. However, the model seems irrecon cilable with electromagnetic and quantum quantities. This makes us think that we are working with faul ty hypothesis from the start . Key words: FRW cosmologies, variable constants. I. INTRODUCTION. In a recent paper (see [1]) a study was carried out of the behaviour of Gand Λ “constants” in the framework described by a cosmological model with flat FRW sym- metries and whose moment-energy tensor was described by a perfect fluid whose state equation is : ( p=ωρ/ ω=const. ) taking into account the conservation princi- ple. In light of coincidences encountered with O’Hanlon and Tam’s work (see [2]) it was decided to demonstrate that we could also reach the same results as those of the authors in the case of a model describing a Universe with predominance of matter. With the hypothesis car- ried out in previous work (see [1]) i.e. the only real con- stants taken into consideration are the speed of light c and the Boltzmann constant kB.In this work the “con- stants” G,/planckover2pi1, a, e, m iand Λ are considered as scalar func- tions dependent on time showing that one of our models reaches the same results of those of O’Hanlon and Tam. It should also be pointed out how the use of Dimensional Analysis (DA) enables us to find in a trivial way, a set of solutions to this type of models with k= 0 y taking into account the conservation principle, as it will be seen that the different equations posed are not trivially integrable, but “without going overboard” as the method has certain limitations. The numerical calculations carried out show that the results obtained are not discordant with those presently observed for cosmological parameters, however, the model seems irreconcilable with electromagnetic and quantum quantities. The idea of being able to unite in one field both gravitation and the rest of the forces has been, since the beginning of the century, a very active field of work. The inconsistencies observed in our mod- el makes us think “ momentaneously ” that the approach to the problem is inadequate, that is to say, that we are working with faulty hypothesis from the start. The paper is organized as follows: In the second sec-tion the equations of the model are presented and some small considerations on the dimensional method followed are made. In the third section, use is made of dimension- al analysis to obtain a solution to the main quantities appearing in the model. In the fourth section two specif- ic models are studied - one with radiation predominance and the other with matter predominance and finally in the fifth section a brief summary and concise conclusions are made. II. THE MODEL. The modified field equations are as follows: Rij−1 2gijR−Λ(t)gij=8πG(t) c4Tij (2.1) and it is imposed that: div(Tij) = 0 where Λ( t) represent the cosmological “constant”. The basic ingredient of the model are: 1. The line element defined by: ds2=−c2dt2+f2(t)/bracketleftbiggdr2 1−kr2+r2/parenleftbig dθ2+ sin2θdφ2/parenrightbig/bracketrightbigg we only consider here the case k= 0. 2. The energy-momentum tensor defined by: Tij= (ρ+p)uiuj−pgij p=ωρ where ωis a numerical constant such that ω∈[0,1] Whit this supposition the equation that govern the model are as follows: 2f′′ f+(f′)2 f2=−8πG(t) c2p+c2Λ(t) (2.2) 13(f′)2 f2=8πG(t) c2ρ+c2Λ(t) (2.3) div(Tij) = 0⇔ρ′+ 3(ω+ 1)ρf′ f= 0 (2.4) integrating equation (2.4) it is obtained the well-known relationship. ρ=Aωf−3(ω+1)(2.5) where frepresent the scale factor that appear in the metric and Aωis the constant of integration that has different dimensions and physical meaning depending on the state equation imposed i.e. depends on ω. Following the Kalligas et al‘s work (see [3]), if we derive equation (2.3) and it is simplified with (2.2) it is obtained the relationship: Gρ′+ 3(1 + ω)ρGf′ f+ρG′+Λ′c4 8π= 0 (2.6) From equations (2.6) and (2.4) we obtain the next equation that relate Gwith Λ G′=−Λ′c4 8πρ(2.7) from all these relationship it is obtained the following differential equation that it is not immediately integrated (see [3]): ρ′ρ′′ ρ2−/parenleftbiggρ′ ρ/parenrightbigg3 = 12π(ω+ 1)2Gρ′ c2(2.8) for this reason we utilize the dimensional method. This equation also we can integrate through similarity and di- mensional method following a well- established way to integrate pde and odes. This last option is studied in other paper (see [4]). In this case we work a naive Di- mensional Analysis. The followed dimensional method needs to make these distinctions. It is necessary to know beforehand the set of fundamental quantities together with one of the un- avoidable constant (in the nomenclature of Barenblatt designated as governing parameters). In this case the only fundamental quantity is the cosmic time tas can be easily deduced from the homogeneity and isotropy sup- posed for the model. The set of unavoidable constant are in this case the speed of light c,the integration constant Aω(obtained from equation (2.5) that depending on the state equation will have different dimensions and physi- cal meaning) and the Boltzman constant kBthat will be taking into account to relate thermodynamics quantities In a previous paper (see [5]) the dimensional base was calculated for this type of models, being this B={L, M, T, θ }where θstands for dimensions of tempera- ture. The dimensional equations of each of the governing parameters is: [t] =T[c] =LT−1[Aω] =L2+3ωMT−2 [kB] =L2MT−2θ−1 All the derived quantities will be calculated in function of these governing parameters, that is say, in function of the cosmic time tand the set of unavoidable constants c, kBandAωwith respect to the dimensional base B= {L, M, T, θ }. III. SOLUTIONS THROUGH D.A. We are going to calculate through dimensional analysis D.A. i.e. applying the Pi Theorem, the variation of G(t) in function on tand temperature θ, G(θ) (see [6]), the Planck’s constant /planckover2pi1(t),the radiation constant a(t),the charge of the electron e(t),the mass of an elementary par- ticlemi(t), the variation of the cosmological “constant” Λ(t),the energy density ρ(t),the matter density ρm(t), the radius of the universe f(t),the temperature θ(t), the entropy S(t) and finally the entropy density s(t) A. Calculation of G (t). As we have indicated above, we are going to ac- complish the calculation of the variation of Gapplying the Pi theorem. The quantities that we consider are: G=G(t, c, A ω).with respect to the dimensional base B={L, M, T, θ }.We know that [ G] =L3M−1T−2 Through a direct aplication of Pi Theorem we obtain a single monomial that leads to the following expression forG G(t)∝t1+3ωc5+3ω Aω(3.1) If we want to relate Gwithθ(see [6]) the solution that DA give us is: G=G(t, c, A ω, a, θ).We need to introduce a new dimensional “constant” a, in this case thermodynamics, to relate the temperature whit the rest of quantities. The same result is obtained if we consider kB. G(θ)∝A1 3(ω+1) ωc4/parenleftbig aθ4/parenrightbigω−1 3(ω+1)(3.2) B. Calculation of the Planck’s constant /planckover2pi1(t) : /planckover2pi1=/planckover2pi1(t, c, A ω) where its dimensional equation is [ /planckover2pi1] = L2MT−1 /planckover2pi1(t)∝Aωc−3ωt1−3ω(3.3) 2C. Calculation of the radiation “constant” a (t): a=a(t, c, A ω, kB) where its dimensional equation is [a] =L−1MT−2θ−4 k−4 Ba(t)∝A−3 ωc9ω−3t9ω−3(3.4) D. Calculation of the electron charge e (t) : e=e(t, c, A ω, ǫ0) where its dimensional equation is/bracketleftbig e2ǫ−1 0/bracketrightbig =L3MT−2 e2(t)ǫ−1 0∝Aωc1−3ωt1−3ω(3.5) E. Calculation of the mass of an elementary particle mi(t) : mi=mi(t, c, A ω) where its dimensional equation is [mi] =M mi(t)∝Aωc−2−3ωt−3ω(3.6) F. Calculation of the cosmological “constant” Λ(t). Λ = Λ( t, c, A ω) where [Λ] = L−2 Λ(t)∝1 c2t2(3.7) it is observed that not depends on Aωi.e. it is not de- pends on state equation. This solution will be valid for both models. G. Calculation of the energy density ρ(t) ρ=ρ(t, c, A ω) with respect to the base Bits dimen- sional equation is: [ ρ] =L−1MT−2 ρ(t)∝Aω(ct)−3(ω+1)(3.8) H. Calculation of the radius of the Universe f(t). f=f(t, c, A ω) where its dimensional equation is [ f] = L=⇒ f(t)∝ct (3.9) it is observed that no depends on Aωi.e. is not depend on state equation. This solution is valid for both models. Then: q=−f′′f (f′)2= 0H=f′ f=1 t dH=ctlim t0→0/integraldisplayt t0dt′ f(t′)=∞ i.e. there is no horizon problem, since dHdiverge when t0→0. I. Calculation of the temperature θ(t). θ=θ(t, c, A ω,a) where [ θ] =θ=⇒ a1 4θ(t)∝A1 4ω(ct)−3 4(1+ω)(3.10) we can too calculate it in function of kBi.e.θ= θ(t, c, A ω,kB) kBθ(t)∝Aωc−3ωt−3ω(3.11) we may check that this relationship is verified: ρ=aθ4=Aω(ct)−3(ω+1)=Aω(f)−3(ω+1) J. Calculation of the entropy S (t). S=S(c, Aω,a) where [ S] =L2MT−2θ−1 S(t)∝/parenleftBig A3 ωa(ct)3(1−3ω)/parenrightBig1 4(3.12) K. Entropy density s (t). s=s(t, c, A ω,a) where [ s] =L−1MT−2θ−1 s(t)∝/parenleftbig A3 ωa/parenrightbig1 4(ct)9 4(1+ω)(3.13) IV. DIFFERENT CASES. All the following cases can be calculated without dif- ficulty. Two specific models are studied: in first place a universe with radiation predominance which corresponds to the imposition of ω= 1/3 in the state equation; and in second place a model describing a universe with matter predominance corresponding to the imposition of ω= 0 as state equation. 3A. Model with radiation predominance In this case the behaviour of the “constants” obtained is the following: G(t)∝A−1 ωt2c6G∝t2 G∝c4(Aωa)−1 2θ−2G∝θ−2 /planckover2pi1∝Aωc−1t0/planckover2pi1∝const. a∝k4 BA−3 ωc0t0a∝const. e2ǫ−1 0∝Aωc0t0e2ǫ−1 0∝const. mi∝Aωc−3t−1mi∝t−1 Λ∝c−2t−2Λ∝t−2 While the result obtained for the rest of quantities is: f∝ct f ∝t ρ∝Aω(ct)−4ρ∝t−4 θ∝k−1 BAωc−1t−1θ∝t−1 S∝/parenleftbig A3 ωa/parenrightbig1 4S∝const. s∝/parenleftbig A3 ωa/parenrightbig1 4(ct)−3s∝t−3 In the first place it should be pointed out that with re- gard to the values obtained for G,Λ, f, ρandθthe same were obtained as those already found in literature (see [3], [7], [8] and [9]) demonstrating in this way that DA is a good tool for dealing with these types of problems. In the same way the result obtained for G(θ)∝θ−2coincides with that obtained by Zee (see [6]). It is proven, amazing- ly, that the result obtained for the remainder of the “con- stants” is that these are constant in the model in spite of considering them as variable, with the exception of the mass of an elemental particle which varies as mi∝t−1. Observe that with regards to the “indissoluble” relation- shipe2ǫ−1 0∝const. it can be said that e2∝ǫ0in such a way that the product e2ǫ−1 0∝const. remains constant. If Moller and Landau et al ‘s observations (see [10]) are tak- en into account, in which the following relation ǫ0≈f(t) is shown , in our case ǫ0≈f(t)∝twe therefore find thate2∝ǫ0∝tof the relation c2= (1/ǫ0µ0) we obtainǫ0∝µ−1 0. In the same way the following coincidences can be observed: /planckover2pi1∝Aωc−1being and a∝k4 B c3h3if substituted the expression obtained though D.A. i.e. a∝k4 BA−3 ωcan be recovered and results consistent. On the other hand, from the relation e2ǫ−1 0∝Aωand/planckover2pi1∝Aωc−1it can be seen that e2ǫ−1 0∝/planckover2pi1ca relation known by all. All these results are coherent with the behaviour of all energies, asE=kBθ∝t−1,E=mc2∝t−1if not this relation would be constant ¡! E=/planckover2pi1γ∝t−1and the total Borh energy ETB=me4 ǫ2 0/planckover2pi12∝t−1. It will now be checked if the results obtained are com- patible with the observational data available. From the equation (2.5) the value of the constant Aωis obtained (as in this model ω= 1/3 thus the denomination henceforth will be as A1). It is known thatρ≈10−13.379Jm−3andf≈1028mwith this da- taA1≈10100.5m3kgs−2is obtained. With this value ofA1it is checked whether the value of Gpredicted by our model is obtained. As G(t)∝A−1 ωt2c6where c≈108.47ms−1andt≈1020s G(t)∝A−1 ωt2c6≈10−10.17m3kg−1s−2 i.e. our model is capable of recovering the value presently accepted of the “ constant ”G. If we proceed in the same way with the formula for G(θ) a value for Gabout G≈ 10−9.562m3kg−1s−2is obtained i.e. a little below that presently observed. With regards to the cosmological “constant” it is observed that : if t≈1020sandc≈ 108.4ms−1⇒Λ≈10−56m−2which corresponds to that presently accepted. If the value obtained for A1is taken into account it can be seen with ease that the following value of the cosmic background radiation temperature is obtained i.eθ≈100.4361Kifthe “constant” atakes a value of a≈10−15.1211Jm−3K−4in the expression a1 4θ(t)∝ A1 4ω(ct)−1i.e. we can also deduce through this result the value presently accepted of the cosmic background radiation temperature also recovering the expression for energy density ρ=aθ4. Finally it should be pointed out that our model is without the nominated problem of hori- zon although it is not yet rid of the problem of entropy, also constant here. For the moment , we can see that the model works well (fantastic ) but we shall now see how it functions with respect to the electromagnetic and quantum con- stants. With the values calculated previously we observe, much to our disappointment, that we do not obtain (with the expressions indicated) any of the values presently ac- cepted for each of these “constants”. For example we see that /planckover2pi1∝A1c−1≈1091.5Js−1while for the radia- tion constant a∝k4 BA−3 ω≈10−300m−1kgs−2K−4and e2ǫ−1 0∝A1≈10100.5m3kgs−2i.e. we are obtaining totally “ preposterous results ”. We can see that we are 4unable to reconcile our results with the present values of the said “constants”. Let us think now in a different way: from the relation e2ǫ−1 0∝A1we obtain the value of the constant A1which we shall now call A′ 1to avoid as far as possible confusion through excessive notation. The value of this new con- stant is in the region of A′ 1≈10−26m3kgs−2.so with this value of A′ 1we can recover the present values both of /planckover2pi1 andabut none of the cosmological parameters such as G, ρetc. how strange. We can see that even if by one path we can perfectly describe the cosmological parameters G, f, ρ, θ and Λ we cannot recover the values of /planckover2pi1, a, e, ǫ 0etc. and vice versa. This makes us think that this approach (I can now dare to qualify it as simplistic as it is indeed a “toy model”) is not correct and that previous hypothesis should be taken into account or perhaps create an adequate theoretical framework capable of describing both worlds ... ( I think we have rediscovered America!). However, the approach displayed here is not totally wild for the following reasons: J.A. Wheeler (see [11]) stated that if the constants of Physics must vary, these would do so in function of universal time. This time is our universal time function which we can define in our model as we have built it through a FRW metric type i.e. that our ST space-time can be foliated in 3-spaces and these are different from one another in the value of slice-labeling i.e. trK. The uranian mine in Gabon has given the evidence necessary to be able to affirm that the masses and charges of particles have changed with time, but which time? Proper or universal? One can suddenly think that we are talking about proper time as changes of masses and charges are proper ones! The col- lapse syndrome puts a limit in the region of 10−37/year on the variation of the charge of the electron e(with re- spect to proper time). A more thorough analysis of this last section shows a series of difficulties “the collapse syn- drome” hindered by Pauli’s exclusion principle. The idea expressed by Wheeler is, therefore, that the “constants” vary with respect to universal time and not proper time. This argument sets universal time in a place of privilege in the argument about the “change” in microphysics on cosmological scales. On the other hand, as previously in- dicated, Moller and Landau et al (see [10])) have shown a relation, now firmly established , by which ǫ0should vary according to the radius of the universe, ǫ0≈f(t) . For all these reasons we have decided to carry out a similar study (i.e. it seems our suppositions are not com- pletely preposterous) however, the results obtained sur- prise us.B. Model with matter predominance. In this case the behaviour of the “constants” obtained is the following: G(t)∝A−1 ωc5t G ∝t /planckover2pi1∝Aωc0t /planckover2pi1∝t a∝k4 BA−3 ωc−3t−3a∝t−3 e2ǫ−1 0∝Aωct e2ǫ−1 0∝t mi∝Aωc−2t0mi∝const. Λ∝c−2t−2Λ∝t−2 While the result obtained for the rest of quantities is: f∝ct f ∝t ρ∝Aω(ct)−3ρ∝t−3 θ∝k−1 BAωc0t0θ∝const. S∝/parenleftbig A3 ωa(ct)3/parenrightbig1 4S∝t3 s∝/parenleftbig A3 ωa/parenrightbig1 4(ct)−9/4s∝t−9/4 In the same way as for the previous model it is seen that with respect to quantities G, f, ρ and Λ the same results found in literature (see [3], [8] and [9]) are ob- tained. With regards to the rest of “constants” studied we see that in this case they do vary. In particular regarding G,/planckover2pi1, eandρthe same results as O’Hanlon et al (see [2]) are obtained. These authors set out from the Dirac model , its LNH, and by means of some pertinent modifications five dimensionless numbers are obtained, in the same way as that of the Dirac model, but this time reaching totally different results. With their five dimensionless numbers together with the hypothesis that the mass of the universe is constant (we do not need to make a similar hypothesis, the model tells us mi∝ const. ) they are brought to the only way in which G, e2 and/planckover2pi1vary. The ρaverage density of the universe mass varies thus ρ∝t−3while G∝t,e2∝tand/planckover2pi1∝twhile energy is conserved. This models responds, at the same time, to the axioms of Milne’s Kinematic Relativity (see [12]). 5Ifρis considered as mass density then Aω(constant denoted by A0) [A0] =Mrepresents the universe mass, and the expression G(t) remains thus: G∝Aωc3t G ∝t verifying the Sciama formula ρGt2≈1 (on inertia). Fur- thermore if we take into account the numeric values of the constant and the quantity twe obtain the present value of G≈10−10.1757m3kg−1s−2i.e.t≈1020s, c≈108.5ms−1 andA0≈1056kg.This result was already obtained by Milne in 1935 (see [12]). In the same way as in the previ- ous case we are capable (with the value obtained of A0) of recovering all the cosmological quantities but not those corresponding to electromagnetism and the Planck “con- stant”. And vice versa. With regards to the cosmological “constant” in the same way as in the previous case we see that if t≈1020sandc≈108.4ms−1⇒Λ≈10−56m−2a value which corresponds to that presently accepted. V. SUMMARY AND CONCLUSIONS. We have resolved through Dimensional Analysis DA a flat FRW model i.e. with k= 0 whose energy-momentum tensor is described by a perfect fluid and taking into account the conservation principle for said tensor i.e. div(Tij) = 0 in which some constants are considered as variable i.e. as scalar functions dependent on time. It has been proven that the dimensional technique used re- solves in a trivial manner the problem posed and that the results reached correspond to those already existing in literature. New solutions have been contributed as our model is more general as the variation of the “constants” /planckover2pi1, e, a, ǫ 0,µ0andmiis contemplated. In the two models studied it is proven that the solu- tions obtained are coherent with reference to cosmolog- ical parameters while for electromagnetic and quantum quantities our model is not capable of adjusting itself to data presently accepted for these. “ At this time” we be- lieve that the approach is erroneous, an unsettling ques- tion under revision. With regards to the model with matter predominance it is seen that it is capable of theoretically justifying the O’Hanlon et al model, as we obtain their same results without having to resort to any assumption or precise numerological coincidence, even if it is based on the Dirac hypothesis.[1]Belinch´ on, J. A., (physics/9811017) [2]O’Hanlon, J. &Tam, K-K. Prog. Theor. Phys. 41, 1596, (1969). O’Hanlon, J. &Tam, K-K. Prog. Theor. Phys.43, 684, (1970). [3]Kalligas et al . Gen. Rel. Grav. 24,351 (1992). [4]Belinch´ on, J.A. submitted to Class. Quant. Grav. [5]Belinch´ on, J.A., (physics/9811016) [6]Zee. Phys. Rev. Lett. 42, 417,(1979) [7]A-M. M. Abdel-Rahman. Gen. Rel. Grav. 22, 655,(1990). [8]M. S. Bermann . Gen. Rel. Grav. 23, 465,(1991). M. S. Bermann . Phys. Rev. D 43, 1075, (1991) [9]Abdussaltar and R. G. Vishwakarma. Class. Quan. Grav.14, 945,(1997) [10]Moller, C. The theory of Relativity. Oxford Clarendom 1952.Landau & Lifshitz. The classical theory of fields. Oxford Clarendom 1975. Sumner, W. Q. Astro. Jour., 429, 491-98, (1994). [11]Qadir, A. and Wheeler, J.A. in From SU(3) to Grav- ity Ed. Gotsman and Tauber. CUP (1988) p. 383-394. Marzke, R. K. & Wheeler, J.A. In Gravitation and Relativity. (Eds Chiu, H.Y. and Hoffmann, W. F.) N.Y. Benjamin 1964. pp.60-61. [12]Milne, E. Relativity, gravitation and world structure. Oxford (1935). Milne, E. Proc. R. Soc. A 165, 351, (1938). Milne, E. Mont. Not. R. Astr. Soc. 106, 180, (1946). Milne, E. Kinematic Relativity. Oxford (1948) 6

arXiv:physics/0002047v1 [physics.optics] 26 Feb 2000JAYNES-CUMMINGS MODEL WITH DEGENERATE ATOMIC LEVELS V. A. Reshetov Department of Physics, Tolyatti Pedagogical Institute, 13 Boulevard Korolyova, 445859 Tolyatti, Russia Abstract The Jaynes-Cummings model describing the interaction of a s ingle linearly- polarized mode of the quantized electromagnetic field with a n isolated two-level atom is generalized to the case of atomic levels degenerate i n the projections of the angular momenta on the quantization axis, which is a usua l case in the exper- iments. This generalization, like the original model, obta ins the explicit solution. The model is applied to calculate the dependence of the atomi c level populations on the angle between the polarization of cavity field mode and that of the laser excitation pulse in the experiment with one-atom micromase r. The Jaynes-Cummings model [1] describes the interaction of a single linearly-polarized mode of the quan- tized electromagnetic field with an isolated two-level atom . The full set of states of the system atom+field is |n,α> =|n>·|α>, ,n = 0,1,..., α =b,c, wherenis the number of photons in the field mode, while bandcdenote the upper and lower atomic levels correspondingly. This model is applied successfully to ana lyse the results of the experiments with one-atom micromasers (see, e.g., [2]). However, the levels of an isol ated atom are degenerate in the projections of the total elctronic angular momenta on the quantization axi s, so that the original Jaynes-Cummings model becomes, in general, invalid. Now, let us take into account the degeneracy of atomic levels . Then, the full set of states of the system becomes |n,Jα,mα>=|n>·|Jα,mα>, n= 0,1,..., m α=−Jα,...,J α, α=b,c, whereJbandJcare the values of the total electronic angular momenta of res onant levels, while mbandmcare their projections on the quantization axis - the cartesian a xis Z, which is directed along the polarization vector of the field mode. The Hamiltonian of the system may be written as ˆH=ˆHF+ˆHA+ˆV , (1) where ˆHF= ¯hωˆa+ˆa is a free-field Hamiltonian, ˆHA=1 2¯hω0(ˆnb−ˆnc) is a free-atom Hamiltonian, ˆV=−(ˆDˆE) 1is an operator of field-atom interaction, while ˆ a+and ˆaare the operators of the creation and annihilation of photons with the frequency ωin the field mode, ˆnα=Jα/summationdisplay mα=−Jα|Jα,mα><J α,mα|, α=b,c, are the operators of total populations of resonant atomic le velsbandc, ω0is the frequency of the optically- allowed atomic transition Jb→Jc, ˆE=eˆa+e∗ˆa+, e=ılz/radicalbigg 2π¯hω V, is the electric field intensity operator, Vandlzbeing the resonator cavity volume and the unit vector of the cartesian axis Z, ˆD=ˆd+ˆd+, ˆd=/summationdisplay mb,mcdJcJb mcmb· |Jc,mc><J b,mb|, is the dipole moment operator of the atomic transition Jb→Jc,which matrix elements are defined through Wigner 3j-symbols (see, e.g., [3]): (dq)JbJc mcmb=d(−1)Jb−mb/parenleftbigg Jb1Jc −mbq m c/parenrightbigg , d=d(JbJc) -being a reduced matrix element and dq(q=−1,0,1) - are the circular components of vector d. In the interaction representation ˆfI= exp/parenleftBigg ıˆH0t ¯h/parenrightBigg ·ˆf·exp/parenleftBigg −ıˆH0t ¯h/parenrightBigg , where ˆH0= ¯hω/braceleftbigg ˆa+ˆa+1 2(ˆnb−ˆnc)/bracerightbigg , the operators ˆ aandˆdobtain the oscillating factors ˆaI= ˆa·exp(−ıωt),ˆdI=ˆd·exp(−ıωt). Then, in the rotating wave approximation, when the terms osc illating with double frequences are neglected, the Hamiltonian (1) becomes ˆHI=ˆH0−¯hˆΩ, where ˆΩ =δ 2(ˆnb−ˆnc) +ıg(ˆaˆp+−ˆa+ˆp), while δ= (ω−ω0) is the frequency detuning, g=/radicalbigg 2πd2ω ¯hV and ˆp=/summationdisplay mαm· |Jc,m><J b,m|, 2αm= (−1)Jb−m/parenleftbigg Jb1Jc −m0m/parenrightbigg . From the equation dˆσ dt=ı ¯h/bracketleftBig ˆσ,ˆH/bracketrightBig for the system density matrix ˆ σfollows the equation dˆρ dt=ı/bracketleftBig ˆΩ,ˆρ/bracketrightBig (2) for the density matrix ˆρ= exp/parenleftBigg ıˆH0t ¯h/parenrightBigg ·ˆσ·exp/parenleftBigg −ıˆH0t ¯h/parenrightBigg in the interaction representation. The formal solution of t he equation (2) is obtained immediately ˆρ= exp/parenleftBig ıˆΩt/parenrightBig ·ˆρ0·exp/parenleftBig −ıˆΩt/parenrightBig , where ˆρ0is the initial density matrix of the system. In order to obtain the average value <ˆf >=Tr/parenleftBig ˆρˆfI/parenrightBig of any operator ˆfit is necessary to calculate the matrix elements <n,J α,m|exp(ıˆΩt)|n1,Jβ,m1>, α,β =b,c, of the evolution operator. The explicit analytical express ions for these matrix elements may be derived with the use expansion exp(ıˆΩt) =∞/summationdisplay n=0(ıˆΩt)n n!, since the operator ˆΩ2=δ2 4ˆ1 +g2·/braceleftBig ˆRcˆn+ˆRb(ˆn+ 1)/bracerightBig , where ˆRβ=/summationdisplay mα2 m· |Jβ,m><J β,m|, β=b,c, is diagonal: ˆΩ2|n,Jb,m>= Ω2 n+1,m|n,Jb,m>, ˆΩ2|n,Jc,m>= Ω2 n,m|n,Jc,m>. Here Ωn,m=/radicalbigg δ2 4+α2mg2n. (3) So, the matrix elements of the evolution operator are: <n,J b,m|exp(ıˆΩt)|n1,Jb,m1>= δn,n1δm,m1/braceleftbigg cos(Ω n+1,mt) +ıδ 2Ωn+1,msin(Ω n+1,mt)/bracerightbigg , (4) <n,J c,m|exp(ıˆΩt)|n1,Jc,m1>= 3δn,n1δm,m1/braceleftbigg cos(Ω n,mt)−ıδ 2Ωn,msin(Ω n,mt)/bracerightbigg , (5) <n,J b,m|exp (ıˆΩt)|n1,Jc,m1>= −δn+1,n1δm,m1gαm√ n+ 1·sin (Ω n+1,mt) Ωn+1,m. (6) In the experiment [2] the average total population nb=Tr/braceleftBig ˆnbexp (ıˆΩT)ρ0exp (−ıˆΩT)/bracerightBig of the upper resonant level bafter the atom passes through the resonant cavity, where T is the time of interaction, was detected. As follows from (4)-(6), nb=/summationdisplay n,mfnnnb mm/braceleftBigg cos2(Ωn+1,mT) +δ2 4Ω2 n+1,msin2(Ωn+1,mT)/bracerightBigg , (7) where the atomic and field subsystems at the initial instant o f time, when the atom enters the cavity, are independent and the initial density matrix of the system is ˆρ0= ˆρA 0·ˆρF 0, while ˆρA 0=/summationdisplay m,m1nb mm1· |Jb,m><J b,m1|, ˆρF 0=/summationdisplay n,n1fnn1· |n><n 1|. The cavity temperature in [2] was low, so that the initial fiel d may be considered to be in its vacuum state: fn,n1=δn,0δn1,0. Then, in case of exact resonance δ= 0 the equation (7) simplifies to nb=/summationdisplay mnb mmcos2(θm), θm=αmgT . (8) Herenb mmis the initial population of the Zeeman sublevel mof the upper level b. The resonant lev- elsbandcin the experiment [2] were the Rydberg states of the rubidium atom with the angular mo- mentaJb= 3/2 andJc= 3/2 orJc= 5/2 . The upper level bwas excited from the ground state awith the angular momentum Ja= 1/2 by the linearly-polarized laser pulse. The evolution of th e atomic den- sity matrix under the action of the excitation pulse in the ro tating-wave approximation is desribed by the equation dˆρA dt=ı ¯h/bracketleftBig ˆρA,ˆVe/bracketrightBig , (9) where ˆVe=−(ˆd+ eee+ˆdee∗ e) is the interaction operator of an atom with the cohernt reson ant laser field, ˆde=/summationdisplay mb,ma(de)JaJbmamb· |Ja,ma><J b,mb|, 4is the dipole moment operator of the atomic transition Jb→Ja,ee=eelis the slowly-varying amplitude of laser field, lis its unit polarization vector, which constitutes the angl eψwith the polarization of the cavity field mode: lq= cosψδq,0+1√ 2sinψ(δq,−1−δq,1). For purposes of simplicity we shall consider the exciting pu lses with small areas θe=|de| ¯h/integraldisplayTe 0ee(t)dt≪1 (10) (though in case of transition 3 /2→1/2 in the experiment [1] the following results do not depend on the exciting pulse area), de=d(JbJa) is a reduced matrix element of the dipole moment operator fo r the transitionJb→Ja,Teis the exciting pulse duration. Under the limitation (10) we obtain from (9) the density matrix of an atom (renormalized to unity trace) ˆρA 0=(ˆd+ el)ˆρA in(ˆdel) Tr/braceleftBig (ˆd+ el)ˆρA in(ˆdel)/bracerightBig (11) at an instant when it enters the cavity. Here ˆρA in=1 (2Ja+ 1)/summationdisplay m|Ja,m><J a,m| is the initial equilibrium atomic density matrix before the incidence of the exciting pulse. As follows from (11) the Zeeman sublevel populations in (8) are nb mm=<Jb,m|ˆρA 0|Jb,m>=amcos2ψ+bmsin2ψ , where am= 3/parenleftbigg Jb1Ja −m0m/parenrightbigg2 , bm=3 2/braceleftBigg/parenleftbigg Jb1Ja −m−1m+ 1/parenrightbigg2 +/parenleftbigg Jb1Ja −m1m−1/parenrightbigg2/bracerightBigg . In case of transitions Jb= 3/2→Ja= 1/2 nb −1/2,−1/2=nb 1/2,1/2=1 2−3 8sin2ψ , nb −3/2,−3/2=nb 3/2,3/2=3 8sin2ψ , and the total population (8) of the upper level after the atom leaves the cavity is nb=/parenleftbigg 1−3 4sin2(ψ)/parenrightbigg cos2(θ) +3 4sin2(ψ) cos2(3θ), θ=gT 2√ 15, for the transitions Jb= 3/2→Jc= 3/2 and nb=/parenleftbigg 1−3 4sin2(ψ)/parenrightbigg cos2(θ) +3 4sin2(ψ)cos2/parenleftBigg/radicalbigg 3 2θ/parenrightBigg , 5θ=gT√ 10, for the transitions Jb= 3/2→Jc= 5/2 . The atom behaves like a two-level system - the population nb= cos2(θ) oscillates with a single Rabi frequency - only in case when th e polarizations of the exciting laser pulse and of the cavity field mode coincide - ψ= 0,otherwise the oscillations with more than one Rabi frequenc ies appear. So, the Jaynes-Cummings model generalized to the case of the atomic levels degenerate in the projec- tions of the angular momenta on the quantization axis is a use ful tools for the description of the polarization properties of one-atom micromasers. References [1] Jaynes E T, Cummings F W 1963 Proc. IEEE 5189 [2] Walther H 1995 Ann.N.Y.Acad.Sci. 755133 [3] Sobelman I I 1972 Introduction to the Theory of Atomic Spectra (New York:Pergamon) 6

- 1 -Irreversibility and Measurement in Quantum Mechanics DOUGLAS M. S NYDER LOS ANGELES , CALIFORNIA In quantum mechanics, in principle a human observer does not require a macroscopic physical instrument to make a measurement. A good deal of evidence supporting this thesis comes from work on macroscopic quantum tunneling (Das Sarma, Kawamura, & Washburn, 1995). It is often maintained that there is nothing unique that a human observer can do that a physical, macroscopic measuring instrument, often of simple design, cannot also accomplish. It is generally argued that when a macroscopic measuring instrument is used, irreversibility characterizes the measurement process. The measurement cannot be reversed even if a human observer does not immediately see or know the result obtained with the measuring instrument.1 This irreversibility has been ascribed to an increase in entropy that occurs when a measurement is made (e.g., van Hove, 1959). It is implied that irreversibility is a common element in quantum mechanical measurement, even characterizing the circumstances where a person might make a measurement unaided by a macroscopic measuring device. In a related manner, Bohr (1935) maintained that quantum mechanical measurement also depended on the interaction between a macroscopic measuring instrument and the physical existent measured. He noted that when a macroscopic physical measuring apparatus is used, there is inevitably some loss of informa tion concerning the measured system due to the resulting physical interaction. Bohr presented a gedankenexperiment involving a particlepassing through a slit in a diaphragm that is part of an experimental apparatus. In one scenario in which the position of the particle passing through the diaphragm is precisely determined, knowledge of the particle's momentum is lost because of the rigid connection between the diaphragm and the rest of the apparatus necessary to establish the position of the diaphragm when the particle passes through. In another scenario, in which the momentum of the particle is precisely determined, knowledge of the position of the particle is lost because of the flexible connection between the diaphragm and rest of the apparatus necessary to determine the momentum of the diaphragm before and after the 1 Generally, this irreversibility means that it is highly unlikely that the physical interaction that is the measurement could occur in the opposite direction of time to the one in which it isoccurring or has occurred.Irreversibility and Measurement - 2 -particle passes through the diaphragm. For Bohr, once the information is lost in the measurement process, the measurement cannot be reversed. It is known that the consideration of a physical system as a macroscopic system instead of as a collection of microscopic quantum mechanical systems is arbitrary. If the physical system is considered in the latter manner, the Schršdinger equation provides the basis for a lawful delineation of the physical existent that was to be measured by the macroscopic apparatus and that microscopic part of the physical system designated the measurement apparatus with which the system that was to be measured interacts. The lawful manner inwhich these microscopic physical systems function, including their interaction, is reversible. Information is not lost, and the concept of entropy, dependent on a macroscopic physical system, is not relevant. Thus when physicists maintain that irreversibility in measurement is a sufficient condition to ensure that a measurement has indeed occurred, the following feature of quantum mechanical measurement is not addressed. Whether a physical system is considered: (1) macroscopic in nature (so as to beavailable to measure a microscopic physical system) or (2) a collection of microscopic physical systems some portion of which interact with the microscopic system of interest in a reversible fashion in accordance with the Schršdinger equation, is an arbitrary decision on the part of the individual considering the overall experimental circumstances. That is, the presence of irreversibility depends on whether a measurement is being made, and whether or not a measurement is being made depends on an arbitrary decision by the experimenter as how the experimental circumstances are structured. It should be noted that the assumption that irreversibility characterizes a human observer who does not require a macroscopic physical instrument to make a quantum mechanical measurement is unwarranted in the absence of specifying the neurophysiological mechanism, presumably functioning like a macroscopic physical apparatus, by which an irreversible measurement is made. Also, given the flexibility in considering a macroscopic physical systemas such or as a combination of many microscopic systems, this result holds when a macroscopic measuring device is used in making a measurement as well. It would be important to specify how this neurophysiological mechanism involves those physical processes that would be responsible for a macroscopic physical instrument's ability to engage in an irreversible measurement. It would be these processes that would presumably underlie the neurophysiological mechanism.Irreversibility and Measurement - 3 -Another concern should be noted. How do irreversible measurement processes, presumably based on physical interaction, account for "negative" observations (Epstein, 1945; Renninger, 1960)? In a negative observation, a measurement occurs in the absence of a physical interaction between a measuring instrument and the existent measured, with an accompanying changein the wave function describing the existent measured. 2 Because of the above considerations, one has to doubt that irreversibility in quantum mechanical measurement has been given an adequate foundation. Indeed, the flexibility in the measurement process as regards the observerÕs ability to consider a physical system as a macroscopic measuring instrument or instead as a large system comprised of many macroscopic systems indicates the importance of a subjective element. REFERENCES Bohr, N. (1935). Can quantum-mechanical description of nature be considered complete? Physical Review , 49, 1804-1807. Cook, R. J. (1990). Quantum jumps. In E. Wolf (Ed.), Progress in Optics (Vol. 28) (pp. 361-416). Amsterdam: North-Holland. Das Sarma, S., Kawamura, T., and Washburn, S. (1995). Resource letter QIMA-1: Quantum interference in macroscopic samples. American Journal of Physics , 8, 683-694. Epstein P. (1945). The reality problem in quantum mechanics. American Journal of Physics , 13, 127-136. Renninger, M. (1960). Messungen ohne Stšrung des Me§objekts [Observations Without Changing the Object]. Zeitschrift fŸr Physik , 158, 417-421. van Hove, L. (1959). The ergodic behaviour of quantum many-body systems. Physica , 25, 268-276. 2 A negative observation depends on the possibility of an alternative observation involving a physical interaction. Indeed, Cook (1990) noted the possibility of a negative observation in which one existent (a light photon) involved in the physical interaction can be registered by the unaided human eye.

- 1 -On the Nature of Measurement in Quantum Mechanics DOUGLAS M. S NYDER LOS ANGELES , CALIFORNIA Abstract A number of issues related to measurement show that self-consistency is lacking in quantum mechanics as this theory has been generally understood. Each issue is presented as a point in this paper. Each point can be resolved by incorporating a cognitive component in quantum mechanics. Measurement inquantum mechanics involves the meaning of the physical circumstances of the experiment. This meaning is in part independent of what traditionally areconsidered purely physical considerations. Text The following five points concern issues in quantum mechanics that indicate that experimenter variables play a role in this theory. Each point is followed by a brief discussion. POINT ONE 1. An experimenter oftentimes may consider a physical interaction either as a measurement involving a macroscopic measuring instrument interacting with an observable or in terms of a set of microscopic systems, including the observable, functioning lawfully in accordance with the Schršdinger equation. The question arises: On the microscopic level, does the abrupt change in the wave function associated with the observable that often occurs in a measurement occur when the observable and the macroscopic apparatus are considered as a set of microscopic physical systems interacting with each other?According to Bohr, the answer apparently is, ÒNo.Ó Bohr emphasized that it was the macroscopic nature of the measuring apparatus that was responsible fora measurement. According to Bohr (1935), there was some loss of control dueto the macroscopic character of the physical apparatus used to make a measurement. This lack of control resulted in the limitation on knowledge described by the uncertainty principle. Thus if the macroscopic system is considered in terms of its microscopic components, a measurement should notOn the Nature - 2 -occur. There should not be an abrupt change in the wave function associated with the observable. Here is how Bohr considered measurement in quantum mechanics. He wrote: It is imperative to realize that in every account of physical experience one must describe both experimental conditions and observations by the same means of communication as one used in classical physics. In the analysis of single atomic particles [inquantum mechanics], this is made possible by irreversible amplification effects [emphasis added]Ðsuch as a spot on a photographic plate left by the impact of an electron, or an electricdischarge created in a counter deviceÐand the observations concern only where and when the particle is registered on the plate or its energy on arrival at the counter. (Bohr, 1955/1958b, p. 88) Bohr (1955/1958a) essentially defined the phenomenon in quantum mechanics as the entire experimental context, specifically the experiment that is conducted on an observable. Bohr maintained that the experiment conducted was described in terms of the physical apparatus used and the observable measured and that the physical world in quantum mechanics could only be understood in terms of the phenomenon. On the lines of objective description, it is indeed more appropriate to use the word phenomenon to refer only to observations obtained under circumstances whose description includes an account of the whole experimental arrangement. In such terminology, the observational problem in quantum physics is deprived of any special intricacy and we are, moreover, directly reminded that every atomic phenomenon is closed in the sense that its observation is based on registrations obtained by means of suitable amplification devices withirreversible functioning such as, for example, permanent marks on a photographic plate, caused by the penetration of electrons into the emulsion. (Bohr, 1955/1958a, p. 73) 1 1 Citing Bohr, Wheeler (1981/1983) wrote: ÒA phenomenon is not yet a phenomenon until it has been brought to a close by an irreversible act of amplification such as the blackening of agrain of silver bromide emulsion or the triggering of a photodetector....We are dealing with anOn the Nature - 3 -An experiment discussed by Feynman, Leighton, and Sands (1965) that will be discussed shortly indicates that the destruction of interference often associated with a measurement in quantum mechanics does not depend on a macroscopic measuring instrument. POINTS TWO AND THREE 2. A measurement can occur on a microscopic level without depending on a macroscopic measuring instrument. 3. The presence or absence of interference may depend on a comparison of different observables in an experiment, even if the observables cannot actually be observed by the experimenter. The Importance of Comparison In demonstrating the point that measurement does not rely on a macroscopic physical apparatus acting as a measuring instrument interactingwith the observable measured, Feynman et al. considered the destruction of interference on the atomic level in the case of neutron scattering by a crystal. This destruction of interference depends on a change in the nucleus in the crystal that interacts with the impacting neutron allowing the experimenter to Òinprinciple, find out which nucleus had done the scattering, since it would be the only one with spin turned overÓ (Feynman et al. , 1965, chap. 3, p. 8). Initially, all the nuclei in the crystal have their spin components along a specificspatial axis set in one direction. It is a change in the spin component along a spatial axis for a nucleus in the crystal impacted by a neutron that distinguishes this nucleus from other nuclei for which the spin component along this same axis has not changed and which thus all remain in the original direction. Note that the destruction in interference depends on the nuclei that are not impacted by the neutron all having spin components that are opposite in orientation to the nucleus that is impacted by the incoming neutron. It is not thephysical interaction between a nucleus in the crystal and the neutron that alone is responsible for the destruction of interference. If, in the interaction, the spin component for the nucleus does not flip and the spin component for the neutron therefore does not flip, interference is not destroyed. In the experiment, destruction of the interference depends on a microscopic event, a spin component flip on the part of the neutron that distinguishes the path of the event that makes itself known by an irreversible act of amplification, by an indelible record, an act of registration.Ó (pp. 184-185, 196)On the Nature - 4 -impacting neutron from alternative pat hs. The different spin components cannot, of course, be observed by the experimenter. It is the information conveyed to the observer that determines whether or not interference is destroyed. The physical interaction alone does not account for the different physical results. Feynman et al. (1965) had a different view: Well, if we can tell which atom [nucleus] did the scattering, what have the other atoms [nuclei] to do with it? Nothing of course. The scattering is exactly the same as that from a single atom. (chap. 3, p. 8) When Feynman et al. wrote: ÒNothing of course,Ó their own description of the experiment indicates their statement is incorrect. Perhaps, they could argue that on a physical level alone the other nuclei are not involved in the scattering, or that if considered in isolation the impacted nucleus would not show interference. Their own description of the destruction of interference, though, in the crystal depends on the difference between the spin component orientations of the impacted nucleus and the other nuclei. In other words, the experimental circumstance involving numerous nuclei is not the same as one involving only one nucleus. POINT FOUR 4. A measurement needs to make which-way information only in principle available to destroy interference. In different works on quantum theory, Feynman distinguished those circumstances in which wave amplitudes associated with an observable are first added and the absolute value of the sum is squared to derive the probability of an event, as opposed to taking the absolute square of each of the wave amplitudes and taking their sum to determine the probability of the same event. Feynman et al. (1965) wrote concerning quantum mechanics: [1] The probability of an event in an ideal experiment is given by the square of the absolute value of a complex number f which is called the probability amplitude... [2] When an event can occur in several alternative ways, the probability amplitude for the event is the sum of the probability amplitudes for each way considered separately...On the Nature - 5 -[3] If an experiment is performed which is capable of determining whether one or another alternative is actually taken, the probability of the event is the sum of the probabilities for each alternative. (chap. 3, p. 10) In their experiment discussed above, Feynman et al. (1965) wrote that the distinction between the different paths need not actually be known to an observer.2 All that is necessary is that their distinction is in principle possible in the experimental circumstances. If you could, in principle , distinguish the alternative final states [of the observables] (even though you do not bother to do so), the total, final probability is obtained by calculating the probability for each state (not the amplitude) and then adding them together. If you cannot distinguish the final states even in principle , then the probability amplitudes must be summed before taking the absolute square to find the actual probability. (Feynman et al., 1965, chap. 3, p. 9) Point four might appear to indicate that the observer is not central to measurement in quantum mechanics. It is important to note that dis tinguishing final paths constitutes a measurement. Remember, oftentimes, the physical interaction occasioning a measurement can be a macroscopic apparatusinteracting with a microscopic observable or it can be an interaction among microscopic observables alone (Point one). Because of these different views one can take of an interaction, the in principle possibility of distinguishing paths supports a cognitive component to measurement in quantum mechanics. In this context, it is important to note negative observation. In a negative observation, a measurement does not involve a physical interaction at all (Bergquist, Hulet, Itano, & Wineland, 1986; Cook, 1990; Epstein, 1945; Nagourney, Sandberg, & Dehmelt, 1986; Renninger, 1960; Sauter, Neuhauser, Blatt, & Toschek, 1986; Snyder, 1996, 1997). A negative observation occurs where an observation is made by deducing that a particular physical event must have occurred because another physical event did not occur 2 In QED: The Strange Theory of Light and Matter , Feynman (1985) maintained the same principles hold in quantum electrodynamics. In sum, where one can distinguish between the possible routes, or developmental sequence, for an observable, interference between the component wave amplitudes characterizing the existent is destroyed.On the Nature - 6 -with subsequent consequences for the functioning of the physical world stemming from the change in knowledge. POINT FIVE 5. Macroscopic devices may either : 1) interact with an observable acting as a measuring instrument where coherence is destroyed and there is an abrupt change in the wave function associated with the observable, or 2) act as a device in the service of separating a wave function into coherent components or recombining coherent component wave functions so that interference is expressed. There is nothing in the nature of a macroscopicdevice itself that necessarily signals a measurement. An example is a double-hole diaphragm that is rigidly attached to a support through which electrons pass on their way to a screen that they impact and that records their position (Bohr, 1949/1969). Coherent component wave functions are developed as a result of the electron passing through the diaphragm, resulting in interference which is detectable when the electrons impact the screen. A macroscopic apparatus in this case does not result in a measurement when it interacts with a microscopic observable. Something else is needed to distinguish whether a measurement is made. If, instead of being fixed, the diaphragm was movable due to momentum transfer between the electron and the diaphragm, a position measurement would occur. Which hole an electron went through on its way tothe screen could be determined by how the diaphragm moved as the electron passed through (Bohr, 1949/1969). What is it about allowing the diaphragm tomove that changes its function so dramatically? The answer lies in the destruction of interference, and interference in quantum mechanics depends on coherent waves that are mathematically complex and do not have a physical existence. One can in fact piece together coherent component wave functions associated with an observable using a macroscopic physical apparatus. For example, one may use a one-half silvered mirror to recombine coherent component wave functions associated with a photon that had originally been separated earlier by another one-half silvered mirror. The macroscopic apparatus used to combine the component wave functions does not destroy interference. It actually implements interference using the coherent wave functions.On the Nature - 7 -CONCLUSION Following is a summary of the points made in this paper. 1. In many experimental circumstances, a macroscopic apparatus appears to be responsible for the occurrence of measurements in quantum mechanics. 2. Feynman et al. Õs experiment works as they discussed, and their position that a macroscopic physical apparatus is not always necessary for a measurement is sound. 3. The occurrence of a measurement may depend on a comparison of observables, even if these observables cannot be observed but can only be thought about. 4. A physical interaction that is considered a measurement, accompanied by an abrupt change of the wave function associated with the observable measured, does not necessarily have to be considered as such. The collapse ofthe wave function need not occur in another view of the same interaction that focuses on the interacting systems microscopically. 5. A measurement can be said to occur when in principle it is possible to distinguish possible measurement outcomes, even if these outcomes are not actually measured. 6. Negative observations are possible in quantum mechanics.7. A macroscopic physical apparatus is not necessary for a measurement, and indeed it can be used to develop interference. Notice the use of the terms view , comparison , and in principle . The inconsistencies noted in this paper indicate that quantum mechanics is not concerned exclusively with the physical. A measurement affecting an observable can be made without a physical interaction. Quantum mechanics, though, always does depend on the mathematically complex waves associated with observables, and interference itself combines these complex waves. Anything that can be known about an observable is derived from these waves. Ascribing a cognitive component to quantum mechanics accounts for the terms indicating cognition that are noted above as well as the result stemmingOn the Nature - 8 -from BohrÕs concept of measurement that depending on oneÕs view there may or not be an abrupt change of the wave function associated with an observable when a physical interaction occurs. One can summarize the points in this paperby characterizing measurement in quantum mechanics as involving the meaning of the physical circumstances of the experiment. This meaning is in part independent of what traditionally are considered purely physical considerations. R EFERENCES Bergquist, J. C., Hulet, R. G., Itano, W. M., and Wineland, D. J. (1986). Observation of quantum jumps in a single atom. Physical Review Letters , 57, 1699-1702. Bohr, N. (1935). Can quantum-mechanical description of nature be considered complete? Physical Review , 49, 1804-1807. Bohr, N. (1969). Discussion with Einstein on epistemological problems in atomic physics. In P. A. Schilpp (Ed.), Albert Einstein: Philosopher- scientist (Vol. 1) (pp. 199-241). La Salle, Illinois: Open Court. (Original work published 1949) Bohr, N. (1958a) Atoms and human knowledge. In N. Bohr, Atomic Physics and Human Knowledge (pp. 67-82). New York: John Wiley & Sons. (Original work published 1955) Bohr, N. (1958b) Unity of knowledge. In N. Bohr, Atomic Physics and Human Knowledge (pp. 83-93). New York: John Wiley & Sons. (Original work published 1955) Cook, R. J. (1990). Quantum jumps. In E. Wolf (Ed.), Progress in Optics (Vol. 28) (pp. 361-416). Amsterdam: North-Holland. Epstein, P. (1945). The reality problem in quantum mechanics. American Journal of Physics , 13, 127-136. Feynman, R. P., Leighton, R. B., &, and Sands, M. (1965). The Feynman Lectures on Physics: Quantum Mechanics (Vol. 3). Reading, Massachusetts: Addison-Wesley. Feynman, R. P. (1985). QED: The Strange Theory of Light and Matter . Princeton, New Jersey: Princeton University Press. Nagourney, W., Sandberg, J., and Dehmelt, H. (1986). Shelved optical electron amplifier: ob servation of quantum jumps. Physical Review Letters , 56, 2797-2799.On the Nature - 9 -Renninger, M. (1960). Messungen ohne Stšrung des Me§objekts [Observations without disturbing the object]. Zeitschrift fŸr Physik , 158, 417-421. Sauter, T., Neuhauser, W., Blatt, R. and Toschek, P. E. (1986). Observation of quantum jumps. Physical Review Letters , 57, 1696-1698. Snyder, D. M. (1996). On the Nature of the Change in the Wave Function in a Measurement in Quantum Mechanics. Los Alamos National Laboratory E-Print Physics Archive (WWW address: http://xxx.lanl.gov/abs/quant- ph/9601006). Snyder, D. M. (1997). Negative Observations in Quantum Mechanics. APS E- print Server (WWW address: http://publish.aps.org/eprint/gateway/ eplist/aps1997jun24_006) and Cogprints Electronic Archive (WWW address: http://cogprints.soton.ac.uk/abs/psyc/199806017). Wheeler, J. A. (1983). Law without law. In J. A. Wheeler & W. H. Zurek (Eds.) Quantum Theory and Measurement (pp. 182-213). Princeton, New Jersey: Princeton University Press. (Original work published 1981)

arXiv:physics/0002050v1 [physics.bio-ph] 28 Feb 2000Scaling in dynamical Turing pattern formation: density of defects frozen into permanent patterns Jacek Dziarmaga Los Alamos National Laboratory, Theory Division T-6, MS-B2 88, Los Alamos, NM 87545, USA and M.Smoluchowski Institute of Physics, Jagiellonian Uni versity, Krak´ ow, Poland dziarmaga@t6-serv.lanl.gov (February 28, 2000) We estimate density of defects frozen into a biological Turi ng pattern which was turned on at a finite rate. A self-locking of gene expression i n individual cells, which makes the Turing transition discontinuous, stabilizes the pattern together with its defects. A defect-free pattern can be obtained by spatially inhomogeneous activation of the genes. Motivation and summary of results Long time ago Turing pointed out [1] that simple reaction-di ffusion (RD) systems of equations can account for formation of biological patterns. The mainstream of resear ch, as reviewed in Ref. [2], is devoted to RD models in continuous space. The continuum RD-patterns are smooth and nonpermanent. On the other hand, it is an empirical fact that even the nearest neighbor cells can differ sharply i n their biological functions and their sets of expressed genes. Moreover many biological patterns are permanent. Ev en the most primitive viruses, like the much studied bacteriophage λ[3], possess genetic switches that discriminate between di fferent developmental pathways and make a once chosen pathway permanent. It is reasonable to assume t hat cells of higher organisms can also lock their distinctive sets of expressed genes. The Turing patterns on figures in the review [2] are contamina ted with defects. If we insist on pattern permanence, we must accept that patterns are permanent together with the ir defects. Sometimes, like for the animal coat patterns, permanent defects can provide an animal with its own charact eristic life-long but not inheritable ”fingerprints”. In other cases, like formation of vital organ structures, a sin gle defect can be fatal. In this situation it is important to understand better the origin of defects. In this paper we use a simple toy model which in principle shou ld give a homogeneous Turing pattern. Defects are particularly manifest on such a simple background. The m odel has two genes AandB. The genes are strong mutual repressors. The strong intracellular mutual inhibi tion is the factor responsible for pattern permanence. Both genes are activated simultaneously in a given cell when a lev el of their common activator aexceeds its critical value ac. Pattern formation in RD models of Ref. [2] was simulated wit h fixed model parameters. In this paper we turn on the activator level aat a finite rate to find a scaling relation between density of de fects and the rate. Strong mutual intracellular inhibition stabilizes the pattern together with its defects. We obtain permanent domains of A-phase and domains of B-phase divided by sharp cell-size boundaries. We also show that an inhomogeneous activation of the genes ca n result in a perfect defect-free homogeneous pattern. At first the activator aexceedsacin a small seed area where, say, the gene Ais chosen. Then aslowly spreads around gradually activating more and more cells. The initial choic e ofAis imposed via intercellular coupling on all newly activated cells. The inhomogeneous activation can be suffici ently characterized by a velocity vwith which the critical a=acsurface spreads. Thanks to the strong mutual intracellular inhibition there is a nonzero threshold velocity vc, such that for v<vcthe formation of defects is completely suppressed. In this w ay the very mutual inhibition which is responsible for stability of defects can be harnessed to g et rid of them. The genetic network that we use in our toy model is functional ly equivalent to the genetic toggle switch which was syntetized by the authors of the recent paper [4]. In that pap er the network is studied experimentally in a single ”cell”. It would be interesting to generalize the experimen t to a ”multicellular” structure. The toy model For the sake of definiteness we take a genetic network with two genesAandB.AandBare mutual repressors. The network is symmetric under exchange A↔B. Expression of both genes is initiated by a common activator a. Let A(t,/vector x) andB(t,/vector x) denote time-dependent protein concentrations in the cell at the position /vector x./vector xbelongs to a discrete 1square lattice with a lattice constant of 1. Evolution of the protein concentrations is described by the stochastic differential equations ˙A(t,/vector x) =RSA(t,/vector x)−A(t,/vector x), (1) ˙B(t,/vector x) =RSB(t,/vector x)−B(t,/vector x). (2) The last terms in these equations are responsible for the pro tein degradation. Ris a transcription rate. SA,B(t,/vector x)∈ {0,1}are dichotomic stochastic processes. They switch on (0 →1) and off (1 →0) transcription of a given gene. For simplicity the processes are assumed to have the same con stant switch-off rate roff. The switch-on rates depend on concentrations ron A(t,/vector x) =a(t)F −W B(t,/vector x) +V/summationdisplay n.n./vector yA(t,/vector y) , (3) ron B(t,/vector x) =a(t)F −W A(t,/vector x) +V/summationdisplay n.n./vector yB(t,/vector y) . (4) W,V are positive coupling constants, a(t) is a concentration of the activator. F[z] is a smooth step-like sigmoidal function; the function F[z] = 103exp(z−2.2)/[1+exp(z−2.2)] was used in our numerical simulations. In this model the genes A and B are mutual repressors ( W > 0). There is a ”ferromagnetic” coupling between nearest-ne ighbor cells (V >0); expression of Ain a given cell enhances expression of Ain its nearest neighbors. The model is motivated by a genetic switch between two mutual repressors like the one studied in the phage λ[3] and in the E. coli switch [4]. The mutual repressors have a common promoter sit e on DNA. A necessary condition for expression of any of them is a binding of an activator molecul e to their promoter site [5]. The concentrations Aand Binfluence its affinity to the promoter site. The gene expressio n is intermittent because of binding and unbinding of activator molecules. The nearest-neighbor coupling is pos sible thanks to signalling through intercellular membrane channels. In an adiabatic limit, when switching of SA,Bis much faster than protein expression and degradation, the processes SA,Bcan be replaced by their time averages, ˙A(t,/vector x) =Ra(t)F/bracketleftBig −W B(t,/vector x) +V/summationtext n.n./vector yA(t,/vector y)/bracketrightBig roff+a(t)F/bracketleftBig −W B(t,/vector x) +V/summationtext n.n./vector yA(t,/vector y)/bracketrightBig−A(t,/vector x), (5) ˙B(t,/vector x) =Ra(t)F/bracketleftBig −W A(t,/vector x) +V/summationtext n.n./vector yB(t,/vector y)/bracketrightBig roff+a(t)F/bracketleftBig −W A(t,/vector x) +V/summationtext n.n./vector yB(t,/vector y)/bracketrightBig−B(t,/vector x). (6) Here we temporarily neglect any noise terms. Attractor structure In a subspace of uniform configurations A(t),B(t) these equations simplify to the dynamical system ˙A=RaF[−W B+ 2dV A] roff+aF[−W B+ 2dV A]−A , (7) ˙B=RaF[−W A+ 2dV B] roff+aF[−W A+ 2dV B]−B , (8) where 2dis the number of nearest neighbors in ddimensions. The RHS’s of these equations define a velocity field on the A−Bplane, which is not a gradient field. The velocity field has attractor structure which depends on the a ctivator level a. There are two critical activator levels ac1< ac2. Fora < ac1there is one attractor at [ A,B] = [γ(a),γ(a)] with an increasing function γ(a). In the range ac1<a<ac2there are three attractors: the old [ γ(a),γ(a)] plus a new symmetric pair of [ α(a),β(a)] and [β(a),α(a)] withα(a)> β(a). Forac2< athere remain only the two broken symmetry attractors [ α(a),β(a)] and [β(a),α(a)]. The functions α(a),β(a) andγ(a) are plotted in Fig.1. 2If we start in the [ A,B] = [0,0] state and slowly increase a-level, the system will stay in the γγ-phase until we reacha=ac2. Ata=a+ c2the system will roll into αβorβα-phase. On the other hand, if we start from ac2<awith the system in, say, αβ-phase, then we will have to decrease adown toa=ac1, whereαβbecomes unstable towards the symmetric γγ-phase. The discontinuous jumps of the concentrations are i llustrated in Fig.1. This hysteresis loop is characteristic for first order phase transitions. In the a diabatic limit, where fluctuations are small, there are no short cuts via bubble nucleation. When ac1(ac2) is approached from above (below), the correlation length o f small fluctuations around this uniform state diverges like in a con tinuous phase transition. The critical regime is narrow in the adiabatic limit so we can rely on the mean field approximat ion. A finite rate Turing transition Let us think again about starting from [ A,B] = [0,0] and continuously increasing a(t) aboveac2. Ata+ c2theγγ state becomes unstable and the system has to choose between t heαβandβαattractors. If a(t) is increased at a finite rate, then there are finite correlated domains which ma ke the choice independently. Despite divergence of the correlation length at a− c2, the critical slowing down results in a certain finite correl ation length ˆξ”frozen” into the fluctuations. This scale defines density of defects in the Tur ing pattern. This effect is well known in cosmology and condensed matter physics as Kibble-Zurek scenario [6]. In t hose contexts the defects disappear rapidly as a result of phase ordering kinetics. We will see that in our gene netwo rk model the defect pattern is permanent. This effect results from a combination of the histeresis loop and the dis creteness of the cell lattice. To be more quantitative we substitute A(t,/vector x) =γ(a(t)) +δA(t,/vector x) andB(t,/vector x) =γ(a(t)) +δB(t,/vector x) into Eqs.(5) and linearize them in δA,δB . The linearized equations can be diagonalized by φ=δA−δBandψ=δA+δB. After Fourier transformation in space φ(t,/vector x) =/integraldisplay ddk˜φ(t,/vectork)ei/vectork/vector x(9) they become ˙φ(t,/vectork) =Rsφ(t,/vectork) +roffRa(t)F′ a [roff+a(t)Fa]2/bracketleftBig W φ(t,/vectork) +V e/vectorkφ(t,/vectork)/bracketrightBig −φ(t,/vectork), (10) ˙ψ(t,/vectork) =Rsψ(t,/vectork) +roffRa(t)F′ a [roff+a(t)Fa]2/bracketleftBig −W ψ(t,/vectork) +V e/vectorkψ(t,/vectork)/bracketrightBig −ψ(t,/vectork), (11) wheree/vectork= 2/summationtextd i=1coskiinddimensions and we skipped the tildas over Fourier transform s.F′[z] =dF[z]/dzand we used the shorthands F(′) a=F(′)[(−W+ 2dV)γ(a(t))].Rsφ,ψare noises which result from fluctuations in RSA,B. In the adiabatic limit they can be approximated by white nois es (both in space and in time) with small magnitude. The next step is to linearize a(t) around its critical value a(t) =ac2+t/τ, whereτis the transition rate. This linearization gives roffRa(t)F′ a [roff+a(t)Fa]2=c0+c1t τ+O[(t/τ)2]. (12) Approximating e/vectork= 2d−/vectork2in Eqs.(10,11) and keeping only leading terms in t/τand ink2we get ˙φ(t,/vectork) =Rsφ(t,/vectork) +/bracketleftbigg (c1 c0)t τ−(c0V)/vectork2/bracketrightbigg φ(t,/vectork), (13) ˙ψ(t,/vectork) =Rsψ(t,/vectork)−[2c0W+c0V/vectork2]ψ(t,/vectork). (14) Here we used the identity c0[W+ 2dV] = 1, which has to be satisfied because, by definition, φ(t,/vector0) is a zero mode atac2. Theψmodes are stable for any /vectork. Theφ-modes in the neighborhood of /vectork=/vector0 become unstable for t>0 (or ac2<a). Eq.(13) is a standard linearized Landau model with the sym metry breaking parameter ( c1/c0)(t/τ) changing sign att= 0. The length scale ˆξfrozen into fluctuations at t >0 can be estimated following the classic argument given by Zurek [6]. For t <0 the model (13) has an instantaneous relaxation time c0τ/c1|t|and an instantaneous correlation length c0/radicalbig Vτ/c 1|t|. They both diverge at t= 0−. The fluctuations can no longer follow the increasing a(t) when their relaxation time becomes equal to the time still r emaining to the transition at a=ac2,c0τ/c1|t| ≈ |t|. At this instant the correlation length is 3ˆξ≈/parenleftBigg V1/2c3/4 0 c1/4 1/parenrightBigg τ1/4. (15) This scale determines the typical size of the αβ- andβα-domains. The scaling relation ˆξ∼τ1/4was verified by numerical simulations illustrated at figures 2 and 3. The dom ain structures generated in the simulations turned out to be permanent. The domain structures are permanent because already at ac2the width of the domain wall interpolating between αβandβαis less then the cell size (lattice spacing). The nearest nei ghbor cells across the wall express different genes. The width (the healing length) is determined by the longest l ength scale of fluctuations around the αβ- orβα-state. These correlation lengths are plotted in Fig.4. For a≥ac2they are substantially less than 1. In the adiabatic limit, where the noises are weak, the domain wall cannot evolve beca use it would have to overcome a prohibitive potential barrier. On a cellular level the barrier originates from the mutual inhibition between AandBin a single cell. Roughly speaking, much above ac1each cell is locked in its gene expression state and insensit ive to its nearest neighbors’ states. Inhomogeneous activation The intracellular mutual inhibition stabilizes the Turing pattern but it also stabilizes the defects frozen into the pattern. With the ˆξ∼τ1/4scaling the number of defects is rather weakly dependent on τ. There may be not enough time during morphogenesis to get rid of the defects by simply increasingτ. However, it is possible to generate a defect-free pattern by spatially inhomogeneous switching of the activator level a. For example, its concentration can exceedac2at one point at first, where the cells happen to pick (or are for ced to pick), say, αβ-phase, and then the activator can gradually spread around so that the initial se ed ofαβ-cells gradually imposes their choice on the whole system. For continuous transitions this effect was describe d in Ref.( [7]). The effect of defect suppression in inhomogeneous activatio n can be most easily studied in a one dimensional version of the model (1). Suppose that a smooth activator front is mov ing across the one dimensional chain of cells with a velocityv,a(t,x)≈ac2+ (vt−x)/vτclose tox=vtwherea=ac2. For definiteness we impose two asymptotic conditions: for vt≪x( wherea < ac2) the cells are in the γγ-state, and for x≪vt(wherea > ac2) they are in αβ-phase. We can expect that as the a-front moves to the right it is followed by the αβfront gradually entering the area formerly occupied by the γγ-phase. If the concentration front is fast enough to move in s tep with the activator front, then the αβ-phase will gradually fill the whole system. If, on the other h and, the concentration front is slower than the activator front then the front of the αβ-phase will lag behind the a=ac2front. The gap between the two fronts will grow with time. The gap will be filled with the unst ableγγ-phase (a > ac2behind the a-front). When the gap becomes wide enough, then γγ-state will be able to decay towards the βα-state. A domain of βα-phase will eventually be nucleated behind the a-front. Now the βα-domain will grow behind the a-front until its front lags sufficiently behind so that a new domain of αβ-phase will be nucleated. In this way the activator front wil l leave behind a landscape of alternating αβ- andβα-domains qualitatively the same as for homogeneous activat ion. The success of the inhomogeneous activation depends on the r elation between the velocity vof thea-front and that of the concentration front. As illustrated in Fig.4 fluctuat ions around the αβ-state have two families of modes each with a different correlation length. For any aeach/vectork-mode within each family has a different diffusion velocity: a ratio of its wavelength to its relaxation time. The lowest of these diffusion velocities, vc(a), is the maximal velocity at which theαβ-phase can spread into the area occupied by the γγ-phase.vc(ac2)≡vc2>0 because at a=ac2theαβ-state is stable (the hysteresis loop again!). vc(a) increases with an increasing a. Ifv<vc2theαβ-front moves in step with the a-front; its tail spreads into the vt<x area imposing an αβ-bias on the fluctuations around γγ-state. The αβ-phase spreads without nucleation of any βα-domains. For v < vc2a defect-free uniform Turing pattern forms behind the activator front. Results from numerical simulations of the inhomogeneous activation are presented in Fig.5. More complicated patterns Finally, it is time to comment on more complicated models whi ch are expected to give more complicated patterns than the (in principle) uniform pattern discussed so far. Le t us pick a zebra pattern for example. For the uniform pattern the first mode to become unstable in Eq.(13) is the /vectork=/vector0 mode. The final pattern has an admixture of /vectork’s in a range ≈ˆξ−1around/vectork=/vector0. In distinction, for the zebra pattern the first unstable mo des are those on the circle|/vectork|= 2π/L, whereLis the spacing between zebra stripes. The final pattern has an admixture of /vectork’s in a ring of thickness ≈ˆξ−1around the circle |/vectork|= 2π/L, compare results for Swift-Hohenberg equation in Ref. [8]. This 4admixture results in defects frozen into zebra pattern. The inhomogeneous activation can be applied in the zebra case too. In addition it can be used to arrange the stripes. An acti vator spreading from an initial point would result (at least close to the initial point) in concentric black and whi te rings. A front of activator moving through the system would comb the stripes perpendicular to the front. Acknowledgements. I would like to thank M.Sadzikowski and W.Zurek for useful co mments on the manuscript. [1] A.M. Turing, Phil.Trans.Roy.Soc.Lond. B237 , 37 (1952). [2] A.J. Koch and H. Meinhardt, Rev.Mod.Phys. 66, 1481(1994). [3] M. Ptashne, A Genetic Switch: Phage and Higher Organisms , Blackwell Science Inc. 1992; A.Arkin, J.Ross, and H.H.McAdams, Genetics 149,1633 (1998). [4] T.S.Gardner, C.R.Cantor, and J.J.Collins, Nature 403, 339 (2000). [5] M.S.H. Ko, J.Theor.Biol. 153, 181 (1991). [6] T.W.B. Kibble, Phys.Rep. 67, 183 (1980); W.H.Zurek, Phys.Rep. 267, 177 (1996). [7] J. Dziarmaga, P.Laguna, and W.H. Zurek, Phys.Rev.Lett. 82, 4749 (1999); N.B. Kopnin and E.V.Thuneberg, Phys.Rev.Let t. 83, 116 (1999). [8] G. Lythe, Phys.Rev.E E53, R4271 (1996). 0.32 0.34 0.36 0.38 0.4a0.20.40.60.81A,B A BA B FIG. 1. The thick lines are: α(a) (top), γ(a) (middle), β(a) (bottom). The vertical lines with arrows illustrate the di scon- tinuous jumps by the concentrations AandBduring the αβ→γγtransition at ac1≈0.34, and the γγ→αβtransition at ac2≈0.37. Model parameters used in this graph are: R= 4, W= 3, V= 1, d= 2, roff= 103. 50 5 10 15 20 25 30051015202530 FIG. 2. Permanent pattern obtained after switching-on the a ctivator on a 32 ×32 periodic lattice. It is a contour plot of A−B; white is A-rich ( αβ) and black is B-rich ( βα). The activator was turned on as a(t) =t/τwithτ= 32 and t∈(0,32). Model parameters were the same as in Fig.1. A discrete time st ep was ∆ t= 10−4. 1 1.5 2 2.5 3 3.51.21.31.41.51.61.7 FIG. 3. log( ˆξ) as a function of log( τ).ˆξwas obtained as an average domain size along a cross section t hrough patterns like that in Fig.2. For any given τthe average was taken over outcomes of many simulations and o ver all the possible vertical and horizontal cross sections. The vertical point size is a trip le standard deviation. The simulations were done on a 1024 ×1024 lattice. The slope was fitted as 0 .24±0.02, which is consistent with the predicted 0 .25. 60.35 0.375 0.4 0.425 0.45 0.475 0.500.511.5 FIG. 4. The correlation lengths of the fluctuations around th e state αβas functions of a. The vertical gridlines mark ac1≈0.332 and ac2≈0.375. The larger correlation length diverges at ac1. These correlation lengths should be compared with the lattice spacing which is 1. The correlation lengths were obtained by expanding A(t,/vector x) =α(a) +δA(t,/vector x) and B(t,/vector x) =β(a) +δB(t,/vector x), Fourier-transforming the fluctuations in space and subse quent diagonalization for small k. 1 2 3 4 5v0.020.040.060.080.1n FIG. 5. Density nof domain walls between αβandβα-states behind an activator front with velocity v. The activator was a(t,x) = (vt−x)/vτforx < vt anda(t,x) = 0 for vt < x . 1/vτ= 0.1 was kept fixed so that the slope of aversus xwas independent of v. The model parameters were the same as in Figs.1,2 but with d= 1 instead of 2 and V= 2 instead of 1 (V d= 2 as before). For these model parameters vc≈0.9 in consistency with the numerical results. 7

arXiv:physics/0002051v1 [physics.atm-clus] 29 Feb 2000Thermal expansion in small metal clusters and its impact on t he electric polarizability S. K¨ ummel1, J. Akola2, and M. Manninen2 1Institute for Theoretical Physics, University of Regensbu rg, D-93040 Regensburg, Germany 2Department of Physics, University of Jyv¨ askyl¨ a, P.O. Box 35, FIN-40351 Jyv¨ askyl¨ a, Finland (February 21, 2014) The thermal expansion coefficients of Na Nclusters with 8≤N≤40 and Al 7, Al− 13and Al− 14are obtained from ab ini- tioBorn-Oppenheimer LDA molecular dynamics. Thermal expansion of small metal clusters is considerably larger th an that in the bulk and size-dependent. We demonstrate that the average static electric dipole polarizability of Na clu sters depends linearly on the mean interatomic distance and only to a minor extent on the detailed ionic configuration when the overall shape of the electron density is enforced by electro nic shell effects. The polarizability is thus a sensitive indica tor for thermal expansion. We show that taking this effect into account brings theoretical and experimental polarizabili ties into quantitative agreement. PACS: 36.40.Cg, 65.70.+y, 33.15.Kr Since electronic shell effects were put into evidence in small metallic systems [1–4], metal clusters have conti- nously attracted great interest both experimentally and theoretically [5–10]. Besides technological prospects, o ne of the driving forces for this research has been the funda- mental question of how matter develops from the atom to systems of increasing size, and how properties change in the course of this growing process. In some cases it has been possible to extract detailed information from ex- periments done at low temperatures [11] and the related theories [12]. In many cases, however, a deeper under- standing is complicated by the finite temperature which is present in most experiments due to the cluster pro- duction process, see e.g. the discussion in [13]. Whereas a lot of theoretical information about finite temperature effects in nonmetallic systems has been gained in the last years [14], only little is known about it in metallic clus- ters. Here, sodium is a particularly interesting reference system because of its textbook metallic properties and the fact that it has been extensively studied within the jellium model, see e.g. [15] for an overview. Aluminum, on the other hand, is of considerable technological inter- est. Some advances to study temperature effects in metal clusters including the ionic degrees of freedom were made using phenomenological molecular dynamics [16], a tight- binding hamiltonian [17], the Thomas-Fermi approxima- tion [18] or the Car-Parrinello method [19]. Recently, it has also become possible to study sodium clusters of con- siderable size [20] using ab initio Born-Oppenheimer, lo- cal spin density molecular dynamics (BO-LSD-MD) [21]. In this work we report on the size dependence of a thermal property which is well known for bulk systems,namely the linear thermal expansion coefficient β=1 l∂l ∂T. (1) For crystalline sodium at room temperature, it takes [22] the value 71 ×10−6K−1, for Al 23 .6×10−6K−1. To the present date, however, it has not been known how small systems are affected by thermal expansion. At first sight, it is not even obvious how thermal expansion can be defined in small clusters. Whereas in the bulk it is no problem to define the length lappearing in Eq. (1), e.g. the lattice constant, it is less straightforward to choose a meaningful lin the case where many different ionic ge- ometries must be compared to one another. For small metal clusters, the latter situation arises because of the many different isomers which appear at elevated temper- atures. We have calculated the thermal expansion coefficients for Na 8, Na10, Na12, Na14, Na20and Na 40in BO-LSD- MD simulations. Results concerning isomerization pro- cesses in these simulations have been presented in [23], and the BO-LSD-MD method is described in detail in Ref. [21]. A meaningful length to be used in Eq. (1) if it is applied to finite systems with similar overall deforma- tion is the mean interatomic distance lmiad=1 N(N−1)N/summationdisplay i,j=1|Ri−Rj|, (2) where Riare the positions of the Natoms in the clus- ter. Obviously, lmiadmeasures the average “extension” of a clusters ionic structure, and we calculated it for all configurations obtained in a BO-LSD-MD run. Two dif- ferent methods were used to calculate β. First, we discuss the heating runs, in which the clusters were thermalized to a starting temperature and then heated linearly with a heating rate of 5K/ps and a time step of 5.2 fms. lmiad was recorded after each time step. In this way, for Na 8 the temperature range from about 50 K to 670 K was covered, corresponding to 24140 configurations, for Na 10 from ca. 150 K to 390 K (9260 configurations), for Na 14 from ca. 50 K to 490 K (17020 configurations), for Na 20 from ca. 170 K to 380 K (8000 configurations), and for Na40from ca. 200 K to 400 K (7770 configurations). Fig. 1 shows how lmiadchanges with temperature for Na8and Na 10. Both curves show large fluctuations, as is to be expected for such small systems. However, one clearly sees a linear rise as the general trend. We there- fore made linear fits to the data for each cluster in two 17.27.47.67.888.28.48.68.899.2 0100200 300 4005006007009.49.6lmiada0 T / KNa10 Na 8 FIG. 1. Mean interatomic distance in a0versus tempera- ture in K for Na 8and Na 10. The dashed lines indicate linear fits to the complete set of data, see text for discussion. Note the different slopes for the two clusters. ways. The first column in the left half of table I gives the linear thermal expansion coefficients which we obtained from fitting the data in the temperature interval between 200 K and 350 K, i.e. around room temperature, where bulk sodium is usually studied. In order to allow for an estimate of the statistical quality of the fits in view of the fluctuations, the second and third column in the left half of Table I list the ratio of the fit parameters, i.e. the axis interception aand the slope b, to their stan- dard deviations. It becomes clear from these results that thermal expansion in the small clusters is considerably larger than that in the bulk. This can be understood as an effect of the increased surface to volume ratio in the finite systems. However, the expansion coefficient also strongly depends on the cluster size. This can even be seen directly from the different slopes in Fig. 1. As we will show below, this size dependence has far reaching consequences for the interpretation of experimental data which is usually measured on hot clusters, as e.g. the static electric polarizability. In addition to the values given in Table I, we calcu- lated the expansion coefficient of Na 12with a different method. In two separate runs, the cluster was thermal- ized to temperatures of about 200 K and 350 K, and then BO-LSD-MD was performed for 5 ps at each tem- perature, i.e. without heating. From the average lmiad found in the two simulations, βNa12= 2.5βbulkwas cal- culated. Thus, also the second method leads to a βthat is larger than that of the bulk, i.e. it confirms the results of the heating runs. The average thermal expansion coefficient for the fullβ/βbulk σ(a)/aσ(b)/bβ/βbulk σ(a)/aσ(b)/b Na8 2.4 0.001 0.04 1.7 <0.001 0.01 Na10 3.6 0.002 0.03 2.8 0.001 0.02 Na14 1.2 0.002 0.07 1.7 <0.001 0.01 Na20 1.9 0.001 0.03 1.9 0.001 0.01 Na40 - - - 1.2 0.001 0.04 TABLE I. Left half, first column: Linear thermal expan- sion coefficient of small Na clusters in the temperature inter - val between 200 and 350 K, given in terms of the bulk value 71×10−6K−1. Columns two and three give the ratio of the axis interception aand the slope bto their standard deviations as obtained from the fits. Right half: Expansion coefficient averaged over 50-670 K for Na 8, 150-390 K for Na 10, 50-490 K for Na 14, 150-460 K for Na 20, and 200-300 K for Na 40. See text for discussion. temperature range covered in each simulation is obtained from a fit to the complete set of data, shown as a dashed line in Fig. 1 for Na 8and Na 10. This average is of inter- est because it covers several hundred K for each cluster in the range of temperatures which are to be expected for clusters coming from the usual supersonic expansion sources [24]. The right half of table I lists these average expansion coefficients and their statistical deviations in the same way as before. As is to be expected, the val- ues differ from the previous ones for the small clusters, because the expansion coefficient is influenced by which isomers are or become accessible at a particular tempera- ture, i.e. especially at low temperatures it is temperature dependent. In Fig. 1 one e.g. sees from comparison with the average dashed line that for temperatures between 50 K and 100 K, the thermal expansion is smaller than that seen for higher temperatures. However, once the cluster has reached a temperature where it easily changes from one isomer to another, the thermal expansion coefficient becomes nearly independent of the temperature. In the case of Na 8, e.g., βchanges only by about 5 % in the interval between 300 K and 670 K. Detailed previous investigations [20,23] have shown that small clusters do not show a distinct melting tran- sition. However, the largest cluster studied here, Na 40, shows a phase transition above 300 K [20]. At the melt- ing point, the octupole and hexadecupole deformation of the electronic density sharply increase. If lmiadis a rele- vant indicator for structural changes, then melting should also be detectable from it. Indeed we find a noticeable increase in lmiadat 300 K, and similar fluctuation pat- terns as in the multipole moments. In our simulation, we could only determine the expansion coefficient for the solid phase, and it is given in the right half of table I. As seen in Fig. 1, Na 8shows thermal expansion already at 50 K. This raises the question at which temperature the expansion actually starts, i.e. where anharmonic ef- fects in the ionic oscillations will start to become impor- tant. In this context we note that one can compare the lmiadat T=0 K found by extrapolation from the heating 2data to the lmiadwhich is actually found for the ground state structure at T=0 K. We have done this for Na 8, Na10and Na 14, where the ground state structures are well established. In all cases, the differences between the two values were less than 1%. This indicates that the an- harmonic effects for Na clusters are important down to very low temperatures. Furthermore, the anharmonici- ties should also be observable in the heat capacities [20], where they will lead to deviations from Dulong-Petit’s law. We have checked this and indeed found deviations between 8 % (Na 20) and 19 % (Na 8) from the Dulong- Petit value. As an example for the considerable influence of ther- mal expansion on measurable physical properties we dis- cuss the average static electric dipole polarizability α, which is defined as the trace of the polarizability ten- sor. It was one of the first observables from which the existence of electronic shell effects in metal clusters was deduced [1], and it has been measured for clusters of var- ious sizes and materials [10]. For Na clusters with up to eight atoms, the polarizability was also calculated in different approaches [5–7,9,10]. These calculations qual- itatively reproduce the experimentaly observed trends, but they all underestimate the measured value. We show that this discrepancy is to a large part due to the fact that the calculations were done for T=0, whereas the measurement is done on clusters having temperatures of about 400 to 600 K [24]. For various, different isomers obtained in our heating runs for Na 8and Na 10, we have calculated the polariz- ability from the derivative of the induced dipole moment with respect to the electric field (finite field method). Since highly unsymmetric isomers from the high temper- ature part of the simulations were taken into account, the full tensor was computed by numerically applying the dipole field in the different directions in seperate cal- culations. We have checked that the used field strength of 5×10−5e/a2 0is large enough to give a numerically stable signal and small enough to be in the regime of linear response. In Fig. 2 we have plotted the thus ob- tained polarizabilities versus lmiad, and show three in- stances of ionic geometries for each cluster that demon- strate how different the structures actually are. Never- theless, within a few percent the polarizabilities are on a straight line. This shows that the average polarizability depends mainly and strongly on the mean interatomic distance, and only to a minor extent on details in the ionic configurations. Of course, the situation might be more complicated for clusters where the overal shape, i.e. the lowest terms in the multipole expansion of the valence electron density, is not stabilized by electronic shell effects. For the present clusters, however, the de- formation induced by the electronic shell effects persists even at elevated temperatures. That αis less sensitive to the detailed ionic configuration than, e.g., the pho- toabsorption spectrum, is understandable because it is an average quantity. The dependence of the polarizability on the meanα Å3 100105110115120125130135140145150 7.47.67.888.28.48.68.89 lmiad/a0 α Å3 130135140145150155160165170175180 8.48.68.8 99.29.49.6lmiad/a0 FIG. 2. Static electric dipole polarizability versus mean i n- teratomic distance for different isomers of Na 8(upper) and Na10(lower). Three examples of different geometries are shown as insets for both sizes. interatomic distance has the consequence that αalso strongly depends on the temperature. From Fig. 2 one deduces that an average bondlength increase of 1 a0in Na8and Na 10leads to an increase in the polarizability of about 25 ˚A3. Thus, neglection of the thermal expan- sion in T=0 calculations leads to polarizabilities which are smaller than the ones measured on clusters coming from supersonic expansion sources [1,10]. Of course, also underestimations of the cluster bond lengths that are due to other reasons will directly appear in the polarizabil- ity. With the Troullier-Martins pseudopotential, e.g. the BO-LSD-MD underestimates the dimer bond length by 4.5%, and it is to be expected that the situation is similar for the bond lengths of larger clusters. Taking this into account, one can proceed to calculate the polarizability for clusters with a temperature corresponding to the ex- perimental one of about 500 K [24]. In the experiments the clusters are spending about 10−4s in the deflecting field from which the polarizability is deduced, i.e. the ex- perimental timescale is orders of magnitude larger than the timescale of the fluctuations in the mean interatomic 3distance (see Fig. 1). Thus, the fluctuations will be av- eraged over and can be neglected. From the average ex- pansion coefficients we obtain a bond length increase of 0.48a0for Na 8and 0.87 a0for Na 10at 500 K, which in turn leads to an increase in the polarizability of 12 ˚A3 and 23 ˚A3, respectively. The resulting polarizabilities of 130˚A3for Na 8and 172 ˚A3for Na 10compare favourably with the experimental values 134 ±16˚A3and 190 ±20˚A3 [1,10]. For all other cluster sizes, the two experiments [1,10] give different values for the polarizability. From th e present work it becomes clear that differences in the ex- perimental temperatures might be the reason for the dis- crepancies. Therefore, an accurate measurement of the clusters’ temperatures is necessary before further quan- titative comparisons can be made. However, a detailed comparison to both experiments showed that the theo- retical T=0 polarizability of all isomers underestimates both experimental results [25]. Thus, the increase in α that is brought about by thermal expansion will lead to better agreement between theory and experiment for all cluster sizes. Thermal expansion is also observed in aluminum clus- ters. For Al 7we performed 5 ps of BO-LSD-MD at each of the fixed temperatures 100 K, 300 K, 500 K and 600 K, for Al− 13at 260 K, 570 K and 930 K, and for Al− 14at 200 K, 570 K and 900 K, in analogy to the procedure for Na 12. From the average lmiadat each temperature, we calculated the expansion coefficients βAl7= 1.3βbulk, βAl− 13= 1.4βbulk,βAl− 14= 1.4βbulk. It should be noted that with Al− 13we have chosen an electronically as well as geometrically magic cluster [26], i.e. a particularly rigi d one, and the fact that it also shows a larger expansion coefficient than the bulk is further evidence for the con- clusion that the increased expansion coefficient is indeed a finite size effects. A noteworthy difference between Al and Na is seen in the temperatures where the expansion sets in. Whereas for Na this temperature is below 50 K, we observe that Al− 13and Al− 14show no expansion below 300 K. In summary, we have calculated thermal expansion co- efficients for small metal cluster and demonstrated that thermal expansion in these systems is larger than that in the bulk. For the case of sodium, the dependence of the expansion coefficient is not monotonous according to the cluster size. We showed that the average static elec- tric dipole polarizability of clusters whose overall shape is fixed by electronic shell effects depends linearly on the mean interatomic distance. Thus, thermal expansion in- creases the static electric polarizability, and we demon- strated that taking this effect into account brings the theoretical values in a close agreement with the experi- mental ones. We thank M. Brack and A. Rytk¨ onen for clarifying discussions. J.A. acknowledges support by the V¨ ais¨ al¨ a Foundation, S.K. by the Deutsche Forschungsgemein- schaft, and all authors by the Academy of Finland.[1] W. D. Knight et al., Phys. Rev. B 31, (1985) 2539. [2] W. Ekardt, Phys. Rev. Lett. 52, (1984) 1925. [3] D. E. Beck, Phys. Rev. B 30, 6935 (1984). [4] M. Manninen, Phys. Rev. B 34, 6886 (1986). [5] I. Moullet et al., Phys. Rev. B 42, 11589 (1990). [6] J. Guan et al., Phys. Rev. B 52, 2184 (1995). [7] A. Rubio et al., Phys. Rev. Lett. 77, 247 (1996). [8] C. A. Ullrich, P.-G. 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1Enhanced photoinduced nematic reorientation in mixture with azo-dye-substituted polymer. A. Parfenov a) *, A. Chrzanowska *Diagnostic Instrumentation and Analysis Laboratory, 205 Research Blvd, Starkv ille, MS 39759, USA; Krakow University of Technology Institute of Physics, Podchorazych 1, Krakow 30- 084, Poland The optically induced reorientation of nematic liquid crystal doped with azo-dye substituted polymer is investigated measuring photoinduced birefringence. These measurement reveal the value (Δn~0.1) of induced birefringence of liquid crystal with dye polymer which significantly exceeds the value of birefringence previously obtained in nematic mixture with the low molecular weightdye. More than one order of enhancement is connected with lower diffusivity of polymer. PACS numbers: 42.70.Df, 42.70.Gl, 78.20.Fm a) on leave from Lebedev Physics Institute, Russia, electronic mail: parfenov@neur.lpi.msk.su2The great interest in light-induced reorientation of nematic liquid crystals 1 is closely connected with large potential of this effect in such applications as nonlinear optics, photonics andoptical data processing 2. Optical reorientation is greatly enhanced when the liquid crystal is doped with a small value of light -absorbing dye 2- 4. It is also observed that this specific effect is strongly sensitive to the molecular structure of the guest-host, and mainly to dye structure 5. It is established that diffusion of dye molecules in liquid crystal host has great influence on the observed reaction 6, 7. Recently the molecular theory 7 of the resonant and viscous torques in trans-cis systems with nematic liquid crystals has explained the main dependencies in terms of diffusional motion andhydrodynamic reorientation of molecules in a three component mixture ( trans-, cis- conformers and nematic liquid crystal molecules). We report the enhanced optically induced reorientation in liquid crystal doped with dye - substituted polymer and corresponding induced birefringence. The experiments are performed witha homeotropic LC cell, containing nematic 5CB from Merck (birefringence Δn=0.2) and azobenzene substituted polymer poly-Orange Tom-1 Isophoronedisocyanate. The LC - polymermixture contains 3-5 –wt.% of the latter substance. The molar part of azobenzene in this polymeris estimated as ca 50% in weight. The spacers fixing the thickness of this cell L are of 34 µm. Induced birefringence is observed in a polarised probe beam of a diode laser. The absorptionspectra for this polymers presented in 8 as verified coincides with the spectra of its solution in the LC (maximum absorption density achieved 1.8-2.0 in the solution). Wavelength of laser dioderadiation (630 nm) is far from absorption band of polymer solution. The optical response of themedia is measured under exposure by linearly polarised light of the wavelength 488 nm locatedwithin the absorption band of the azobenzene group. The diameter of the illuminated area and lightintensity are of 4-9 mm and of 2-7 mW/cm 2 respectively. The polarization plane of diode laser3radiation is set at 45 degrees to the green beam polarization plane (polarization planes are denoted in Fig.1 by arrows). The LC cell transmission for the λ=630 nm light is observed between polarisers. The change of the transmission is attributed to the birefringence arising due to the declination of nematic LC from the homeotropic alignment at the angle θ. This angle determines the induced birefringence as follow 9 : oe o n n n n − + =−2/12 2 2 2)/sin/(cos)( θ θ θΔ ,( 1 ) with onand en, refractive indices for ordinary and extraordinary optical beams in liquid crystal media. The cell transmission in birefringence regime is determined by induced phase shift ΔΦ=2π Δn( θ)L/λ between two polarization components (chosen at 45 degrees to the polarizer axis): I=I0 T0 sin2(ΔΦ /2)+I scat, (2) with I0, intensity of incident light; T 0, transparency of polarizer for light polarized at 0 degrees; I scat, light intensity scattered and partially depolarized by LC cell. The change of the LC cell transmission (Equation 2) is measured in a simple scheme (optical scheme of the experiment is presented in Fig.1), which included the LC cell itself, green Ar laser(488 nm) for excitation and red diode laser (630 nm). A photodiode registers the transmitted powerof the red beam. The signal from the photodiode is recorded on a plotter with swiping speed 1-6cm/min, well suitable for realized response time. The excitation beam is incident almost normally,red readout beam - at a small angle of 5 degrees.4The induced birefringence Δn(θ)L is determined from induced phase shift π⋅Δn(θ)L /λ ~ kπ, where k is a number of the specific oscillations (shown in the inset in Fig.1) appeared at ascend anddecay of the optical response activated by a pulse of green light. The rise and decay of the inducedphase shift can be well fitted by exponential dependencies of a time (Fig.2). The maximum value of induced birefringence Δn(θ) is estimated as 0.08-0.10 for light intensity 5 mW/cm2. This value corresponds to declination from the initial state at the angel θ ~45 degrees (determined by means of Equation 1 as a value averaged over the LC layer thickness). The LC texture created under theuniform illumination is not quite flat. Considerable part of the light power (up to 30% in somecases) is scattered; thereby diminishing the contrast of mentioned oscillations. An aggregation ofpolymer in a solution could be a reason for irregularity leading to this scattering. Known results 10 are giving much lower induced birefringence ~0.005 for LC doped with azobenzene D2 dye in the comparable conditions (LC cell thickness ~ 37.5 µm, only 2-3x larger light intensity ~ 10 mW/cm 2). The rise/decay time published is of 1 sec. The time constant of visco –elastic relaxation [9] of LC is KL/~2γτ , with γ, viscosity; K, elastic coefficient of LC and L- its thickness. With the given values of thickness and material constants it is of order of 1 second as in mentioned experiments with monomers 10. In our experiments the process of reorientation is characterized by the rise/decay times of 40 seconds andthus cannot be explained exclusively by visco-elastic relaxation. Dependencies of rise/decay timeon the light intensity are presented in Fig.3 as well as the maximum phase shift achievable forgiven light intensity determined from specific oscillations at the optical response as mentionedabove. The polymer dye in LC imposes longer excitation relaxation than in a case of monomer dyeadditive (40 sec compared to 1 sec in 10), but it is much shorter than for light induced reorientation in azobenzene-containing polymer liquid crystals 11. The measurements in the lower intensity5region have obstacles in slowing of the LC - mixture response, while at higher intensities (5-6 mW/cm2) the visibility of oscillations at the front of response becomes inferior. The observed difference explanation is probably concluded in a role of polymer binding the dye molecules. According to the theory 7 the resonant contribution in optical torque is strongly dependent on the diffusion coefficients of trans and cis conformers of dye molecules (and their difference). At the same time the doping dye molecules can be introduced in the LC material indifferent forms- as a stand-alone monomer or in the polymer compound as in our experiments. Inthe last case the dye group are included as substitute side-groups. These groups beingcomparatively weakly bound with the main polymer chain retain much of the optical properties ofgenuine dye molecules, including absorption strength of oscillator for resonant wavelength. Theazobenzene inserted in LC in the same molar portion as a monomer and as a polymer substituteacts optically in the same way (absorbs the light and perpetuates conformal transitions while thelight is on). At the same time the diffusion coefficient for polymer dye must be significantly lowerthan for monomer. Actually the rotational diffusion is efficiently dumped due to interactionbetween neighboring side groups, while translation one is practically excluded for dye moleculeattached to the polymer main chain. The dumping of rotational diffusion must strongly depend onthe distance between side-groups attached to main polymer chain. This dependence can be testeddirectly if different homologues of the same polymer with different separation of substitute becomeavailable. The lower diffusion coefficient for dye -substituted polymer (compared to the monomers) leads to a lesser migration of dye-initiated excitations and as a result to a larger value of theresonant torque per a mixture volume in accordance with the mentioned theory. Knowledge ofdiffusion coefficients for polymer and corresponding dye monomer could provide betterquantitative comparison. Unfortunately these coefficients are still to be measured for both polymerand monomer dissolved in LC.6Komitov 12 reports on the planar-homeotropic transition in the cell containing azobenzene- based LC with ability of photoconformation, when the adsorption of cis-conformers on the LC- substrate interface prevails over trans- conformer adsorption due to the larger dipole moment. We can not exclude similar behavior of the dye in our case, which could induce some stabilization ofthe homeotropic alignment. At the same time such an adsorption is originated from the interactionwith the substrates. In the case of ITO coated substrates it is logical to assume that this interactionhas the nature of electrostatic attraction to the conductive surface. As we are taking uncoatedglasses this stabilizing effect must be effectively reduced. Also in LC-dye mixture the role of thissurface adsorption should not be as large as for the cited case for the used concentrations of dye(ca 5 weight % and only small of them are statistically under illumination in cis -form, which causes the stabilization of homeotropic orientation). Thus to our opinion this stabilization effectshould not contribute significantly in observed LC reorientation. In conclusions the azo-dye polymer is used as a photoactive additive to the nematic LC. Under the resonant illumination this mixture exhibits induced birefringence. Comparison with thepublished results on the LC mixtures containing monomer azobenzene dyes shows significant gainin the value of induced birefringence. This observation of the enhanced induced birefringence inLC-polymer mixture qualitatively proves the theory 7. Practically the sensitivity enhancement for LC-polymer mixture (e.g. the threshold of reorientation is only of 2 mW/cm2) directly leads to applications in optical technology. For example lower light power requirements open an applicationfor fiber optics switches and routers as well as optical image processing via nonlinear filtering 13 or dynamic holography. The Association for Super-Advanced Electronics Technologies (ASET) and the New Energy Development Organization (NEDO) supported in part this study. We thank ElectrotechnicalLaboratory and Dr. H.Yokoyama for s upport.7Figures captions: Figure 1. Optical scheme of the experiment: P –polarizer, PD – photodiode.Figure 2. Phase shift changes under exposure for the selected intensity of light (5 mW/cm 2). Both rise and decay are well fitted by the exponential dependence with time constant 40 s. Figure 3. Intensity dependencies of rise/decay times vs light intensity determined as a time constant τ with approximation by exp(-t/ τ) and maximum phase shift achieved at given illuminating light intensity.8References. 1.B.Ya. Zel'dovich, N.F. Pilipetskij, A.V. Sukhov, N.V. Tabiryan, JEPT Lett. 32, 263 (1980). 2. I.Janossy, L.Csillag, A.D. Lloyd, Phys.Rev. A 44, 8410 (1991). 3. I. Janossy, Phys. Rev. E 49, 2957 (1994). 4. M.I.Barnik, A.S.Zolot'ko, V.G.Rumyantsev, D.B.Terskov, Crystallogr.Rep., 40, 691 (1995). 5. E.Santamato, G.Abbate, P.Maddalena, L.Marrucci, D. Paparo, E. Massera Mol.Cryst.Liq.Cryst., 302, 111 (1997). 6. B.Saad, M.M.Denariez-Roberge, T.V.Galstyan, Optics Letters 23, 727 (1998). 7. A.Chrzanowska, A.Parfenov, H. Yokoyama, Yu. Tabe. Phys.Rev. E, (submitted) (1999). 8. K.Harada, M.Itoh, H.Matsuda, S.Ohnishi, A.Parfenov, N.Tamaoki, T.Yatagai, J.Phys.D:Appl.Phys. 31, 463 (1998). 9. L.M.Blinov, V. G. Chigrinov, Electrooptic effects in liquid crystal materials (Springer-Verlag, New York, 1994). 10. B.Saad, T.V. Galstyan, M. M. Denariez-Roberge, M. Dumont, Optics Communications 151, 235 (1998). 11. Y. Wu, A.Kanazawa, T. Shiono, T. Ikeda., Q. Zhang, Polymer 40, 4787 (1999) 12. L.Komitov, (personal communication, 1998).13. M. Itoh, K. Harada, S.Kotova, A.Naumov, A.Parfenov, T.Yatagai, Laser and Optics Technology 30, 147 (1998)9LC cell ttransmission PDreadout PExcitation Fig.1. A.Parfenov, A.Chrzhanowska Enhanced photoinduced …..100 50 10001234561-exp(-t/40) exp(-t/40) 5 mW/cm2 decay risePhase shift, π Time, sec Fig.2. A.Parfenov, A.Chrzhanowska Enhanced photoinduced …..112.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0020406080100120140 rise decayTime, sec Light intensity, mW/cm202468 phase shift Maximum phase shift, π Fig.3. A.Parfenov, A.Chrzhanowska Enhanced photoinduced …..

arXiv:physics/0002053v1 [physics.bio-ph] 29 Feb 2000SAMPLING PROPERTIES OF THE SPECTRUM AND COHERENCY OF SEQUENCES OF ACTION POTENTIALS M. R. Jarvis(1)and P. P. Mitra(2) February 2, 2008 (1)Division of Biology, California Institute of Technology, P asadena, 91125 USA (2)Bell Laboratories, Lucent Technologies, Murray Hill, New J ersey, 07974 USA Abstract The spectrum and coherency are useful quantities for charac terizing the temporal correlations and functional relations within and between point processes. This paper begins with a review of these quantiti es, their inter- pretation and how they may be estimated. A discussion of how t o assess the statistical significance of features in these measures i s included. In ad- dition, new work is presented which builds on the framework e stablished in the review section. This work investigates how the estima tes and their error bars are modified by finite sample sizes. Finite sample c orrections are derived based on a doubly stochastic inhomogeneous Pois son process model in which the rate functions are drawn from a low varianc e Gaussian process. It is found that, in contrast to continuous process es, the variance of the estimators cannot be reduced by smoothing beyond a sca le which is set by the number of point events in the interval. Alternativ ely, the degrees of freedom of the estimators can be thought of as bounded from above by the expected number of point events in the interval. Further new work de- scribing and illustrating a method for detecting the presen ce of a line in a point process spectrum is also presented, corresponding to the detection of a periodic modulation of the underlying rate. This work demo nstrates that a known statistical test, applicable to continuous process es, applies, with little modification, to point process spectra, and is of util ity in studying a point process driven by a continuous stimulus. While the mat erial discussed is of general applicability to point processes attention wi ll be confined to sequences of neuronal action potentials (spike trains) whi ch were the moti- vation for this work. 1Keywords and phrases: Spectrum, coherency, coherence, multitaper, lag window, spike train analysis, point processes, finite size e ffects in spectra, doubly stochastic Poisson process 1 Introduction The study of spike trains is of central importance to electro physiology. Often changes in the mean firing rate are studied but there is increa sing interest in characterising the temporal structure of spike trains, and the relationships between spike trains, more completely (Gray et al., 1989; Gerstein e t al., 1985; Abeles et al., 1983). A natural extension to estimating the rate of n euronal firing is to estimate the autocorrelation and the cross-correlation fu nctions1. This paper will discuss the frequency domain counterparts of these quantit ies. Auto- and cross- correlations correspond to spectra and cross spectra respe ctively. The coherency, which is the normalised cross spectrum, does not in general h ave a simple time domain counterpart. The frequency domain has several advantages over the time do main. Firstly often subtle structure can be detected with the frequency do main estimators which is difficult to observe with the time domain estimators. Secon dly, the time domain quantities have problems which are associated with sensiti vity of the estimators to weak non-stationarity and the non-local nature of the err or bars (Brody, 1998). These problems are greatly reduced in the frequency domain. Thirdly, reasonably accurate confidence intervals may be placed on estimates of t he second order properties in the frequency domain which permits the statis tical significance of features to be assessed. Fourthly, the coherency provides a normalised measure of correlations between time series, in contrast with time dom ain cross-correlations which are not normalisable by any simple means. This paper begins by reviewing the population spectrum and c oherency for point processes and motivating their use by describing some example applications. Next direct, lag window and multitaper estimators of the spe ctrum and coherency are presented. The concept of degrees of freedom is introduc ed and used to obtain large sample error bars for the estimators. Many elements of the work discussed in the review section of this paper can be found in the referen ces (Percival and Walden, 1993; Cox and Lewis, 1966; Brillinger, 1978; Bartle tt, 1966). Most of the material in these references is targeted at either spectral analysis of continuous processes or at the analysis of point processes but with less emphasis on spectral analysis. Building on this framework corrections, based on a specific model, will be given for finite sample sizes. These corrections are cast i n terms of a reduction in the degrees of freedom of the estimators. For a homogeneou s Poisson process the modified degrees of freedom is the harmonic sum of the the a symptotic de- 1Definitions of these quantities will be given in section 2.3. 2grees of freedom and twice the number of spikes used to constr uct the estimate. Modifications to this basic result are given for structured s pectra and tapered data. A section is included on the treatment of point process spectra which con- tain lines. A statistical test for the presence of a line in a b ackground of coloured noise is given, and the method for removal of such a line descr ibed. An example application to periodic stimulation is given. 2 Population measures and their interpretation 2.1 Counting representation of a spike train A spike train may be regarded as a point process. If the spike s hapes are neglected, it is completely specified by a series of spike times {ti}and the start and end points of the recording interval [0 , T]. It is convenient to introduce some notation which enables the subsequent formulae to be written in a compact fo rm (Brillinger, 1978). The counting process N(t) is defined as the number of spikes which occur between the start of the interval ( t= 0) and time t. The counting process has the property that the area beneath it grows as tbecomes larger. This is undesirable because it leads to an interval dependent peak at low frequen cies in the spectrum. To avoid this a process N(t) =N(t)−λt, where λis the mean rate, which has zero mean may be constructed. Note that dN(t) =N(t+dt)−N(t) which is either 1−λdt or−λdtdepending on whether or not there is a spike in the interval dt. Thus dN(t)/dtis a series of delta functions2with the mean rate subtracted. Figure 1 illustrates the relationship between N(t),N(t), and dN(t)/dt. 0time time time TN(t) 0 T0N(t) dN(t) dt T Figure 1: Example illustrating how the processes N,NanddN/dt relate to each other. The vertical lines in the process dN/dt depict delta functions. 2A delta function is a generalized function. It has an area of o ne beneath it but has zero width and therefore infinite height. 32.2 Stationarity It will be assumed in what follows that the spike trains are se cond order stationary. This means that their first and second moments do not depend on the absolute time. In many electrophysiology experiments this is not the case. In awake behaving studies the animal is often trained to perform a hig hly structured task. Nevertheless it may still be the case that over an appropriat ely chosen short time window, the statistical properties are changing slowl y enough for reasonable estimates of the spectrum and coherency to be obtained. As an example, neurons in primate parietal area PRR exhibit what is known as memory a ctivity during a delayed reach task (Snyder et al., 1997). The mean firing rat e of these neurons varies considerably during the task but during the memory pe riod is roughly constant. The assumption of stationarity during the memory period is equivalent to the intuitive notion that there is nothing special about 0 .75s into the memory period as compared to say 0.5s. Second order stationarity im plies that the mean firing rate ( λ) is independent of time and additionally that the autocovar iance depends only on the lag ( τ) and not on the absolute time. 2.3 Definitions Equations 1 - 4 give the first and second order moments of a sing le spike train for a stationary process. The spectrum S(f) is the Fourier transform of the autocovariance function ( µ(τ) +λδ(τ)). E{dN(t)} dt=λ (1) E{dN(t)} dt= 0 (2) µ(τ) +λδ(τ) =E[dN(t)dN(t+τ)] dtdτ(3) S(f) =λ+/integraldisplay∞ −∞µ(τ) exp(−2πifτ)dτ (4) Where Edenotes the expectation operator. The autocovariance measures how likely it is that a spike wil l occur at time t+τ given that one has occurred at time t. Usually µ(τ) is estimated rather than the full autocovariance which includes a delta function at zero lag3. However, in order to take the Fourier transform the full autocovariance is req uired. The inclusion of this delta function leads to a constant offset of the spectr um. This offset is an 3When estimating the autocovariance using a histogram metho d one generally omits the spike at the start of the interval which would always fall in t he bin nearest zero. 4important difference between continuous time processes and point processes. The population coherency γ(f) is defined in equations 5 - 7. µab(τ) =E[dNa(t)dNb(t+τ)] dtdτ(5) Sab(f) =/integraldisplay∞ −∞µab(τ) exp(−2πifτ)dτ+λaδab (6) γ(f) =S12(f)/radicalBig S11(f)S22(f)(7) Where indices 1 and 2 denote simultaneously recorded spike t rains from different cells. Unlike the spectrum, which is strictly real and positive, th e coherency is a complex quantity. The modulus of the coherency, which is kno wn as the coher- ence4, can only vary between zero and one. This makes coherence par ticularly attractive for detecting relationships between spike trai ns as it is insensitive to the mean spike rates. 3 Examples and their interpretation Before discussing the details regarding how to estimate the spectrum and co- herency it will be helpful to motivate them further by consid ering some simple examples. 3.1 Example population spectra For a homogeneous Poisson process of constant rate λthe autocovariance is simply λδ(τ) and hence the spectrum is a constant equal to the rate λ. At the opposite extreme consider the case where the spikes are spaced by inte rvals ∆ τ. This is not a stationary process but if a small amount of drift is permitt ed, so that over an extended period there is nothing special about a given time, it becomes stationary. The spectrum of this process contains sharp lines at integer multiples of f=1 ∆τ. Due to the drift the higher harmonics will become increasing ly blurred and in the high frequency limit the spectrum will tend towards a consta nt value of the mean rateλ. As a final example consider the case where µ(τ) is a negative Gaussian centered on zero τ. This form of µ(τ) is consistent with the probability of firing being suppressed after firing5. The spectrum of this process will be below λat 4Some authors define coherence as the modulus squared of the co herency. 5This need not necessarily correspond to the biophysical ref ractive period but, it could arise, rather from a characteristic integration time. 5low frequencies and will go to a constant value λat high frequencies. Figure 2 illustrates these different population spectra. S(f) λ f fλS(f) λ fS(f) (a) (b) (c) Figure 2: Example population spectra for different types of u nderlying process. (a) Homogeneous Poisson process with rate λ. (b) Regularly spaced spikes with jitter. (c) Spike trains in which the probability of firing is suppressed immediately after firing. 3.2 Example population coherency The population coherency of two homogeneous Poisson proces ses is zero. In con- trast if two spike trains are equal then the coherence is one a nd the phase of the coherency is zero at all frequencies. If two spike trains are identical but offset by a lag ∆ τthen the coherence will again be one but the phase of the coher ency will vary linearly with frequency with a slope proportional to ∆τand given by φ(f) = 2πf∆τ. 4 Estimating the spectrum Section 3 demonstrated that the population spectrum may pro vide insights into the nature of a spike train. In this section the question of ho w to estimate the spectrum from a finite section of data will be introduced. In w hat follows the population quantity λin the definition of N(t) is replaced by a sample estimate N(T)/T. 4.1 Direct Spectral Estimators 4.1.1 Definition A popular, though seriously flawed, method for estimating th e spectrum is to take the modulus squared of the Fourier transform of the data dN(t). This estimate is known as the Periodogram and is the simplest example of a di rect spectral estimator. More generally, a direct spectral estimator is t he modulus squared of the Fourier transform of the data but with the data being mu ltiplied by an envelope function h(t), known as a taper (Percival and Walden, 1993). Equations 68 - 10 define the direct estimator. On substituting N(t) into equation 9 a form amenable to implementation on a computer is obtained (equat ion 11). In this form the Fourier transform may be computed rapidly and witho ut the need for the binning of data. Note that equation 10 results in h(t) scaling as 1 /√ Tas the sample length is altered. This ensures proper normaliza tion of the Fourier transformation as sample size varies. ID(f) =|JD(f)|2(8) JD(f) =/integraldisplayT 0h(t)e−2πiftdN(t) (9) Where, /integraldisplayT 0h(t)2dt= 1 (10) JD(f) =N(T)/summationdisplay j=1h(tj)e−2πift j−N(T)H(f) T(11) andH(f) is the Fourier transform of the taper. The direct estimator suffers from bias and variance problems , described below, and is of no practical relevance for a single spike train samp le. 4.1.2 Bias It may not be immediately apparent why the above procedure is an estimate of the spectrum, especially when one is permitted to multiply the d ata by an arbitrary, albeit normalized, taper. The relation between ID(f) and the spectrum may be obtained by taking the expectation of equation 8. E{ID(f)}=E{/integraldisplay∞ −∞/integraldisplay∞ −∞h(t)h(t′)e−2πif(t−t′)dN(t)dN(t′)} (12) Assuming that the integration and expectation operations m ay be interchanged and substituting equation 3 yields6, E{ID(f)}=/integraldisplay∞ −∞/integraldisplay∞ −∞h(t)h(t′)e−2πif(t−t′){µ(t−t′) +λδ(t−t′)}dtdt′(13) Which may be rewritten in the Fourier domain as, E{ID(f)}=/integraldisplay∞ −∞S(f′)|H(f−f′)|2df′(14) 6For the moment, we assume that the population quantity λis known. This is of course not the case in practice, and one employs the estimate N(T)/Tas stated before. The effect of this extra uncertainty is given in equation 15. 7The expected value of the direct estimator is a convolution o f the true spectrum and the modulus squared of the Fourier transform of the taper . The normaliza- tion condition on the taper (equation 10) is equivalent to th e requirement that the kernel of the convolution has unit area underneath it. Sharp features in the true spectrum will be thus be smeared by an amount which depends on the width of the taper in the frequency domain. If the taper is well locali zed in the frequency domain the expected value of the direct estimate is close to t he true spectrum but if the taper is poorly localized then the expected value of th e direct estimator will be incorrect i.e. the direct spectral estimator is biased. T here is a fundamental level beyond which the bias cannot be reduced, due to the unce rtainty relation for- bidding simultaneous localization of a function in the time and frequency domains below a given limit. Since the maximum width of the taper is T t he minimum frequency spread is 1/T which is known as the Raleigh frequen cy. Figure 3 shows the smoothing kernel for a rectangular taper and a T of 0.5s. N ote that this kernel has large sidelobes which is the primary motivation for usin g tapering. −6−4−20246−3−2.5−2−1.5−1−0.50 frequency (Hz)log10|h(f)|2 Figure 3: The smoothing kernel |H(f)|2. This is the expected direct estimate of the spectrum in the case of a population spectrum which has a delta function (very sharp feature) at the center frequency. A rectangular taper of length 0.5s was used. Solid vertical lines are drawn at ±the Raleigh frequency. In the above argument equation 3 was used in spite of the appea rance of the population quantity λrather than the sample estimate N(T)/Tfor which equation 12 was defined. A more careful treatment, which includes this correction, leads to an additional term at finite sample sizes in the expectation o f the direct spectral estimator at low frequencies. The full expression is given b elow, E{ID(f)}=/integraldisplay∞ −∞S(f′)|H(f−f′)|2df′− |H(f)|2S(0)/T (15) In the case of the periodogram, where h(t) = 1/√ T, the effect is clear since in this case JD(0) = 0 and hence ID(0) = 0 for any set of spike times and any T. 4.1.3 Asymptotic variance In the previous section it was shown that provided the taper i s sufficiently local in frequency the expected value of the direct spectral estimat or will be close to the 8true spectrum. However, the fact that the estimate is on aver age close to the true spectrum belies a serious problem with direct spectral esti mators, namely that the estimates have very large fluctuations about this mean. The u nderlying source of this problem is that one is attempting to estimate the value o f a function at an infinite number of points using a finite sample of data. The pro blem manifests itself in the fact that direct spectral estimators are inconsistent estimators of the spectrum7. In fact it may be shown that, under fairly general assumptio ns, the estimates are distributed exponentially (or equivalently asS(f)χ2 2/2) for asymp- totic sample sizes (i.e. T→ ∞ ) (Brillinger, 1972). Figure 4 illustrates that direct spectral estimators are noisy and untrustworthy, a f act emphasised by the observation that the χ2 2distribution has a standard deviation equal to its mean. In the next three subsections methods for reducing the varia nce of direct spectral estimators using different forms of averaging will be discus sed. 02468100.001 0.01 0.1 1 10 100 1000 frequency (Hz)spectrum Figure 4: An example of a direct spectral estimate. A 40% cosi ne taper was used. A sample of duration 20s was drawn from a homogeneous Poisson process with a constant rate of 50 Hz. The population spectrum for this pro cess is flat and is shown by the solid horizontal line. The direct spectral esti mate is clearly noisy although on average the correct spectrum is obtained. 4.2 Trial averaging If there are a number of trials ( NT) available then the variance of the direct estimator may be reduced by trial averaging. IDT(f) =1 NTNT/summationdisplay n=1ID n(f) (16) Where ID n(f) is the direct spectral estimate based on the nthtrial. In the large T limit taking the average entails summing NTindependent sam- ples from a χ2 2distribution the result of which is distributed as χ2 2NT. The reduc- tion in variance is inversely proportional to the number of t rials corresponding to a reduction in standard deviation which is the familiar fact or of 1 /√NT. 7Inconsistent estimators have a finite variance even for an in finite length sample. 9At first sight it appears one would be getting something for no thing by breaking a single section of data into NTsegments and treating them as separate trials. This is, of course, not the case. The reason is that if the data is se gmented into short length samples, there is loss of frequency resolution propo rtional to the inverse of segment length. Lag window and multitaper estimators use th e information from these independent estimates without artificially segmenti ng the data. 4.3 Lag Window Estimates A powerful property of the frequency domain is that, unless t wo frequencies are very close together, direct estimates of the spectrum of a st ationary process at different frequencies are nearly uncorrelated. This proper ty arises when the covari- ance between frequencies falls off rapidly. If the true spect rum varies slowly over the width of the covariance then the large sample covariance of a direct spectral estimator is given by equation 17. cov{ID(f1), ID(f2)} ≃E{ID(f)}2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞ −∞h(t)2e−2πi∆ftdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 (17) Where f= (f1+f2)/2 and ∆ f=f1−f2 For ∆ f= 0, this expression reduces to the previously mentioned res ult that the variance of the estimator is equal to the square of the mea n. For ∆ f >> 1/T, |/integraltext∞ −∞h(t)2e−2πi∆ftdt|2→0, since h(t)2is a smooth function with extent T. This implies that cov{ID(f1), ID(f2)} ≈0 for|f1−f2|>>1/T. The approximate independence of nearby points means that, if the true spectr um varies slowly enough, then closely spaced points will provide several ind ependent estimates of the same underlying spectrum. This is the motivation for the lag window estimator which is simply a smoothed version of the direct spectral est imator (Percival and Walden, 1993). The lag window estimator is defined in equatio ns 18 and 19. ILW(f) =/integraldisplay∞ −∞K(f−f′)ID(f′)df′(18) Where,/integraldisplay∞ −∞K(f)df= 1 (19) Averaging over trials may be included by using the trial aver aged direct spec- tral estimate IDT(see equation 16) in place of IDin the above expression. It is assumed that K(f) is a smoothing kernel with reasonable properties. 4.3.1 Bias The additional smoothing of the lag window kernel modifies th e bias properties of the estimator from those expressed in equation 15. The exp ected value of the 10lag window estimator is given by, E{ILW(f)}=/integraldisplay∞ −∞K(f−f′)|H(f′−f′′)|2S(f′′)df′df′′−S(0) T/integraldisplay∞ −∞K(f−f′)|H(f′)|2df′ (20) 4.3.2 Asymptotic Variance The large sample variance of this estimator is readily obtai ned using equation 17. var{ILW(f)}=ξ NTE{ILW(f)}2(21) Where, ξ=/integraldisplay∞ −∞/integraldisplay∞ −∞K(f)K(f′)|H(f−f′)|2dfdf′(22) and, H(f) =/integraldisplay∞ −∞h(t)2e−2πiftdt (23) Equation 21 includes the reduction in variance due to trial a veraging. 1 /ξ can be interpreted as the effective number of independent est imates beneath the smoothing kernel, as demonstrated by the following qual itative argument. If ∆fis the frequency width of the smoothing kernel K(f) and δfis the fre- quency width of the taper H(f) then since K(f)∼1/∆fit follows that ξ∼ 1/(∆f)2/integraltext ∆f/integraltext ∆f|H(f−f′)|2dfdf′and hence that ξ∼δf/∆f. 4.4 Multitaper Estimates While the lag window estimator is based on the idea that nearb y frequencies provide independent estimates, the estimation is not very s ystematic, since, one should be able to explicity decorrelate nearby frequencies from the knowledge of the correlations introduced by a finite window size. This is a chieved in multitaper spectral estimation. The basic idea of multitaper spectral estimation is to average the spectral estimates from several orthogonal tapers. The orthogonality of the tapers ensures that the estimates are uncorrelated for larg e samples (consider substituting h1(t)h2(t) forh(t)2in equation 17). A critical question is the choice of a set of orthogonal tapers. A natural choice are the discre te prolate spheroidal sequences (dpss) or Slepian sequences, which are defined by t he property that they are maximally localised in frequency. The dpss tapers m aximize the spectral concentration defined as; λ=/integraltextW −W|H(f)|2df/integraltext∞ −∞|H(f)|2df(24) 11Where in the time domain h(t) is strictly confined to the interval [0,T]. For given values of W and T there are a finite number of tapers wh ich have concentrations ( λ) close to one, and therefore have well controlled bias. This number is known as the Shannon number and is 2 WT. This sets an upper limit on the number of independent estimates that can be obtained f or a given amount of spectral smoothing. A direct multitaper estimate of the spectrum is defined in equ ation 25. IMT(f) =1 KK−1/summationdisplay k=0ID k(f) (25) The eigenspectra ID kare direct spectral estimates based on tapering the data with the kthdpss function. As previously trial averaging can be include d by using IDTrather than ID. More sophisticated estimates involve adaptive (rather th an constant) weighting of the data tapers (Percival and Walden , 1993). Multitaper spectral estimation has been recently shown to be useful for analysing neurobio- logical time series, both continuous processes (Mitra and P esaran, 1999) and spike trains (Pesaran et al., 2000). 4.4.1 Bias The bias for the multitaper estimate is given by equation 15 b ut with |H(·)|2 replaced by an average over tapers1 K/summationtextK−1 k=0|Hk(·)|2. 4.4.2 Asymptotic Variance The asymptotic variance of the multitaper estimator, inclu ding trial averaging, is given by equation 26. var{IMT(f)}=1 NTKE{IMT(f)}2(26) 4.5 Degrees of freedom At this point it is useful to introduce the concept of the degr ees of freedom ( ν0) of an estimate. The degrees of freedom is twice the number of i ndependent esti- mates of the spectrum. Degrees of freedom is a useful concept as it permits the expressions for the variance of the different estimators to b e written in a common format. var{IX(f)}=2E{IX(f)}2 ν0(27) Where, 12X D DT LW MT ν0 2 2NT 2NT/ξ 2NTK Degrees of freedom is also a useful framework in which to cast both finite size corrections and the confidence limits for the spectra and coh erence. The variance of estimators of the spectrum can be estimated u sing internal methods such as the bootstrap or jackknife (Efron and Tibshi rani, 1993),(Thom- son and Chave, 1991). Jackknife estimates can be constructe d over trials or over tapers. If ν0is large ( >20), then the theoretical and Jackknife variance are in close agreement. If distributional assumptions can be vali dly made about the point process, theoretical error bars have an important adv antage over internal estimates since they enable the understanding of different f actors which enter into the variance in order to guide experimental design and data a nalysis. However Jackknife estimates are less sensitive to failures in distr ibutional assumptions, and this provides them with statistical robustness. It is conventional to display spectra on a log scale. This is b ecause taking the log of the spectrum stabilizes the variance and leads a di stribution which is approximately Gaussian. 4.6 Confidence intervals The expected values of the estimators and also their varianc e have been discussed for several different spectral estimators but it is desirabl e to put confidence inter- vals on the spectral estimates rather than standard deviati ons. As previously mentioned in section 4.2 the averaging of dire ct spectral esti- mates from different trials yields, in the large sample limit , estimates which are distributed as χ2 2NT. In general for the other estimates a well known approxima- tion (Percival and Walden, 1993) is to assume that the estima te is distributed as χ2 ν0. Confidence intervals can then by placed on estimates on the b asis of this χ2 ν0 distribution. The confidence interval applies for the popul ation spectrum S(f) and is obtained from the following argument. P/bracketleftBig q1≤χ2 ν0≤q2/bracketrightBig = 1−2p (28) Where Pindicates probability, q1is such that P[χ2 ν0≤q1] =pandq2is such that P[χ2 ν0≥q2] =p. It follows that, P/bracketleftBig q1≤ν0IX(f)/S(f)≤q2/bracketrightBig = 1−2p (29) Hence an approximate 100% ×(1−2p) confidence interval for S(f) is given by, P/bracketleftBig ν0IX(f)/q2≤S(f)≤ν0IX(f)/q1/bracketrightBig (30) 13For large ν0(>20) these confidence intervals do not differ substantially fr om those based on a Gaussian ( ±2 standard deviations) but at small ν0the difference can be substantial as for these values the χ2 ν0distribution has long tails. 4.7 High Frequency limit The population spectrum goes to a constant value equal to the rateλin the high frequency limit. In practice spectra calculated from a finit e sample will go to a value close to λbut fluctuations in the number of spikes in the interval will l ead to an error in this estimate. For a given sample the spectrum w ill go the value given by equation 31. I(f→ ∞) =1 NTKK−1/summationdisplay k=0NT/summationdisplay n=1Nn(T)/summationdisplay j=1hk(tn j)2(31) Where tn jis the jthspike in the nthtrial and Nn(T) is the total number of spikes in the nthtrial. In the case of direct and lag window estimators the ave raging over tapers need not be performed. This expression yields a value which is typically very close to the sample esti- mate of the mean rate8. It is significant departures from this high frequency limit which are of interest when interpreting the spectrum as thes e indicate enhance- ment or suppression relative to a homogeneous Poisson proce ss. 4.8 Choice of estimator, taper and lag window The preceding section discussed the large sample statistic al properties of direct, lag window and multitaper estimates of the spectrum. The choice of which estimator to use remains a contentious one (Percival and Walden, 1993) . The multitaper method is the most systematic of the estimators but the lag wi ndow estimators should perform almost as well for those spike train spectra w hich have reasonably small dynamic ranges9. However, it is possible to construct spike trains with widely different time scales, which can possess a large dynam ic range. In addition, the multitaper technique leads to a simple jackknife proced ure by leaving out one data taper in turn. A further important property of the multi taper estimator is that it gives more weight to events at the edges of the time int erval and thus ameliorates the arbitrary downweighting of the edges of the data introduced by single tapers. 8It is exactly the sample estimate of the mean rate for a rectan gular taper. 9Dynamic range is a measure of the variation in the spectrum as a function of frequency and is defined as 10 log10(max fS(f) min fS(f)). 14If using the lag window estimator there are many choices avai lable for both the taper and the lag window. The choice of taper is generally not critical provided that the taper goes smoothly to zero at the start and end of the interval. A rectangular taper has particularly large sidelobes in the f requency domain which can lead to significant bias. The choice of lag window is also u sually not critical and typically a Gaussian kernel will be satisfactory. 5 Estimating the Coherency Sample coherency, which may be estimated using equation 32, may be evaluated using any of the previously mentioned spectral estimators. The principle difference is that the direct estimator, in terms of which the other esti mators are expressed, is given by equations 33 and 34 rather than 8 and 9. C(f) = IX 12//radicalBig IX 11IX 22 (32) ID 12(f) = JD 1(f)JD 2(f)∗(33) JD a(f) =/integraldisplay∞ −∞h(t)e−2πiftdNa(t) (34) Where the N1(t) and N2(t) are simultaneously recorded spike trains from two different cells and Xdenotes the type of spectral estimator. Possible choices of estimator Xinclude; Ddirect, DTtrial averaged direct, LWlag window or MT multitaper. Lag window and multitaper coherency estimates may be constr ucted by sub- stituting ID 12(·) in place of ID(·) in equations 18 and 25. The estimates are biased over a frequency range equal to the width of the smoothing alt hough the exact form for the bias is difficult to evaluate. 5.1 Confidence limits for the Coherence The treatment of error bars is somewhat different between the spectrum and the coherency, since the coherency is a complex quantity. Usual ly one is interested is in establishing whether there is significant coherence in a g iven frequency band. In order to do this the sample coherence should be tested agains t the null hypothesis of zero population coherence. The distribution of the sampl e coherence under this null hypothesis is given below. P(|C|) = (ν0−2)|C|(1− |C|2)(ν0/2−2)0≤ |C| ≤1 (35) A derivation of this result is given in (Hannan, 1970). In out line the method is to rewrite the coherence in such a way that it is equivalent to a multiple correlation 15coefficient (Anderson, 1984). The distribution of a multiple correlation coeffient is then a known result from multivariate statistics. In the cas e of coherence estimates based on lag window estimators the appropriate ν0may be used although this is only approximately valid because this method of derivation assumes integer ν0/2. It is straightforward to calculate a confidence level based o n this distribution. The coherence will only exceed/radicalBig 1−p1/(ν0/2−1)inp×100% of experiments. In addition it is notable that the quantity ( ν0/2−1)|C|2/(1− |C|2) is distributed asF2,ν0−2)under this null hypothesis. It is useful to apply a transform ation to the coherence before plotting it which aids in the assessmen t of significance. The variable q=/radicalBig −(ν0−2)log(1− |C|2) has a Raleigh distribution which has density p(q) =qe−q2/2. This density function does not depend on ν0and furthermore has a tail which closely resembles a Gaussian. For certain values of a fitting parameter10 β, a further linear transformation r=β(q−β) leads to a distribution which closely resembles a standard normal Gaussian for r >2. This means that for r >2 one can interpret ras the number of standard deviations by which the coherence exceeds that expected under the null hypothesis. 5.2 Confidence Limits for the Phase of the Coherency If there is no population coherency then the phase of the samp le coherency is distributed uniformly. If, however, there is population co herency then the distri- bution of the sample phase is approximately Gaussian provid ed that the tails of the Gaussian do not extend beyond a width 2 π. An approximate 95% confidence interval for the phase (Rosenberg et al., 1989; Brillinger, 1974) is given below. ˆφ(f)±2/radicaltp/radicalvertex/radicalvertex/radicalbt2 ν0/parenleftBigg1 |C(f)|2−1/parenrightBigg (36) Where ˆφ(f), the sample estimate of the coherency phase, is evaluated u sing tan−1{Im(C)/Re(C)}. 6 Finite Size Effects In the preceding sections error bars were given for estimato rs of the spectrum and the coherence. However these error bars were based on large s ample sizes (they apply asymptotically as T→ ∞). Neurophysiological data are not collected in this regime and, particularly in awake behaving studies whe re data is often sparse, corrections arising at small T are potentially important. I n order to estimate the size of these corrections a particular model for the poin t process is required. 10A reasonable choice for βis 23/20. 16The model studied was chosen primarily for its analytical tr actability while still maintaining a non-trivial spectrum. The model and the final results will be presented here but the d etails of the analysis are reserved until appendix A. The model is a doubly stochastic inho- mogeneous Poisson process with a Gaussian rate function. A s pecific realization of a spike train is generated from the model in the following m anner. Firstly a population spectrum SG(f) is specified. From this a realization of a zero mean Gaussian process λG(f) is generated. A constant λ, the mean rate, is then added to this realization. This function is then considered to be t he rate function for an inhomogeneous Poisson process. A realization of this inh omogeneous Poisson process is then generated. A schematic of the model is shown i n figure 5. G ft t tλ λ SG (t) (t) (a) (b) (c) (d) Figure 5: Schematic illustrating the model for which finite s ize corrections to the asymptotic error bars will be evaluated. (a) A spectrum SG(f) is defined (b) A realization λG(t) is drawn from a Gaussian process with this spectrum (c) The mean rate λis added to λG(t) to obtain λ(t) (d) This rate function is used to generate a realization of an inhomogeneous Poisson process yielding a set of spike times. Technically this is not a valid process because the rate func tionλ(t) may be negative. However, if the area underneath the spectrum is sm all enough then the fluctuations about the mean rate seldom cross zero and correc tions due to this effect are negligible. In addition large violations of this a rea constraint have been tested by Monte Carlo simulation and the results still apply to a good approxi- mation. An important feature of this model is that the population spe ctrum of the spike trains is simply the spectrum of the inhomogeneous Poisson p rocess rate function plus an offset equal to the mean rate11. The spectrum of the rate function is a positive real quantity and therefore within this model the p opulation spectrum cannot be less than the mean rate at any frequency. Intuitive ly, the reason for this is that the process must be more variable than a homogeneous P oisson process at all frequencies. To make the nature of the result clear a simplified version is g iven in equation 11This result does not depend on the Gaussian assumption. 1737. This version is for the particular case of a homogeneous P oisson process (which has a flat population spectrum) and a rectangular taper12. var{IX(f)}=λ2/bracketleftbigg2 ν0+1 NTTλ/bracketrightbigg (37) Where λis the mean rate. A sample based estimate of NTTλis the total number of spikes over all trials. It is to be noted that finite size effects reduce the degrees of f reedom. This result implies that there is a point beyond which additional smooth ing does not decrease the variance further and this point is approximately when ν0is equal to twice the total number of spikes. The full result is given in equations 38 - 43. var{IX(f)}=2E{IX(f)}2 ν(f)(38) 1 ν(f)=1 ν0+CX hΦ(f) 2TNTE{IX(f)}2(39) Where, CX h=  /integraltext1 0f(t)4dt If X = LW ,D or DT 1 K2/summationtext k,k′/integraltext1 0fk(t)2fk′(t)2dt If X = MT(40) f(t/T) =√ Th(t) (41) and, Φ(f) =λhf+ 4[E{IX(f)} −λhf] + 2[E{IX(0)}−λhf] + [E{IX(2f)}−λhf] (42) λhf=E{IX(f→ ∞)} (43) CX his a constant of order unity which depends on the taper. When a taper is used to control bias some of the spikes are effectively disreg arded and this has an effect on the size of the correction. The function f(t) has the same form as the taper h(t) but is defined for the interval [0 ,1].CX his the integral of the fourth power of fand obtains its minimum value of one for a rectangular taper. It is typically between 1 and 2 for other tapers. In the multitaper case cross terms between tapers are included. Equation 42 describes how the finite size correction depends on the structure of the spectrum. Φ( f) is the sum of four terms. The first term is the only 12The expression also holds approximately for the multitaper estimate provided all tapers up to the Shannon limit are used. 18term which is present for a flat spectrum. The second term is a c orrection which depends on the spectrum at the frequency being considered. T he next two terms depend on the spectrum at zero frequency and the spectrum at t wice the frequency being considered. The latter three terms all depend on the di fference between the spectrum at some frequency and the high frequency limit. Equ ation 42 applies provided that the spike train is well described by the model. However, this is not necessarily the case and a suppression of the spectrum, whic h cannot be described by the model, often occurs at low frequencies13. In the event that there is a significant suppression of the spectrum Φ( f) may become small or even negative. To avoid this a modified form for Φ( f) may be used which prevents this. Φ(f) =λhf+ 4max([E {IX(f)} −λhf],0) + 2max([E {IX(0)} −λhf],0)... + max([E {IX(2f)} −λhf],0) (44) The above modification to the result is somewhat ad hoc so Monte Carlo sim- ulations of spike trains with enforced refractory periods h ave been performed to test its validity. These simulations demonstrated that, al though the correction de- rived using 44 was significantly different from that obtained from the Monte Carlo simulations in the region of the suppression, equation 44 pr ovided a pessimistic estimate in all cases studied. This increases confidence tha t applying finite size corrections using equation 44 will provide reasonable erro r bars for small samples. Equation 39 gives the finite size correction in terms of a redu ction in ν0. The newν(f) may be used to put confidence intervals on the results, as des cribed in section 4.6, although the accuracy of the χ2 νassumption will be reduced. In the case of the coherence an indication of the correction to the c onfidence level can be obtained by using the smaller of the two ν(f) from the spike train spectra to calculate the confidence level using equation 35. In all case s if the effect being observed only achieves significance by an amount which is of t he same order as the finite size correction then it is recommended that more da ta be collected. 7 Experimental Design Often it is useful to know in advance how many trials or how lon g a time interval one needs in order to resolve features of a certain size in the spectrum or the coherence. To do this one needs to estimate the asymptotic de grees of freedom ν0. This depends on the size of feature to be resolved α, the significance level for which confidence intervals will be calculated pand the fraction of experiments which will achieve significance P. In addition the reduction in the degrees of freedom due to finite size effect depends on the total number of spikes Nsand also Ch(see section 6). 13Note that any spike train spectra displaying significant sup pression below the mean firing rate can immediately rule out the inhomogeneous Poisson pro cess model. 19An estimate of v0may be obtained in two stages. Firstly α,pandPare specified and used to calculate a degrees of freedom ν. Secondly the asymptotic degrees of freedom ν0is estimated using ν,NsandCh. The feature size α= (S−λ)/λis the minimum size of feature which the experimenter is cont ent to resolve. For example, a value of 0.5 indicates that where the population spectrum exceeds 1 .5λthe feature will be resolved. The significance level should b e set to the same value that will be used for calculating the confidenc e interval for the spectrum, typically be 0 .05. For a given pthere is some probability Pthat an experiment will achieve significance. To calculate νone begins with a guess νg. Then q1is chosen such that P/bracketleftBig χ2 νg≥q1/bracketrightBig =p/2. On the basis of this one then evaluates P′= 1−Φ [q1/(1 +α)] where Φ is the cumulative χ2 νgdistribution14. If P′is equal to the specified fraction Pthenν=νgotherwise a different νgis chosen. This procedure is readily implemented as a minimization of ( P − P ′(νg))2on a computer. Having obtained νone can estimate ν0using the following expression. 1 ν0=1 ν−Ch[1 + 4α] 2Ns[1 +α]2 (45) Where the 4 αis omitted from the numerator if α <0. Figure 6 illustrates example design curves generated using this method. These curves show the asymptotic degrees of freedom as a function o f feature size for different total numbers of spikes. −0.5 00.5 11.5 2020406080100 ν0 α Figure 6: Example design curves for the case when p= 0.05,P= 0.5 and Ch= 1.5. The three curves correspond to Ns=∞(solid), Ns= 100 (dashed), Ns= 50 (dotted). The existence of a region bounded by vertical asymptotes imp lies that as long as the total number of measured spikes is finite, modulations in the spectrum 14These formulae apply for α >0. Ifα <0 then P/bracketleftBig χ2 νg≤q1/bracketrightBig =p/2 and P′= Φ [q1/(1 +α)] should be used. 20below a certain level cannot be detected no matter how much th e spectrum is smoothed. These curves may be used to design experiments cap able of resolving spectral features of a certain size. In the case of the coherence one calculates how many degrees o f freedom are required for the confidence line to lie at a certain level as de scribed in section 5.1. 8 Line Spectra One of the assumptions underlying the estimation of spectra is that the population spectrum varies slowly over the smoothing width ( Wfor multitaper estimators). While this is often the case there are situations in which the spectrum contains very sharp features which are better approximated by lines t han by a continuous spectrum. This corresponds to periodic modulations of the u nderlying rate, such as when a periodic stimulus train is presented. In such situa tions it is useful to be able to test for the presence of a line in a background of col ored noise (i.e. in a locally smooth but otherwise arbitrary continuous popula tion spectrum). Such a test has been previously developed, in the context of multi taper estimation, for continuous processes (Thomson, 1982) and in the following s ection the analogous development for point processes is presented. 8.1 F-test for point processes A line in the spectrum has an exactly defined frequency and con sequently the process N(t) has a non-zero first moment. The natural model in the case of a single line is given by equation 46. E{dN(t)}/dt=λ0+λ1cos(2πf1t+φ) (46) A zero mean process ( N) may be constructed by subtraction of an estimate ofλ0t. Provided that the product of the line frequency( f1) and the sample dura- tion(T) is much greater than one the sample quantity N(T)/Tis an approximately unbiased estimate of λ0. The resultant zero mean process Nhas a Fourier trans- form which has a non-zero expectation. Jk(f) =/integraldisplay∞ −∞hk(t)e−2πiftdN(t) (47) E{Jk(f)}=c1Hk(f−f1) +c∗ 1Hk(f+f1) (48) Where, c1=λ1eiφ/2 (49) In the case where f >0 and f1> W, E{Jk(f)} ≃c1Hk(f−f1) (50) 21The estimates of Jk(f1) from different tapers provide a set of uncorrelated estimates of c1Hk(0). It is hence possible to estimate the value of c1by complex regression. ˆc1=/summationtext kJk(f1)Hk(0) /summationtext k|Hk(0)|2(51) Under the null hypothesis that there is no line in the spectru m (c1= 0) it may readily be shown that E{ˆc1}= 0 and var{ˆc1}=S(f1)//summationtext k|Hk(0)|2. The residual spectrum15, which has the line removed, may be estimated using equation 52. ˆS(f) =1 K/summationdisplay k|Jk(f)−ˆc1Hk(f−f1)|2(52) In the large sample limit the distributions of both ˆ c1andˆS(f1) are known (Percival and Walden, 1993) and may be used to derive relatio n 53. |ˆc1|2/summationtext k|Hk(0)|2(K−1) /summationtext k|Jk(f1)−ˆc1Hk(0)|2.=F2,2(K−1) (53) Where.= denotes ‘is distributed as’. The null hypothesis may be tested using this relation and, if rejected, the line can be removed using equation 52 to estimate the residua l spectrum. It is worth noting that although relation 53 was derived for large samples the test is remarkably robust as the sample size is decreased. Numerica l tests indicate that the tail of the F distribution is well reproduced even in situ ations where there are as low as 5 spikes in total. 8.2 Periodic Stimulation A common paradigm in neurobiology where line spectra are par ticularly important is that of periodic stimulation. When a neuron is driven by a p eriodic stimulation of frequency f1the spectrum may contain lines at any of the harmonics nf1. Provided that f1>2Wthe analysis of section 8.1 applies with each harmonic being separately tested for significance. The first moment of the process, which has period 1 /f1, is given by equation 54 and may be estimated using ˆ cn. λ(t) =λ0+/summationdisplay nλncos(2πnf1t+φn) (54) Where λn= 2|cn|,φn= tan−1{Im(cn)/Re(cn)}, the sum is taken over all the significant coefficients. 15It is also possible to estimate a residual coherency. In orde r to do this one uses a residual cross-spectrum ˆSxy(f) =1 K/summationtext k(Jx k(f)−ˆcx 1Hk(f−f1))∗(Jy k(f)−ˆcy 1Hk(f−f1)), together with the residual spectra to evaluate the usual expression for co herency. 22This rate function λ(t) is the average response to a single stimulus or impulse response. The coefficients cnare the Fourier series representation of λ(t). 8.3 Error Bars It is possible to put confidence intervals on both the modulus and the phase of the coefficients ˆ cn. For large samples( >10 spikes) the real and imaginary parts of ˆ cnare distributed as independent Gaussians each with standar d deviation σn=/radicalBig S(nf1)/(2/summationtext k|Hk(0)|2). For cn/σn>3 the distribution of |ˆcn|is well approximated by a Gaussian centered on |cn|and with standard deviation σn. In addition the estimated phase angle ( ˆφn) is also almost Gaussian with mean φnand standard deviation σn/|cn|. Approximate error bars or confidence intervals may be obtained using a sample based estimate of σn, ˆσn=/radicalBig ˆS(nf1)/(2/summationtext k|Hk(0)|2). Estimating error bars for the impulse response function is m ore involved due to their non-local nature (if one of the Fourier coefficients i s varied the impulse response function changes everywhere). It is therefore of i nterest to estimate a global confidence interval, defined as any interval such that the probability of the function crossing the interval anywhere is some predefined probability. A method for estimating a global confidence band is detailed in (Sun an d Loader, 1994) and outlined here. First a basis vector Φ( t) is constructed. Φ(t) = ˆσ1cos(2πf1t) ... ˆσNcos(2πfNt) ˆσ1sin(2πf1t) ... ˆσNsin(2πfNt) (55) Where Nis the total number of harmonics. The elements of this vector have unit variance and a standard approximation may be applied. P(sup|λ(t)−E{λ(t)}|> c||Φ(t)||)≤2(1−N(c)) + (k/π)e−c2/2(56) Where supis the maximum value of its operand, ||Φ(t)||denotes the length of vec- tor Φ( t),N(c) is the cumulative standard normal distribution and kis a constant. kmay be evaluated by constructing the 2 ×Nmatrix X(t) = [Φ( t)dΦ(t)/dt], forming its QRdecomposition (Press et al., 1992) and then evaluating k=/integraltextT 0|R22(t)/R11(t)|dt. Confidence intervals for the residual spectrum are calculat ed in the usual man- ner (using χ2 ν) although at the line frequencies the interval is slightly b roadened due to the loss of a degree of freedom incurred by estimation o fcn. Section 11 contains an example application of the methods described in this section. 239 Example Spectra Figure 7 is a spectrum calculated from data collected from a s ingle cell recorded from area PRR in the parietal cortex of an awake behaving monk ey during a de- layed memory reach task (Snyder et al., 1997). The spectrum i s calculated over an interval of 0.5s during which the firing rate is reasonably stationary and is av- eraged over 5 trials. The spectrum shows two features which a chieve significance. There is enhancement of the spectrum in the frequency band 20 -40 Hz indicating the presence of an underlying broad band oscillatory mode in the neuronal firing rate. In addition there is suppression of the spectrum at low frequencies. As dis- cussed previously a suppression of this sort is consistent w ith an effective refactory period during which the neuron is less likely to fire. Care mus t be taken at low frequencies since at frequencies comparable to the smoothi ng width the spectrum is particularly sensitive to any non-stationarity in the da ta. 0 1 202040 time (s)rate (Hz) 050100 15010203050100150 frequency (Hz)spectrum 050100 15010203050100150 frequency (Hz)spectrum (a) (b) (c) Figure 7: (a) Gaussian kernel (100 ms width) smoothed firing r ate with 2 σerror bars based on a stationarity assumption. The vertical lines indicate the period over which the spectrum was calculated. A light is flashed at t ime zero and the spectrum is evaluated over the interval when the monkey is re quired to remember the target location. (b) The spectrum evaluated over this in terval using a lag window estimator with a 40% cosine taper and a Gaussian lag wi ndow of width 3.5 Hz. 95% confidence limits are shown with the finite size cor rection included (this typically resulted in a decrease in ν(f) from about 50 to 36). The horizontal line indicates the high frequency limit.(c) The same spectr um evaluated using a multitaper estimator. A bandwidth (W) of 5 Hz was used allowi ng 5 tapers. Both estimators have the same degrees of freedom. 2410 Example Coherency To illustrate the estimation of coherency simulated spike t rains were generated from a coupled doubly stochastic Poisson process. For a give n trial a pair of rate functions were drawn from a Gaussian process. The realizati ons share a coherent mode which is linearly mixed into the rates of both cells. The se coupled rate functions are then used to independently draw a realization of an inhomogeneous Poisson process for each cell. Using this method 15 trials of duration 0.5s were generated. The coherent mode was set such that the populatio n coherence was a Gaussian of height 0.35 and standard deviation 5 Hz centere d on 20 Hz. The phase of this mode was set to 180o. Figure 8 indicates that this coherent mode is reasonably estimated. 02040608010000.20.40.60.81 frequency (Hz)coherence −360−270−180−900phase (degrees) 020406080100−2−101234 frequency (Hz)standardized coherence (a) (b) Figure 8: (a) Coherence (left axis) and the phase of the coher ency (right axis). Fifteen trials of 0.5s duration were simulated using a doubl y stochastic Poisson process as described in the text. A multitaper estimator wit h a smoothing width of 7 Hz was used. Finite size corrections were used and resulted in 25% reduction in the degrees of freedom. A horizontal line has been drawn at th e 95% confidence level under the null hypothesis of no coherency. Where the nu ll hypothesis is rejected the phase of the coherency is estimated and shown wi th an approximate 95% confidence interval. (b) The standardized coherence is a transformation which maps the null distribution onto an approximately standard n ormal variate (as described in section 5.1). The estimated coherence at 20 Hz w ould therefore lie at three standard deviations if there were no population coh erence. 11 Example Periodic Stimulation An example of an analysis of a periodic stimulus paradigm is s hown in figure 9. The data is a single cell recording collected from the barrel cortex of an awake 25behaving rat during periodic whisker stimulation at 5.5 Hz ( Sachdev et al., 1999). There is a single trial of duration 50s. 0 50 100 150 20045678910 frequency (Hz)spectrum 1369121518212427303336−π/4 0 π/4 π/23π/2 0 0.05 0.1 0.150.183050100150Rate (Hz) time (s)136912151821242730333601234567 (a) (b) (c) (d) Figure 9: Response to a periodic stimulation of frequency 5. 5 Hz. (a) Impulse response function with global 95% confidence interval (b) |ˆcn|versus index n with 95% confidence interval. Dots indicate points which achieve d significance in the F-test. (c) Residual spectrum with finite size corrected con fidence interval. A multitaper spectrum with 100 tapers and a bandwidth of 1.5 Hz was used initially to avoid overlap of harmonics. This spectrum was then furthe r smoothed using a Gaussian lag window with standard deviation 9 Hz. (d) The coe fficient phases ˆφn (in radians) versus index n after subtraction of a fitted stra ight line of gradient 2π/3±0.01. The black dashed lines are a 95% confidence interval about zero. The estimated impulse response function ˆλ(t) is seen to have two distinct sharp peaks outside of which the response does not differ significan tly from zero. The moduli of the Fourier coefficients are significant out to n= 25. This automatically sets the smoothing of ˆλ(t) as structure on a time scale of less than 1 /(25×5.5) = 7 ms does not achieve significance. Note that the coefficients a re enhanced at multiples of 6 (i.e. ∼33 Hz) which comes from having two peaks in the time 26Jkn a(f) =/integraltextT 0hk(t)e−2πiftdNn a(t) Ikn ab(f) =Jkn a(f)Jkn∗ b(f) Table 1: The basic direct spectral estimator in terms of whic h the other estimators can be written. For clarity the superscript Don the direct spectral estimate has been omitted. The index nlabels trials, index klabels tapers, and indices aand blabel cells. domain λ(t) which are separated by ∼30 ms. The phase of the coefficients closely follows a straight line but there is a small periodic deviati on from this line which is again at index multiples of 6. The gradient of the straight li ne depends on the time delay of the response. The residual spectrum was calculated by first evaluating a multitaper estimate from which the significant harmonics w ere removed. This spectrum had a bandwidth of 1.5 Hz chosen to avoid overlap of t he harmonics leading to the multitaper estimate being undersmoothed. A f urther smoothing was performed using a lag window16and the resultant spectrum, displays a slight but significant suppression relative to a Poisson process ou t to almost 200 Hz. Such a spectrum is characteristic of a short time scale refra ctive period. The residual spectrum is particularly useful because rate non- stationarity has been removed. 12 Summary It is the belief of the authors that spectral analysis is a fru itful and under exploited analysis technique for spike trains. In this paper an attemp t has been made to collect the machinery necessary for performing spectral an alysis on spike train data into a single document. Starting from the population de finitions the statis- tical properties of estimators of the spectrum and coherenc y have been reviewed. Estimation methods for both continuous spectra and spectra which contain lines have been included. In addition new corrections to asymptot ic error bars have been presented which increase confidence in applying spectr al techniques in prac- tical situations where data is often sparse. Tables 1 to 5 sum marize the important formulae. Matlab software implementing the methods discus sed in this paper is available from xxx.lanl.gov/archive/neuro-sys. 16The previous theory developed for lag window estimators app lies to this hybrid esti- mator with |H(·)|2replaced by1 K/summationtextK−1 k=0|Hk(·)|2in equation 20 and |H(·)|2replaced by 1 K/summationtextK−1 k=0|Hk(·)|2in equation 22. 27X IX ab(f) Eq. ν0 D I01 ab(f) [8 ] 2 DT1 NT/summationtextNT n=1I0n ab(f) [16 ] 2NT LW1 NT/summationtextNT n=1/integraltext∞ −∞K(f−f′)I0n ab(f′)df′[18 ] 2NT/ξ MT1 NTK/summationtextNT n=1/summationtextK−1 k=0Ikn ab(f) [25 ] 2NTK Table 2: The different estimators and the large sample degree s of freedom ν0of estimates of the spectrum ( ab= 11). The indices on the Ikn abare as follows. ab label the cells from which the estimates are constructed. Th e index klabels the taper and nlabels the trial. Equation Eq. Comment useν0for asymptotic Variance var{IX aa(f)}=2E{IX aa(f)}2 ν(f)[27] orν(f) if using finite size correction Degrees1 ν(f)=1 ν0+CX hΦ(f) 2TNTE{IX(f)}2[39] See text for definitions of freedom ofCX hand Φ( f) Confidence/bracketleftBig νIX(f)/q2, νIX(f)/q1/bracketrightBig [30] q1s.tP[χ2 ν≤q1] =p (1−2p)×100% q2s.tP[χ2 ν≥q2] =p Table 3: Main formulae required for estimating spectral err or bars. Refer to section 4 for additional information. Equation Eq. Comment Coherency CX(f) =IX ab√ IXaaIX bb[32] Distribution P(|C|) = (ν−2)|C|(1− |C|2)(ν/2−2)[35] Under nullfor coherence hypothesis γ= 0 Confidence ˆφ(f)±2/radicalbigg 2 ν/parenleftBig 1 |C(f)|2−1/parenrightBig [36] Approx. for phase 95% Table 4: Main formulae required for coherency estimation. R efer to section 5 for additional information. 28Equation Eq. Comment Complex amplitude ˆc1=/summationtext kJk(f1)Hk(0)/summationtext k|Hk(0)|2 [51] of line F-test to access the|ˆc1|2/summationtext k|Hk(0)|2(K−1)/summationtext k|Jk(f1)−ˆc1Hk(0)|2.=F2,2(K−1) [53] Null significance of a line c1= 0 Residual spectrum ˆS(f) =1 K/summationtext k|Jk(f)−ˆc1Hk(f−f1)|2[52] Table 5: Main formulae required for the detection and remova l of a line from the spectrum. Refer to section 8 for additional information. A Derivation of Finite Size Correction The following is an outline derivation of the finite size corr ections described in section 6. Firstly the characteristic functionals (Bartle tt, 1966) for the processes Nand the inhomogeneous Poisson process rate λ(t) are related. CN(θ(t)) =E{exp(i/integraldisplayT 0θ(t)dN)}=Eλ{exp(/integraldisplayT 0λ(t)b(θ(t))dt)} (57) b(θ(t)) =exp[iθ(t)−i T/integraldisplayT 0θ(t′)dt′] (58) Under the Gaussian process assumption for λ(t) this integral may be done. CN=exp[1 2/integraldisplayT 0/integraldisplayT 0b(t)Λ(t, t′)b(t′)dtdt′+λ/integraldisplayT 0b(t)dt] (59) Λ(t, t′) =Eλ{(λ(t)−λ)(λ(t′)−λ)} (60) Note that λdenotes the mean rate. Taking the log of the characteristic f unctionals now yields the following relation between the resultant cum ulant functionals. KN=lnE{exp(i/integraldisplayT 0θ(t)dN)}=1 2/integraldisplayT 0/integraldisplayT 0b(t)Λ(t, t′)b(t′)dtdt′+λ/integraldisplayT 0b(t)dt(61) Next θ(t) is chosen appropriately and substituted into KN. The form for θ(t) which is required to obtain the covariance of multitaper est imators is; iθ(t) =θ1hk(t)e−2πif1t+θ2hk(t)e2πif1t+θ3hk′(t)e−2πif2t+θ4hk′(t)e2πif2t(62) 29Substituting into the cumulant functional for Nyields; KN=lnE{exp(θ1JD k(f1) +θ2JD∗ k(f1) +θ3JD k′(f2) +θ4JD∗ k′(f2))} (63) Where JD kis the Fourier transform of the data tapered by a function ind exed by k. Application of the cumulant expansion theorem (Ma, 1985) t hen leads to; KN=E{exp(θ1JD k(f1) +θ2JD∗ k(f1) +θ3JD k′(f2) +θ4JD∗ k′(f2))−1}C (64) This may then be differentiated and set to zero. Klmno=∂KN ∂θl 1∂θm 2∂θn 3∂θo 4/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle θ1=θ2=θ3=θ4=0=E{JDl k(f1)JDm∗ k(f1)JDn k′(f2)JDo∗ k′(f2)}C (65) Moments of the estimators may be expressed in terms of these c umulant deriva- tives. The expressions are simplified by the fact that all cum ulant derivatives which have indices summing to an odd number are zero because Nis a zero mean process. E{ID k(f)}=K1100 (66) var{IMT(f)}=1 K2K−1/summationdisplay k=0K−1/summationdisplay k′=0cov{ID k(f), ID k′(f)} (67) cov{ID k(f), ID k′(f)}=K1010K0101+K1111+K1001K0110 (68) The problem has now been reduced to that of calculating these derivatives within the model. This is done by substituting the expression for θ(t) into the RHS of equation 61. Considerable algebra then leads to the followi ng exact result. Klmno=KA lmno+KB lmno (69) Where, KA lmno=1 2/summationdisplay li,mi,ni,oil!m!n!o! Πli!Πmi!Πni!Πoi!/bracketleftBigg−H1(f1) T/bracketrightBiggl2+l4/bracketleftBigg−H1(f1)∗ T/bracketrightBiggm2+m4 ... /bracketleftBigg−H1(f2) T/bracketrightBiggn2+n4/bracketleftBigg−H1(f2)∗ T/bracketrightBiggo2+o4 Il1,m1,n1,o1 l3,m3,n3,o3(70) Where/summationtext ili=land cases where l1+l2=lorl3+l4=lare excluded (and also forn, m, o ). Il1,m1,n1,o1 l3,m3,n3,o3=/integraldisplay∞ ∞Sλ(f)Hl1+m1+n1+o1[f1(l1−m1) +f2(n1−o1)−f] ... H∗ l3+m3+n3+o3[f1(l3−m3) +f2(n3−o3)−f]df(71) 30Where Sλ(f) is the spectrum of the Gaussian process and Hlis; Hl(f) =/integraldisplay∞ −∞hl(t)exp(−2πift)dt (72) H0(f) =Texp(−iπfT)sinc(πfT) (73) KB lmno=λl/summationdisplay p=0m/summationdisplay q=0n/summationdisplay r=0o/summationdisplay s=0[lp][mq][nr][os]Hp+q+r+s[f1(p−q) +f2(r−s)]... /bracketleftBigg−H1(f1) T/bracketrightBigg(l−p)/bracketleftBigg−H1(f1)∗ T/bracketrightBigg(m−q)/bracketleftBigg−H1(f2) T/bracketrightBigg(n−r)/bracketleftBigg−H1(f2)∗ T/bracketrightBigg(o−s) (74) The preceding result is somewhat cumbersome but readily eva luated computation- ally for a given spectrum. The expression simplifies greatly when only frequencies above the smoothing width are considered and many of the term s may be ne- glected. Restricting attention to the second order propert ies there are only a few remaining dominant terms. Terms from K1001lead to the previously discussed asymptotic results but there are corrections which arise fr om the term K1111. As- suming that the population spectrum varies slowly over the w idth of the tapers leads to the result given by equations 38 - 43. The validity of this assumption has been tested computationally and was found to be very accurat e even for spectra with sharp peaks. Acknowledgment The authors thank C. Buneo, and R. Sachdev for providing exam ple datasets, Clive Lauder for help with the calculation of global error ba rs and D. R. Brillinger and D. J. Thomson for comments which substantially improved the manuscript. M. Jarvis is grateful to R. A. Andersen both for his continued support of the- oretical work in his lab and also for his careful reading of th e manuscript. M. Jarvis acknowledges the generous support of the Sloan found ation for theoretical neuroscience. References Abeles, M., Deribaupierre, F., and Deribaupierre, Y. (1983 ). 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arXiv:physics/0002054v1 [physics.bio-ph] 29 Feb 2000Evolution of Differentiated Expression Patterns in Digital Organisms Charles Ofria⋆, Christoph Adami†, Travis C. Collier‡, and Grace K. Hsu Beckman Institute California Institute of Technology, Pasadena, California 91125, USA †W.K. Kellogg Radiation Lab 106-38, Caltech, Pasadena, CA 91 125 ‡Division of Organismic Biology, Ecology, and Evolution, UCLA, Los Angeles, CA 90095 ⋆To whom correspondence should be addressed. E-Mail: charle s@gg.caltech.edu Abstract. We investigate the evolutionary processes behind the devel - opment and optimization of multiple threads of execution in digital or- ganisms using the avida platform, a software package that implements Darwinian evolution on populations of self-replicating co mputer pro- grams. The system is seeded with a linearly executed ancesto r capable only of reproducing its own genome, whereas its underlying l anguage has the capacity for multiple threads of execution (i.e., si multaneous expression of sections of the genome.) We witness the evolution to multi- threaded organisms and track the development of distinct ex pression pat- terns. Additionally, we examine both the evolvability of mu lti-threaded organisms and the level of thread differentiation as a functi on of envi- ronmental complexity, and find that differentiation is more p ronounced in complex environments. 1 Introduction Evolution has traditionally been a formidable subject to st udy due to its grad- ual pace in the natural world. One successful method uses mic roscopic organisms with generational times as short as an hour, but even this app roach has diffi- culties; it is still impossible to perform measurements wit hout disturbing the system, and the time-scales to see significant adaptation re main on the order of weeks, at best1. Recently, a new tool has become available to study these problems in a computational medium—the use of populations o f self-replicating computer programs. These “digital organisms” are limited i n speed only by the computers used, with generations in a typical trial taking a few seconds. Of course, many differences remain between digital and simpl e biochemical life, and we address one of the critical ones in this paper. In nature, many chem- ical reactions and genome expressions occur simultaneousl y, with a system of 1Populations of E.coli introduced into new environments begin adaptation immedi- ately, with significant results apparent in a few weeks [3].gene regulation guiding their interactions. However, in di gital organisms only one instruction is executed at a time, implying that no two se ctions of the pro- gram can directly interact. Due to this, an obvious extensio n is to examine the dynamics of adaptation in artificial systems that have the ca pacity for more than one thread of execution (i.e., an independent CPU with i ts own instruction pointer, operating on the same genome). Work in this direction began in 1994 with Thearling and Ray us ing the pro- gram tierra [7]. These experiments were initialized with an ancestor th at creates two threads each copying half of its genome, thereby doublin g its replication rate. Evolution then produces more threads up to the maximum allowed [11]. In subsequent papers [12,9] this research extended to organ isms whose threads are not performing identical operations. This is done in an e nhanced version of the tierra system ( “Network Tierra ” [8]), in which multiple “islands” of digital organisms are processed on real-world machines acr oss the Internet. In these later experiments, the organisms exist in a more compl ex environment in which they have the option of seeking other islands on which t o place their off- spring. The ancestor used for these experiments reproduces while searching for better islands using independent threads. Thread different iation persists only when island-jumping is actively beneficial; that is, when a m eaningful element of complexity is present in the environment. In experiments reported on here, we survey the initial emerg ence of multiple threads and study their subsequent divergence in function. We then investigate the hypothesis that environmental complexity plays a key ro le in the pressure for the thread execution patterns to differentiate. 2 Experimental Details We use the avida platform to examine the development of multi-threading in populations exposed to different environments at distinct l evels of complexity, comparing them to each other and to controls that lack the cap acity for multiple threads. 2.1 The AvidaPlatform Avida is an auto-adaptive genetic system designed for use as a plat form in Arti- ficial Life research. The avidasystem comprises a population of self-reproducing strings of instructions that adapt to both an intrinsic fitne ss landscape (self- reproduction) and an externally imposed (extrinsic) bonus structure provided by the researcher. A standard avida organism is a single genome composed of a sequence of instructions that are processed as commands to the CPU of a vi rtual computer. This genome is loaded into the memory space of the CPU, and the execution of each instruction modifies the state of that CPU. In additio n to the memory, a virtual CPU has three integer registers, two integer stack s, an input/output buffer, and an instruction pointer. In standard avidaexperiments, an organism’sgenome has one of 28 possible instructions at each line. The v irtual CPUs are Turing-complete, and therefore do not explicitly limit the ability for the popu- lation to adapt to its computational world. For more details onavida, see [5]. To allow different sections of a program to be executed in para llel, we have implemented three new instructions. A new thread of executi on is initiated with fork-th . This thread has its own registers, instruction pointer, an d a single stack, all initialized to be identical to the spawning threa d. The second stack is shared to facilitate communication among threads. Only t he new thread will execute the instruction immediately following the fork-th ; the original will skip it enabling the threads to act and adapt independently. If, f or example, a jump instruction is at this location, it may cause the new thread t o execute a differ- ent section of the program ( segregated differentiation ), whereas a mathematical operation could modify the outcome of subsequent calculati ons (overlapping dif- ferentiation ). On the other hand, a no-operation instruction at this posi tion allows the threads to progress identically ( non-differentiated ). We have also im- plemented kill-th , an instruction that halts the thread executing it, and id-th , which places a unique thread identification number in a regis ter, allowing the organism to conditionally regulate the execution of its gen ome. We performed experiments on three environments of differing complexity, with both the extended instruction set that allows multiple expression patterns and the standard instruction set as a control. As individual trials can differ extensively in the course of their evolution, each setup was repeated in two hundred trials to gain statistical significance. The experi ments were performed on populations of 3600 digital organisms for 50,000 updates2. Mutations are set at a probability of 0.75% for each instruction copied, and a 5 % probability for an instruction to be inserted or removed in the genome of a new offspring. The first environment (I) is the least complex, with no explic it environmen- tal factors to affect the evolution of the organisms; that is, the optimization of replication rate is the only adaptive pressure on the popu lation. The next environment (II), has collections of numbers that the organ isms may retrieve and manipulate. We can view the successful computation of an y of twelve log- ical operations that we reward3as beneficial metabolic chemical reactions, and speed-up the virtual CPU accordingly; more complex tasks re sult in larger speed- ups. If the speed increase is more than the time expended to pe rform the task, the new functionality is selected for. The final environment (III) studied is the most complex, with 80 logic operations rewarded. 2An update represents the execution of an average of 30 instru ctions per program in the population. 50,000 updates equates to approximately 90 00 generations and takes about 20 hours of execution on a Pentium Pro 200. The data and c omplete genomes are available at http://www.krl.caltech.edu/avida/pubs/ecal99/ . 3The completion of a logic operation involves the organism dr awing one or more 32- bit integers from the environment, computing a bitwise logi cal function using one or morenandinstructions, and outputting the result back into the envir onment.A record is maintained of the development of the population, including the genomes of the most abundant organisms. For each trial, t hese dominant genomes are analyzed to produce a time series of thread use an d differentiation. 2.2 Differentiation Metrics The following measures and indicators keep track of the func tional differentiation of codes. We keep this initial analysis manageable by settin g a maximum of two threads available to run simultaneously. The relaxation of this constraint does lead to the development of more than two threads with charact eristically similar interactions. Thread Distance measures the spatial divergence of the two instruction pointers. This measurement is the average distance (in units of instructions) between the execution positions of the individual threads. If this value becomes high relative to the length of the genome, it is an indication that the threads are segregated, executing different portions of the genome at an y one time, whereas if it is low, they likely move in lock-step (or sightly offset) with nearly identical executions. Note, however, that if two instruction pointer s execute the code offset by a fixed number of instructions, but otherwise identically , the thread distance is an inflated measure of differentiation because the tempora l offset does not translate into differing functionality. Code Differentiation distinguishes execution patterns with differing be- havior . A count is kept of how often each thread executes each portio n of the genome. The code differentiation is the fraction of instruct ions in the genome for which these counts differ between threads. Thus, this met ric is insensistive to the ordering of execution. Execution Differentiation is a more rigorous measure than code differ- entiation. It uses the same counters, taking into considera tion the difference in the number of times the threads execute each instruction. Th us, if one thread executes a line 5 times and the other executes it 4 times, it wo uld not con- tribute as much towards differentiation as an instruction ex ecuted all 9 times by one thread, and not at all by the other. This metric totals the se differences in execution counts at each line and then divides the sum by the t otal number of multi-threaded executions. Thus, if the threads are perfec tly synchronized, there is zero execution differentiation, and if only one thread exc lusively executes each line, this metric is maximized at one. An execution different iation of 0.5 indicates that half of the instructions did not have matched execution s in each thread. 3 Evolution of Multi-Threaded Organisms For our initial investigations, we focus on the 200 trials in environment III (the most complex), with the extended instruction set, allowing for multi-threading.01234500.20.40.60.81 Updates [x103] Frac. runs multi−threaded A 01234500.20.40.60.81 Updates [x103] Frac. time w/ 2 threads B Fig.1. The time progression of organisms learning to use multiple t hreads av- eraged over 200 trials. (A) The fraction of trials which thre ad at all, and (B) The average fraction of time organisms spend using both thre ads at once. The data displayed here is for the first 5000 updates of 50,000 upd ate experiments in environment III. 3.1 Emergence of Multiple Execution Patterns Describing a universal course of evolution in any medium is n ot feasible due to the numerous random and contingent factors that play key rol es. However, there are a number of distinct trends, which will be discussed furt her. Let us first consider the transition of organisms from a purel y linear execution to the use of multiple threads. In Fig. 1A, we see that most pop ulations do develop a secondary thread near the beginning of their evolu tion. Secondary threads come into use as soon as they grant any benefit to the or ganisms. The most common way this occurs is by having a fork-th and akill-th appear around a section of code, which the threads thereby move thro ugh in lock-step, performing computations twice. Multiple completions of a t ask provide only a minor speed bonus, but this is often sufficient to warrant a dou ble execution. Once multiple execution has set in, it will be optimized with time. Smaller blocks of duplicated code will be expanded, and larger secti ons will be used more productively, sometimes even shrinking to improve effic iency. Once multiple threads are in use, differentiation follows. 3.2 Execution Patterns in Multi-threaded Organisms A critical question is “What effect does a secondary thread ha ve on the process of evolution?” The primary measure to denote a genome’s leve l of adaptation to an environment is its fitness . The fitness of a digital organism is measured as the number of offspring it produces per unit time, normalized to t he replication rate of the ancestor. In all experiments, the fitness of the domina nt genotype starts at one and increases as the organisms adapt. Fitness improve ments come in two forms: the maximization of CPU speed by task completion, and the minimization012345100101102103 Updates [x104] Fitness A 012345020406080100 Updates [x104] Genome Length B Fig.2. (A) Average fitness as a function of time (in updates) for the 2 00 en- vironment III trials. Most increases to fitness occur as a mul tiplicative factor, requiring fitness to be displayed on a logarithmic scale. (B) Average sequence length for the linear execution experiments (Solid line) an d the multiple execu- tion experiments (dashed line). of gestation time. As all tasks must be computed each gestati on cycle to maintain a reward, this gestation time minimization includes the optimization of tasks in addition to speed-ups in the replication process. The ave rage progression of fitness with time is shown in Fig. 2A for both the niche with the expanded instruction set that allows multiple threads, and the stand ard, linear execution niche as a control. Contrary to expectations, the niche that has additional thr eads available gives rise to a slower rate of adaptation. However, the avera ge length of the genomes (Fig. 2B) reveals that the code for these marginally less fit organisms is stored using 40% fewer instructions, indicating a denser encoding. Indeed, the very fact that multi-threading develops spontaneously imp lies that it is bene- ficial. How then can a beneficial development be detrimental t o an organism’s fitness? Inspection of evolved genomes has allowed us to determine th at this code compression is accomplished by overlapping execution patt erns that differ in their final product. Fig. 3A displays an example genome. The i nitial thread of execution (the inner ring) begins in the D“gene” and proceeds clockwise. The execution of Ddivides the organism when it has a fully developed copy of its elf ready. This is not the case for this first execution, so the gen e fails with no effect to the organism. Execution progresses into gene C0where computational tasks are performed, increasing the CPU speed. Near the center of C0, afork-th instruction is executed initiating secondary execution (o f the same code) at line 27, giving rise to gene C2. The primary thread continues to line 55, the Sgene, where genome size is calculated and the memory for its offspri ng is allocated. Next, the primary instruction pointer runs into gene R, the copy loop, where replication occurs. It is executed once for each of the 99 ins tructions in theC002 558089 27 54D S IA B C D EFGH JK A BR A*C CI0 11 298 72 Fig.3. A: Execution patterns for an evolved avida genome. The inner ring displays instructions executed by the initial thread, and t he outer ring by the secondary thread. Darker colors indicate more frequent exe cution. B:Genome structure of the phage ΦX174. The promoter sequence for gene A∗is entirely within gene A, causing the genes to express the same series of amino acids f rom the portion overlapped. Genes B,E, and Kare also entirely contained within others, but with an offset reading frame, such that different a mino acids are produced. genome (hence its dark color in the figure). When this process is complete, it moves on through gene I0shuffling numbers around, and re-enters gene Dfor a final division. During this time, the secondary thread executes gene C2computing a few basic logical operations. C2ends with a jump-f (jump forward) instruction that initially fails. Passing through gene I1, numbers are shuffled within the thread and the jump at line 72 diverts the execution back to the begin ning of the organism. From this point on, its execution loops through C1andC2for a total of 10 times, using the results of each pass as inputs to the next, computing different tasks each time. Note that for this organism, the secondary t hread is never involved in replication. Similar overlapping patterns app ear in natural organisms, particularly viruses. Fig. 3B exhibits a gene map of the phag eΦX174 containing portions of genetic code that are expressed multiple times, each resulting in a distinct protein [13]. Studies of evolution in the overlapp ing genes of ΦX174 and other viruses have isolated the primary characteristic hampering evolution. Multiple encodings in the same portion of a genome necessita te that mutations be neutral (or beneficial) in their net effect over allexpressions or they are selected against. Fewer neutral mutations result in a reduc ed variation and in turn slower adaptation. It has been shown that in both viruse s [4] and Avida organisms [6], overlapping expressions have between 50 and 60% of the variation of the non-overlapping areas in the same genome, causing gen otype space to be explored at a slower pace.01234502468 Updates [x104] Thread Distance A 01234500.050.10.150.2 Updates [x104] Frac. Thread Distance B 01234500.51 Updates [x104] Code Differentiation C 01234500.10.20.30.4 Updates [x104] Execution Differentiation D Fig.4. Differentiation measures averaged over all trials for each e xperiment. (A) Thread Distance, (B) Fractional Thread Distance, (C) Co de Differentiation, (D) Expression Differentiation. Experiments from environm ent III (solid line), environment II (dashed line), and environment I (dotted lin e) In higher organisms, multiple genes do develop that overlap in a portion of their encoding, but are believed to be evolved out through gene duplication and specialization, leading to improved efficiency [2]. Unfo rtunately, viruses and avidaorganisms are both subject to high mutation rates with no err or correction abilities. This, in turn, causes a strong pressure to compre ss the genome, thereby minimizing the target for mutations. As this is an immediate advantage, it is typically seized, although it leads to a decrease in the adap tive abilities of the population in the long term. 3.3 Environmental Influence on Differentiation Now that we have witnessed the development of multiple threa ds of execution inavida, let us examine the impact of environmental complexity on th is process. Populations in all environments learn to use their secondar y thread quite rapidly, but show a marked difference in their ability to diverge the th reads into distinct functions. In Fig 4A, average Thread Distance is displayed f or all trials in each environment showing a positive correlation between the div ergence of threads and the complexity of the environment they are evolving in. More complex environments provide more information to be st ored within the organism, promoting longer genomes [1], and possibly bi asing this measure. To account for this, we consider this average thread distanc e normalized to the length of the organisms, displayed in Fig 4B. When threads fu lly differentiate,they often execute neighboring sections of code, regardles s of the length of the genome they are in, biasing this measurement in the opposite direction. Longer genomes need their threads to be further spatially different iated in order to obtain an equivalent fractional thread distance. Thus, the fact that more com- plex environments give rise to a marginally higher fraction al distance is quite significant. Interestingly, Code Differentiation (Fig 4C) does not firmly distinguish the environments, averaging at about 0.5. In fact, the distribu tion of code differ- entiation turns out to be nearly uniform. This indicates tha t the portion of the genomes that are involved with the differentiated thread s are similarly dis- tributed between complexity levels. Execution Differentia tion (the measure of the fraction of executions that occurred differently betwee n threads, shown in Fig 4D), however, once again positively correlates environ ments with thread di- vergence. The degree of differentiation between the executi on patterns is much more pronounced in the more complex environments. 4 Conclusions We have witnessed the development and differentiation of mul ti-threading in dig- ital organisms, and exhibited the role of environmental com plexity in promoting this differentiation. Although this is an inherently comple x process, the ability to examine almost any detail and dynamic within the framewor k ofavida pro- vides insight into what we believe are fundamental properti es of biological and computational systems. The patterns of expression (lock-step, overlapping, and sp atial differentia- tion) are selected by balancing the “physiological” costs o f execution and differ- entiation against the implicit effects of mutational load. C learly, multiple threads executing single regions of the genome provides for additio nal use of that region. The benefit is in the form of additional functionality and a re duction in the mu- tational load required for that functionality. Within the c ontext of this thinking, the correlation between environmental complexity and the u sage of multiple threads makes a great deal of sense: multiple threads are adv antageous only if they can provide additional functionality. However, we have witnessed the cost side in this equation: wh en a gene or gene product is used in multiple pathways, variations are reduce d as the changes to each gene must result in a net benefit to the organism. We obser ved a negative correlation between rates of adaptation and use of multiple threads. Further- more, the ability to analyze the entropy of each site in the ge nome quantifies the loss in variability predicted by this hypothesis. This entr opy analysis has been carried out in a biological context by Schneider [10], openi ng up opportunities to verify our results. Implications of this work with potentially far reaching con sequences for Com- puter Science involve the study of howthe individual threads interact and what techniques the organisms implement to obtain mutually robu st operations. The internal interactions within computer systems lack the rem arkable stability ofbiological systems to a noisy, and often changing environme nt. Life as we know it would never have reached such vast multi-cellularity if eve ry time a single com- ponent failed or otherwise acted unexpectedly, the whole or ganism shut down. Clearly, we are still taking the first steps in developing sys tems of computer programs that interact on similarly robust levels. Here we h ave performed ex- periments on a simple evolutionary system as a step towards d eciphering these biological principles as applied to digital life. In the fut ure, we plan to add ex- plicit costs for multi-threading that depend on the localavailability of resources for thread execution. Systems at levels of integration anyw here near that of bio- logical life are still a long way off, but more concrete concep ts such as applying principles from gene regulation to develop self-schedulin g parallel computers may be much closer. Acknowledgements This work was supported by the National Science Foundation u nder Grant No. PHY-9723972. G.H. was supported in part by a SURF fellowship from Caltech. Access to a Beowulf system was provided by the Center for Adva nced Computing Research at the California Institute of Technology. We woul d like to thank an anonymous referee for useful comments. References 1. Adami C, Introduction to Artificial Life (Telos Springer-Verlag, New York, 1998). 2. Keese P and Gibbs A, Origins of genes: “Big bang” or continu ous creation?, Proc. Natl. Acad. Sci. 89: 9489-9493 (1992). 3. Lenski RE and Travisano M, Dynamics of adaptation and dive rsification: A 10,000 generation experiment with bacterial populations, Proc. Nat. Acad. Sci. 91: 6808- 6814 (1994) 4. Miyata T and Yasunaga T, Evolution of overlapping genes, Nature 272: 532 (1978). 5. Ofria C, Brown CT, and Adami C, The Avida User’s Manual , in [1] 297-350 (1998). 6. Ofria C and Adami C, Evolution of genetic organization in d igital organisms, Proc. of DIMACS Workshop on Evolution as Computation , Jan 11-12, Princeton Univer- sity, Landweber L and Winfree E, eds. (Springer-Verlag, NY, 1999). 7. Ray TS, An approach to the synthesis of life, in Proc. of Artificial Life II , Langton CG, Taylor C, Farmer JD, and Rasmussen S, Eds. (Addison Wesle y, Redwood City, CA, 1992), p. 371. 8. Ray TS, A proposal to create a network-wide biodiversity r eserve for digital organ- isms, ATR Technical Report TR-H-133 (1995). 9. Ray TS and Hart J, Evolution of differentiated multi-threa ded digital organisms, inProc. of Artificial Life VI , Adami C, Belew RK, Kitano H, and Taylor CE, eds. (MIT Press, Cambridge, MA, 1998), p. 295. 10. Schneider TD, Stormo GD, Gold L, and Ehrenfeucht A, Infor mation content of binding sites on nucleotide sequences J. Mol. Biol. 188: 415-431 (1986). 11. Thearling K and Ray TS, Evolving multi-cellular artifici al life, in Proc. of Artificial Life IV , Brooks RA and Maes P, Eds. (MIT Press, Cambridge, MA, 1994), p. 283. 12. Thearling K and Ray TS, Evolving parallel computation, Complex Systems 10: 229-237 (1997).13. Watson JD, et al., Molecular Biology of the Gene (Fourth Ed., Benjamin Cum- mings, Menlo Park, 1987).

arXiv:physics/0003001v1 [physics.gen-ph] 1 Mar 2000DYNAMIC MODEL OF WAVE - PARTICLE DUALITY AND SUPERUNIFICATION by: Alex Kaivarainen University of Turku, JBL, FIN-20520, Finland http://www.karelia.ru/˜alexk H2o@karelia.ru SUMMARY The new Dynamic Model of Wave-Particle Duality elaborated i s devoted to analysis of the internal (hidden) parameters of e lemen- tary de Broglie waves (waves B) and their interrelation with external, observable ones. Dynamic model is based on assumption of alt erna- tive pulsation of sub-elementary particles, between the co rpuscular [C] and wave [W] phase, each of them representing correspond ing semiperiod of wave B. The triplets of sub-elementary partic les form the elementary particles. The energy and impulse of sub-ele mentary particles and antiparticle, composing coherent pair, comp ensate each other. The resulting properties of each triplet: mass, spin , charge is determined by uncompensated sub-elementary particle. The model considers the positive and negative vacuum as two ’oceans’ o f su- perfluid quantum liquid, composed from virtual quanta of opp osite energies. This interacting positive and negative vacuum, t ermed BI- VACUUM, is an infinitive source of bi-vacuum bosons (BVB). Th e spatial image of BVB is a pair of [rotor + antirotor] of opposi te di- rection of rotation in the energetic planes of positive and n egative vacuum with the radius, close to Compton radius of the electr on, muon or tau-electron, depending on the resonant energetic s lit value of bi-vacuum. Due to impulse and energy compensation there a re no size limitations for BVB. The minimum radius of BVB may be det er- mined by Plank’s length. Such small BVB may serve as a ”molecu les” of bi-vacuum as a quantum superfluid liquid. The nonlocal pro perties of BVB are the consequence of their zero resulting impulse, r esponsi- ble for their infinitive virtual wave B length and correspond ing scale of PRIMORDIAL bi-vacuum Bose condensation. In SECONDARY vacuum, with symmetry, perturbed by presence of matter, the im- pulse of BVB becomes non zero. As a result, the infinitive virt ual Bose condensate ’dissociate’ to huge superfluid bi-vacuum d omains. The gradient of difference of concentration of rotors and ant irotors with opposite direction of rotation (virtual spin equilibr ium shift), originated under the influence of rotating atoms, molecules or macro- scopic bodies - is responsible for TORSION field. The transmi ssion of signal in form of energetic slit symmetric pulsation of bi -vacuum, representing the vacuum amplitude waves (VAW) in the membra ne of 1bi-vacuum Bose condensate is instant. It is not related with energy- impulse propagation. The Corpuscular and Wave phase of part icles are considered in our model as two alternative phase of wave B , which are in dynamic equilibrium. The wave [W]- PHASE in form of cumulative virtual cloud (CVC) originates as a result of q uan- tum beats between real and mirror states of corpuscular [C] p hase of sub-elementary particles, composing elementary partic les. It is shown that CVC has a spatial image of half of parted (two-cavi ty) hyperboloid. This half is in realm of positive or negative va cuum for sub-particles and sub-antiparticles, correspondingl y. The CVC is composed from virtual density waves (VDW), responsible for elec- tric component of resulting electromagnetic charge and fro m virtual symmetry waves (VSW), related to magnetic component of resu lting charge. The product of two components is equal to resulting charge sq uared. The VAW interference with standing VDW and VSW (scalar waves ), excited by symmetric pairs of sub-particle and sub-antipar ticle may be responsible for Informational field origination. It leads from equations obtained, that small part of CVC ener gy determined by fine structure constant is responsible for ele ctromag- netism and much smaller part of CVC - for gravitation. Our mod el unifies electromagnetism and gravitation in natural way. Th e restora- tion of [C]-PHASE in form of [real+mirror] mass-dipole is a r esult of binding of CVC to BVB, accompanied by excitation of BVB. The spatial image of sub-particle in [C]-phase is a correlated p air [mirror rotor + real vortex]. The energies of both phase [C] and [W] ar e equal. Propagation of fermion in bi-vacuum is a jump-way process, b e- cause the [W] is luminal in contrast to [C] phase. The frequen cy of [C-W] pulsation is equal to frequency of quantum beats betwe en real and mirror corpuscular states. Our dynamic duality model elucidates the quantum backgroun d of non-locality, principle of least action and Golden mean, unifies the quantum and relativist theories. Tending of the open sys tems to conditions of Golden mean is supposed to be a driving force of their self-organization. It is shown also that the pace of time - ch anges with opposite sign with pace of kinetic energy for each selected c losed sys- tem. It leads from formulae obtained, that pace of time is int errelated with electromagnetic and gravitational energy change. The notions of TEMPORAL waves and field has been introduced. Our approach serves as a key to Superunification. Introduction Einstein never accepted the Bohr’s philosophy, that proper ties of particles cannot be analyzed without direct experimental control. Bo hr’s objection of 2EPR paradox was based on this point. David Bohm was the first one, who made an attempt to explain who leness of the Universe, without loosing the causality principle. Exp erimental discovery: ”Aharonov-Bohm effect” (1950) pointing that electron is abl e to ”feel” the pres- ence of a magnetic field even in a regions where the probabilit y of field existing is zero, was stimulating. For explanation of nonlocality Bo hm introduced in 1952 the notion of quantum potential, which pervaded all of s pace. But unlike gravitational and electromagnetic fields, its influence did not decrease with dis- tance. All the particles are interrelated by very sensitive to any perturbations quantum potential. This means that signal transmission bet ween particles oc- curs instantaneously. The idea of quantum potential or acti ve information is close to notion of pilot wave, proposed by de Broglie at the So lvay Congress in 1927. In our model instead quantum potential we introduce d the notion of informational vacuum amplitude waves (VAW). These waves ha ve a concrete interpretation in the framework of our model. Actually Bohm develops the de Broglie idea of pilot wave, app lying it for many-body system. In 1957 Bohm published a book: Causality a nd Chance in Modern Physics. Later he comes to conclusion, that Universe has a properties of giant, flowing hologram. Taking into account its dynamic n ature, he prefer to use term: holomovement. In his book: Wholeness and the Imp licate Order (1980) he develops an idea that our explicated unfolded real ity is a product of enfolded (implicated) or hidden order of existence. He cons ider the manifesta- tion of all forms in the universe as a result of enfolding and u nfolding exchange between two orders, determined by super quantum potential. According to Bohm, manifestation of corpuscle - wave dualit y of particle is dependent on the way, which observer interacts with a syst em. Both of this properties are always enfolded in a quantum system. It is a ba sic difference with our model, assuming that the wave and corpuscle phase are rea lized alternatively with high frequency during two different semiperiods of de Br oglie wave (wave B). Bohm, like Einstein, rejected the statement, that particle s can not be consid- ered until they are observed. In his last book, written with B asil Hiley: ”THE UNDIVIDED UNIVERSE. An ontological interpretation of quan tum theory” (1993), he considered the electron as a particle with well- d efined position and momentum which are, however, under influence of special wave (quantum po- tential). Particle in accordance with this authors is a sequ ence of incoming and outgoing waves, which are very close to each other. However, particle itself does not have a wave nature after Bohm. Interference pattern in do uble slit exper- iment is a result of periodically ”bunched” character of qua ntum potential in Bohm’s view. In accordance to first version of our model (see next section) , one of two standing neutrino (2 ν0) in the coherent triplet [2 ν0+/tildewideν0],composing the electron as a coherent dynamic system, is always in the corpuscular st ate in each of two semiperiods. The pair of standing neutrino and antineutrin o are pulsing between Corpuscular [C] and Wave [W] states in-phase, compensating the influence of 3energy, spin and charge of each other on vacuum symmetry. The bunched character of the electron’s trajectory can be a result of imp ulses, produced by uncompensated standing neutrino in a course of its [ C⇋W] pulsations, accompanied by outgoing and incoming Cumulative Virtual Cl oud (CVC). At this point our model has some similarity with idea of Bohm. However, our duality model can explain the nonlocality and d ouble slit experiment without using the notion of quantum potential or pilot-wave, but by the internal (hidden) dynamics of the components of eleme ntary particles. The important point of Bohmian philosophy, coinciding with our theory, is that everything in the Universe is a part of dynamic continuu m. Neurophysiologist Karl Pribram made the next exciting step in the same di- rection as Bohm: ”The brain is a hologram enfolded in a hologr aphic Universe”. The good popular description of Bohm and Pribram ideas are pr esented in books: The Bell’s theorem and the curious quest for quantum r eality (1990) by David Peat and ”The Holographic Universe” (1992) by Michael Talbot. Such original concepts are interesting and stimulating, in deed, but should be considered as a first attempts to transform intuitive perc eption of duality and quantum wholeness into clear geometrical and mathemati cal models. In 1950 John Wheeler and Charles Misner published Geometrod ynamics, a new description of space-time properties, based on topolo gy. Topology is more general than Euclidean geometry and deeper than non-Eu clidean, used by Einstein in his General theory of relativity. Topology do es not deal with distances, angles and shapes. Drawn on a sheet of stretching rubber, a circle, triangle and square are indistinguishable. A ball, pyramid and a cube also can be transformed into the other. However, objects with holes i n them can never be transformed by stretching and deforming into objects wit hout holes. For example black hole can be described in terms of topology. It means that massive rotating body behave as a space-time hole. Whee ler supposed that elementary particles, their spins, positive and negative c harges can be presented as interconnected black and white holes. Positron and elect ron pair correspond to such model. The energy, directed to one of the hole, goes th row the connecting tube -”handle” and reappears at the other. The connecting tube exist in another space-time than holes i tself. Such a tube is undetectable in normal space and the process of energ y transmission looks as instantaneous. In conventional space-time two end s of tube, termed ’worm holes’ can be a vast distant apart. It gives an explanat ion of quantum nonlocality. Like Bohm’s quantum potential, the Wheeler’s quantum topology remains fascinating but unproved hypothesis. The most serious attack on problem of quantum nonlocality wa s performed by Roger Penrose (1989) from Oxford University with his twis ter theory of space-time. His ideas are quite close to those, developed in my dynamic wa ve-particle duality model. For example, in accordance with Penrose, qua ntum phenomena can generate space-time. The twisters, proposed by Penrose , are lines of infinite extent, resembling twisting light rays. Interception or co njunction of twistors 4lead to origination of particles. In such a way the local and n onlocal properties and particle-wave duality are interrelated in twistor geom etry. In our model the coherent triplets of standing neutrinos and antineutrinos, representing fermions, could be resulted from free neutrin o (or antineutrino) strings interception with virtual [neutrino+antineutrin o] pairs, represented by symmetric bi-vacuum excitations (excited bi-vacuum boson s). Corresponding vacuum symmetry breach is responsible for [mass–velocity- space-time] origina- tion, pertinent only for Corpuscular [C] phase. The analysis of main quantum paradoxes was presented by Ashe r Peres (1992) and Charles Bennett et.al, (1993). One of the most important question is related with possibili ty of existing of hidden parameters. Searching of such parameters was strong ly discouraged by a theorem of Von Neumann (1955), claiming to show their to be u nnecessary for explanation the known quantum phenomena (see also N. Mer min, 1990). Bohm proved his disagreement with formal statistical inter pretation of quantum theory and with conclusions of Von Neuman (Bohm & Hiley, 1993 ), concerning nonexistence of hidden parameters. We assume in our model that hidden (internal) parameters of e lementary particles are existing. In corpuscular phase they are inter related in definite way with external, experimentally detectable parameters. Basic notions of new model This work is devoted to analysis of the internal (hidden) par am- eters of elementary de Broglie waves termed waves B and their interrelation with external, observable ones, in the frame work of new wave-particle duality model. The corpuscular [C] and the wa ve [W] states are considered in our model as two alternative phase o f wave B, which are in dynamic equilibrium. We discuss two possible interrelated model versions of elem en- tary particles formation and their high frequency [C ⇋W] pulsa- tions. Each of them is based on consideration of positive and nega- tive vacuum as two ”oceans’ of superfluid quantum liquid, com posed from virtual quanta of opposite energies. This interacting positive and negative vacuum, termed BI-VACUUM, is composed from in- finitive number of bi-vacuum bosons (BVB) in form of virtual [rotor+antirotor] pairs. The resulting energy and momentum of BVB at the conditions of their virtual spins equilibrium are equal to zero due to their opposite direction of rotation in the ene rgetic planes of positive and negative vacuum. The dimensions of BV B are determined by the values of opposite impulses of rotor and an tirotor in accordance with principle of uncertainty in coherent for m. The notion of bi-vacuum allows the dimensions of rotor and antir otor to vary for many orders from each other as far their momentum and 5energy compensate each other. Elementary and subelementar y par- ticles in their [C] and [W] phase may be considered as a result of hierarchical self-organization of BVB in their symmetrica l (ground) and asymmetrical (excited) states. The nonlocality of bi-vacuum bosons interaction is determi ned by their zero resulting impulse, which determines the infiniti ve virtual wave B length and the infinitive by scale PRIMORDIAL bi-vacuu m Bose condensation. The symmetric primordial bi-vacuum exi sts in the total absence of matter and at the wave fronts, correspon ding to [W] phase of matter. The primordial bi-vacuum symmetry sh ift, induced by presence of mater as a [C] phase of waves B, means or ig- ination of SECONDARY bi-vacuum, composed from BVB with very small, but nonzero resulting impulses. It means the fragmen tation of the infinitive bi-vacuum Bose condensate to huge, but finit e bi- vacuum domains. The dimensions of these domains are determi ned by corresponding virtual de Broglie wave (wave B) length of bi- vacuum bosons, composing them. It is assumed that any kind of Bose condensate: real or virtual one - has nonlocal properties, corresponding to instant virtual signal transmission between particles, pu l- sating [C ⇋W]in-phase. The signal transmission is medi- ated by oscillation of energetic slit between ground states of positive and negative vacuum (Vacuum Amplitude Waves - ”VAW”). Due to symmetry as respect to positive and negative vac- uum the VAW and their interference are not related with impul se- energy transmission. However, they may be responsible for s torage of information in form of corresponding standing waves, i.e . Infor- mational field origination in bi-vacuum. The resulting impulse of bi-vacuum bosons (BVB), equal to ze ro in [W] phase and very small in [C] phase, as well as correspondin g virtual wave B length - remain unchanged in a course of VAW excitation . The BVB Bose condensate, produced by infinitive number of pai rs [rotor+antirotor] of BVB, serves as a background for [ C⇋W]duality realization, as will be shown below. The virtual vortexes, rotors and BVB, as a components of sub- elementary particles, introduced in our model, are the resu lt of col- lective excitations of virtual microparticles with simila r angle velocity, composing bi-vacuum as a quantum superfluid liquid. In the first model’s version it is assumed that just a neutrino, antineutrino and their pairs are the primordial sub-elemen tary par- ticles, building the elementary particles, like electrons , positrons, quarks, nucleons, etc. It is assumed that at certain conditi ons the free neutrino turns to the standing one, leading to fusion of regular elementary particles. This could happen when trajectories of free neutrinos and antineutrinos ( e, µ, τ )change to the closed ones, cor- 6responding to their standing waves length conditions. It ma y occur, for example, in a course of strong vacuum symmetry fluctuatio ns, accompanied by [origination ⇋evaporation] of mini-black holes with Plank’s parameters. The spatial image of each standing neutrino in [C] phase as a [real+mirror] mass dipole is asymmetric pair of [real vortex + mir- ror rotor] with radiuses, determined by Compton length of re al and mirror masses. The realization of [C - W] duality is a consequ ence of quantum beats between real [C+]and mirror [C−]corpuscular states, accompanied by ejection of cumulative virtual cloud (CVC), repre- senting [W] phase. Spatial image of [W] phase in both model versions is a half of parted hyperboloid, i.e. hyperbolic vortex in positive reg ion of bi- vacuum for particles and in negative - for antiparticles. Bosons are considered as a coherent group, formed by the inte ger number of pairs of fermions like standing [neutrino + antine utrino]. For example, it is proposed that photons are composed from th ree such pairs. The triplets of coherent standing neutrino (ν0)and antineutrino (/tildewideν0)in ratio 2:1 and 1:2 - form the fermions: electron (2ν0+/tildewideν0)and the positron ( ν0+ 2/tildewideν0)correspondingly. Such fermions, containing one uncompensated (ν0)or (/tildewideν0)can originate also due to high-energy photons ”splitting” under conditions of strong bi-vacuum s ymmetry shift. A new fundamental Coherent Neutrino/Antineutrino Interac tion, responsible for stabilization of elementary particles is i ntroduced. The exchange of virtual quanta in form of CVC between standin g neutrinos (sub-elementary particles) in a course their cou nterphase [C⇋W]pulsation, electromagnetic and gravitational interactio n be- tween sub-elementary particles - contribute to stabilizat ion of ele- mentary particles. The u-quark is considered as a superposition of 2 positron- l ike structures u ˜ [e++e+]u.The d-quark can be composed from two elec- trons and one positron - like structures: d ˜ [2e−+1e+]d. Each of exces- sive standing neutrino and antineutrino has an electric cha rge, equal to +1/3 and -1/3 correspondingly. In such a model the proton: p = [2u+d]contains more standing antineutrino (12 /tildewideν0)than neutrino (9 ν0).The neutron: n = [d+2u]is composed from the equal number of standing neutrino and anti neu- trino (12ν0)and (12/tildewideν0).Each proton contains three excessive standing antineutrinos with resulting spin and charge, opposite to t hat of the electron. In second version of model of wave B we do not use the notions of standing neutrino and antineutrino as a sub-elem entary particles. The real and mirror mass origination is assumed t o be a 7result of BVB symmetry breach, accompanied by origination o f the same spatial images of [C] phase as the [real vortex + mirror r otor] dipole and that of [W] phase, as the real hyperbolic vortex. F or sub-elementary particle real vortex is located in positive part of bi- vacuum and mirror rotor in zero-point energetic plane of neg ative part of bi-vacuum. For sub-elementary antiparticle, the si tuation is opposite. In this model version, in contrast to the first one, elementar y par- ticles are secondary phenomena, resulted from bi-vacuum sy mmetry certain violations. Three generations of standing neutrino, electron and quark cor- respond to three different and stable values of energy slit be tween rotor and antirotor of (BVB) e,µ,τ.Free neutrino of each generation is defined as the collective nonlocal excitation of correspo nding kind of BVB Bose condensate in form of (VAW )e,µ,τ. The difference in two model versions does not influence the mai n results of our theory. GENERAL DESCRIPTION OF DUALITY MODEL Corpuscle-Wave duality is supposed to be a result of high- frequency oscillations (quantum beats) between the real an d mirror asymmetric states of particles. In the course of [C ⇋W]pulsations the part of real and mirror corpuscular mass reversibly transforms to number of positi ve and neg- ative virtual quanta, forming cumulative virtual cloud (CV C). This virtual cloud corresponds to [W] - phase of particle and exci tes the secondary virtual waves in bi-vacuum in accordance with Hug ence principle. The asymmetry of real and mirror mass-energy dis tri- bution in bi-vacuum is a primary reason for electromagnetis m and gravitation in our model. We consider each of standing neutrino or two other described above bi-vacuum excitations, composing elementary partic les in the CORPUSCULAR [C] - phase as a Mass-Dipole , represented by real/parenleftbig m+ C/parenrightbig and ”mirror”/parenleftbig m− C/parenrightbig masses, corresponding to correlated excita- tion of positive and negative vacuum sublevels. These masse s can be positive and negative in general case: (±m+ C)and (±m− C).However, their product: (±m+ C)·(±m− C) =m2 0=const (1) is always positive and equal to the electron’s mass of rest (m0)squared. In general case (m0)e,µ,τis equal to mass of rest of µandτelectrons. The frequency and energy of quantum beats between positive a nd negative vacuum in a course of [C ⇋W]pulsations determine the 8frequency and energy of wave B. It will be shown below, that at con- ditions of Golden mean realization (22b) this frequency is d etermined by (m0): ωS=m0c2//planckover2pi1= 9.03·1020s−1(1a) In accordance to relativist mechanics and (1), the real and m irror masses - change with external group velocity ( v≡vgr)of particles in the counterphase manner: m+ C=±m0/[1−(v/c)2]1/2and (2) m− C=±m0·[1−(v/c)2]1/2(3) Dividing eq.(3) to (2), we get: 1−m− C m+ C= (v/c)2(4) or:m− C m+ C= 1−(v/c)2(5) Eqs. 2 and 3 can be transformed to shape, close to that, obtain ed by Dirac /parenleftbig E+ C/parenrightbig2= (m+ C)2c4=m2 0c4+ (m+ Cv)2c2(6) /parenleftbig E− C/parenrightbig2= (m− C)2c4=m2 0c4−(m0v)2c2(7) where:E+ CandE− Care the real and mirror energy of wave B. Adding (6) and (7) we got the formulae for zero-point resulti ng energy, taking into account both real and mirror corpuscula r masses of one of standing neutrino in composition of elementary par ticle, like electron: 2E2 0= 2m2 0c4=E2 tot−/parenleftbig P± C/parenrightbig2·c2= (8) =c4[(m+ C)2+ (m− C)2]−c2v2[(m+ C)2−m2 0] It is possible after some reorganizations of (8) and using (5 ), to get the formulae for the resulting (hidden) impulse of mass- dipole of standing neutrino or that of rotor-dipole (9). The resultin g impulses of [C] and [W] phases are equal to each other and less than real impulse of particle (9a), detectable in experiment. P± C=m+ C·(v2/c) = (m+ C−m− C)c=P± W (9) P+ C=P+ W=m+ C·v[P+ C>P± C] (9a) 9Consequently, the real wave B length ( λ+ C=h/P+ C)is shorter, than that of mass-dipole ( λ± C=h/P± C). From (4) we can find out the difference between total energies o f real and mirror states of [C] phase. It is equal to energy of qu antum beats between these states, which determines the energy of w ave B in both phase [C] and [W] and that of cumulative virtual cloud (C VC) [EB=EC=EW=ECV C] : EW=/planckover2pi1ωB= (m+ C−m− C)·c2=m+ C·v2= 2T+ kin=ECor: (10) EW=ECV C=/vextendsingle/vextendsinglem+ C−m0/vextendsingle/vextendsingle·c2+/vextendsingle/vextendsinglem− C−m0/vextendsingle/vextendsingle·c2=E+ V DW+E− V SW (10a) where :m+ C>m0>m− Cand m2 0=m+ C·m− C (10b) We subdivide the total energy of [W] phase in form of CVC - to virtual Vacuum Density Waves (VDW) of positive vacuum and to virtual Vacuum Symmetry Waves (VSW) of negative vacuum. The mirror VSW are not related to the impulse-energy transmissi on and can be superluminal: E+ V DW =/vextendsingle/vextendsinglem+ C−m0/vextendsingle/vextendsingle·c2∼T+ kin(11) E− V SW=/vextendsingle/vextendsinglem− C−m0/vextendsingle/vextendsingle·c2∼V−(11a) The cumulative virtual cloud (CVC), corresponding to the Wa ve [W] phase, propagates in bi-vacuum with luminal velocity. T he com- ponent of CVC, corresponding to VDW, propagates in the posit ive vacuum in the case of particles and in the negative vacuum in t he case of antiparticles. The velocity of propagation of parti cle in real [C] phase is limited by particle’s external group velocity a nd can be much lower than light velocity of CVC, corresponding to [W] p hase. It means that propagation of particle in space in a course of its [C⇋W] pulsation has a jump-way character. The energies of both pha se are equal and can be presented in a few forms: E± W=c2[m+ C−m− C] =/planckover2pi1[ω+ C−ω− C] =/planckover2pi1ωB=m+ C·v2=E± C(12) or:E± C=/vextendsingle/vextendsinglem+ C−m0/vextendsingle/vextendsingle·c2+/vextendsingle/vextendsinglem− C−m0/vextendsingle/vextendsingle·c2=T+ k+V−=E± W(12a) or:E± W=/planckover2pi1[ω+ C−ω0] +/planckover2pi1[ω− C−ω0] =E+ V DW+E− V SW (12b) The ratio of the effective mass of cumulative virtual quanta o f [W]- phase (m+ C−m− C)to the real mass of [C]- phase ( m+ C)is equal to : 1− m− C/m+ C= (v/c)2(see also eq.4). The characteristic frequencies of real and mirror corpuscu lar states are defined as: ω+ C=m+ C·c2 /planckover2pi1;ω− C=m− C·c2 /planckover2pi1;ω0=m0·c2 /planckover2pi1(13) 10The frequency of beats is equal to ω± W= [ω+ C−ω− C]and the resulting frequency of beats is equal to: ωR=ω+ C+ω− C 2(14) T+ kandV−are the kinetic and potential energy of CVC, corre- spondingly. The period of quantum beats is equal to: TB= 2π/ ω B= 2π/[ω+ C−ω− C] (14a) This phenomena reflects the pulsation of wave B in a course of i ts [W⇋C]reversible transitions and energy exchange with bi-vacuum . From the known formulae, interrelated the energy of the rela tivist wave B with its external group ( v=vgr)and phase ( vph)velocities: EB=EC=V+Tk=m+ C·v2=m+ C·vgrvph (14b) and: 2vph vgr−1 =V Tk one can see, that for CVC the condition under consideration: 2Tk= EB=V+Tkcorresponds to that of harmonic oscillator or standing wave: V=Tkat v gr=vph (14c) SPATIAL IMAGES The spatial images of elementary wave B in [C] and [W] phase ca n be analyzed in terms of the wave numbers or energy distributi on, if we transform the basic equations for real and mirror energy, squared (6 and 7) to forms: for real[C+]state:/parenleftbiggm+ C·c /planckover2pi1/parenrightbigg2 −/parenleftbiggm+ C·v /planckover2pi1/parenrightbigg2 =/parenleftBigm0c /planckover2pi1/parenrightBig2 (15) for mirror [C−]state:/parenleftbiggm− C·c /planckover2pi1/parenrightbigg2 +/parenleftBigm0·v /planckover2pi1/parenrightBig2 =/parenleftBigm0c /planckover2pi1/parenrightBig2 (15a) The spatial image of energy distribution of real corpuscula r state [C+], defined by equation (15), corresponds to equilateral hyper- bola (Fig.1a): [C+] :X2 +−Y2 +=a2(15b) 11where:X+=/parenleftbig k+ C/parenrightbig tot=m+ C·c//planckover2pi1;Y+=m+ C·v//planckover2pi1;a=m0c//planckover2pi1 The spatial image of mirror [C−]state (15a) corresponds to circle (Fig. 1b), described by equation: X2 −+Y2 −=R2(15c) where:X−= (k− C)tot=m− C·c//planckover2pi1;Y−= (k0)kin=m0v//planckover2pi1. The radius of mirror circle: R=k0=m0c//planckover2pi1is equal to the axe length of equilateral hyperbola: R=aof real [C+]state. In fact this circle represents the half of bi-vacuum boson (BVB). Fig. 1a. Equilateral hyperbola, describing the energy dis- tribution for real corpuscular state [C+]of sub-elementary particle (positive region) and sub-elementary antipartic le (negative region). The rotation of equilateral hyperbola around common axe of symmetry leads to origination of parted hyperboloid or conjugated pair of paraboloids of rev - olution. The direction of this rotation as respect to vector of particle propagation in space may be responsible for spin . This excited state of bi-vacuum is responsible also for real mass and electric component of electromagnetic charge. Fig. 1b. Circle, describing the energy distribution for the mirror (hidden) corpuscular state [C−].It is located near 12zero-point level of negative realm of bi-vacuum for sub- elementary particles and near zero-point level of positive region of bi-vacuum for corresponding antiparticles. Circ u- lation of virtual quanta in the ground energetic planes is re - sponsible for magnetic properties of elementary particle i n accordance to our model. Such a rotor is a part of secondary bi-vacuum bosons (BVB) as a pair [rotor+antirotor]. The [W] phase in form of cumulative virtual cloud (CVC) origi- nates as a result of quantum beats between real and mirror sta tes of [C] phase (see 12) of elementary wave B. Consequently, the spatial image of CVC energy distribution can be considered as a geo- metric difference between energetic surfaces of real [C+]state as an equilateral hyperbola and that of [C−]state as a mirror circle. After substraction of left and right parts of (15b and 15c) and some reor- ganization, we get the energetic spatial image of [W]phase or [CVC] as a geometrical difference of Equilateral hyperbola and cir cle: (m+ C)2 m2 0+(m− C)2 m2 0c2 v2−(m+ C)2 m2 0c2 v2=−1 (16) This equation in dimensionless form describes the parted (two- cavity) hyperboloid (Fig. 2): x2 a2+y2 b2−z2 c2=−1 The(c)is a real semi-axe; a and b −the imaginary ones. 13Fig. 2. The parted (two-cavity) hyperboloid (in arbitrary scale) is a spatial image of CVC, corresponding to [W] phase of elementary wave B. The positive half of this parted hy- perboloid corresponds to CVC of [W] phase of elementary particle and the negative one - to CVC of antiparticle. The whole picture may characterize the twin CVC of positive and negative energy, produced by pair of sub-elementary [particle+antiparticle] or standing [neutrino+antineut rino] as a part of electron, positron, photon and quarks. If we consider the real and the mirror states of [C] phase as a t wo rotors of different shape and resulting frequency, equal, co rrespond- ingly, toω+ Candω− C,then the difference of rotors of corresponding fields of velocities:− →V+ C(r)and− →V− C(r)can be presented as doubled energy of CVC: rot[/planckover2pi1− →V+ C(r)]−rot[/planckover2pi1− →V− C(r)] =2− →n·/planckover2pi1(ω+ C−ω− C) =2− →n·/planckover2pi1ωCV C (16a) where:− →n is the unit-vector, common for both states; ωCV C= (ω+ C−ω− C)is frequency of beats between real and mirror rotors. All the virtual microparticles of bi-vacuum as a quantum liquid , forming each of rotors, should have the same angle frequency ( ω+ Candω− C). 14The spatial image of BVB is a pair of [rotor + antirotor] with opposite circulation in the ground energetic planes of posi tive and negative vacuum, forming bi-vacuum. Their surfaces are equ al, cor- respondingly to: S+ BV B=π/parenleftbig L+ 0/parenrightbig2=π(/planckover2pi1/m+ 0c)2;S− BV B=π/parenleftbig L− 0/parenrightbig2=π(/planckover2pi1/m− 0c)2(16b) For the case of totally symmetric BVB: S+ BV B=S− BV B, the resulting surface of BVB is SBV B=S+ BV B+S− BV B= 2S+ BV B= 2S− BV B (16c) The oscillations of SBV Bas a result of symmetric oscillations of m+ 0andm− 0at condition ( m+ 0-m− 0) = 0,are related to excitations of vacuum amplitude waves VAW and torsion field. Restoration of [C] phase is a result of binding of CVC on BVB, serving as a anchor site. Let us consider the elementary wave B as a quantum harmonic oscillator, corresponding to conditions (14c) with energy quantiza- tion in the realms of positive and negative bi-vacuum: En=/planckover2pi1ωB=±/planckover2pi1ω0/parenleftbigg n+1 2/parenrightbigg (17) where quantum number: n= 0; 1; 2...and /planckover2pi1ω0=m0c2. Two sublevels, with n= 0are:E± 0=±1 2/planckover2pi1ω0correspond to positive and negative zero-point states of bi-vacuum. They are gener al for particles, antiparticles and bi-vacuum bosons (BVB). The additional third sublevel of positive vacuum at n= 1:E+ V= +3 2/planckover2pi1ω0characterize the asymmetry of energy distribution, accomp a- nied the sub-elementary particle origination (Fig.3). The additional sublevel of negative vacuum: E− V=−3 2/planckover2pi1ω0is perti- nent for sub-elementary antiparticles origination. Parti cles and an- tiparticles have the opposite symmetry of energy distribut ion, how- ever, with the same absolute values. 15Fig.3. The spatial image of [C] phase of elementary wave B [re al vortex+mirror rotor], corresponding to [real+mirror] mas s-dipole. The real total energy of wave B as quantum oscillator in [C] an d [W] phases can be defined as: EC=3 2/planckover2pi1ω0+ (−1 2/planckover2pi1ω0) =/planckover2pi1ω0total energy of [C] states: real and mirror (17a) andEW=3 2/planckover2pi1ω0−1 2/planckover2pi1ω0=/planckover2pi1ω0energy of beats between [C] states, equal to energy of CVC The length of [real +mirror] mass-dipole, equal to that of CV C: − →λ±=h/(m+ Cv2/c) =h/(m+ C−m− C)c (17b) of the fermions is determined by dynamic properties of ”unco m- pensated” standing neutrino in form of [vortex+ rotor] dipo le in com- position of the electron’s triplets. 16THE EXTERNAL AND HIDDEN PARAMETERS OF WAVE B. HIDDEN HARMONY AND GOLDEN MEAN Our model includes the notions of detectable in experiment - ex- ternal and internal (hidden) parameters. It postulates the equality of internal kinetic energies of real and mirror corpuscular states. In accordance to introduced in our theory definition of time (see eq. 61), this condition means t hat the pace of time for both of these states is zero and their life-ti me is infinitive. At the different real and mirror masses ( m+ C>m− C)such condi- tion can be achieved by corresponding difference of hidden ve locities (v+ in<v− in): (2T+ k)in=m+ C(v+ in)2=m− C(v− in)2= (2T− k)in(18) It is easy to show that the internal kinetic energies are equal to resulting one, determined by the mass of rest of the electron : (T+ k)in= (T− k)in=T0=1 2m0·c2(18a) The relativist increasing of m+ Cand decreasing of m− Cwith the enhancement of external group velocity ( v)is compensated by the opposite change of hidden group and phase velocities, define d as: v+ in≡vin grand v− in≡vin ph Like for external group (v)and phase ( vph)velocities, the product ofinternal velocities is equal to the light velocity squared: vin gr·vin ph=v·vph=c2, (18b) however, in general case :vin gr/negationslash=v and vin ph/negationslash=vph The resulting hidden impulse of sub-elementary wave B squar ed is equal to P2 0=P+·P−= (m+ C·vin gr)·(m− C·vin ph) =m2 0·c2=const (18c) From (18) and (4) we get the important relation between the in - ternal and external velocities: m− C m+ C=/parenleftBigg vin gr vin ph/parenrightBigg2 =/parenleftBigg vin gr c/parenrightBigg4 = 1−/parenleftBigv c/parenrightBig2 (19) Taking into account (18b), formula (19) can be transformed t o: vin gr vin ph=(vin gr)2 c2=/bracketleftbigg 1−/parenleftBigv c/parenrightBig2/bracketrightbigg1/2 =/bracketleftbigg 1−/parenleftbiggv vph/parenrightbigg/bracketrightbigg1/2 (20) 17In more convenient for us shape the above formula looks like: c vingr=1 [1−(v/c)2]1/4 We introduce the important condition of Hidden Harmony : the equality of internal (hidden) and external group and phase velocities : /bracketleftbig vin gr=vext grand vin ph=vext ph/bracketrightbig (21) Realization of this primary condition determines the value of Golden mean, as a definite ratio of group and phase velocities (inter nal and external): S=vin gr vin ph=/bracketleftbiggv vph/bracketrightbiggext =/parenleftbiggv2 c2/parenrightbiggext, in =A B(21a) Using (21a), we get from (20) the simple quadratic equation: S2+S−1 = 0 (22) or:S (1−S)1/2= 1 (22a) The positive solution of (22) gives the numerical value of Go lden mean: S≡Phi= (A/B) = 0.618 (23) Golden mean is one of the most intriguing, important and univ ersal number of Nature. Lot of very different natural phenomena - fr om ratio of parameters of planetary orbits to the law of multipl ication of microbes and rabbits ( Fibonacci row ) follow Golden mean. Our intuitive perception of beauty and harmony also is related t o Golden mean. However, in contrast to number ( π),it origin was entirely obscure. The deep quantum roots of Golden mean, reflecting realizatio n of conditions of Hidden Harmony (21), leading from our dynamic model of wave-particle duality, explain the universality of this number (S = 0.618). We put forward a hypothesis, that any kind of selected system , enable to self-assembly, self-organization and evolution : from atoms to living organisms, galactics and the Universe - are tendin g to con- ditions of Hidden Harmony as a background of Golden Mean real - ization. The less is deviation of ratio of characteristic pa rameters of system from [ S≡Phi], the more advanced is evolution of this system. 18We have to keep in mind that all forms of matter are composed from hierarchic system of de Broglie waves. Our statement that tendency of any system (from elementary particle to the Universe) to Hidden Harmony is a driving forc e and final goal of evolution - can be confirmed on the lot of examples . One of them is evaluation of energy of wave B (i.e. electron), corre- sponding to Golden mean condition. The formula (4) at condition (c/v)2=S,taking into account (22a), can be easily driven to the following expression, meaning th e equality of mass symmetry shift, produced by particle ( ∆mS C)to mass of rest of this particle ( m0): ∆mS C= (m+ C−m− C)S=m+ C·S=m0·S (1−S)1/2=m0 (23b) The energy of wave B, equal to that of CVC, at condition of Hidd en harmony and Golden mean, is equal to: ES B= ∆mS C·c2=m0·c2=m0·vgrvph=/planckover2pi1ω0=Tk+V (24) where the angle frequency of [ C⇋W]pulsations of ”ideal” electron with external and internal group velocities, determined by condition (21) is equal to: ω0=m0c2//planckover2pi1= 9.03·1020s−1 We can see, that at condition of Hidden Harmony, the total ene rgy of wave B is described by Einstein’s formula, unifying the en ergy and mass. This means that each cycle of [ C⇋W]pulsation is accompanied by reversible annihilation of each of standing neutrino (el ementary wave B), composing elementary particles. From the known relation between kinetic ( Tk)and potential ( V)energy of wave B and that of group and phase velocities (14b), at the c on- dition of Golden mean: S= [vgr/vph]ext= [vgr/vph]in,we get the ratios between different energy contributions of wave B: /bracketleftbiggV Tk/bracketrightbiggS =2 S−1 = 2.236or S =/bracketleftbigg2Tk EB/bracketrightbiggS = 0.618 (25) We may introduce here the ”Dead mean” conditions, correspon d- ing to thermal equilibrium. At this conditions any system re presents the number of independent oscillators, unable, consequent ly to self- organization: /bracketleftbiggV Tk/bracketrightbiggD = 1/bracketleftbigg2Tk EB/bracketrightbiggD =/bracketleftbiggTk+V EB/bracketrightbiggD = 1 (25a) 19In section: ”Electromagnetism” the important results will be pre- sented, pointing that principles of hydrogen and other atom s/molecules assembly follow the rule of Golden mean, based on Hidden harm ony. It is shown also that the famous Sri Yantra diagram contains t he features, compatible with our model of duality. Scenario of free asymmetric particle (fermions) propagati on in vacuum in a course of its [ C⇋W]pulsation For the end of simplification we consider the behavior of sub- elementary particle, like a single standing neutrino with △mC= (m+ C− m− C)>0in Corpuscular [C] phase. Such standing neutrino can exist, in accordance to our model, only in composition of triplets, like the electron [ 2ν0+/tildewideν0]. We can subdivide the process of this particle dynamics in a co urse of propagation to following stages: 1. The particle (wave B) in [C] phase, representing mass-dip ole, with real mass ( m+ C)and mirror mass ( m− C)with linear dimension (λ±)moves in vacuum with external group velocity ( v < c ),the re- sulting energy ( E±),resulting impulse ( P±),the external real impulse (Pext C)and corresponding external wave B length ( λext C),equal corre- spondingly to: E±=m+ C·v2= (m+ C−m− C)·c2(26) P±=/bracketleftbig (m+ C)·(− →v2 gr/c)/bracketrightbig C= (m+ C−m− C)c and λ±=h/P±(27) Pext C=/bracketleftbig (m+ C)·v/bracketrightbig Cand λext C=h/(m+ C·v)<λ±(27a) 2. The [C] phase is unstable due to its specific asymmetric com po- sition from elements of positive and negative vacuum: [real +mirror] mass - dipole, corresponding to spatial image of pair of [real vor- tex+ mirror rotor] dipole. This instability is a ”driving force” of [C ⇋W] pulsations in form of quantum beats between real and mirror states. There are three interrelated consequences of the first semip eriod of these pulsations, corresponding to transition from Corp uscular to the Wave phase: a) The excessive energy of real corpuscular state [C+]as respect to mirror one [C−] of corpuscular phase - turns to the energy of cumulative virtual cloud (CVC), corresponding to that of th e [W] phase. The CVC propagates in bi-vacuum with light velocity. The 20energy of CVC can be subdivided to energy of Vacuum Density Wa ves (VDW) and Vacuum Symmetry Waves (VSW): EW= (m+ C−m− C)·c2=m+ C·v2=EC (28) or:ECV C=EV DW+EV SW=EW (28a) where :EV DW =/vextendsingle/vextendsinglem+ C−m0/vextendsingle/vextendsingle·c2;EV SW=/vextendsingle/vextendsinglem− C−m0/vextendsingle/vextendsingle·c2 The hidden resulting impulse of CVC, composed from these two kinds of virtual quanta ( P± CV C),determines the resulting radius of CVC (L± CV C),equal to that of corpuscular mass-dipole: P± CV C=c(m+ C−m− C) =c△mCand L± CV C=/planckover2pi1 (m+ C−m− C)·c(29) The spatial image of CVC is a half of two-cavity parted hyper- boloid (see eq. 16 and Fig.2), each of cavity, corresponding to posi tive for particles and negative for antiparticles components of CVC. The resulting energy and the external impulse of secondary b i- vacuum boson (BVB) - turns to zero: EBV B=1 2/planckover2pi1ω0+ (−1 2/planckover2pi1ω0) =1 2(m+ 0−m− 0)·c2→0 (30) and P BV B=1 2(m+ 0−m− 0)·c→0λBV B=h/PBV B→ ∞. (31) The radius of BVB, with spatial image of pair of 2D rotors, is determined by the electron’s Compton radius, equal to that o f CVC at condition of Hidden harmony (21): LBV B=/planckover2pi1/m0c The conditions (31) means that the bi-vacuum bosons (BVB), c or- responding to [W] phase of coherently pulsating particles, may form nonlocal virtual Bose condensate with infinitive dimension s. The secondary curved bi-vacuum, corresponding to [C] phase , dis- plays nonlocality only in the volume of finite, but huge clust ers in contrast to unperturbed flat primordial bi-vacuum, which no nlocal- ity has no limitations. The probability of BVB origination and their Bose condensat ion is very high for both: primordial and secondary vacuum. 3. At the next, the reverse stage of our scenario, the ”ejecte d” in a course of [C →W]transition cumulative virtual cloud (CVC), repre- senting [W] phase of particle - restores in bi-vacuum the asy mmetric [real-mirror] mass dipole (C-phase) in form of [real vortex +mirror rotor] pair as a result of CVC absorption by one of BVB. The most probable distance of restoration of [C] phase, i.e. [W→ C]transition - from the point of previous [C →W]transition is de- termined by the external wave B length of the electron (27a). 21The in-phase [C⇋W]pulsations of symmetric pair of standing neutrino and antineutrino [ ν0+/tildewideν0]in composition of the electron [2ν0+/tildewideν0]is also accompanied by [ejection ⇋absorption ]of twin CVC in form of parted hyperboloid (Fig. 2). These correlated CVC of positive and negative vacuum - totally compensate each othe r and do not contribute to the impulse - energy of uncompensated stan ding neutrino of triplet. However, the periodic [ emission ⇋absorption ]of CVC doublet may generate the nonlocal (instant) Vacuum Ampl itude Waves (VAW), as oscillations of bi-vacuum zero-point energ y slit. Our model do not needs the Bohmian ”quantum potential” or ”pilot wave” for explanation of two-slit experiment . For the com- mon case of ensembles of particles, the explanation can be ba sed on interference of their [W] phase in form of cumulative virtua l clouds (CVC). The ability of CVC to activate secondary virtual wave s and wavefronts in bi-vacuum could be responsible for interfere nce, pro- duced even by single electron or photon. Scattering of photo ns on the free electrons will affect their impulse, mass, wave B len gth and, consequently, the interference picture. Only [C] phase of p article, but not its [W] phase can be registered by detectors of particles . Such a consequences of our dynamic duality model can explain all de tails of well known and still mysterious double slit experiment. The absence of dissipation in the process of [C⇋W]transitions in bi-vacuum Bose condensate as in superfluid quantum liquid , makes them totally reversible. Propagation of fermion in 3D space in a course of its [C⇋W] pulsation can be considered as a periodic jumping of particl e in form of CVC of [W] phase with light velocity between wavefronts of bi- vacuum, corresponding to [C] phase, moving with group veloc ity lower than luminal one. The wavefronts of [C] and [W] phase ar e normal as respect to direction of particles propagation. The separation between coherent wavefronts, generated by r eal [C] phase, is determined by real external wave B length (eq.27a) . The energy ( ECV C)of cumulative virtual cloud (CVC) is deter- mined by difference in energy of quantum beats: ∆mC·c2= (m+ C− m− C)·c2in a course of [C⇋W]and energy of secondary bi-vacuum symmetry shift ( ∆mV·c2): ECV C= (∆mC−∆mV)·c2(32) The total energy of wave B is a sum of CVC energy and energy of vacuum symmetry shift: EB=ECV C+ ∆mV·c2= ∆mC·c2(33) The nonlocal quantum interaction can occur between [C] phas e particles via secondary bi-vacuum Bose condensate. The nonlocal 22interaction between [C] phase of particles in the huge but li mited volume of secondary bi-vacuum Bose condensate, may be reali zed as a coherent change of vacuum symmetry shift: ∆mV=1 2/vextendsingle/vextendsinglem+ 0−m− 0/vextendsingle/vextendsingle=/vextendsingle/vextendsinglem+ V−m− V/vextendsingle/vextendsingle (34) in the shell of secondary superfluid Bi-vacuum Bose condensa te with the radius of curvature: LBV C=/planckover2pi1/(∆mV·c) (34a) The vacuum symmetry shift oscillation may be induced by the mass symmetry shift ( ∆mC)oscillation, resulted from change of ve- locity in a coherent system of particles and in-phase [ C⇋W]pulsation (see ”Gravitation”): ∆mC=/vextendsingle/vextendsinglem+ C−m− C/vextendsingle/vextendsingle˜ ∆mV (34b) Such kind of nonlocal interaction display itself in oscilla tion of LBV C.(34a),i.e. radius of nonlocality . The oscillation of real mass of uncompensated standing neut rino (ν0)of elementary particle due to its alternating acceleration - is ac- companied by corresponding oscillations of real masses of p air [ν0+/tildewideν0] in composition of the electrons [2 ν0+/tildewideν0],protons and neutrons. These symmetric oscillation of [ ν0+/tildewideν0], in turn, excite the nonlocal Vacuum Amplitude Waves (VAW), accompanied by oscillations of bi-v acuum Bose condensate (BC) energy slit ( EBV B), equal to that of BVB: EBV B+ ∆EBV B=1 2[(m+ 0+ ∆m+ 0) + (m− 0+ ∆m− 0)]·c2= (m0+ ∆m0)·c2 (34c) AV AW= ∆EBV B= ∆m+ 0+ ∆m− 0= ∆m± 0 (34d) Such mechanism of nonlocal interaction via virtual BC can ex plain the instant interaction between coherent twin photons in ex periments of Aspect et all and realization of principle of least action . Our theory relates the pace of time for any selected closed sy stem to thepace of the external real kinetic energy Tkinchange (see below). The notion of real time is existing only on realm of wave- fronts, corresponding to [C] phase of wave B. The pace of real time is determined by the pace of real kinetic energy change. At the [W] wavefronts and bi-vacuum energetic surface, the notion of real time is absent as far the notion of real kinetic energy is absent. However, the virtual time could coexist with change of virtual kinetic energy. 23Spatial stability of complex systems: atoms, molecules and that of solids means that in these systems superposition of asymm etrical cumulative virtual clouds, representing [W] states of elem entary par- ticles - forms hologram - like 3D standing waves superpositi on with location of nodes, corresponding to the most probable posit ions of corpuscular phase of the nucleons, electrons, atoms and mol ecules in condensed matter. The binding of CVC by bi-vacuum bosons res tore the [C] phase of particles in positions, close to the most pro bable ones in accordance with the value of corresponding wave function squared (see below). ELECTROMAGNETISM The fine structure constant ( α= 7.29735·10−3)in our model can be related to ratio of minimum zero-point external group veloc ity of the electron (v=v0),to the light velocityas follows: α=e2 /planckover2pi1c=/parenleftBigv0 c/parenrightBig2 (35) The new notions of external and internal (hidden) electric (i;iin)and magnetic ( η;ηin)components of resulting electromagnetic charge (e )are interrelated with each other as (i·η) =/parenleftbig iin·ηin/parenrightbig =e2(36) In accordance to our theory, the real and mirror states of [vo r- tex+rotor] dipole of [C] phase - reflect the internal electric ( iin)and magnetic ( ηin)components of electromagnetic charge, correspond- ingly. It may be shown, that the ratio of Bohr magneton ( µB)to internal magnetic moment of the electron ( µη): µB=e·(/planckover2pi1/2m0c);µin η=ηin·(/planckover2pi1/2m0c) (37) at very low external group velocity, equal to that of zero-po int: v→v0,when the eqs. (35 and 19) are valid, tends to: µin η µB→/parenleftbiggηin iin/parenrightbigg1/2 0=/parenleftBigg vin ph vingr/parenrightBigg3 0=1 [1−α]1/6(38) This value is very close to experimental one: (µe/ µB)exp= 1.00115965221 The notion of Dirac’s magnetic monopole is replaced in our mo del by the notion of internal magnetic component of electromagn etic charge (ηin). 24The Lienor-Vihert’s vector (− →A)and scalar ( φ)potential, produc- ing by moving elementary charge with velocity (− →v)(Landau and Lifshiz, 1988) are equal to: − →A=e− →v c/parenleftBig R−− →vR c/parenrightBig (39) and φ =e/parenleftBig R−− →vR c/parenrightBig (40) Dividing (39) to (40) and taking into account (19), we get: /parenleftBig− →A/φ/parenrightBig2 = (− →v/c)2= 1−m− C m+ C(41) The electromagnetic energy, corresponding to maximum poten- tial of the electron, is expressed in our model as an internal interaction energy between elementary internal electric (iin)and magnetic (ηin) charges, separated by the radius (L±)of [real+mirror] mass-dipole. This value is equal to radius of cumulative virtual cloud (CV C),pro- duced by quantum beats between real and mirror states of the e lec- tron’s (2ν0+/tildewideν0)uncompensated standing neutrino ( ν0): [Emax el]e=/parenleftBigg [i·η]in L±=e2 L±/parenrightBigg e=α/vextendsingle/vextendsinglem+ C−m− C/vextendsingle/vextendsingle e·c2=α/parenleftbig m+ C·v2/parenrightbig e(42) whereα=e2//planckover2pi1c=e2/Q2is a fine structure constant, which deter- mines the isotropic component of CVC; Q2=/planckover2pi1cis a total charge squared;vis the external group velocity of particle in [C] state. The total electromagnetic potential of proton with positive charge, determined by fractional charges of three uncompensated an tineutri- nos of three quarks, is equal to that of the electron, but with opposite sign. It can be expressed in similar way: [Emax el]P=/parenleftbiggi·η L±=e2 L±/parenrightbigg P=α/vextendsingle/vextendsinglem+ C−m− C/vextendsingle/vextendsingle P·c2=α/parenleftbig m+ C·v2/parenrightbig P(42a) From eq.9 the electron’s and proton’s mass-dipole characte ristic dimensions, equal to radius of their CVC, should be equal to e ach other: /bracketleftbig L±=/planckover2pi1/P±/bracketrightbig e,P=/bracketleftbig /planckover2pi1/(m+ C−m− C)c=/planckover2pi1/(m+ C·v2/c)/bracketrightbig e,P(43) [L±]e,Pcharacterizes the distance between real and mirror masses o f mass-dipole and between magnetic and electric components o f elec- tromagnetic charge, distributed over CVC for electron and p roton. 25As far the mass of proton is much bigger, than that of the elect ron: mP>> m e, it leads from the right part of eq.43 that the external group velocity of proton should be much smaller than that of t he electron. The dependence of electromagnetic potential on distance ( r)is like: /bracketleftbigg Eel(r) =Emax el·− →r r/bracketrightbigg e,P(43a) where:− →ris the unitary radius-vector. From condition: [Etot el]e= [Etot el]P,and equality of real kinetic en- ergies of the electron and proton (see 42 and 42a) 2Te k=/parenleftbig m+ C·v2/parenrightbig e=/parenleftbig m+ C·v2/parenrightbig P= 2TP k,we got the following interrelation between their real external impulses, masses and corresponding group velocit ies: P+ P P+e=/bracketleftBigg/parenleftbig m+ C/parenrightbig P/parenleftbig m+ C/parenrightbig e/bracketrightBigg1/2 =ve vP(44) At the conditions of Golden mean , when:/bracketleftbig ∆mC=m+ C−m− C/bracketrightbigS e= m0(see eq. 22b), eqs. 42 and 43a turns to: ES el(r) =Emax el·− →r r=e2 L0·− →r r=α·m0c2·− →r r(44a) where :L0=/planckover2pi1/m0c= 3.86×10−13mis a Compton radius of the electron As far the Compton radius is in fact experimental parameter, ob- tained from analysis of scattering of photons on the electro ns, we may conclude, that the Nature follows the principle of Hidden ha rmony or Golden mean, indeed. The Compton radius of the electron is an averaged value of the real internal real ( L+)vortex and mirror ( L−)rotor radiuses, as it leads from our model: L0= (L+·L−)1/2=/bracketleftbig /planckover2pi1//parenleftbig m+ C·vin gr/parenrightbig ·/planckover2pi1//parenleftbig m− C·vin ph/parenrightbig/bracketrightbig1/2=/planckover2pi1/m0c equal to the radius of cumulative virtual cloud (CVC) and tha t of bi-vacuum bosons (BVB) at conditions of Hidden Harmony: /parenleftbig L±/parenrightbigS=/planckover2pi1/(m+ C−m− C)Sc=L0=/planckover2pi1/m0c − →ris radius-vector; r≥ |− →r|=L0is the distance from the electron. The averaged hidden impulse of the electron is determined by product of internal real and mirror impulses: P+ in=m+ C·vin grand 26P− in:=m− C·vin ph.Taking into account that m+ C·m− C=m2 0andvin gr·vin ph= c2:we get: P0=/parenleftbig P+ in·P− in/parenrightbig1/2=±m0c From the principle of uncertainty in coherent form: L2 0·P2 0=/parenleftbigg/planckover2pi1 2/parenrightbigg2 we find out that the internal mechanic moment of the electron i n the units of Plank constant, equal to its spinis: s=±1 2 Two possible projections of momentum of real vortex of [C+] state in form of paraboloid of revolution, for particles and antip articles to selected direction, determines the sign of spin of fermio ns, like electron or positron. The total electromagnetic energy of the electron ( Eel)can be con- sidered as a part of total energy of wave B, equal to that of cum ula- tive virtual cloud (CVC), determined by the fine structure co nstant (α=e2//planckover2pi1c)as a factor. This means that the notion of the electric and magnetic components of virtual quanta, responsible for int eraction between charged particles, looks to be pertinent for the wav e [W] phase of particle only. However, the electric and magnetic c ompo- nents of charge are related to [real vortex + mirror rotor] di pole of corpuscular [C] phase, correspondingly. The notions of spin, real mass and time also are pertinent onl y for [C] phase of particle only. THE HYDROGEN ATOM One more evidence in proof of our model is that the Bohr radius of the Hydrogen atom is equal to radius of CVC of the electron a nd proton, at conditions of Golden mean [L0= (L±)Sat(m+ C−m− C) =m0]: aB=/planckover2pi1 α·m0c=L0 α= 0.529177249 ×10−10m (44b) Corresponding condition of the electron’s standing wave wi th the electron’s group velocity on the orbit, equal to: v=αcis: λB= 2π·aB=h m0(αc)(44c) 27One can see from (44c), that the Bohr radius can be expressed a lso via total electromagnetic energy of the electron in ”ideal c ondition” (eq.44a) as: aB=/planckover2pi1c ES el=Q2 ES el whereQ2=/planckover2pi1cis a total charge of the electron, squared, related to the total energy of CVC (see 42). The energy of electrostatic attraction between electron an d proton at the hydrogen atom, compensated in Bohr’s model by energy o f centripetal energy, is proportional to the total electroma gnetic energy of the electron at condition of hidden harmony: EH=e2 aB=α·e2 L0=α2·m0c2=α·ES el (44d) The biggest part of energy of cumulative virtual cloud (CVC = [vacuum density + vacuum symmetry waves] ≡VDW + VSW), re- sulted from quantum beats of the electron’s unpaired standi ng neu- trino, is nonparticipating in electromagnetism and is resp onsible for realization of [C⇋W]duality and stabilization of elementary parti- cles. The part of energy of CVC of [W] phase of particle is char ac- terized by radius: L0=/planckover2pi1/m0c=α·aB. The radius of component of CVC, responsible for interaction be- tween electron and proton, is equal to the Bohr radius (44b). The part of CVC, nonparticipating in electromagnetism may b e responsible for coherent exchange interaction between two standing neutrinos of the electron [ 2ν0+/tildewideν0]with opposite spins and counter- phase [C⇋W]pulsation. The energy of this part is equal to: EW CV C= (EV DW+EV SW)−α(EV DW+EV SW) = (EV DW+EV SW)·(1−α) (45) where:Emax el=α·(EV DW+EV SW)≪EV DW+EV SW The energy of vacuum density waves is EV DW =/vextendsingle/vextendsinglem+ C−m0/vextendsingle/vextendsingle·c2and the energy of vacuum symmetry waves is: EV SW=/vextendsingle/vextendsinglem− C−m0/vextendsingle/vextendsingle·c2 The total CVC moment of the particle in [W] phase, introduced here, is the invariant: dtot= (m+ C−m− C)·L±=/planckover2pi1 c(46) Using (13), the quantization rule for electromagnetic ener gy (42) can be expressed as: nEel=α·n/planckover2pi1[ω+ C−ω− C] (47) 28At the Hidden harmony/Golden mean condition, we have for the electron’s frequency of [ C⇋W]pulsation: ω0= [ω+ C−ω− C] =m0c2//planckover2pi1= 9.03·1020s−1(47a) and n(Eel)e=α·n/planckover2pi1ω0=α·nm0c2=α·n/vextendsingle/vextendsinglem+ C−m− C/vextendsingle/vextendsingle P·c2=n(Eel)P(47b) where:ω0= [ω+ C−ω− C]P=/vextendsingle/vextendsinglem+ C−m− C/vextendsingle/vextendsingle P·c2//planckover2pi1, and/vextendsingle/vextendsinglem+ C−m− C/vextendsingle/vextendsingle P≪ (m0)P From this formula one can see that the electromagnetic energ y emergence is a result of quantum beats between real and mirro r cor- puscular states of one uncompensated [vortex+rotor] dipol e of the electron and three of them in proton’s quarks with fractiona l charge (+1 3). The formula (47) and our presentation of elementary wave B as a dynamic system of [real vortex+mirror rotor] dipole get su pport from the known expression of vector analysis (48). We can exp ress the divergency of Pointing vector: P = (c/4π)[EH]via difference of contributions, related to real and mirror rotors: div[EH] =4π cdivP=HrotE−ErotH (48) where H and E are the magnetic and electric components of virt ual photons, radiated by electron in course of quantum beats bet ween real and mirror phase of uncompensated bi-vacuum excitation in a course of its [C⇋W]pulsation. The analogy between (47) and (48), illustrating the dynamic [vor- tex+rotor] dipole background of (47), is evident. THE MECHANISM OF ELECTROMAGNETIC INTERACTION In accordance to model, the mechanism of electromagnetic re - pulsion and attraction between charged particles is a resul t of re- alization of principle of least action in our formulation. I t means tendency of system to minimum of resulting density of energy of cumulative virtual cloud (CVC) (see eq.60), interrelated w ith vac- uum symmetry shift ( ∆mV=/vextendsingle/vextendsinglem+ V−m− V/vextendsingle/vextendsingle)and mass symmetry shift (∆mC=/vextendsingle/vextendsinglem+ C−m− C/vextendsingle/vextendsingle)as one can see from eq. 52. Therepulsion is a consequence of streaming of two particle with similar charge to increase the distance between them, as far it mini- mize the density of energy of CVC of the same sign . The electro mag- neticattraction is a consequence the same principle, because of two CVC of the opposite energy are tending to compensate the influ ence 29of each other, decreasing the resulting energy of CVC as a cluster of virtual photons. The closer are opposite charges to each o ther, the more symmetric becomes energy distribution and resulting v acuum and mass symmetry shift. In the case of the electron and positron scattering, their oppo- site by energy cumulative virtual clouds: CVC−and CVC+-trans- form to the high energy photon structure after overlapping, leading to annihilation of e−and e+. The energy of such photon is: hνph= [(m+ C−m− C)e−+ (m+ C−m− C)e+]c2 At the Golden mean condition this formula change to: hνph=2m0c2 In general case the energy of photon as a result of annihilati on: [e−+e+]→[(2ν0+/tildewideν0)+(2/tildewideν0+ν0)]can be more than 2m0c2,depending on velocity of their colliding. If the energy of photon is more t han2m0c2, it can split to [electron+positron] pair again at certain co nditions. Alternative Corpuscle-Wave model of atom Besides planetary model of hydrogen atom of Bohr-Sommerfel d, our duality approach allows to propose the new one It is assum ed that [C⇋W]pulsations of the electrons and nuclear of atom are in-phase. We suppose also, that the electrons in the [C] phase has an abi lity for limited jumps along line tangent to orbit around the nucl ear. The length of one jump ( l0)is determined by period of the electron [C⇋W]pulsation: T0= 2π·ω0(47a) and its external group velocity (v=αc): llb=T0·v=αh m0c=αL0 (48a) This real [C] - jump of the electron is much smaller than the le ngth of Bohr’s orbit (44c). Their ratio is equal to fine structure c onstant squared: llb/(2πaB) =α2(48b) The stage of the electron-proton interaction, correspondi ng to [W] phase of atom, represents a superposition of part of their CV C with equal wave B length, determined by Bohr radius (44b) in form o f virtual 3D standing wave . Such 3D standing wave, reflecting [W] 30phase of atom is not dissipating. It is formed by compensatin g each other virtual photons of opposite energy. The total absolut e value of such energy of electromagnetic interaction in Hydrogen ato m is: EH el= 2α2·m0c2(48c) At the following semiperiod the atom returns to its [C] phase and the electron makes its next corpuscular jump around nucl ear. As far the real time in virtual [W] phase is absent (see sectio n: ”The principle of least action and problem of time”), this phase i s out of perception and it looks that the electron in [C] phase is in th e process of permanent rotation along the orbit. The density of charge is oscillating in a course of [ W⇋C]pulsa- tions of the [electron+proton] and its movement around nucl ear. It means that the interpretation of Van-der-Waals interactio n as a re- sult of coherently flickering charge of atoms/molecules rem ains valid in our model. In atoms, containing one or integer number of the electrons p airs with opposite spins of the electrons, their counterphase [C -W] cy- cles of each selected [electron+proton] pair - are accompan ied by 3D standing waves formation, which are more symmetric and stab le, than in atoms with unpaired valent electrons. Unification of atoms in a course of different reactions, accom panied by unification of unpaired valent electrons and creation of a dditional symmetric standing waves B, is energetically favorable. Mo lecules could be considered as a highly orchestrated dynamic system s, where the [W⇋C]pulsations of all protons, neutrons and electrons are coherent with frequency (47a), corresponding to Hidden har mony condition for the electron. New interpretation of Coulomb interaction between macrosc opic bodies. The spinning effect Let us proceed from assumption, that the interaction (attra ction or repulsion) between charged macroscopic bodies is a resul t ofaver- aging of their electromagnetic potentials ( EC= (E1·E2)2,depending on their average charge, squared ( q2 1,2)and the distance between cen- ters of bodies ( r1,2): q2 1,2=±(q1·q2) r1=r2=r1,2 Consequently, the Coulomb interaction between these bodie s, tak- ing into account dielectric permeability ( ε), can be expressed as: EC=−q1·q2 εr1,2=−q2 1,2 εr1,2(48d) 31The analog of this formula, leading from our theory (see eq.4 2): E∗ C=−− →r r1,2E1,2=−− →r r1,2[E1·E2]1/2=−− →r r1,2α·c2 Ni/summationdisplay i=1(m+ C−m= C)i·Nj/summationdisplay j=1(m+ C−m= C)j 1/2 (49) whereNiandNjare the numbers of elementary uncompensated charges in the volume of first and second body; (− →r)is the unitary radius-vector. If we denote the total mass symmetry shift (difference betwee n real, inertial and mirror, inertialess mass) for each of bod ies as a sum of contributions of all elementary charges in its volume: ∆m1=Ni/summationdisplay i=1(m+ C−m= C)iand ∆m2=Nj/summationdisplay j=1(m+ C−m= C)j (49a) we get from (49): EC=−− →r r1,2αc2·∆m1/2 1∆m1/2 2=−− →r r1,2α∆m1,2·c2(49b) where the averaged mass symmetry shift of uncompensated ele - mentary charges of two bodies is: ∆m1,2=±(∆m1∆m2)1/2. From the comparisons of 49b and 48d, we find the expressions fo r total charges of two bodies in terms of our model: q1= (− →r ε)1/2α1/2∆m1/2 1·c q1= (− →r ε)1/2α1/2∆m1/2 2·c The averaged resulting charge, squared of two interacting b odies is q2 1,2= (− →r ε)α∆m1,2·c2(49c) As far, in accordance to our theory (see eq. 42), the mass symm e- try shift of elementary charge ( ∆mC)is related to its kinetic energy as: ∆mC·c2=m+ C·v2,the eq.(49c) for resulting charge may transformed to: q2 1,2= (− →r ε)αm+ 1,2·v2(49d) where:m+ 1,2= (m+ 1·m+ 2)1/2is the resulting/averaged real corpus- cular mass of resulting charge of two bodies; vis the resulting group velocity of elementary charges, depending on their coheren t thermal oscillations and relative macroscopic movement of two bodi es, with- out the change of distance between centrums of their mass ( r1,2). The 32simplest kind of such movement is the relative spinning/rot ation of bodies. The biggest contribution of resulting kinetic energy to (49 d) is mostly due to in-phase microscopic atomic/molecular oscil lations in the volume of coherent clusters, resulting from high-tempe rature Bose condensation, in accordance to our Hierarchic theory o f con- densed matter (see: www.egroups.com/docvault/antigrav) . This con- tribution depends on temperature and the external fields ten sion, affecting the dynamics of lattice of bodies. Substitution of 49c or 49d to 48d, will lead to eq.49e, unifyi ng our approach and Coulomb interaction. Taking into account that q2 1,2=n2e2(nis the integer number) and α=e2//planckover2pi1c,this formula can be driven to: n2=q2 1,2 /planckover2pi1α= (− →r ε)x,y,z/parenleftbigg∆m1,2·c /planckover2pi1/parenrightbigg x,y,z= (− →r ε)x,y,z(k1,2)x,y,z(49e) where (ε)x,y,zis a tensor of dielectric permeability and (k1,2)x,y,z= (∆m1,2·c//planckover2pi1)x,y,zis a resulting for system of two bodies the Compton’s wave number tensor, determined by the sum of their uncompens ated elementary charges mass symmetry shift. At the conditions of Golden mean, when:/vextendsingle/vextendsinglem+ C−m= C/vextendsingle/vextendsingle=m0we have: m1=N1m0; ∆m2=N2m0and ∆m1,2=m0(N1N2)1/2,whereN1and N2are the numbers of uncompensated elementary charges in the fi rst and second bodies, correspondingly. For the case of isotropic dielectric permeability, we get fr om 49b: [n2/(− →r ε)]3= (c//planckover2pi1)3·(∆m1,2)x·(∆m1,2)y·(∆m1,2)z=const (49f) For Newtonian gravitation we get the similar formulae, repl ac- ingαtoβ= (m0/MPl)2.This phenomena, responsible for electro- gravitational interaction , will be discussed at the next section. Using again relation between (49c) and (49d), we get from (49 f): [n2/(− →r ε)]3=c//planckover2pi13·(mv2 1,2)x·(mv2 1,2)y·(mv2 1,2)z=const (49g) These formulae leads to important conclusion: the increasing of velocity of relative rotation of two charged bodies in plane (x,y), nor- mal to (z, parallel to− →r) will be accompanied by relativist enhance- ment of product [ (∆m1,2)x·(∆m1,2)y] ˜ [(mv2 1,2)x·(mv2 1,2)y]and corre- sponding decreasing of(∆m1,2)zand(mv2 1,2)z,responsible for Coulomb and gravitational interaction between these two bodies. GRAVITATION The total energy of gravitation of particle is introduced in our model as the energy of gravitational attraction between rea l and mir- ror corpuscular masses, separated by wave B dipole length (L±= 33/planckover2pi1/P±see eq. 43 )of uncompensated standing neutrino in composition of particle. The formula obtained for gravitation are very s ymmet- ric to those, obtained for electromagnetism (42–47). The ma ximum of gravitational potential, produced by one standing neutr ino (ν0)or antineutrino (/tildewideν0)as a part of elementary particle: Emax G=G·m+ C·m− C L±=G·m2 0 L±(50) =β·/vextendsingle/vextendsinglem+ C−m− C/vextendsingle/vextendsingle ν0,/tildewideν0·c2=β·(m+ C·v2)ν0,/tildewideν0 where β = (m0/MPl)2(50a) is the new dimensionless Gravitational Fine Structure , intro- duced in our theory, containing the Plank mass squared M2 Pl=/planckover2pi1c/G; E± B=/vextendsingle/vextendsinglem+ C−m− C/vextendsingle/vextendsingle·c2=m+ Cv2= 2Text kis a doubled external kinetic en- ergy of the electron,.equal to the resulting energy of wave B (E± B).The total real energy of wave B is equal to E+ B=m+ Cc2. The decreasing of maximum of gravitational potential with d is- tance, like that of electromagnetic one can be expressed as: EG(r) =Emax G·− →r r(50b) where |− →r|is a radius-vector of gravitating particle or body; r≥ |− →r| is a distance from the particle. In case of the electron |− →r|=/planckover2pi1/(m0c). Using (13), the quantization rule of gravitational energy ( 50), resulted from beats between positive ( m+ V)and negative ( m− V)ground states of secondary bi-vacuum, can be expressed as: nEG=β·n/planckover2pi1·/vextendsingle/vextendsingleω+ C−ω− C/vextendsingle/vextendsingle ν0,/tildewideν0=n/vextendsingle/vextendsinglem+ V−m− V/vextendsingle/vextendsingle ν0,/tildewideν0·c2(51) =n·hνG=n·h(c/λG) where:n= 1,2,3...is the integer number; νGis frequency of grav- itational waves (GW), equal to frequency of quantum beats be tween positive and negative vacuum states; λG=h/[/vextendsingle/vextendsinglem+ V−m− V/vextendsingle/vextendsingle ν0,/tildewideν0·c] (51a) is the length of gravitational wave, produced by one element ary par- ticle/antiparticle. For particle the difference:/vextendsingle/vextendsingleω+ C−ω− C/vextendsingle/vextendsingle˜/vextendsingle/vextendsinglem+ V−m− V/vextendsingle/vextendsingleis positive and for antiparticles it is negative. However, just the absolute va lue of this difference determines the frequency and energy of gravitati onal field. It means that particles and antiparticles with identical ma ss generate the equal gravitational potential. 34From the right part of eqs. 42 and 50, it is easy to show after differentiation, that: dlnEel=dlnEG=dlnm+ C+ 2dlnv (51b) It is one of the convincing formula, demonstrating unificati on of electromagnetism and gravitation in the framework of our th eory in very simple way. At the conditions of Hidden harmony (21), necessary for Gold en mean realization (21a), when condition (22b) is fulfilled, t he eqs. (49- 51) transforms to: ES G(r) =n·β·m0c2·− →r r=n·β·/planckover2pi1ω0·− →r r(51c) where zero-point frequency is defined from the mass of rest of the electron: ω0=m0c2//planckover2pi1,equal to frequency of quantum beats between real and mirror states of [C] phase at conditions of Hidden ha rmony. The gravitational waves, could be considered as a result of q uan- tum beats between positive and negative bi-vacuum ground st ates: m+ Vandm− V, pertinent to [C] phase. We assume that asymmetry of real and mirror corpuscular masses of uncompensated stan ding neutrinos or antineutrinos of fermions - the mass symmetry s hift: ∆mC=/vextendsingle/vextendsinglem+ C−m− C/vextendsingle/vextendsingleinduce corresponding bi-vacuum symmetry shift: △mV=/vextendsingle/vextendsinglem+ V−m− V/vextendsingle/vextendsingle=β/vextendsingle/vextendsinglem+ C−m− C/vextendsingle/vextendsingle=β·∆mC (52) where:m+ V=/vextendsingle/vextendsingleE+ V/c2/vextendsingle/vextendsingle=β·m+ Candm− V=/vextendsingle/vextendsingleE− V/c2/vextendsingle/vextendsingle=β·m− Care the absolute effective masses of positive and negative groun d vacuum states correspondingly; The new fundamental constant: Gravitational fine structure [by analogy with electromagnetic fine structure α= (e/Q)2=e2//planckover2pi1c)]is equal to: β=△mV ∆mC=/parenleftbiggm0 MPl/parenrightbigg2 = 1.7385·10−45(53) where the Plank mass: MPl= (/planckover2pi1c/G)1/2=Q/G1/2(Qis a total charge ). The Gravitational fine structure may be represented as a rati o of surfaces of bi-vacuum bosons (BVB), with Plank’s ( SPl=πL2 Pl= π(/planckover2pi1/MPl·c)2and electron’s Compton’s ( S0=πL2 0=π(/planckover2pi1/m0·c)2radiuses: β=SPl S0 It follows from (53) that the sign and value of mass symmetry s hift △mC=±(m+ C−m− C)for particle/antiparticle are interrelated with the 35sign and value of vacuum symmetry shift: △mV=±(m+ V−m− V)in the point of particle localization. At the Golden mean condition:/bracketleftbig △mC=/vextendsingle/vextendsinglem+ C−m− C/vextendsingle/vextendsingle/bracketrightbigS=m0,we get from (53)the vacuum symmetry shift: ±[△mV]S=m3 0 M2 Pl(53a) The corresponding to this condition curvature of bi-vacuum bosons (BVB) Bose condensate: [LBV C]S=/planckover2pi1 ±[△mV]S·c=1 β·/planckover2pi1 m0c=L0 β= 2.22·1032m (54) The analogy is existing between the ”polaron” - quantum exci ta- tion in ionic crystals and particle in [C] state. Polaron in c ondensed matter represents the mobile pair with dipole properties [e lectron + lattice polarization]. For the other hand, the standing neu trino of el- ementary particle in [C] phase we consider as a ”mass - dipole ” with resulting charge ( △mC), coexisting with vacuum polarization, termed in our model - bi-vacuum symmetry shift (△mV). The mass and mobility of system: [(C-phase) + vacuum symmetr y shift] depends on the absolute value of vacuum polarization :△mV. Equalizing the wave B dipole length, equal to radius of CVC, p re- sented in form: Ld=/planckover2pi1c/(m+ Cv2)and gravitational Schwarzschild ra- dius:rg= 2Gm+ C/c2, corresponding to radius of black hole, we get the condition of black hole emergency for relativist partic le like elec- tron, when Ld→rg: m+ Cv=m0v/[1−(v c)2]1/2=MPl·c (55) At this limit condition the de Broglie wave length of particl e, i.e. electron, as a mini black hole is determined by the Plank’s ma ss: LPl=/planckover2pi1/(MPl·c) = 1.61605×10−35m (55a) Corresponding to black hole radius of secondary bi-vacuum B ose condensate curvature is: [LBV C]g=/planckover2pi1 β·MPl·c=LPl β∼109m (55b) It is still pretty big even under condition of mini-black hol e origi- nation. THE MECHANISM OF GRAVITATIONAL INTERACTION 36It can be similar to hydrodynamic Bjorkness interaction bet ween pulsing particles in liquids, radiating acoustic waves. We suppose that gravitational waves, resulted from quantum beats betw een pos- itive and negative ground states of vacuum, decreasing the v acuum symmetry shift, are decreasing also the virtual quanta pres sure be- tween particles more than outside of them. This leads to exce ssive outside vacuum pressure, providing the gravitational attr action be- tween bodies. In accordance with the existing theory of Bjorkness force, i t is dependent on distance between pulsing bodies or microbubbl es in liquid - as (1/r2). It is important that this force could be positive and negativ e, de- pending on difference of phase of pulsations, generating den sity waves. In turn, this phase shift is dependent on relation of distanc e between bodies to acoustic (or gravitational in our case) wave lengt h. If the length of acoustic (gravitational) waves, excited by bodie s, is less or comparable with the distance between bodies, the Bjorkness (gravi- tational) force is attractive. If the distance is much bigger than wave length, then the attraction of bodies turns to repulsion. Th is mean origination of antigravitation. The large-scale honey-comb structure of the Universe, its h uge voids, could be explained by the interplay of gravitational attraction and repulsion between clusters of galactics, depending on t he distance between them. Recently a strong evidence appears, pointing to accelerati on of the Universe expansion. This phenomena could be explained by in creas- ing the antigravitation factor with increasing the distanc e between galactics. This confirms our hydrodynamic model of mechanis m of gravitation. Like electromagnetic interaction, the gravitational one c an be con- sidered as a result of principle of least action realization (see eq. 60). For this end we have to assume, that the resulting vacuum symm etry shift, represented by sum of contributions of ( ∆mV)between bodies and outside of them - decreases with decreasing the distance between them. For the other hand, if the starting separation between bodie s (L) is big enough:. L>LBV C=/planckover2pi1/[∆mV·c] (55c) then the increasing of this separation minimize the resulti ng vac- uum symmetry shift, i.e. we have a gravitational repulsion. The gravitational interaction is related to the very low-en ergy vir- tual quanta pressure oscillation between positive and nega tive zero- point vacuum, which may be termed as under-zero-point virtual quanta ( -1 2/planckover2pi1ω0< E SV<+1 2/planckover2pi1ω0),in contrast to virtual quanta with energy, bigger than absolute value of zero-point energy:/vextendsingle/vextendsingle±1 2/planckover2pi1ω0/vextendsingle/vextendsingle.2 37The gravitational interaction is displayed itself as a resu lt of bi- vacuum symmetry shift and corresponding virtual pressure e mer- gency: ΠV˜ ∆mV·c2 The quantum beats between bi-vacuum sublevels, exciting th e gravitational waves (GW) with frequency νGW=c2 h/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsinglem+ V−1 2m+ 0/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsinglem− V−1 2m− 0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg decreases the ∆mVandΠV. Consequently, theGW represents the density/pressure osci llations of theunder zero-point virtual quanta , excited as aconsequenceof beats between positive and negative vacuum states, formed b y bi- vacuum bosons Bose condensate. In another terms, GW can be co n- sidered as a result of interference between vacuum symmetry waves (VSW+and VSW−).In the framework of our model there are no evidence, pointing to luminal limitation of gravitational waves prop- agation. It contrast to gravitational field, the electromagnetic fiel d is a result of quasi-real vacuum density waves (VDW) interferen ce, enable to impulse and energy transmission. Comparing our formulae for total electromagnetic (42) and g ravi- tational (49) energies, we get the relation between them: Eel−m EG=α β=e2 Gm2 0= 4.1975·1042(56) These results and presented below point out, that our model m ay serve as a natural and clear background for Superunification . The interrelation with general theory of relativity It is possible to demonstrate a relation between Einstein’s idea concerning curving the geometry of space in the presence of g ravitat- ing body and our vacuum symmetry shift parameter: ∆mV=/vextendsingle/vextendsinglem+ V−m− V/vextendsingle/vextendsingle=β/vextendsingle/vextendsinglem+ C−m− C/vextendsingle/vextendsingle (57) Einstein postulates that gravitation changes the trajecto ry of probe body from the right to geodesic one due to curving conventional tw o-dimensional sur- face in 3D space. For example, trajectories of planets round the sun corresponds to geodesic lines. 38Instead Euclid geometry on flatsurface ,the Lobachevsky geometry on curved surface was used in Einstein’s classic theory of gravitatio n. The criteria of sur- face curvature for sphere - is a difference between sum of angl es in triangle on the flat surface equal to π= 1800, and that on curved surface: Σ =π+S/R2 where:Sis a square of triangle ( πR2on the flat surface); Ris a sphere radius, or a curvature radius in general case: R=/radicalbigg S Σ−π(57a) when (Σ −π)>0, the curvature ( R>0) is positive; when (Σ −π)<0,the curvatureRis imaginary and corresponding space is negative. If the spa ce (surface) is flat, then R=∞and Σ =π= 1800. In our Wave - Corpuscle Duality Model of Gravitation instead space-time curvature R we introduce a Bi-Vacuum Symmetry Cu r- vature, defined as: ±Lvac=±λvac 2π=/planckover2pi1 ±∆mV·c=/planckover2pi1 ±β∆mC·c(57b) where: ±∆mV=±(/vextendsingle/vextendsinglem+ V/vextendsingle/vextendsingle−/vextendsingle/vextendsinglem− V/vextendsingle/vextendsingle) =±β∆mCis a vacuum symmetry shift, positive for particles and negative for antiparticles, rel ated directly to mass symmetry shift. It is possible to calculate, using (2.9) that vacuum curvatu re, in- duced by particle with mass, equal to that of the electron (me= 9.1095·10−31kg)is:Le V= 3.2288·1035m. For the particle with mass of proton (mP= 1.6726·10−27kg)we have: LP V= 5.212·1025m. Energy of gravitational field, produced by one proton, calcu lated from (2.9) is equal to: ǫP G= 8.8904·10−52J. The analogy between RandLvac(2.26 and 2.27) is obvious. The more is energy of gravitational field ǫG, the more is vacuum symmetry shift (∆ mV) and bi-vacuum curvature. The bigger is bi-vacuum curvature , the less is radius (Lvac). In condition of black hole origination, when ∆ mV→βMPlthe bi-vacuum curvature radius tends to that, determined by gravitationa l radius (rg)of black hole and Plank length (see 55b): L∗ Vac=/planckover2pi1 βMPl·c=rg/2β (57c) 39On the other hand, in the absence of gravitation, when the pos itive and negative vacuum ground states are in state of ideal symmetry and equil ibrium: |m+ V|=|m− V|=1 2m0=1 2m+ C(57d) and ∆mV= 0, then the bi-vacuum is flat: LVac=∞. The photons trajectory reflects the bi-vacuum curvature in 3 D space. It is a consequence of our model of photon as a superpos ition of three pairs of coherent standing [neutrino + antineutrin o], moving in bi-vacuum without its symmetry perturbation. The trajectory of photon follows the bi-vacuum Bose condens ate radius and in general case deviate from the straight line, correspondi ng to ”flat” primordial bi-vacuum in the absence of matter. Near the black holes it tu rns to the closed one as a result of corresponding bi-vacuum symmetry violati on. As well as General theory of relativity our theory can explai n the red shift of photons in gravitational field. The RED, low-fre quency shift: ∆ω1,2 p=ω(1) p−ω(2) p (57e) of photons in gravitation field is a result of deviation of the ir trajec- tory from the right line and is a consequence of increasing th e vacuum symmetry curvature and corresponding length of its path. In accordance to our model, red shift has a simple relation wi th difference of vacuum symmetry shiftsat point of photon radia tion ∆m1 V= (|m+ V| − |m− V|)1and at point of it registration ∆m(2) V= (|m+ V| − |m− V|)(2): ∆∆m1,2 V= ∆m(1) V−∆m(2) V in a form: /planckover2pi1∆ω1,2 p= ∆∆m1,2 V·c2(57f) or: ∆ω1,2 p=∆∆m1,2 V·c2 /planckover2pi1 It is easy to see that if ∆∆m1,2 V= 0, i.e. bi-vacuum is flat, then ω(1) p=ω(2) pand red shift is absent. We may conclude, that our Duality Model of Gravitation ex- plains the same phenomena, as do the General theory of relati v- ity, but in terms of vacuum symmetry shift with tensor proper ties, instead of curved space-time. The tensor properties of bi-v acuum symmetry shift is related directly to that of mass symmetry s hift: (∆mV=β∆mC)x,y,z, produced by asymmetry of relativist real mass dependence on the external group velocity in 3D space. 40New interpretation of Newtonian interaction between macro scopic bodies. The spinning and Biefeld-Brown effects Let’ compare the formulae of Newtonian gravitational attra ction between two macroscopic bodies and gravitational potentia ls of these bodies, leading from our theory (49, 50): EG=Gm1·m2 r1,2=Gm2 1,2 r1,2(58) Emax G=− →r r1,2β·1,2/summationdisplay e,p,n/vextendsingle/vextendsinglem+ C−m− C/vextendsingle/vextendsingle e,p,n·c2=− →r r1,2β·1,2/summationdisplay e,p,n(m+ C·v2)e,p,n (58a) where :r1,2isthedis tancebetweencentersof bodies ; m1,2= (m1m2)1/2istheave argedmassof twobodies ;β= (m0/MPl)2 Using the same considerations, as in section ”Electromagne tism”, eq. 58 turns to 58a at conditions: m1= (− →r)1/2β1/2∆m1/2 1·c/G1/2= (− →r)1/2β1/2·(m+ C)1/2 1v1,2/G1/2(58b) m2= (− →r)1/2β1/2∆m1/2 2·c/G1/2= (− →r)1/2β1/2·(m+ C)1/2 2v1,2/G1/2 where :v1,2= (v1v2)1/2istherelativeresultingvelocity ; ∆m1=1/summationdisplay e,p,n/vextendsingle/vextendsinglem+ C−m− C/vextendsingle/vextendsingle e,p,n∆m2=2/summationdisplay e,p,n/vextendsingle/vextendsinglem+ C−m− C/vextendsingle/vextendsingle e,p,n The masses of each body are the result of all elementary parti - cles mass summation. In the Golden mean conditions, as was sh own earlier:/vextendsingle/vextendsinglem+ C−m− C/vextendsingle/vextendsingle e,p,n= (m0)e,p,n.It looks, that the existing of such invariant as Avogadro number - confirms our hypothesis, that the ele- mentary particles in composition of atoms and molecules and atoms/molecules itself in any phase of matter are tending to certain ratio (25 ), provid- ing realization of Hidden harmony condition (21), as a backg round of Golden mean rule (21a). For the averaged mass, squared, we get from 58b: m2 1,2= (− →r)·(m0/MPl)2·∆m1,2·c2/G= (− →r)·(m0/MPl)2·(m+ C)1,2·v2 1,2/G (58c) If we assume that ( m2 1,2/m2 0) =N2,then, taking into account that G=/planckover2pi1c/M2 Pl,(58c) may be transformed to important formula, meaning 41that the resulting volume in 3D energetic space is permanent : /parenleftbigg1 − →rN2·/planckover2pi1c/parenrightbigg3 = (∆m1,2·c2)3 x,y,z=/bracketleftbig (m+ C)1,2·v2 1,2/bracketrightbig3 x,y,z=const (58d) One can see from this formula that if due to any external fac- tors (electric or magnetic fields, relative rotation of inte racting bod- ies, etc., the value of (∆m1,2·c2)2 x,y=/bracketleftbig (m+ C)1,2·v2 1,2/bracketrightbig2 x,yis increasing, the ”vertical” component of gravitational interaction (∆m1,2·c2)z=/bracketleftbig (m+ C)1,2·v2 1,2/bracketrightbig zshould decrease. Such effect corresponds to decreasing of gravitational attraction between bodies. In the case of charged condensers, the mechanism described, may be responsible for Biefeld-Brown effect. The explanation of this effect is related to polarization of charge and mass of di elec- tric molecules in electric field in such a way, that massive po sitive nuclears are shifted towards negative plate of condenser an d their oscillations become more unharmonic and asymmetric in 3D sp ace. In accordance to our Hierarchic theory of condensed matter, signif- icant fraction of atoms/molecules in solids are compositio n of co- herent clusters, formed as a result of high-temperature Bos e con- densation (see my: ”Hierarchic theory of condensed matter. ..” at: www.egroup.com/docvault/antigrav) This is important fac tor, mak- ing the influence of external field on the atoms/molecules cha rge and mass polarization - cumulative. Correspondingly, the viri al coeffi- cient, equal to ratio between the resulting doubled kinetic and po- tential energy of coherent fraction of positively charged nuclears of dielectric between plates, as a tensor, becomes also more asym- metric. For the other hand, in accordance to this theory, bet ween tensors of mass symmetry shift, responsible for electromag netism, and tensor of vacuum symmetry shift, responsible for gravit ation, - the direct correlation exists: (∆mV=β∆mC)x,y,z. The decreasing of resulting component of tensor in directio n of positively charged plate of condenser (+): (∆m1,2·c2)+ z=/bracketleftbig (m+ C)1,2·v2 1,2/bracketrightbig+ z(58e) corresponds to decreasing of gravitational attraction and electro- magnetic interaction presumably in this direction (+z) as r espect to directions, opposite (-z) and normal ( ±x,±y) to this one. This may explain the asymmetric Biefeld-Brown effect. If both plates of condenser have the same charge, then inde- pendently of sign of charge , the amplitude and kinetic energy of [nuclears+electronic shells] oscillations will be more li mited in direc- tion normal to plates, than in directions of the same plane, d ecreasing selectively the normal component of gravitational interac tion of mass of dielectric between plates with Earth. 42Charging the dielectric disk positively or negatively with exter- nal source of charge - will lead to charge distribution near i ts sur- face due to Coulomb repulsion. In this case the effects will be the same, as in the case of unipolar condensers, described above . The electro-gravitational effect will increase with relative r otation of discs in accordance to mechanism proposed. Our mechanism of Biefeld-Brown and related effects - ex- plains their dependence on mass of dielectric between plate s, dielec- tric permeability, as far it is related with polarizability and density of dielectric, the density of electromagnetic energy betwe en plates, related to proximity of plates and voltage. The mechanism predicts also, that the increasing of tempera ture, declining the coherence of unharmonic nuclears oscillatio ns at the permanent other conditions (taking into account possible d ecreas- ing of dielectric permeability) should decrease the values of electro- gravity effects. It.is a way for experimental verification of suggested mechanis m, based on our duality model. THE PRINCIPLE OF LEAST ACTION AND PROBLEM OF TIME It can be shown, that the Principle of Least Action is one, re- flecting the nonlocal Wave - quality of the World and its feedb ack reaction with local Corpuscular - quality. After Lagrange, the action for particle, like electron could be expressed as: S=t2/integraldisplay t12Tkdt (59) IfTkinis the averaged kinetic energyof particle (or system of particles) during the time interval: t=t2−t1,we have from (59): S= 2Tkint (59a) Representing the time interval (t)as an integer number of [ C⇋W] pulsation period ( T0)we have: t=nTC⇋W=n·h/[(m+ C−m− C)c2], where :n= 1,2,3,... (59b) At condition of Golden mean, when: (m+ C−m− C) =m0,we get: t=nT0.This means that time is a discreet parameter and can be quantizated. 43Using expression for doubled kinetic energy of particle in [ C] and [W]-phase eq.(12), we get the following expressions for act ion: S=m+ Cv2·t= (m+ C−m− C)c2·t= [(m+ V−m− V)c2/β]·t (60) We can see from (60) that the fundamental principle of least a ction: δS= 0,leads to new formula, interrelating the positive pace of tim e for particle or body (dlnt=dt/t)with the decreasing of its kinetic energy, including real mass of particle (body) and its veloc ity: dlnt=−dlnTkin=−[dlnm+ C+ 2dlnv] (61) The similar result we can get from principle of uncertainty i n co- herent form. Condition (61) unify the pace of time for particle or system o f particles with pace of this system’s mass change ( dlnm+ C=dm+ C/m+ C) and with pace of its velocity change ( dlnv=dv/v). Another version of this formula interrelates the pace of tim e with particles mass - symmetry shift △mC= (m+ C−m− C)and corresponding vacuum symmetry shift △mV=m+ V−m− V=β△mC: △lnt=−dln(m+ C−m− C) =−dln(m+ V−m− V) (62) or:△lnt=△lnEel=△lnEG Comparisons of this formula with those, describing the elec tromag- netic (42) and gravitational (51) energy leads to important conclusion: oscillations of electromagnetic and gravitational fields s hould be ac- companied by time oscillations, i.e. TEMPORAL WAVES & FIELD origination. Like electromagnetic and gravitational, the TEMPORAL waves may form a standing, hologram-like pattern. The TEMPORAL FIELD, consequently, is related directly and may be influenced by electromagnetic and gravitational field s. At the condition of Hidden Harmony and Golden mean, when △mC= (m+ C−m− C) =m0=const, the pace of time for such system is zero: △lnt=−dlnm0= 0 (62a) This condition means the achievement of top of evolution/se lf- organization of selected system. Our approach to problem of time, based on eq.(61) leads to def - inition: ”Time for any closed nonequilibrium or oscillating physica l system is a parameter, characterizing the pace of this syste m kinetic energy (mass and velocity) change”. In accordance to our model, the characteristic time for any c losed system (ti),including the Universe, is positive if the kinetic energy, 44including the real mass of this system Miand its velocity is decreasing and negative in the opposite case: ti=−Tkin dTkin/dt=−/bracketleftbiggMi dMi/dt+ 2v dv/dt/bracketrightbigg (63) This relation, derived from eq. (61) between the time and pac e of kinetic energy change is valid for any selected closed syste m. When the pace of change of such system’s mass and velocity - tends to zero, it determines the delay of different physico-c hemical processes. The course of time for any system could be characterized by th e time decrement, introduced as: Di= exp/parenleftbigg −ti T0/parenrightbigg (64) where:T0= 1/ν0=h/(m0c2)is period of wave B [C ⇋W]oscilla- tions, corresponding to Hidden harmony condition. Hierarchy of systems from atom to universe determines the co r- responding hierarchy of time-scales. If the decreasing of mass of system and its cooling, (kT →0andv→ 0)is irreversible, for example as a result of IR photons radiat ion, the time for this system is positive and irreversible also. This process corresponds to second law of thermodynamics realization. The Principle of least action in form (62) for the minimum tim e in- terval ( △t= min) means that charged and neutral particles ”choose” the trajectory, corresponding to the minimum change of gene rated by their propagation electromagnetic ( ES el˜ ∆mC, eq.44a)and gravita- tional (ES G˜ ∆mV, eq.51a)energy. This happens, for example, at the conditions of Hidden harmony for selected system. Such ability of particles (including photons) to ”seek out” this optimal trajectory can be only a consequence of feedback rea ction be- tween their nonlocal due to vacuum amplitude waves (VAW) and local properties of particles in a course of their [ C⇋W]pulsations. This is a new quantum explanation of the fundamental Principle of Least Action, based on our model . For the other hand, formula (62) in combination of eqs.(42 an d 49) means that the pace of time and, consequently, the dynami cs of different process should be in-phase with oscillations of gr avitational and electromagnetic fields. On macroscopic scale such oscillations can correspond to pe riods of: a) Earth rotation around its own axe; 45b) rotation of Earth around Sun and c) rotation of Moon around Earth. The experimental evidence of macroscopic oscillations of v ery different dynamic processes (physical, chemical and biological) of the mentioned above periods has been obtained in long term syste matic observations by team of S. Shnol from Moscow university. NEW INTERPRETATION OF THE WAVE FUNCTION Our dynamic duality model makes it possible to modify the in- terpretation of the wave B function of Schr¨ odinger equatio n. We can present wave functions ψandψ∗in dimensionless form, using the ef- fective mass of the electron [2 ν0+/tildewideν0], determined by uncompensated standing neutrino ( ν0) : Ψ+=/vextendsingle/vextendsinglem+ C−m0/vextendsingle/vextendsingle m0(65) Ψ−=/vextendsingle/vextendsinglem− C−m0/vextendsingle/vextendsingle m0(65a) On the microscopic level the wave function squared is depend ent on the product of instant values of real and mirror corpuscul ar mass shifts: ( ∆m+·∆m−), related to fraction of time (ft C), which particle spend in corpuscular phase: |Ψ|2= Ψ+·Ψ−=/vextendsingle/vextendsinglem+ C−m0/vextendsingle/vextendsingle m0·/vextendsingle/vextendsinglem− C−m0/vextendsingle/vextendsingle m0=(∆m+·∆m−) m2 0(66) ft C=τC τC+τW=1 1 +K[W⇋C](66a) where:mt 0= (m+ C·m− C)1/2is the resulting zero-point corpuscular mass. The product of real and mirror mass symmetry shift: [(∆m+·∆m−)>0]C in Corpuscular phase is positive. In the Wave phase, when ∆m+= ∆m−= 0,it is zero. One can see from (66), that the mass symmetry shifts - changes in-phase with probability of location of particle in [C] pha se in the any given volume of space, equal to the wave function squared |Ψ|2. Our theory predicts that finding the way to shift the [C⇋W]equilibrium could be a way to change the real mass of body. For example, the application of strong magnetic or electric fields, interact ing with CVC of the charged particles - can influence the vacuum symmet ry shift and change the real mass of body. This effect may be used a lso for propulsion of one matter throw another or big space-jump s. 46The elementary particles of positive and negative charge, l ike elec- trons and protons have the opposite influence on the vacuum sy mme- try shift, i.e. gravitation. In accordance to our Alternati ve corpuscle- wave model of atom, at some conditions, the distance between their electron’s and proton’s can become very small as a result of v iolation of Golden mean condition: (m+ C−m− C)> m 0(see eq.44b). As condi- tions of strong overlapping of their cumulative virtual clo uds (CVC), their opposite bi-vacuum shifts can totally or partially co mpensate each other without annihilation. Our gravitation theory pr edicts, that this effect should be accompanied by decreasing of the eff ective mass of such atoms and their gravitational potential. INTERRELATION BETWEEN MICROSCOPIC [WAVE ⇋CORPUSCLE ]EQUILIBRIUM AND MACROSCOPIC [VACUUM S⇋MATTER] EQUILIBRIUM Physical vacuum in our theory is subdivided to Primordial BI - VACUUM, existing without matter, and Secondary one [VACUUM S], coexisting with matter and fields, produced by matter. Secondary vacuum reflects the interference of the nonlocal vac- uum amplitude waves (VAW), vacuum density waves(VDW) and va c- uum symmetry waves (VSW) as a components of cumulative virtu al cloud (CVC), resulted from [C⇋W]pulsation of elementary par- ticles. Secondary vacuum is a result of perturbations of sup erfluid Primordial vacuum properties (energy slit and symmetry) by huge number of particles. It follows from our model, that coherent over the large-scal e mi- croscopic [Wave ⇋Corpuscle] dynamic equilibrium of elementary par- ticles - can lead to macroscopic [VACUUM S⇋MATTER] dynamic equilibrium. Theasymmetrical state of the electron is related to semiperiod of wave B, when one (unpaired) standing neutrino (v0)is in Corpus- cular phase and pair (ν0+/tildewideν0)is in the Wave phase. In contrast to symmetrical [C⇋W]pulsation of (ν0+/tildewideν0)pairs, generating nonlocal vacuum amplitude waves (VAW), the asym met- rical pulsation of uncompensated standing neutrino ( ν0), accompa- nied by [emission ⇋absorption] of cumulative virtual cloud (CVC), can be accompanied by energy (electromagnetic and gravitat ional) transfer as far the resulting Pointing vector is nonzero. Su ch kind of CVC can’t be superluminal or nonlocal. The same is true for bo sons, like photons, when one of [neutrino + antineutrino] pair hav e nonzero resulting momentum and their spins have the same sign. At the 1st semiperiod of wave B, when pair (ν0+/tildewideν0)as a part of the electron (2ν0+/tildewideν0)or quark is in the [W] phase and forms posi- tive and negative virtual quanta of (VACUUM S)ONwith minimum 47slit of bi-vacuum , the second standing neutrino (ν0)of the electron is in Corpuscular state and forms the (MATTER )ON. This instant situation on macroscopic scale corresponds to our percepti ble ”ON”- World. At the 2nd semiperiod of wave B pulsation, when standing [neutrino- antineutrino] pair (ν0+/tildewideν0)is in [C] phase and the energy slit of bi-vacuum (VAW) is increasing , the unpaired standing neutrino (ν0)is in the [W] phase. At this moment our (VACUUM S)ONcol- lapses to hidden (MATTER )OFF.Simultaneously our explicated Cor- puscular ”ON” World transforms to hidden implicated (VACUUM S)OFF. Such enfolded, [VACUUM S+ MATTER] OFFsubsystem of the part of the UNIVERSE with in-phase [ C⇋W]pulsation - is alternative to unfolded [MATTER + VACUUM S]ONsubsystem. The perception of World by bio-receptors and equipment norm ally is limited by [ON] in-phase subsystem, corresponding to the first of two described above semiperiods only and fields, resulting f rom [ON ⇋OFF] transitions. The dynamics of conversions between [ON]and [OFF]−coherent Worlds due to their ultrahigh frequency cannot be registered experimentally. The result of dynamic equilibr ium be- tween these subsystems in form of fields is detectable only. The feedback reaction between [ON]and[OFF]coherent Worlds subsystems as a consequence of nonlocality of VAW may exist as a conditio n for the UNIVERSE self-organization. The coherent molecular dynamics of real MATTER ON, including living organisms, with alternating acceleration and mass o f parti- cles (m+ C)- is related with corresponding dynamics of VDW, VSW and nonlocal VAW, beats between them and wave packets forma- tion. Changing the part of this complicated interference pi cture with hologram properties, induced by change of matter propertie s means holomovement after Bohm. Such holomovement can be imprinted in [VACUUM S]OFFand, con- sequently, in [VACUUM] ONas the Informational Dynamic Replica of our World. The deviation of Informational Replica from ”T her- mal Noise” of [VACUUM S]ONis dependent on the scale of coherent molecular/atomic excitations, responsible for Virtual Re plica origi- nation. PROPERTIES OF BI-VACUUM BOSONS (BVB), VACUUM AMPLITUDE, VACUUM DENSITY AND SYMMETRY WAVES (VAW; VDW and VSW), AS A SOURCE OF INFORMATIONAL FIELD In corpuscular (collapsed) state the real corpuscular mass (m+ C) of standing neutrino or antineutrino, in another terms: [re al vor- tex+mirror rotor] or [mirror vortex+real rotor] dipoles de termines 48the real, measurable mass of particle or antiparticle. The m irror mass (m− C)is hidden. As a result of beats between real and mirror states, the both mass are equalizing and tending to mass of re st (m0) of particle: m+ C→m− C→m0, where :/vextendsingle/vextendsinglem+ C/vextendsingle/vextendsingle>m0>/vextendsingle/vextendsinglem− C/vextendsingle/vextendsingle It happens due to partial conversion of m+ Candm− Cto cumulative cloud of positive and negative virtual quanta (CVC), corres ponding to [W]-phase of particle. From eq.(9) it leads that at this phase, the mass symmetry shi ft (△mC), the external impulse and external group velocity of BVB (binding or anchor site of CVC ) turns to zero: △mC= (m+ C−m− C) = 0;PC= 0and v →0 (67) Putting the external group velocity: v= 0to formula (19) one can see, that at this condition the internal group and phase velocities becomes equal to each other and to that of light velocity. vin gr=vin ph=c at m+ 0=m− 0=m0 (67a) It is a conditions of symmetric BVB existing as a pair of virtu al [rotor + antirotor]. The radius of BVB is equal to that of the e lec- tron’s cumulative virtual cloud (CVC) and the Compton radiu s at the condition of Hidden harmony: LBV B=/planckover2pi1/m0c= 3.86×10−13m (68) The surfaces of positive and negative components of BVB and their sum are equal to (16d). Our notion of BI-VACUUM did not make limitations on the value ofm0=m+ 0=m− 0and, consequently radiuses of rotor and antiro- tor, composing bi-vacuum bosons (BVB) because of compensat ion of energy and impulse in the realm of bi-vacuum .The radiuses of ro- tor (L+)and antirotor ( L−)of BVB from principle of uncertainty in coherent form: L±·P±=/planckover2pi1/2may be expressed like: L+=+/planckover2pi1 2P+and L−=−/planckover2pi1 2P−(68a) In the case of symmetric BVB with zero resulting momentum, we haveL+=L−andP++P−= 0. The values of impulses in the framework of our model may vary from|P+|=|P−|=m0cto|P+|=|P−|=MPl·c, wherem0is a rest-mass of the electron and MPlis the Plank mass. In principle, the variation in mass and corresponding variation in diameter of rotor and antiro- tor, forming different BVB may be even bigger. However, judgi ng 49from discreet values of mass of elementary particles, the ”r esonant” values of BVB parameters: mass, impulse, radius must exist. The minimum by dimensions BVB with resonant properties may form the ”molecules” of bi-vacuum as a quantum superfluid liquid. It m eans that a scale fractal hierarchy of BVB may exist, which determ ines the selected properties of bi-vacuum and that of elementary par ticles. The idea of Neil Boyd about subquantum particles, composing vacuum and elementary particles, is compatible with our mod el, if we assume that they represent the smallest resonant BVB in th eir ground (symmetric) and .excited (asymmetric) states. The quantum beats between different resonant conditions of s ym- metric BVB lead to excitation of nonlocal vacuum amplitude w aves (VAW). The vacuum amplitude waves (VAW) display themselves as the oscillation of energetic slit ( ∆AV AW)between positive and negative ground levels of bi-vacuum, equal to: ∆AV AW= (∆m+ V+ ∆m− V)·c2(69) For the case of symmetric BVB excitation: ∆m+ V= ∆m− V= ∆mVand ∆AV AW= 2∆mV where two resonant energies of bi-vacuum are: E0=A0=m0c2andE1= (m0+ ∆m0)c2(70) The radius of corresponding symmetrical BVB is equal to: L0 BV B= /planckover2pi1/m0candL1 BV B=/planckover2pi1/(m0+ ∆m0)c. The frequency of VAW is: νV AW= ∆AV AW//planckover2pi1 (71) After squaring the left formula in (70) and differentiation, taking into account, that m2 0=m+ C·m− Cwe get after simple reorganizations: dlnAV AW=1 2[dlnm+ C+dlnm− C]·c2(72) This formula points that acceleration of particles and corr espond- ing change of m+ Candm− Cshould influence on the energy of vacuum amplitude waves (VAW). The excitation of VAW is accompanied by oscillation of BVB de n- sity, virtual pressure and correlated oscillation of energ y in realms of positive and negative vacuum. If we assume that between two c on- ducting plates the probability of big transitions, between different co- herent states of bi-vacuum and VAW excitation is less, than o utside of such ”condenser”, this explains Casimir effect. Such prob ability 50decreasing may happen due to stabilization of selected bi-v acuum ex- citations (VAW) as a result of correlation of [C ⇋W]transitions of the electrons in two conducting plates. The correlation, in turn is dependent on distant Van der Waals interaction between plat es and decrease with their separation. In formula (69) the positive and negative vacuum symmetry sh ifts, produced by standing neutrino ( ν0)and antineutrino ( /tildewideν0)of coherent pairs [ν0+/tildewideν0]in composition of elementary particles are defined as: /bracketleftbig ∆m+ V=m+ V−m− V/bracketrightbig ν0(73) /bracketleftbig ∆m− V=m− V−m+ V/bracketrightbig /tildewideν0 The oscillations of bi-vacuum slit AV AWmay be a result of peri- odic acceleration of pair of standing [neutrino+antineutr ino] or cor- responding sub-elementary [particle+antiparticle] in [C ] phase un- der the influence of fields or in a course of vibrations of atoms and molecules. In the case if VAW-source has asymmetric shape, the ∆AV AWmay be considered as asymmetric scalar potential of informatio nal field, then the corresponding component of informational field is t he vector one, defined as − − − →IV AW=grad(∆AV AW) (74) The instant propagation of VAW is related to nonlocal proper ties of virtual Bose condensate of primordial BVB and BVB∗,perturbed by VAW. The radius of perturbed by VAW bi-vacuum bosons is equal to: LBV B∗=/planckover2pi1/[(m0+ ∆m± V)·c] (75) The corresponding scalar potential of BVB∗: EBV B∗= (m0+ ∆m± V)·c2(75a) From (9) it follows also that at conditions of primordial bi- vacuum, i.e. in the absence of matter when: ∆mC=/vextendsingle/vextendsinglem+ C−m− C/vextendsingle/vextendsingle→0and/parenleftbig vext gr=v/parenrightbig → 0, the external wave B length of BVB and corresponding scale of their virtual Bose condensation tends to infinity: λBV B=h/PBV B=h//bracketleftbig (m+ 0−m− 0)·c/bracketrightbig → ∞ (76) The length of secondary BVB∗in contrast to primordial BVB has a huge, but limited dimension due to small difference of en - ergy/impulses between virtual rotor and antirotor: λBV B∗=h/[(m+ 0+∆m+ V)·c−(m− 0+∆m− V)·c]<∞ (77) 51The results, obtained above, points, that the ability of som e stable particles, like free neutrino, photons, etc. to move with lu minal ve- locity is a consequence of the basic condition: the equality of the real and mirror masses in the case of free neutrino ( m+ C=m− C=m0), or symmetric distribution of real masses for coherent pairs of [neutrino + antineutrino], forming photons. The gradient of difference of concentration of rotors (n+)and an- tirotors (n−)of BVB with opposite direction of rotation (virtual spin equilibrium shift), originated under the influence of rotat ing atoms, molecules or macroscopic bodies - is responsible for TORSIO N field, introduced as: − →Tn=grad(n+/n−) (78) The other torsion field components, in accordance to our mode l, may display itself also in the gradient of following propert ies of BVB Bose condensate: a) the internal radius of individual BVB as a pair of [rotor+a ntirotor]: TR=grad(LBV B∗) (78a) b) the external wave B length of secondary BVB∗, which deter- mines the spatial scale of virtual Bose condensate: Tλ=grad(λBV B∗) (78b) c) the amplitude of torsion waves, determined by values of ∆TR and∆Tλoscillations. The[C⇋W]pulsations of two standing neutrinos (2 ν0)with op- posite half-integral spins, forming part of the electron (2 ν0+/tildewideν0)are counterphase. It means that when one of them is in [C] phase, t he other is always in the [W] phase. This makes these neutrinos a s well as two electrons of opposite spins spatially compatible and enable to coherent exchange interaction by means of CVC. Such a mechan ism is used for explanation of Pauli principle. The[C⇋W]pulsations of standing antineutrino ( /tildewideν0)are in-phase with one of these two neutrinos of triplet (2 ν0+/tildewideν0). They form the coherent symmetric pair (ν0+/tildewideν0),stabilized by special kind of dipole- dipole exchange interaction, introduced in our model. The c harge, energy and impulse of standing neutrino and antineutrino in pair (ν0+/tildewideν0)of electron, positron, quark or other fermions compensate each other. The in-phase [C⇋W]pulsations of such pairs are re- sponsible for nonlocal VAW, virtual VDW and VSW standing wav es 52as a components of CVC. The standing CVC waves did not transfe r the energy. Their electromagnetic resulting Pointing vect or is equal to zero: − →Pν0+/tildewideν0=− →Pν0+− →P/tildewideν0= 0 (79) where: − →Pν0=/bracketleftBig− →E×− →H/bracketrightBig (80) − →P/tildewideν0=/bracketleftBig− →H×− →E/bracketrightBig =−− →Pν0 (81) Standing electromagnetic, gravitational, torsion and non local (in- stant) VAW, as a quantum collective excitations of bi–vacuu m Bose condensate (BC), excited in a course of [C⇋W]pulsation of pairs [ν0+/tildewideν0]in composition of elementary particles - can be responsi- ble for Informational field, wholeness and self-organizati on process of the Universe. This means that formation of dynamic coherent sys- tem of standing waves: VDW and VSW, modulated by VAW, with properties of 3D virtual hologram - is possible. The unified s ystem: [Secondary bi-vacuum + Matter] can evolve in a course of ”hol omove- ment” the - notion, introduced by Bohm. Each elementary particle, atom, molecule or macroscopic bo dy is a source of nonlocal resulting informational field with sy mmetry, depending on their shape, dynamics and mass distribution. Superposition of nonlocal vectorial informational fields, gener- ated by spatial combination, mass and dynamics of all elemen ts in corpuscular phase represents the virtual Informational Dynamic Replica of our real World. All four kinds of virtual collective excitations, listed ab ove, are the oscillations of density, energy and symmetry of corresp onding kinds of virtual quanta. Consequently, their superpositio n and in- terference, especially in form of virtual standing waves, m ay be ac- companied by the ”ordering” of bi-vacuum, decreasing its en tropy and increasing the information. It is known that the entropy and information are simple related to each other as: S= (kBln 2)·I (82) Each of 24 quantum excitations, pertinent for condensed mat ter, in accordance to our Hierarchic theory of condensed matter ( see: www.karelia.ru/˜alexk [New articles], may have own contri bution to the informational entropy (82). Hypothesis of Informational Vacuum Replica of living organisms and its consequences 53In any living organism: from microbe to elephant the more ord ered quasi-crystallin and sensitive structure - is a fraction of water in hol- low core of MTs with diameter about 140 ˚A. The bigger is number of MTs of coherently interacting cells, the bigger is correspo nding frac- tion of ordered water, very sensitive to nerve excitation. T he spatial and dynamic properties of MTs and internal water structure f ollow the Golden mean rule. I have a special work, related to role of MTs in Hierarchic model of consciousness (see: www.karelia.ru/˜ alexk [Ab- stracts and New articles]. Just this fraction of virtual waves, excited by water in MTs, is responsible, in accordance to my hypothesis, for special vi rtual Infor- mational field (IF), characterizing the individuality as a p roduct of brain and peripheral nerve system activity. In some cases, when corresponding [IF] form a complex system of virtual standing waves with properties, close to condition s of Hidden harmony: vin gr=vext gr (83) vin ph=vext ph it may remain stable even after the real source (living organ ism) becomes destroyed or dead. The Hidden harmony conditions mean the equality of pairs: in - ternal (hidden) and external group velocities; internal (h idden) and external phase velocities (83). This hypothesis, based on our theory, points to possibility of ex- istence of ’SOUL’. Some of corresponding systems of such vir tual Informational replica of living organism really may be more or less stable, depending on their properties as a systems of standi ng waves. During the life of animal, plant or human being, the direct an d back reaction should exist between ’soul’ and organism, gen erating this soul. This interaction may have a character of quantum b eats. If so, the important contribution to consciousness or MIND is p rovided by acts of interaction between brain and ’soul’ in form of qua ntum beats. Each individual soul is formed by interference of non local Vacuum Amplitude Waves (VAW) with standing vacuum density a nd vacuum symmetry waves (VDW and VSW) of the all nerve cells of living organism. The nonlocal character of each individual soul makes it possible souls interaction and even unification. The idea of collective informational soul is close to idea of NOO- SPHERE, proposed by Vernadsky at the beginning of this centu ry. If we make one more step ahead, we can suppose that unification of all souls of human beings, animals, plants of all planets w ith any forms of life over the Universe may form virtual dynamic Supe r- Consciousness. 54Due to mentioned above feedback reaction, the evolving of Su per- Consciousness is dependent on evolution of the intelligent part of biosphere of the Universe and vice-verse. The influence of Golden mean - based 3D rigid geometrical structures (Harmonizators) on surrounding medium To explain this effect, we assume that 3D structures, like cro ss, pyramids, double cones, etc. with proportions, following G olden mean (GM) rule works as effective generator of virtual image i n bi- vacuum, i.e. provide the ordering action on virtual quanta. It may happens in a course of [ C⇋W]pulsations of particles, com- posing GM-based structures. Pulsations of standing [neutr ino+antineutrino] pairs in composition of elementary particles (fermions) li ke electrons, positrons and quarks - are accompanied by vacuum amplitude w aves (VAW) three-dimensional (3D) superposition with standing virtual density and virtual symmetry waves (VDW±and VSW±). VAW, in accordance to our model, represent symmetrical osci lla- tions of energy slit between positive and negative vacuum wi thout vacuum symmetry shift, when: ∆mV=m+ V−m− V= 0 (84) If we use instead notion of standing neutrino/antineutrino , the notion of [vortex+rotor] dipole, nothing will change in our consider- ations. The 3D interference pattern, formed by VDW±and VSW±, mod- ulated by vibrations of atoms and molecules and VAW, with non local properties, modulated by [C] phase of standing [neutrino+a ntineutrino] pairs in similar way, may serve Informational Replica of GM- based structures. The VDW±and VSW±,in contrast to VAW, are the parts of two cumulative virtual clouds (CVC±),related to [W] phase of the same [ν0+/tildewideν0]pairs. In accordance to our model (eq.52), between the tensors of ma ss symmetry shift and vacuum symmetry shift of uncompensated s tand- ing neutrino or antineutrino: /parenleftbig β/vextendsingle/vextendsinglem+ C−m− C/vextendsingle/vextendsingle=/vextendsingle/vextendsinglem+ V−m− V/vextendsingle/vextendsingle/parenrightbig 1,2,3(85) and similar parameters of [ ν0+/tildewideν0]pairs - the strong correlation exists. It means that 3D spatial distribution of electromagnetic, g ravitational and torsion potential of any solid body, determined by uncom pen- sated neutrinos and antineutrinos, rotors and antirotors o f BVB - should be reflected in corresponding properties of 3D comple x holo- gram, produced by all kind of standing waves ,modulated by VAW as a pilot wave. It was shown before, that electro-gravitation al contri- bution to secondary vacuum is related to temporal field. 55The important consequence of our model is that the closer are properties of medium (object), as ”recipient” of action of H armo- nizator to conditions of Hidden harmony (see eq.21), the mor e is influence of Informational Replica of Harmonizator as ”indu ctor”. It is a condition of resonant kind of informational interactio n via bi- vacuum between ”inductor” and ”recipient”. Telepathy, as informational exchange between living organ isms with resonant properties of their microtubules and interna l water - producing individual Informational Replicas, may be expl ained in a similar way. We have to assume in the mechanism proposed, that properties of bi-vacuum virtual quanta has a fractal structure and tends t o Hidden harmony conditions (eq. 21) under the influence of Harmoniza tor. It is the intermediate effect, stimulating the next stage of a ction of Harmonizator on liquids, solids, biosystems, etc. as a resu lt of cor- responding bi-vacuum perturbations influence on target par ticles in [W] phase. The effect of Harmonizators on open systems looks to be relate d with their ability to stimulate postulated in our work self- organization as a natural spontaneous process, leading systems to condit ions of Hidden harmony. It is accompanied by the entropy decreasing and storage of information. This process is opposite to tending the sys- tems to thermal equilibrium or thermal death, correspondin g to con- ditions (25a). Experimentally the influence of Harmonizators on virtual In forma- tional Replica may be detected by the change of Casimir attra ction between two metallic sheets. If our mechanism, proposed, is right, it means that most effec tive Harmonizators under certain conditions may serve as the pur e energy generators, shifting the open systems from thermal equilib rium. The interrelation is predictable between the scale of real a tomic/molecular Bose condensation in condensed matter and influence of this m atter on nonlocal properties of bi-vacuum virtual Bose condensat e. The vacuum amplitude waves (VAW) may serve as the instant carrie r of information. The future instant transmitters/receiver s of infor- mation, containing cells of macroscopic Bose condensate (B C), like superconductors, superfluids and crystals with modulated B C param- eters, could be based on this principle. The more shape of the se cells follows the Golden mean rules, the bigger is predictable inf ormational effect. CONCLUSION 56The formulae, obtained in our work, interrelate electric an d mag- netic elementary charges, electromagnetic and gravitatio nal interac- tions, theory of relativity and quantum mechanics, [mass - v elocity - space - time]. The dynamic model of duality provides the dee per understanding of Pauli and Heizenberg principles. It is abl e also to explain the quantum roots of Golden mean, as a result of revea led Hidden Harmony conditions realization, principle of least action and. The two slit experiment also can be explained without introd ucing the notion of pilot wave. The Alternative Corpuscle-Wave model of atom based on the in - phase [C⇋W]oscillations of the electrons and nuclear and 3D virtual standing waves properties of their [W] phase is proposed. The new fundamental principle of self-assembly of ”simple” sys- tems physical systems, like elementary particles, atoms, m olecules, self-organization of complex open systems, like condensed matter, star systems, galactics and evolution of very complex syste ms like biopolymers (proteins, DNA, microtubules), cells, organi sms - can be formulated as: ”The different selected systems on each lev el of temporal and spatial hierarchy are tending spontaneously t o condi- tion of Hidden Harmony (equality of most important internal and external parameters of de Broglie waves), which determines the Golden Mean”. If we define the nonequilibrium system as beau- tiful, when it follows the Golden mean rule, then the formula ted principle of evolution is a ”driving force (hidden will)”, l eading our World to Hierarchical Beauty. REFERENCES Aspect A., Dalibard J. and Roger G. Phys.Rev.Lett. 1982,49, 1804. Aspect A. and Grangier P. 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arXiv:physics/0003002v1 [physics.data-an] 1 Mar 2000IASSNS-HEP-00/11 February, 2000 THE EQUILIBRIUM DISTRIBUTION OF GAS MOLECULES ADSORBED ON AN ACTIVE SURFACE Stephen L. Adler Institute for Advanced Study Princeton, NJ 08540 Indrajit Mitra Department of Physics, Jadwin Hall Princeton, NJ 08544 Send correspondence to: Indrajit Mitra imitra@princeton. edu 1Abstract We evaluate the exact equilibrium distribution of gas molec ules adsorbed on an active surface with an infinite number of attac hment sites. Our result is a Poisson distribution having mean X=µPP s Pe, withµthe mean gas density, Psthe sticking probability, Pethe evap- oration probability in a time interval τ, and PSmoluchowski’s exit probability in time interval τfor the surface in question. We then solve for the case of a finite number of attachment sites using the me an field approximation, recovering in this case the Langmuir isothe rm. 21 Introduction One of the models aimed at explaining the collapse of the wave function [1] predicts that the wave function of every system collapses to an eigenstate of the Hamiltonian in the energy basis in a time which depends on the energy spread of the wave packet. For a system including the measuri ng apparatus, relevant sources of energy fluctuations are thermal energy fl uctuations and energy (mass) fluctuations coming from fluctuations in the nu mber of surface adsorbed molecules. Our aim in this paper is to derive formul as for the equilibrium distribution of adsorbed molecules on an activ e surface S, from which the root mean square mass fluctuation can be calculated . Our results could also be relevant in other contexts - e.g. in surface cat alysis. Since our work is based largely on the classical colloidal st atistics problem [2] solved by Smoluchowski, we will review his result first. C onsider a gas chamber of volume Vwhich has Ngas molecules distributed randomly inside. Assuming uniform occupancy, the probability that a single m olecule is found inside a small subvolume visv Vand of not being found inside isV−v V. So the probability U(n) of some nparticles being found inside vis given by the binomial distribution 3U(n) =/parenleftBiggN n/parenrightBigg (v V)n(1−v V)N−n(1) The mean number of particles µfound inside the small volume vis just the mean of this binomial distributionNv V. In terms of µthen, the distribution U(n) becomes U(n) =/parenleftBiggN n/parenrightBigg (µ N)n(1−µ N)N−n(2) For most practical cases NandVare both very large, but the ratio of N/V is finite so that the mean µis finite. In this limit, the binomial distribution of Eq.(2) reduces to the Poissonian form U(n) =e−µµn n!(3) The interpretation of this equation is the following: If we f ocus on a small subvolume vinside a much larger volume V, then the frequency with which different numbers of particles will be observed in the s maller volume will follow a Poisson distribution. It should be noted that i n addition to the assumption of all positions in the volume having equal a prio ri probability of occupancy, we also assume that the motions of individual p articles are 4mutually independent. In the surface adsorption generaliz ation discussed in Sec. 2, this is the case for an infinite number of attachment si tes, but would not be the case for a finite number of attachment sites. Let us now define Pto be the probability that a particle somewhere inside the small volume vwill have emerged from it during the time interval τ. The “probability after-effect factor” Pwill depend on physical parameters such as the velocity distribution and mean free path of the partic les, as well as the geometry of the surface boundary. In terms of P, the probability that starting with an initial situation of nmolecules inside v,iof them escape in timeτis A(n, i) =/parenleftBiggn i/parenrightBigg Pi(1−P)n−i(4) LetEidenote the probability of the volume vcapturing iparticles during timeτ.Eiclearly is independent of the number of molecules already in side. But, under equilibrium conditions, the a priori probabilit ies for entrance and exit must be equal. For each nthere is a contribution to the exit probability; summing over all of them and equating to Eiwe get 5Ei=∞/summationdisplay n=iU(n)A(n, i) (5) Inserting the expressions for U(n) and A(n, i) from Eqs.(3) and (4) we get Ei=∞/summationdisplay n=ie−µµn n!/parenleftBiggn i/parenrightBigg Pi(1−P)n−i=e−µ(µP)i i!∞/summationdisplay n=iµn−i(1−P)n−i (n−i)!=σ(i, µP) (6) where from here on we denote a Poisson distribution with mean Xbyσ(n, X) with σ(n, X) =e−XXn n!(7) 2 Adsorption of gas molecules To make our analysis intuitively clear, let us draw an imagin ary surface I just outside the active surface area S. The following notations will be used: (i)Ei= Probability for imolecules to enter the volume enclosed by Iin the time inverval τ. Since this is the same as in the case where the surface S inside is absent, this probability is just as in Eq.(6), 6Ei=σ(i, µP) (8) (ii)U(n) = Probability to observe nmolecules sticking to S. (iii)Ps= Probability of a molecule to stick to Safter crossing I. (iv)Pe= Probability for a molecule that is stuck to Sto evaporate off in a time interval τ. (v)B(n, i) = Probability that starting with an initial situation with npar- ticles stuck to S,iof them evaporate in time τ. By Smoluchowski’s reasoning leading to Eq.(4) above, we hav e B(n, i) =/parenleftBiggn i/parenrightBigg (Pe)i(1−Pe)n−i(9) At equilibrium, the detailed balance condition holds. This is just the condi- tion that the probability that iparticles stick in a time interval τis equal to the probability that iparticles evaporate in the same time interval τ. The probability for imolecules to stick to Sis /summationdisplay j≥iEj/parenleftBiggj i/parenrightBigg Pi s(1−Ps)j−i 7Using Smoluchowski’s expression for Ejfrom Eq.(8) this becomes e−µP(µPP s)i i!/summationdisplay j≥i[µP(1−Ps)]j−i (j−i)!=σ(i, µPP s) (10) The other part of the detailed balance condition, the probab ility that out ofnmolecules on S,iof them evaporate in time interval τis /summationdisplay n≥iU(n)B(n, i) =/summationdisplay n≥iU(n)/parenleftBiggn i/parenrightBigg Pi e(1−Pe)n−i. (11) Equating these two probabilities, we have σ(i, µPP s) =/summationdisplay n≥iU(n)/parenleftBiggn i/parenrightBigg Pi e(1−Pe)n−i. (12) Our task now is to determine the equilibrium distribution U(n) from this equation. We start with the ansatz that U(n) is a Poisson distribution σ(n, X) with a mean Xwhich is to be determined, U(n) =σ(n, X) =e−XXn n!(13) 8Substituting Eq.(7) into Eq.(12) and using the sum evaluate d in Eq.(6), we get the condition σ(i, µPP s) =σ(i, XP e) (14) which is satisfied when X=µPP s Pe(15) Eqs.(13) and (15) are our result for the equilibrium distrib ution of adsorbed molecules. We note that, as intuitively expected, the mean n umber of ad- sorbed molecules increases with increasing gas density µand increasing stick- ing probability Ps, but decreases with increasing evaporation probability Pe. As a check on our reasoning, let us calculate the transition p robability W(n, m) formparticles to be stuck to the surface at time T+τwhen n particles were stuck to the surface at time T, and then check that W(n, m) andU(n) have the requisite Markoff property. The transition probab ility is 9given by W(n, m) =/summationdisplay x+y=mW(n) 1(x)W2(y) (16) where W(n) 1(x) is the probability that xparticles remain at time T+τwhen initially there were nat time T, W(n) 1(x) =/parenleftBiggn x/parenrightBigg (1−Pe)xPn−x e (17) andW2(y) is the probability for yadditional particles to adhere to the surface in time τas given by Eq.(10) W2(y) =σ(y, µPP s) (18) The Markoff property requires that U(m) =/summationdisplay nU(n)W(n, m) (19) withU(m) the equilibrium distribution of Eqs.(13) and (15). Evalua ting the sum on the right-hand side of Eq.(19), we find as required that /summationdisplay x+y=m(/summationdisplay nU(n)W(n) 1(x))W2(y) =/summationdisplay x+y=mσ(x, X(1−Pe))σ(y, µPP s) =σ(m, X(1−Pe) +µPP s) =σ(m, X) =U(m) 103 Finite number of attachment sites - mean field approach Let us now proceed to calculate the equilibrium distributio n of the number of molecules attached to Swhere Shas a finite (although very large) number of attachment sites M. Clearly, our discussion of the previous section breaks down since the sticking probability is no longer a constant, but depends on the number nof molecules already attached to S. In the following discussion, let us use Psto denote the probability for a molecule to stick to Sif no site is occupied, and let us denote the mean number of occupied sites bym. Then the mean sticking probability is just Ps=Ps(1−m M) (20) and the corresponding distribution of stuck molecules is σ(n,X), with X=µPPs Pe(21) 11Since the mean of this distribution is m=X, we get the mean field consis- tency condition m= (1−m M)µPP s Pe(22) with solution m=µPP s/Pe (1 +µPP s PeM)(23) Thus, the mean fractionm Mof total available sites occupied has the form of the Langmuir isotherm [3]. The mean field approximation is valid as long as the mean numbe r of vacant sites M−mis much larger than the width√ mof the distribution of adsorbed molecules, M−m=M(1−m M)>>√ m=√ M/radicalBigg m M(24) Close to saturation, whenm M≈1, substituting Eq.(23) into Eq.(24) gives the 12condition 1 +µPP s PeM<<√ M (25) which when X=µPP s Pe>> M simplifies to X << M3 2 (26) 4 Acknowledgments One of us (S.L.A) was supported in part by the Department of En ergy under Grant No. DE-FG02-90ER40542. He also wishes to thank J. Lebo witz, S. Redner, and R. Ziff for helpful e-mail correspondence. 5 References [1] See, e.g., N. Gisin, Helv. Phys. Acta 62, 363 (1989); I. C. Percival, Proc. R. Soc. Lond. A447, 189 (1994); L. P. Hughston, Proc. R. Soc. L ond. A452, 953 (1996). 13[2] M. v. Smoluchowski, Physik. Zeits. 17, 557 (1916) and 17, 585 (1916).We follow the exposition given in the review of S. Chandrasekha r, Rev. Mod. Phys. 15, 1 (1943). [3] R. H. Fowler, “Statistical Mechanics”, Cambridge, 1936 , pp. 828-830; R. H. Fowler and E. A. Guggenheim, “Statistical Thermodynam ics”, Cam- bridge, 1939, pp. 426-428. 14

arXiv:physics/0003003v1 [physics.atom-ph] 1 Mar 2000Suppression of Magnetic State Decoherence Using Ultrafast Optical Pulses C. Search and P. R. Berman Physics Department, University of Michigan, Ann Arbor, MI 4 8109-1120 (February 2, 2008) 32.80.Qk, 34.50.Rk, 34.20.Cf It is shown that the magnetic state decoherence produced by c ollisions in a thermal vapor can be suppressed by the application of a train of ultrafast opti cal pulses. In a beautiful experiment, Itano et al. demonstrated the Quantum Zeno effect [1]. A radio frequency pi pulse having a duration on the order of 250 ms was applied to a ground state hyperfine transition. At the same time, a series of radiation pulses was used to drive a strongly coupled ground to excited state uvtransition. The rfand strong transitions shared the same ground state level. Itano et al. showed that excitation of the rftransition could be suppressed by the uvpulses. They interpreted the result in terms of collapse of the wave function - spontaneous emission from the excited state during the uvpulses is a signature that the uvpulse projected the atom into its ground state; the lack of such spontaneous emission implies projection into the final state of the rf transition. This paper triggered a great deal of discus- sion, especially with regards to the interpretation of the results [2]. A necessary condition for a quantum Zeno effect is a perturbation of a state amplitude on a time scale that is short compared with the correlation time of the process inducing the transition . In the experiment of Itano et al., this time scale is simply the duration of the pi pulse, 256 ms. On the other hand, if one wished to inhibit particle decay or spontaneous emission [3], it would be necessary to apply perturbations on a time scale that is short com- pared with the correlation time of the vacuum, an all but impossible task. In this paper,we consider the inhi- bition of collisional, magnetic state decoherence, by the application of a train of ultrafast, optical pulses. This correlation time of the collisional perturbations resulti ng in magnetic state decoherence is of order of the dura- tion of a collision and is intermediate between that for the coherent pi pulse applied by Itano et al. and the almost instantaneous, quantum jump-like process pro- duced by the vacuum field. It should be noted that re- lated schemes have been proposed for inhibiting decoher- ence in systems involving quantum computation [4], but the spirit of these proposals differs markedly from the one presented herein. The rapid perturbations of the system are a necessary, but not sufficient, condition for a mechanism to qualify as a Quantum Zeno effect. The perturbations must involve some ”measurement” on the system for the ”Quantum Zeno” label to apply. The suppression of magnetic state coherence discussed in this paper is not a Quantum Zeno effect in this sense. We will return to this point below.We envision an experiment in which ”active atoms” in a thermal vapor undergo collisions with a bath of foreign gas perturbers. A possible level scheme for the active atoms is depicted in Fig. 1. At some initial time, an ultrashort pulse excites an atom from its ground state, having angular momentum J= 0,to the m= 0 sub- level of an excited state having J= 1. The duration of the excitation pulse τpis much shorter than the dura- tionof a collision τc(τcis typically of order 1 ps). As a result of elastic collisions with the ground state per- turbers, population in the J= 1 sublevels equilibrate at a rate Γ colthat is typically of order 107−108s−1per Torr of perturber pressure. The transfer to the m= 1 substate is probed by a circularly polarized pulse act- ing on the J= 1, m= 0→J= 0 excited state transi- tion, applied at a time Γ−1 colfollowing the initial excitation pulse. For the sake of definiteness, we assume that the perturber pressure is such that equilibration occurs in a time Γ−1 col= 0.1−1.0 ns. The question that we address in this paper is ”How can one inhibit this magnetic state decoherence by subjecting the active atoms to additional external radiation fields?” J = 0J = 1J = 0 pumpprobe pulse train FIG. 1. Energy level diagram. The collisional interaction couples the magnetic sublevels in the J= 1 state. 1As was mentioned above, the key to any Zeno-type ef- fect is to disrupt the coherent evolution a system from its initial to final state. In our case, the coherent evolution from the initial m= 0 states to the final m=±1 states is driven by the collisional interaction. Thus it is neces- sary to disturb the system on a time scale that is short compared with the collision duration τc. To do this, we apply a continuous train of ultrashort pulses that cou- ple the m= 0 level to the excited state having J= 0 shown in Fig. 1. The pulses are assumed to have dura- tionτp≪τcand are assumed to be off-resonance; that is the atom-field detuning is large compared with τ−1 p. As such, each pulse simply produces an acStark shift of the m= 0 sublevel of the J= 1 state, resulting in a phase shiftof this state amplitude. As a consequence, the exter- nal pulses break the collision-induced, coherent evolutio n of the atom from its initial m= 0 state to the m=±1 states. If the pulse strengths are chosen such that the phase shift is a random number, modulo 2 π,and if many pulses occur during the collision duration τc, then the atom will be frozen in its initial state. It is interesting to note that collisions, which are normally viewed as a decohering process, must be viewed as a coherent driving mechanism on the time scales considered in this work. To obtain a qualitative understanding of this effect, it is sufficient to consider a model, two-level system, con- sisting of an initial state |0/an}bracketri}ht(corresponding to the J= 1, m= 0 state) and a final state |1/an}bracketri}ht(corresponding to the J= 1,m= 1 state, for example). The Hamiltonian for this two-state system is taken as H=Vc(t)(|0/an}bracketri}ht/an}bracketle{t1|+|1/an}bracketri}ht/an}bracketle{t0|) + ¯h/summationdisplay i∆s(ti)τpδ(t−ti)|0/an}bracketri}ht/an}bracketle{t0|, (1) where Vc(t) is a collisional perturbation that couples the two, degenerate states, and ∆ s(ti) is the ac Stark shift of state |0/an}bracketri}htproduced by the external pulse occurring at t=ti. For simplicity, we take Vc(t) to be a square pulse, Vc(t, b) = ¯hβ(b),for 0≤t≤τc. The quantity b is the impact parameter of the collision. Without loss of generality, we can take the collision to start at t= 0. The collision duration τccan be written in terms of the impact parameter bcharacterizing the collision and the relative active atom-perturber speed uasτc(b) =b/u. Moreover, to simulate a van der Waals interaction, we setβ(b) = (C/b6 0)(b0/b)6,where Candb0are constants chosen such that 2 C/(b5 0u) = 1 .The quantity b0is an effective Weisskopf radius for this problem. An average overbwill be taken to calculate the transition rate. The external pulse train is modeled in two ways. In model A, the pulses occur at random times with some average separation Tbetween the pulses. In model B, the pulses are evenly spaced with separation T.In both models, the pulse areas ∆ s(ti)τpare taken to be random numbers between 0 and 2 π.A quantity of importance is the average number of pulses, n0=τc(b0)/T=b0/(uT), for a collision having impact parameter b0.A. Randomly-spaced pulses The randomly spaced, radiative pulses act on this two- level system in a manner analogous to the way collisions modify atomic electronic-state coherence. In other word, the pulses do not affect the state populations, but do modify the coherence between the levels. The pulses can be treated in an impact approximation, such that dur- inga collision, the time rate of change of density matrix elements resulting from the pulses is ˙ ρ00= ˙ρ11= 0 and ˙ρ10/ρ10= ˙ρ01/ρ01=−Γ/angbracketleftBig 1−e−i∆s(ti)τp/angbracketrightBig =−Γ,(2) where Γ = T−1is the average pulse rate and we have used the fact that the pulse area is a random num- ber between 0 and 2 π. Taking into account the colli- sional coupling Vc(t, b) between the levels, one obtains evolution equations for components of the Bloch vector w=ρ11−ρ00= 2ρ11−1,v=i(ρ10−ρ01) as dw/dx =U(y)v;dv/dx =−U(y)w−n(y)v,(3) where x=t/τc(b) is a dimensionless time, y=b/b0 is a relative impact parameter ,andU(y) =y−5and n(y) =n0yare dimensionless frequencies. These equa- tions are solved subject to the initial condition w(0) = −1;v(0) = 0, to obtain the value ρ11(x= 1, y, n 0) = [w(x= 1, y)+1]/2.The relative transition rate Sis given by S(n0) = 2πNub2 0/integraldisplay∞ 0y dy ρ 11(x= 1, y, n 0)/2,(4) where Nis the perturber density. A coefficient, R(n0), which measures the suppression of decoherence, can be defined as R(n0) =/integraldisplay∞ 0y dy ρ 11(x= 1, y, n 0)//integraldisplay∞ 0y dy ρ 11(x= 1, y,0) (5) Solving Eqs. (3), one finds ρ11(x= 1, y, n 0) =/bracketleftbigg 1−r1 r2−r1/parenleftbigg e−r1−r1 r2e−r2/parenrightbigg/bracketrightbigg /2; (6a) r1,2=/parenleftBig −n0y±/radicalbig (n0y)2−4y−10/parenrightBig /2. (6b) It is now an easy matter to numerically integrate Eqs. (5) to obtain R(n0). Before presenting the numerical results, we can look at some limiting cases which provide insight into the physical origin of the suppression of decoherence. A plot of ρ11(x= 1, y, n 0) as a function of y=b/b0is shown in Fig. 2 for several values of n0.With decreasing y,ρ11increases monotonically to some maximum value ρ11(ym) and then begins to oscillate about ρ11= 1/2 2with increasing amplitude. One concludes from such plots that twoeffects contribute to the suppression of coherence. The first effect, important for large n0, is a reduction in the value of ym. The second effect, impor- tant for n0of order unity, is a decrease in the value of ρ11(ym). Let us examine these two effects separately. 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 y=b/b00.00.10.20.30.40.50.60.70.80.91.0ρ11n0=0 n0=2 n0=10 n0=50 FIG. 2. Graph of ρ11as a function of y=b/b0for several values of n0. For values 0 ≤y≤0.45 not shown on the graph, ρ11oscillates about an average value of 1/2. For n0/ne}ationslash= 0,the oscillation amplitude increases with decreasing y. For very large n0, n5/66 0≫1,one can approximate ρ11 over the range of ycontributing significantly to the in- tegral (4) as ρ11(x= 1, y, n 0)∼/parenleftBig 1−e−y−11/n0/parenrightBig /2.By evaluating the integrals in (5), one finds a suppression of decoherence ratio given by R(n0) = 0.95/n2/11 0. (7) Then−2/11 0 dependence is a general result for a colli- sional interaction that varies as the interatomic sepa- ration to the minus 6th power. It can be understood rather easily. The pulses break up the collision into n0ysegments, each having a (dimensionless) time du- ration xb= 1/(n0y). Each segment provides a perturba- tive contribution to ρ11of order y−10(n0y)−2, provided y < y w, where ywis to be determined below. The to- tal population from the entire collision interval varies as ρ11∼y−10(n0y)−2n0y=y−11/n0. Of course, ρ11can- not exceed unity. One can define an effective relative Weisskopf radius, yw,as one for which ρ11= 1, namely yw=bw/b0=n−1/11 0.The total transition rate varies as y2 w∼n−2/11 0, in agreement with (7). As n0∼ ∞, the atom is frozen in its initial state. For values of n0of order unity, the dominant cause of the suppression of decoherence is a decrease in the value ofρ11(ym), rather than the relatively small decrease in ymfrom its value when n0= 0. For values n0≤3, approximately 45% of the contribution to the transition rateS(n0) originates from y > y m,and, for these val- ues of n0,ym∼π−1/5andρ11(ym)∼(1 +e−n0/2π1/5)/2.This allows us to estimate the suppression of decoherence ratio as R(n0) = [0 .55+.45(1+ e−n0/2π1/5)/2],such that R(1) = 0 .93,R(2) = 0 .88,R(3) = 0 .84.These values are approximately 70% of the corresponding numerical results, indicating that the decrease in ρ11(ym) accounts for approximately 70% of the suppression at low n0, with the remaining 30% coming from a decrease in ym. The first few collisions are relatively efficient in suppressing decoherence. With increasing n0, the suppression process slows, varying as n−2/11 0. In Fig. 3, the suppression of de- coherence ratio R(n0), obtained by a numerical solution of Eq. (5), is plotted as a function of n0. 0 2 4 6 8 10 12 14 16 18 20 n00.30.40.50.60.70.80.91.0R(n0) Equally spaced pulses Randomly spaced pulses FIG. 3. Graph of the suppression of decoherence ratio Ras a function of n0for randomly and uniformly spaced pulses. B. Uniformly Spaced Pulses We consider now the case of equally spaced pulses, hav- ing effective pulse areas that are randomly chosen, mod- ulo 2π. The time between pulses is T, andn0=τc(b0)/T. For a relative impact parameter y=b/b0, with m≤ n(y) =n0y≤m+ 1, where mis a positive integer or zero, exactly morm+ 1 pulses occur. The effect of the pulses is calculated easily using the Bloch vector. At x= 0,w=−1 and v= 0. The Bloch vector then un- dergoes free evolution at frequency U(y) =y−5up until the (dimensionless) time of the first pulse, xs=ts/τc(b). The pulse randomizes the phase of the Bloch vector, so that the average Bloch vector following the pulse is pro- jected onto the waxis. From x=xstoxs+T/τc(b) =xs +1/n(y), the Bloch vector again precesses freely and ac- quires a phase UT=y−5/n(y) =y−6/n0, at which time the next pulse projects it back onto the waxis. Taking into account the periods of free precession and projec- tion, and averaging over the time xsat which the first pulse occurs, one finds w(y) = [1 −n(y)] cos[y−5] +n(y)/integraldisplay1 0dxscos[y−5xs] cos[y−5(1−xs)]; 0≤y≤1/n0, 3w(y) = [m+ 1−n(y)][(m+ 1)/n(y)−1]−1 ×/integraldisplay1/n(y) 1−m/n(y)dxscos[y−5xs] cosm−1[y−6/n0] ×cos[y−5{1−xs−(m−1)/n(y)}] +[n(y)−m] [1−m/n(y)]−1 ×/integraldisplay1−m/n(y) 0dxscos[y−5xs] cosm[y−6/n0] ×cos[y−5{1−xs−m/n(y)}]; m/n0≤y≤(m+ 1)/n0form≥1. (8) In the limit that n0≫1, for all impact parameters that contribute significantly to the transition rate, approxi- mately n(y) pulses occur at relative impact parameter y, implying that w(y)∼cosn(y)[y−5/n(y)] and R(n0) =/angbracketleftbig 1−cosn0y[y−6/n0]/angbracketrightbig /an}bracketle{t1−cos[y−5]/an}bracketri}ht(9) ∼/angbracketleftbig 1−[1−y−12/2n2 0]n0y/angbracketrightbig /an}bracketle{t1−cos[y−5]/an}bracketri}ht ∼/angbracketleftBig 1−e−y−11/2n0/angbracketrightBig /an}bracketle{t1−cos[y−5]/an}bracketri}ht=0.84 n2/11 0, which is the same functional dependence found for the randomly spaced pulses. Note that the form {1− cosn(y)[y−15/n(y)]}is identical to that found in theories of the Zeno effect [1]. The suppression of decoherence ratio R(n0), obtained from Eqs. (5) and (8) [using ρ11= (1 + w)/2], is plotted in Fig. 3. The fact that it lies below that for randomly spaced pulses is connected with the difference in the av- erage collisional phase shift acquired between radiation pulses for the two models. The oscillations in R(n0) ap- pear to be an artifact of our square pulse collision model. In the absence of the pulses, the first maximum in the transition cross section occurs for ymax= (π)−1/5, corre- sponding to a πcollision pulse. With increasing n0, the pulses divide the collision duration into approximately n(y) equal intervals. If these pulse intervals are odd or even multiples of π,one can enhance or suppress the con- tribution to the transition rate at specific impact param- eters. Numerical calculations carried out for a smooth interatomic potential do not exhibit these oscillations. C. Discussion Although the collisional interaction has been modeled as a square pulse, the qualitative nature of the results is unchanged for a more realistic collisional interaction, including level shifts. In fact, for a smooth interatomic potential that allows for an increased number of radiation pulses over the duration of the collisional interaction, th e suppression is slightly enhanced from the square pulse values. Although the pulses are assumed to drive onlytheJ= 1, m= 0→J= 0,excited state transition, it is necessary only that the incident pulses produce different phase shifts on the J= 1, m= 0 and J= 1, m= 1 state amplitudes. To observe the suppression of decoherence, one could use Yb as the active atom and Xe perturbers. The Weis- skopf radius for magnetic decoherence is about 1.0 nm [5],yielding a decoherence rate of ≃1010s−1at 500 Torr of Xe pressure at 300◦C, and a collision duration τc(b0) ≃2.5 ps. Thus, by choosing a pulse train having pulses of duration τp=100 fs, separated by 0.5 ps, it is possible to have 5 pulses per collision. If an experiment is carried out with an overall time of 100 ps (time from initial exci- tation to probing of the final state), one needs a train of about 200 pulses. To achieve a phase shift ∆ sτpof order 2πand maintain adiabaticity, one can take the detuning δ= 3×1013s−1and the Rabi frequency Ω ≃1×1014s−1 on the J= 1, m= 0→J= 0,excited state transition [6]. The corresponding, power density is ≃1.5×1011W/cm2, and the power per pulse is ≃150µJ (assuming a 1 mm2 focal spot size). This is a rather modest power require- ment. With 5 pulses/collision duration, one can expect a relative suppression of magnetic state decoherence of order 40%. Finally, we should like to comment on whether or not the effect described in this work constitutes a Quantum Zeno effect. Normally, the Quantum Zeno effect is pre- sented as a projection of a quantum system onto a given state as a result of a measurement on the system. In the experiment of Itano et al., this ”measurement” is re- flected by the presence or absence of spontaneously emit- ted radiation during each uv”measurement” pulse. The measurement pulse must be sufficiently long to produce a high likelihood of spontaneous emission whenever the atom is ”projected” into the initial state by the pulse. Following each measurement pulse, the off-diagonal den- sity matrix element for the two states of the rftransition goes to zero. In our experiment involving off-resonant pulses, the number of Rayleigh photons scattered from theJ= 0 level during each applied pulse is much less than unity. As such, there is no Quantum Zeno effect, even if suppression of magnetic state decoherence occurs. On average , each pulse having random area destroys the coherence between the J= 1, m= 0 and J= 1, m=±1 state amplitudes, but does not kill this coherence for a single atom. With an increasing number of radiation pulses, n0, however, both the average value and the stan- dard deviation of the transition probability tends to zero asn−1 0for each atom in the ensemble. The experiment of Itano et al. could be modified to allow for a comparison with the theory presented herein, and to observe the transition into the Quantum Zeno regime. If the pulses that drive the strong transition are replaced by a sequence of off-resonant pulses, each pulse having a duration τpmuch less than the time, Tπ, re- quired for the pi pulse to drive the weak transition, and each pulse having an effective area, ∆ sτp= (Ω2/4δ)τp, that is random in the domain [0,2 π],then the pulses will 4suppress the excitation of the weak transition (it is as- sumed that Ω /δ≪1). If the upper state decay rate is γ, then the average number of Rayleigh photons scattered during each pulse is n= (Ω/4δ)2γτp.Forn <1, there is suppression of the transition rate as in our case, while, for n>∼1, there is suppression anda Quantum Zeno effect. There is no average over impact parameter, since exactly [Tπ/T] or ([Tπ/T]+1) pulses in each interval between the pulses, where [ x] indicates the integer part of x. D. Acknowledgments PRB is pleased to acknowledge helpful discussions with R. Merlin, A. Rojo and J. Thomas. This research is sup- ported by the National Science Foundation under grant PHY-9800981 and by the U. S. Army Research Office under grants DAAG55-97-0113 and DAAH04-96-0160.[1] W. M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. Wineland, Phys. Rev. A 41, 2295 (1990). [2] See, for example, D. Home and M. A. B. Whitaker, Ann. Phys.258, 237 (1997), and references therein. [3] B. Misra and E. C. G. Sudarshan, J. Math. Phys. 18, 756 (1977). [4] See, for example, L. Viola and S. Lloyd, Phys. Rev. A 58,2733 (1998), and references therein; D. Vitali and P. Tombesi, Phys. Rev. A 59, 4178 (1999), and references therein; G. G. Agarwal, Phys. Rev. A 61, 013809 (2000). [5] J. C. Keller and J. L. LeGou¨ et, Phys. Rev. A 32, 1624 (1985). [6] In the adiabatic limit, the phase is equal to (1/2)/integraltext {/radicalbig [Ω(t)]2+δ2−δ}dt. 5

arXiv:physics/0003004v1 [physics.atom-ph] 2 Mar 2000Coherent and incoherent atomic scattering: Formalism and application to pionium interacting with matter T A Heim †, K Hencken †, D Trautmann †and G Baur ‡ †Institut f¨ ur Theoretische Physik, Universit¨ at Basel, 40 56 Basel, Switzerland ‡Institut f¨ ur Kernphysik, Forschungszentrum J¨ ulich, 524 25 J¨ ulich, Germany Abstract. The experimental determination of the lifetime of pionium p rovides a very important test on chiral perturbation theory. This qua ntity is determined in the DIRAC experiment at CERN. In the analysis of this experim ent, the breakup probabilities of pionium in matter are needed to high accura cy as a theoretical input. We study in detail the influence of the target electrons. They contribute through screening and incoherent effects. We use Dirac-Hartree-Foc k-Slater wavefunctions in order to determine the corresponding form factors. We find that the inner-shell electrons contribute less than the weakly bound outer elect rons. Furthermore, we establish a more rigorous estimate for the magnitude of the c ontribution from the transverse current (magnetic terms thus far neglected in th e calculations). PACS numbers: 36.10.-k, 34.50.-s, 13.40.-f 1. Introduction In experiments such as atom–atom scattering, atom–electro n scattering, nuclear scattering at high energy, or photo-production of e+e−pairs on atoms one faces the situation of complex systems undergoing transitions betwe en different internal states. For an atomic target, excitation affects the electrons indiv idually. Thus the “form factor” for the target-inelastic process takes the form of a nincoherent sum over all electrons, as opposed to the coherent action of the electrons (and the nucleus) in the target-elastic case. From this observation it is immediate ly obvious that the target- inelastic cross section is proportional to Z, the number of electrons in the target atom, whereas the target-elastic process scales with Z2. Since incoherent scattering off the excited target electrons increases the cross section from its value due to coherent scattering off the atom, the effect is sometimes referred to as “anti-screening” [1, 2]. At large momentum transfer the anti-screening correction can be accurately approximated by increasing the coherent scattering cross section by the f actor (1+1 /Z), see e.g. [3, 4, 5]. We will demonstrate in this article, however, that this simp le re-scaling argument is not sufficiently accurate in general. At the same time we will show how to obtain far superior results in a Dirac-Hartree-Fock-Slater approach with mana geable numerical effort.Coherent and incoherent atomic scattering: Formalism and a pplication 2 As an application of the general formalism discussed in this paper, we put our focus on the particular situation pertaining to experiment DIRAC . This experiment, currently being performed at CERN [6], aims at measuring the lifetime o f pionium, i.e., a π+π− pair forming a bound state as an exotic, hydrogen-like atom. The pionium is formed from pions produced in the collision of a high energy proton beam o n a (heavy element) target foil. The kinematical conditions of the experiment imply an nihilation of the pionium through strong interaction ( π+π−→π0π0) still within the target foil. The pionium lifetime will thus be extracted from the exact measurement o f the electromagnetic breakup into π+andπ−while the pionium interacts with the target material. The paper is organized as follows: In section 2 we review the w ell established formalism for one-photon exchange, applied e.g. in electro n–hadron scattering. In section 3 the cross sections coming from the transverse phot ons are estimated with the help of the long-wavelength limit. The formulas for the e valuation of atomic form factors and scattering functions in the framework of Di rac-Hartree-Fock-Slater theory are derived in section 4, followed by an analysis of si mple alternative models in section 5. Although the results presented in section 6 have b een obtained in the context of experiment DIRAC, many of the conclusions drawn in sectio n 7 are also valid more generally in atomic scattering. 2. Formalism In [7] we applied the semiclassical approximation to calcul ate the coherent (target- elastic) cross section for pionium–atom scattering. Here w e are instead interested in those processes, where the atom is excited together with the pionium. We will treat the problem within the PWBA. For our derivation we will follo w closely the standard formalism of electron scattering as found, e.g., in [8, 9, 10 ]. We only calculate the lowest order result. The relevant Feyn man diagram is shown in figure 1. The cross section for this process is given by σ=1 4I1 (2π)22MA2MΠ/integraldisplay d4q(4πe2)2Wµν AWµνΠ (q2)2, (1) where Wµν AandWµν Πare the electromagnetic tensors describing the electromag netic interaction of atom and pionium with the photon. As we are not interested in the specific final state of the atom, we can average over the possib le initial spin and sum over all final states and directions corresponding to a speci fic final state momentum P′. For the atom we have Wµν A=1 4πMA/summationdisplay X/angbracketleftbig 0, PA|Jµ†|X, P′ A/angbracketrightbig/angbracketleftbig X, P′ A|Jν|0, PA/angbracketrightbig (2π)4δ4(PA−q−P′ A) (2) and similarly for the pionium if its final state is not resolve d either. The electromagnetic tensor can only be a function of PandP′, or equivalently, of Pandq. Gauge invariance or current conservation restricts the po ssible tensor structure ofWµνeven more. It is a well known result that the electromagnetic tensor in this caseCoherent and incoherent atomic scattering: Formalism and a pplication 3 PΠ(ππ) PAAP'Π(ππ)* P'ΑA*q Figure 1. The lowest order Feynman diagram for the simultaneous excit ation of projectile (pionium) and target (atom). The atomic momenta arePAandP′ Abefore and after the collision, those of the pionium PΠandP′ Π. The momentum of the exchanged photon is q=P′ Π−PΠ=−(P′ A−PA). only depends on two scalar functions W1andW2that are functions of q2andPqalone. The electromagnetic tensor is then given by Wµν=/parenleftbigg −gµν+qµqν q2/parenrightbigg W1(q2, Pq) +/parenleftbigg Pµ−Pq qµ q2/parenrightbigg/parenleftbigg Pν−Pq qν q2/parenrightbiggW2(q2, Pq) M2.(3) Since the cross section depends on the product of the tensor f or the atom and the pionium, we calculate this product in terms of W1andW2: Wµν AWµνΠ= 3W1,ΠW1,A+/parenleftbigg −1 +∆2 q2/parenrightbigg W1,ΠW2,A+/parenleftbigg −1 +ω2 q2/parenrightbigg W2,ΠW1,A +/parenleftbigg γ+ω∆ q2/parenrightbigg2 W2,ΠW2,A, (4) where γis the (relative) Lorentz factor between atom and pionium, ∆ =−PAq/M A, ω=PΠq/M Πare (minus) the energy of the exchanged photon in the atom and pionium rest frame, respectively. Following the argument of [11, 7] we expect the cross section to be dominated by the charge operator (termed “scalar inter action” in [7]). Thus we neglect at this point all terms containing a W1. We note also that W1vanishes in the elastic case for spin 0 particles. However, the contributio n to the cross section coming fromW1will be discussed in section 3 below in the analysis of the tra nsverse part of the current operator j. Keeping only the last term and assuming that the pre-factor is dominated by γ(γis in the range 15–20 in the DIRAC experiment) we get Wµν AWµνΠ≈γ2W2,ΠW2,A. (5) Although this coincides with the naive estimate obtained by superficially identifying the leading power of γ(assumed to be dominant), one should keep in mind that (as wil l be shown below) the range of q2starts at about ( ω/γ)2or (∆/γ)2, where the last three pre-factors are of the same order. The magnitude of γshould therefore not be mistaken as a justification for neglecting the terms containing W1. A critical assessment of the relative importance of W1as compared to W2is postponed until section 3.Coherent and incoherent atomic scattering: Formalism and a pplication 4 In our application to pionium–atom scattering the masses MAandMΠwill be much larger than the momentum transfer qof the photon. Therefore we will neglect recoil effects on the atom and pionium. We can then identify ∆ and ωas the excitation energy of the atom and pionium in their respective rest frames. Ther eforeq0andqzare fixed by the values of ωand ∆. In the following we will always denote the spatial part of the photon momentum in the rest frame of the atom by kand in the rest frame of the pionium by s. In the rest frame of the atom we then have q0,A =−∆ qz,A=kz=−∆ β−ω γβ,(6) and in the rest frame of the pionium q0,Π =ω qz,Π=sz=−∆ γβ−ω β.(7) Alsoq2is given by q2=−/parenleftbigg∆2 β2γ2+ω2 β2γ2+2ω∆ β2γ+q2 ⊥/parenrightbigg =:−/parenleftbig q2 l+q2 ⊥/parenrightbig . (8) Replacing the integration over d4q=1 γβdωd∆ d2q⊥, we obtain σ=/integraldisplay dωd∆ d2q⊥4α2 β2W2,Π(ω, q2)W2,A(∆, q2) (q2 l+q2 ⊥)2. (9) One of the advantages of this derivation is that the W2are scalar functions; we can therefore evaluate them in the respective rest frames, even though the relative motion of atom and pionium is relativistic. We now establish a relat ion between W2and the electromagnetic transition currents. As was already done i n [7] we assume the charge operator to be the dominant contribution. In the atom’s rest frame W2is related to the ‘00’-component of the tensor through W2,A=q4 k4W00 rf,A, (10) see, e.g., [12, 13], with k2=k2= ∆2−q2. Here we have added an index “rf” as a reminder that the ‘00’-component in the rest frame has to be t aken. From the definition ofWµνwe get W00 A=1 4πMA/summationdisplay X/angbracketleftbig 0|J0†(q)|X, P′ A/angbracketrightbig/angbracketleftbig X, P′ A|J0(q)|0/angbracketrightbig (2π)4δ4(PA−q−P′ A). (11) Rewriting this expression in terms of the (non-relativisti c) density operator and again neglecting recoil effects we find W00 A=/summationdisplay X/an}bracketle{t0|ρ(q)|X, E X/an}bracketri}ht/an}bracketle{tX, E X|ρ(q)|0/an}bracketri}htδ(E0,A+ ∆−EX) (12) =/summationdisplay X|FX0,A(q)|2δ(E0,A+ ∆−EX). (13)Coherent and incoherent atomic scattering: Formalism and a pplication 5 (Replacing the energy Eby the rest mass Min these weakly bound systems only introduces an error of the order α2). Finally we obtain for the cross section: σ=/integraldisplay dωd∆ d2q⊥4α2 β2q4 s4k4/bracketleftBigg/summationdisplay X|FX0,A(k)|2δ(E0,A+ ∆−EX)/bracketrightBigg × /bracketleftBigg/summationdisplay X′|FX′0,Π(s)|2δ(E0,Π+ω−EX′)/bracketrightBigg . (14) As a verification of this formalism we reproduce the result of [7] for target-elastic scattering. In this case ∆ = 0 and k2=q2, and the only possible final state is X= 0 (assuming no degeneracy of the ground state). We find σ=/integraldisplay dωd2q⊥4α2 β21 s4|F00,A(q)|2/bracketleftBigg/summationdisplay X|FX0,Π(s)|2δ(E0,Π+ω−EX)/bracketrightBigg , (15) andF00,A(q) is the elastic form factor of the atom. This is identical to t he equation derived in [7] in the Coulomb gauge ( k·A= 0). At this point we should add a few comments regarding the gauge invariance and the factor q4/k4(and likewise q4/s4) in (10): In determining the general tensor structure of Wµνin (3) the gauge invariance (or alternatively current conse rvation) was used. The magnitude of the component of jalong the direction of qis then fixed by ρ. Only the transverse parts of the current remain independent quan tities. Our result therefore agrees with the one in Coulomb gauge, as in this gauge the comp onent of Aalong the direction of qvanishes. An alternative approach might start directly fro mWµνwithout decomposition into W1andW2. Assuming that in the rest frame of either the atom or the pionium W00dominates, one can approximate the product of the electroma gnetic tensors by Wµν AWµνΠ≈γ2W00 rf,AW00 rf,Π (16) where the factor γ2comes from the Lorentz transformation of one tensor into the rest frame of the other. Now using (16) instead of (10), the cross s ection reads σ=/integraldisplay dωd∆ d2q⊥4α2 β21 q4/bracketleftBigg/summationdisplay X|FX0,A(k)|2δ(E0,A+ ∆−EX)/bracketrightBigg × /bracketleftBigg/summationdisplay X′|FX′0,Π(s)|2δ(E0,Π+ω−EX′)/bracketrightBigg . (17) This differs from (14) by a factor q4/k4·q4/s4. In the elastic case (17) corresponds to the result of [7] in the Lorentz gauge. However, the approximati on (16) and therefore also (17) is not gauge invariant, whereas (14) is by construction . We will therefore prefer in the following the form as given in (14). In [7] the differenc e between the two results was interpreted as an indicator for the magnitude of the “mag netic terms”, that is, the contribution proportional to j. In the next section we will estimate the contribution of the transverse photons, making use of the long-wavelength l imit. It remains to be seen,Coherent and incoherent atomic scattering: Formalism and a pplication 6 however, which one of the two possible schemes (longitudina l/transverse decomposition versus scalar/magnetic terms) will be better suited for exp licit calculations. In order to determine total cross sections we would like to si mplify the summation over all possible states. The expression (14) depends on ωand ∆ in two ways: Through the energy-conserving delta functions and through the expression for q2, where q2 l= ∆2/(β2γ2) +ω2/(β2γ2) + (2 ω∆)/(β2γ) depends on both ωand ∆. Replacing ω and ∆ in qlby some average values ω0and ∆ 0, we can perform the closure over all final states to get σ=/integraldisplay d2q⊥4α2 β2q4 s4k4Sinc,A(k)Sinc,Π(s) (18) withq2, s2, and k2now the ones using ω0and ∆ 0and Sinc,A(k) =/summationdisplay X|FX0,A(k)|2(19) Sinc,Π(s) =/summationdisplay X′|FX′0,Π(s)|2. (20) In section 6 the dependence of the cross section on the choice of ∆ 0andω0will be studied by varying both parameters over a reasonable range. 3. Contribution of transverse photons In section 2 we have only calculated the effect coming from the longitudinal photons (that is, coming from the charge operator). Already in [11, 7 ] the effect of the so- called “magnetic interaction”, that is, the effect of the cur rent operator was estimated to be of the order of 1%. As the DIRAC experiment requires an ac curacy of 1%, these contributions need to be considered more carefully. The par t of the current operator jin the direction of qis already included in the above calculation. Therefore we n eed to study only the contribution coming from the transverse pa rt of the current operator, that is, the effect coming from the transverse photons. Here w e estimate them with the help of the long-wavelength approximation. Following [ 12], see also [13], W1and W2can be expressed in terms of the Coulomb and transverse elect ric (magnetic) matrix elements: W1= 2π/parenleftbig |Te|2+|Tm|2/parenrightbig (21) W2=q4 s42π/parenleftbigg 2|MC|2−s2 q2/parenleftbig |Te|2+|Tm|2/parenrightbig/parenrightbigg , (22) where we use the usual definitions for MCandTe,m:W00= 4π|MC|2andWλλ′= δλλ′2π(|Te|2+|Tm|2) with λ, λ′denoting the two transverse directions. The Tmcan be safely neglected in our case. The long-wavelength limit r elates TetoMC. In the multipole expansions Te=/summationdisplay J≥1/summationdisplay MTe JM, MC=/summationdisplay J≥0/summationdisplay MMC JM, (23)Coherent and incoherent atomic scattering: Formalism and a pplication 7 we have for a given multipole [8, 14] Te JM≈ω s/parenleftbiggJ+ 1 J/parenrightbigg1/2 MC JM, forJ≥1. (24) In the case of the pionium only odd multipoles contribute and there is thus no MC 00term. As higher multipoles are strongly suppressed we can safely s et the factor ( J+ 1)/Jto its maximum value at 2 (i.e. J= 1) for the purpose of an analytical estimate for the upper limit of |Te|. However, since our computer program already contains a multipole expansion, it is of course straightforward to per form a more refined numerical calculation. For the atom, on the other hand, MC 00/ne}ationslash= 0; it is in fact the dominant contribution in the elastic process. Although generalizin g the relation (24) with the maximum factor√ 2 for all multipole orders still provides an upper limit for Te, it may be a less useful over-estimate for the atom. The relation bet ween|MC|2and|Te|2is then approximately |Te(q)|2≈2ω2 s2|MC(q)|2(25) ≈1 2πω2 s2W00 rf. (26) One sees that the transverse photons contribute to both W1andW2. Denoting their contribution by WT 1andWT 2, they can be expressed with the help of (21), (22), and (26) as WT 1=ω2 s2W00 rf, (27) WT 2=−q2ω2 s4W00 rf. (28) First we look at the elastic case (for the atom). In this case w e have ∆ = 0, W1,A= 0 and W00 rf,A=|F00,A(q)|2. The product of the electromagnetic tensors is then Wµν AWµνΠ=−WT 1,ΠW2,A+γ2WT 2,ΠW2,A (29) =ω2 s2/parenleftbigg −1 +γ2−q2 s2/parenrightbigg W00 rf,Π|F00,A(q)|2. (30) Again summing over all excited states of the pionium and negl ecting the dependence on ωinq2ands2we get the total cross section σT el=/integraldisplay d2q⊥4α2 β2ω2 s2/parenleftbigg −1 γ2q4+1 s2(−q2)/parenrightbigg Sinc,Π(s)|F00,A(q)|2. (31) We see that this cross section differs from (15) by the replace ment 1 s4→ω2 s2/parenleftbigg −1 γ2q4+1 s2(−q2)/parenrightbigg (32) The estimate for the reduction of these terms compared to the charge contribution can be seen easily in this equation, if one assumes that the domin ant contributions come from the range of q2(and therefore also of s2) of the order of k2 Π, where kΠdenotes the Bohr-momentum of the pionium, kΠ= 1/aBohr,Π≈136.566/aBohr. This momentum isCoherent and incoherent atomic scattering: Formalism and a pplication 8 of the order ω/α. Therefore the factor ω2/s2would give a reduction of the order α2, making this contribution completely negligible. However, a discussion of the relevant range of q2to be given in section 6 will show that this estimate is too cru de. In the inelastic case (on the atom side) we can approximately set ∆ 0≈0 again, as the atomic binding energy is small compared to the other en ergies. Then the contribution from the transverse current of the atom will be suppressed even more than for the pionium, since the ratio ∆ /kis even smaller than ω/s. The only difference compared to (31) is then the replacement of |F00,A(q)|2bySinc,A(k). We get σT inel=/integraldisplay d2q⊥4α2 β2ω2 s2/parenleftbigg −1 γ2q4+1 s2(−q2)/parenrightbigg Sinc,Π(s)Sinc,A(k). (33) We will discuss the contribution to the cross section in sect ion 6. 4. Dirac-Hartree-Fock-Slater model We now turn to the question how to evaluate the form factors an d scattering functions derived in the general formalism of the preceding sections. For our specific application of pionium scattering off atomic targets, we use the pionium f orm factors as described in [7]; here we shall only discuss the calculation of the atomic form factors. 4.1. Atomic ground state elastic form factors Within the framework of (Dirac-)Hartree-Fock-Slater theo ry, the atomic ground state wavefunction entering the expression for the form factor, F00(k) =/an}bracketle{tΨ0|Z/summationdisplay j=1exp(ik·rj)|Ψ0/an}bracketri}ht, (34) is given by a single Slater determinant constructed from pro ducts of independent particle orbitals, Ψ0=1√ Z!/summationdisplay psign(p) Φp(1)(r1)···Φp(Z)(rZ). (35) HereZdenotes the nuclear charge (and the number of electrons), wh ilepdenotes the permutations of orbital indices, and Φ jsignify single particle orbitals. Each of the Z exponential terms exp(i k·rj) in (34) acts as a one-particle operator. Orthogonality of the orbitals effectively cancels the summation over permu tations in the bra and ket vectors leading to F00(k) =Z/summationdisplay j=1/an}bracketle{tΦj(rj)|exp(ik·rj)|Φj(rj)/an}bracketri}ht. (36) So far we have not considered any angular momentum coupling o f the independent- particle orbitals, that is, our Ψ 0is determined by a set of quantum numbers (nj, lj, mj), j= 1. . .Z, without coupling to a total L(orJ) and M. Furthermore we assumed the ground state wavefunction in the bra and ket sy mbols to be identical.Coherent and incoherent atomic scattering: Formalism and a pplication 9 This need not actually be the case, as different z-projections Mof the total angular momentum Jof the ground state cannot be distinguished (without extern al fields). The angular momentum coupled ground state wavefunction wou ld then be obtained by summing and averaging over MandM′in the bra and ket vectors, respectively. Likewise on the level of independent particle labels mj, the orthogonality argument applies strictly only to all orbitals except j, that is, we should distinguish between mj andm′ jfor the bra and for the ket vector, respectively. Since the on e-particle operator does not affect the spin part of the wavefunction, we should al so insert an orthogonality factor for the spin orbitals in bra and ket, χj, χ′ j. Of course the principal and azimuthal quantum number njandljcoincide in the bra and in the ket symbol. Expanding the exponential in spherical harmonics we find imm ediately F00(k) =Z/summationdisplay j=1(2lj+ 1)δχj,χ′ j/summationdisplay λ,µiλ√ 4πY∗ λ,µ(ˆk)√ 2λ+ 1/parenleftBigg ljljλ 0 0 0/parenrightBigg × (−1)m′ j/parenleftBigg ljljλ mj−m′ jµ/parenrightBigg Rλ jj(k), (37) with the radial form factor defined by Rλ ij(k) =/integraldisplay∞ 0dr r2Rnili(r)jλ(kr)Rnjlj(r), (38) where the Rnl(r) denote radial wavefunctions for the orbitals, and jλ(kr) is a spherical Bessel function. In a next step averaging |F00(k)|2over all directions ˆ qand using the orthogonality relation of the spherical harmonics yields |F00(k)|2:=1 4π/integraldisplay dˆk|F00(k)|2 =/summationdisplay λ,µ(2λ+ 1)/braceleftBiggZ/summationdisplay j=1(−1)m′ j(2lj+ 1)δχj,χ′ j/parenleftBigg ljljλ 0 0 0/parenrightBigg × /parenleftBigg ljljλ mj−m′ jµ/parenrightBigg Rλ jj(k)/bracerightBigg2 . (39) Obviously all electrons contribute coherently to the form f actor, as expected. Due to the first 3 j-symbol only even multipoles contribute to the sum. In the LS-coupling scheme, the atomic ground state is characterized by a specific totalLand a (possibly averaged) total M. Instead of coupling the individual electrons’ angular momenta to total Land then averaging over M, we average directly over individual mj, m′ jfor the orbitals occupied in accordance with the Pauli princ iple. This amounts to neglecting energy differences between fine struct ure levels. Hund’s rules (see e.g. [15]) state that sub-shells are to be filled by addin g as many electrons with different mjand the same spin projection as possible. The critical multi plicity is thus that for a half filled sub-shell, (2 l+ 1). For half filled or completely filled sub-shells, Hund’s rules imply that both mjandm′ jrun over all possible values from −ljto +lj.Coherent and incoherent atomic scattering: Formalism and a pplication 10 Consequently the angular part for these spherical sub-shel ls reduces to selecting only monopole contributions. A completely filled sub-shell ( nj, lj) thus contributes 2(2lj+ 1)R0 jj(k) to the full form factor. (The leading factor 2 indicates the s pin multiplicity.) For sub-shells that are neither completely nor half filled th e averaging procedure yields different multiplicities depending on the multipole order. For the dominating monopole contribution, this factor is simply given by the oc cupation number of the sub-shell, whereas for higher multipoles this factor is pro portional to the product of occupation number and the number of holes in a sub-shell with identical spin projections. For a given number of electrons in an open sub-shell we averag ed the m-dependent part in (39) over all possible distributions of mjandm′ jvalues. Note that the coherent (elastic) form factor F00(k) derived above describes only the effect of scattering off the atomic electrons. The complete el astic form factor for the atom reads FAtom(k) =Z−F00(k), (40) assuming a point-like nucleus. 4.2. Atomic inelastic scattering functions Besides the elastic form factor F00(k) treated in the preceding section, we also need to consider the contributions due to excitations of the atomic electron cloud. Nuclear excitations will not be considered here because the much lar ger excitation energy required (typically on the order of MeV) exceeds the energy r ange relevant for our application to pionium–atom scattering. We demonstrated i n [7] that deviations from a point-like nucleus are negligible for the electromagneti c processes considered here. Thus the nucleus’ internal structure with its excited state s is equally irrelevant as the experiment DIRAC cannot probe this structure. In analogy to the elastic form factor, a transition form factor is written in the form FX0(k) =/an}bracketle{tΨX|Z/summationdisplay j=1exp(ik·rj)|Ψ0/an}bracketri}ht, (41) corresponding to scattering with excitation of the atomic e lectrons from the ground state to some excited state X. A similar expression was studied in [16] in the context of the equivalent photon approximation. The total inelastic s cattering function is defined as the incoherent sum over all states Xother than the ground state Sinc(k) =/summationdisplay X/negationslash=0|FX0(k)|2 =/summationdisplay allX|FX0(k)|2− |F00(k)|2 =Z+Z/summationdisplay i=1/summationdisplay j/negationslash=i/an}bracketle{tΨ0|exp(ik·[rj−ri])|Ψ0/an}bracketri}ht − |F00(k)|2. (42)Coherent and incoherent atomic scattering: Formalism and a pplication 11 Here we have added the ground state in order to exploit the com pleteness of the set of states {X}: Expanding the squared modulus of FX0introduced a second (primed) set of variables r′ jwhich has been removed again by virtue of the completeness of the set of states {X}. Furthermore we evaluated the sum over the diagonal terms i=j separately, obtaining the term Z(since exp(i k·[rj−ri])≡1 in this case). Using the same Slater determinant wavefunctions for Ψ 0as in the preceding section, expanding the double sum corresponding to |F00(k)|2, and combining terms with the last sum, we find (in terms of the single-electron orbitals) Sinc(k) =Z−Z/summationdisplay i=1Z/summationdisplay j=1|/an}bracketle{tΦi|exp(ik·r)|Φj/an}bracketri}ht|2. (43) The new terms required in the determination of Sincare the matrix elements /an}bracketle{tΦi|exp(ik·r)|Φj/an}bracketri}ht=δχiχj(−1)mi/radicalBig 4π(2li+ 1)(2 lj+ 1)/summationdisplay λ,µY∗ λ,µ(ˆk)iλ√ 2λ+ 1× /parenleftBigg liljλ 0 0 0/parenrightBigg/parenleftBigg liljλ −mimjµ/parenrightBigg Rλ ij(k), (44) some of which (namely, those with i=j) have already been used in calculating F00(k). As before, χi, χjdenote the spin projections. Averaging over the directions ˆkwe obtain immediately Sinc(k) :=1 4π/integraldisplay dˆk Sinc(k) =Z−Z/summationdisplay i=1Z/summationdisplay j=1δχiχj(2li+ 1)(2 lj+ 1)/summationdisplay λ(2λ+ 1)/parenleftBigg liljλ 0 0 0/parenrightBigg2 × /parenleftBigg lilj λ mi−mjmj−mi/parenrightBigg2/bracketleftbig Rλ ij(k)/bracketrightbig2. (45) Considering again the non-averaged incoherent scattering function in the special case of completely filled sub-shell li, lj, the summations over miandmjremove the angular dependence on ˆk. The filled sub-shells thus contribute 2(2li+ 1)(2 lj+ 1)/summationdisplay λ(2λ+ 1)/parenleftBigg liljλ 0 0 0/parenrightBigg2/bracketleftbig Rλ ij(k)/bracketrightbig2 to the incoherent form factor, with a factor 2 for the spin mul tiplicities in both sub-shells (rather than a factor 4, because the cross terms between sub- shells with opposite spin projections drop out due to the orthogonality of the spin orb itals). Except for Z= 58 (Ce, not of interest to us) all atoms with Z≤90 have in their ground state only one sub-shell that is neither half nor comp letely filled [17]. (In atoms with two open sub-shells, one of them has l= 0.) Thus we need not consider the m- averaging procedure for cases where miandmjcome from two different partially filled sub-shells with li>0 and lj>0. In the cases of our interest averaging over miandCoherent and incoherent atomic scattering: Formalism and a pplication 12 mjis then straightforward. The m-averaged contribution from two different sub-shells reads 1 2νiνj/summationdisplay λ(2λ+ 1)/parenleftBigg liljλ 0 0 0/parenrightBigg2/bracketleftbig Rλ ij(k)/bracketrightbig2 where νiandνjdenote the occupation numbers of the two sub-shells and the f actor 1/2 stems from the orthogonality of spin orbitals. If iandjboth refer to the same sub-shell the multiplicity factor is slightly more complic ated, depending on whether the sub-shell is less than, or more than, half filled. For this lim ited number of cases, we again determined the m-dependent part by suitably averaging over all possible dis tributions ofmvalues in a partially filled sub-shell. 5. Some other models For comparison we briefly discuss in this section some simple alternative models for evaluating the coherent and incoherent form factors, F00(k) andSinc(k), respectively, for application in complex atomic scattering. Specifically, we will use analytical screening models [18, 19] to derive the elastic form factors. The inela stic scattering functions can then be obtained from the elastic form factors either in the n o-correlation limit, or from an argument due to Heisenberg and based on the Thomas-Fermi m odel. The simplest possible model to describe the effect of incoher ent scattering off the atom’s electrons would merely divide the cross section for c oherent scattering (scaling withZ2) byZ, on the grounds that everything remains the same except that each electron contributes incoherently to the cross section (co mplete “anti-screening”). We found that this approach underestimates the incoherent sca ttering cross section by as much as 50%. For typical targets like Ti or Ni this implies an e rror of roughly 2% in the target-inclusive cross section, clearly beyond the req uired limit of 1%. 5.1. Elastic form factors In order to simplify the atomic structure calculation, one m ight use the Thomas-Fermi model to replace the density ρ(r) occurring in F00(k) =/integraldisplay d3r ρ(r) exp(i k·r). (46) Expressing the potential due to the charge distribution of t he electrons in the form V(r) =−Z rχ(r), (47) the corresponding charge distribution is given by the secon d derivative of the screening function: ρ(r) =Z 4πrχ′′(r). (48)Coherent and incoherent atomic scattering: Formalism and a pplication 13 Here and in the following the prime denotes differentiation w ith respect to r. For a spherical charge distribution the coherent form factor red uces to the monopole term. Using Moli` ere’s [19] parameterization for χ, χ(r) =3/summationdisplay i=1Biexp(−βir/b); (49) B1= 0.1;B2= 0.55;B3= 0.35; (50) β1= 6.0;β2= 1.2;β3= 0.3; (51) withb=aBohr(9π2/128)1/3Z−1/3, the coherent form factor reads F00(k) =Z3/summationdisplay i=1Bi1 1 + (b k/β i)2. (52) for the electronic part, and again FAtom(k) =Z−F00(k). The same analytical form with different parameters Biandβidetermined by fitting expectation values of powers of r to exact Dirac-Hartree-Fock-Slater results is used in [18] where the fitting parameters for all Z≤92 may be found as well. 5.2. Inelastic form factors: No-correlation limit Inserting the elastic form factor of the previous subsectio n into (42) we are left with the evaluation of Z/summationdisplay i=1/summationdisplay j/negationslash=i/an}bracketle{tΨ0|exp(ik·[rj−ri]|Ψ0/an}bracketri}ht=Z(Z−1)/integraldisplay d3rd3r′N2(r,r′) exp(i k·[r−r′]) (53) where we have already integrated over all variables not pert aining to the orbitals iandj. Here the function N2(r,r′) describes the probability of finding any two of the properly anti-symmetrized electrons at positions randr′. The no-correlation limit now replaces this two-particle density by the product of the single-part icle probabilities ρ(r)/Zand ρ(r′)/Z. The integral in the last expression then reduces to |F00(k)/Z|2, i.e., the square of the elastic form factor normalized per electron. As there areZ(Z−1) such terms in the double sum over i, j, we finally find Sinc(k) =Z− |F00(k)|2/Z. (54) Using the elastic form factor of the previous subsection, th is result provides a simple expression for Sinc. However, we have made here the crucial assumption that ther e is no correlation between the electrons in different orbitals. In a single-particle picture this amounts to assuming that all single-particle states ar e available for all electrons simultaneously. Pauli blocking, i.e., the fact that due to t he Pauli exclusion principle some states Xcannot be excited for a given electron because they are occup ied by other electrons, is then disregarded completely. Since in this li mit the (incoherent) summation overXalso includes the Pauli blocked states, the expression (54) clearly overestimates the correct scattering function.Coherent and incoherent atomic scattering: Formalism and a pplication 14 5.3. Thomas-Fermi model for incoherent scattering Following Heisenberg [20] we can find a simple expression for theincoherent atomic form factor, going beyond the no-correlation limit but rema ining in the spirit of the Thomas-Fermi model (see also [21]). Expanding the squared m odulus in the incoherent sum (43) we write Sinc(k) =Z−/integraldisplay d3rd3r′exp(ik·(r−r′))|Z/summationdisplay j=1Φ∗ j(r)Φj(r′)|2. (55) In the Thomas-Fermi model, the density is related to the volu me of a sphere in momentum space, Z/summationdisplay j=1|Φj(r)|2=2 (2π)3/integraldisplay κ≤kF(r)d3κ=1 3π2[2|V(r)|]3/2. (56) Heisenberg generalized this expression to obtain the two-p article density as an integral in momentum space (with r1= (r+r′)/2) Z/summationdisplay j=1Φ∗ j(r)Φj(r′) =2 (2π)3/integraldisplay κ≤kF(r1)d3κexp(iκ·(r−r′)) (57) which reduces to the Thomas-Fermi expression for r=r′. (55) now contains additional six-fold integration over d3κand d3κ′. Performing these integrations as well as the integration over ( r−r′), we find (following Heisenberg’s argument about the common volume of two intersecting spheres in momentum space, and pa ying attention to the spin multiplicity) Sinc(k) =Z−4 3π/integraldisplayr0 0dr1r2 1(/radicalbig 2V(r1)−k/2)2(/radicalbig 2V(r1) +k/4),(58) where the integration over r1is restricted to the region of coordinate space with kF(r1)≥k/2 since otherwise the two spheres in momentum space do not ove rlap. Using V(r) = (Z/r)χ(r) the incoherent form factor turns into Sinc(k) =Z−4 3π/integraldisplayr0 0dr√r[2Zχ(r)]3/2+2 πZ k/integraldisplayr0 0dr rχ(r)−1 36π(k r0)3(59) where the upper limit of integration r0must satisfy Z r0χ(r0) =1 8k2. (60) Within the frame work of the Thomas-Fermi model, the screeni ng function satisfies the differential equation Z rd2 dr2χ(r) =4 3π[(2Z/r)χ(r)]3/2; χ(0) = 1 , (61) thus enabling us to replace the first integral in (59). The sec ond derivative of χcan then be removed by integrating by parts, yielding Sinc(k) =−Zr0χ′(r0) +2 πZ k/integraldisplayr0 0dr rχ(r) +1 8r0k2−1 36π(k r0)3. (62)Coherent and incoherent atomic scattering: Formalism and a pplication 15 From (59) and (62) we note that this simple model does not repr oduce the correct limit ask→0: In this limit r0→ ∞, and the integral in (62) assumes a constant non-zero value. Thus Sincgrows linearly withk. By contrast the expression in the no-correlation limit (containing F00(k)) grows with k2, as does the Hartree-Fock-Slater result derived in the preceding section (since the term linear in kdrops out of the expansion in (42) due to the symmetry under interchange i↔j). 6. Numerical method and results In our application to scattering of pionium on atomic target s, we use hydrogenic wavefunctions for the pionium system π+π−. The corresponding form factors F00,Π andSinc,Πare evaluated analytically as described in [7]. Section 4 provides expressions for the evaluation of cohere nt atomic form factors and incoherent scattering functions in the framework of the Dirac-Hartree-Fock-Slater formalism. In our calculations we start from a simple analyt ical charge distribution for the electrons as given e.g. in [18] or similarly in [19]. Taki ng the electronic structure of the elements from [17] we solve either the Schr¨ odinger or th e Dirac radial equation for each occupied orbital, treating exchange effects by Latter’ s approximation (see e.g. [18]). The resulting charge density is then used to obtain improved radial orbitals, iterating the process until self-consistency is reached. Even for hea vy elements with electrons in some twenty different orbitals and requiring several ten ite rations, this calculation is readily performed with the help of the program package RADIA L [22]. Using these orbitals we evaluate the radial integrals in (39 ) and (45) on a reasonably dense mesh of kvalues with the help of an integration routine developed for integrals containing spherical Bessel functions and powers [23]. To t his end, the numerical solutions for the orbitals obtained on a grid of rvalues are replaced by piecewise splines. The angular parts for the partially filled sub-shells are det ermined by averaging over all distributions of magnetic quantum numbers min accordance with the Pauli principle, as described in section 4. Let us first investigate the range of q⊥relevant for the cross section as given by (9) or (18). Figure 2 demonstrates the interplay of different mom entum scales associated with the photon, with the atom, and with the pionium, respect ively. The figure shows the integrand from (9) in singly differential form: dσ dq⊥= 2πq⊥4α2 β2(Photon) ×(Atom) ×(Pionium) , (63) together with its decomposition into photon, atom and pioni um parts (cf. (8)): (Photon) =1 q4=1 (q2 l+q2 ⊥)2, (64) (Atom) = W2,A=/braceleftBigg |F00,A(q)|2for coherent scattering, (q4/k4)Sinc,A(k) for incoherent scattering,(65) (Pionium) = W2,Π= (q4/s4)Sinc,Π(s). (66)Coherent and incoherent atomic scattering: Formalism and a pplication 16 10-410-2100102104 q⊥ [a.u.]10-1610-8100108 Coherent IncoherentPhoton Atom Pionium d σ / d (ln q⊥)ql kTF kΠ ZZ2 Figure 2. Various contributions to the integrand for the cross sectio n vs. q⊥, on a log-log scale. To compensate for the logarithmic q⊥-axis, the integrand is represented as dσ/d lnq⊥. The cross sections correspond to target-elastic (solid li nes) and target- inelastic (dashed lines) pionium scattering off Ni ( Z= 28) at a projectile energy E= 5 GeV, summed over all final states of the pionium. The arrows at the upper edge indicate the relevant momentum scales. See text for det ails. The solid lines refer to the total cross section (i.e., summe d over all pionium final states) for target-elastic scattering off Ni ( Z= 28) for pionium in its ground state. The projectile energy is 5 GeV. The dashed lines correspond to th e same setting for the target-inelastic process. Here the integrations over ωand ∆ in (9) have been replaced by setting an average excitation energy for the pionium at ω0= 1.858 keV (ground state binding energy), and for the atom we set ∆ 0= 0 (target-elastic) and ∆ 0=100 eV (target-inelastic), respectively. The arrows indicate the relevant momentum scales: qlfor the photon, kTF= Z1/3/aBohrfor the atom, and kΠfor the pionium. The arrow on the left under the label “ql” corresponds to target-elastic scattering, while the one o n the right corresponds to incoherent scattering (with non-zero ∆ 0and thus with a larger ql). For q⊥≪ql, the photon momentum is essentially given by the constant ql. When q⊥≫ql, on the other hand, this part displays a 1 /q4 ⊥behavior. The atomic part shows an increase between qland several inverse Bohr radii (indicated by kTF). As expected, |F00,A(k)|2grows roughly with q4 ⊥, whereas Sinc,A(k) grows only with q2 ⊥. Atq⊥≈5kTFthe atomic part reaches its asymptotic value ( Z2orZ, respectively). In this regime the pionium part only just starts to contribute appreciably. It grows quadra tically with q⊥to saturate at a few multiples of the pionium scale kΠ. The product of the three factors clearly demonstrates that the main contributions to the cross secti ons come from the range ofCoherent and incoherent atomic scattering: Formalism and a pplication 17 0 0.5 1 1.5 2 ω0 / Ebind0.990.99511.005 a) σ(ω0) / σ(0) n=1n=2n=3n=4 0 0.5 1 1.5 2 ∆0 / Ebind4681012 b)σ(∆0) per electron [barn] KLMN Average total Figure 3. a) Closure cross sections (normalized to ω0= 0) versus ω0(in units of Ebind,Π) forcoherent scattering of pionium in various initial states as indicate d in the figure. Target material is Ni, projectile energy is 5 GeV. Sol id lines: initial sstates; dashed lines: initial pstates of the pionium. b)Contributions of individual target electrons to the closure cross section for incoherent scattering of pionium in its ground state, plotted versus average atomic excitation energy ∆ 0(normalized to the average binding energy of the respective atomic shells, ranging fro m some 10 eV for the N-shell to∼5 keV for the K-electrons). Also shown is the average contribution of all s hells (total incoherent cross section divided by Z). Target material is Ti, projectile energy is 5 GeV. q⊥between kTFandkΠ. In figure 2 we set ω0=Ebind,Πfor the pionium, as well as a non-zero value for ∆0in the case of incoherent scattering. The specific choice for these average excitation energies is guided by the following observations. For coher ent (target-elastic) scattering ∆0≡0, and any ambiguity in calculating total cross sections fro m (18) is limited to the choice of an appropriate ω0. From a comparison [7] of total cross sections for coherent scattering in the closure approximation with the “exact” to tal cross sections obtained by accumulating partial cross sections for bound-bound and bo und-free pionium transitions (summation/integration over all final states), we note that bound-bound transitions (excitation and de-excitation) account for the major part o f the total cross section. Typically, breakup (ionization) accounts for some 30%–40% of the total cross section in the ground state, decreasing roughly by a factor n2for pionium in the initial state ( n, l). Furthermore, most of the breakup cross section from a given i nitial state comes from the range of continuum energies from 0 to about Ebind,Πabove the continuum threshold. Therefore the average energy difference between initial and all final states—weighted by their contribution to the total cross section—is of the or der of the binding energy Ebind,Π. Figure 3a) shows the total cross section for elastic scatter ing in the closure approximation as a function of ω0(in units of Ebind,Π). The total cross sections have been normalized to their values at ω0= 0. As can be seen from the figure, only the ground state total cross section varies appreciably over a reasona ble range of ω0values. We also find [7] that the closure cross sections at ω0=Ebind,Πcoincide with the converged accumulated partial cross sections. We therefore set ω0=Ebind,Πin the following.Coherent and incoherent atomic scattering: Formalism and a pplication 18 10-1100101102103104 q⊥ [a.u.]05e-071e-061.5e-062e-06 No-correlation Incoherent (DHFS) Coherent / Z KL NMd σ / d ln q⊥ Figure 4. Integrand for incoherent (target-inelastic) cross sectio n of ground state pionium scattering off Ti at energy 5 GeV, versus q⊥(on a log-scale). Calculations withω0= 1.858 keV and ∆ 0= 0. In frame b) of the same figure we show the dependence on the aver age atomic excitation energy ∆ 0for the target-inelastic scattering process. Here the pion ium energy has been fixed at ω0≈1.858 keV. Since we calculate Sinc(k) using individual orbitals, we can easily determine the contributions resolved with respe ct to atomic shells, or even per individual electron as in figure 3b). The average excitation energy ∆ 0for each curve has been normalized to the (average) binding energy of the indiv idual shells. We note that the individual contributions are roughly proportional to t he principal quantum number of each electron. Combined with the fact that the outer shell s typically accommodate many more electrons than the inner shells, we find that the tar get-inelastic (incoherent) scattering process is clearly dominated by the loosely boun d outer electrons. At the same time figure 3b) also demonstrates that the incoherent sc attering cross section is almost independent of ∆ 0. Thus we may safely set ∆ 0= 0 in our calculation. As a verification and further illustration of these findings, figure 4 displays dσinc/d lnq⊥for ground state pionium scattering incoherently off Ti ( Z= 22) at 5 GeV projectile energy. The solid line corresponds to the integr and for incoherent scattering (18) calculated using (45). In these calculations we set ω0= 1.858 keV, the binding energy of the pionium, and ∆ 0= 0. As can be seen from the dashed line in the figure, the simplest approximation consisting of scaling the target-e lastic (coherent) cross section by 1/Zclearly underestimates the correct result. At the same time the dash-dotted (chain) curve shows that the approximation using the cohere nt form factor (39) in the no-correlation limit obviously overestimates the correct result. Furthermore, the dotted lines show the contributions to the incoherent scattering c ross section resolved according to atomic shells. A direct comparison of the areas enclosed b etween the individual lines and the abscissa demonstrates that the ten M-electrons dominate the cross sections, followed by the eight L-electrons. Also, the two N-electrons contribute considerably more strongly than the K-electrons, whose influence is limited to rather large q⊥asCoherent and incoherent atomic scattering: Formalism and a pplication 19 0 0.5 1 1.5 2 ω0 / Ebind00.20.40.60.81σT / σC [in %]n=1 n=2 n=3 Figure 5. Ratio (in per cent) between transverse electric and Coulomb contributions to the total cross section of pionium scattering off Ni at 5 GeV , as a function of ω0. Solid line: target-elastic process; dashed line: target-i nclusive process. Pionium initial states (with li= 0) are indicated in the figure. expected. Figures 2 and 4 show clearly that the principal contribution s to the cross sections come from the region kTF< q⊥< kΠ. Thus the long-wavelength limit applies for the pionium (but not for the atom). Furthermore we noted that the relevant excitation energy ∆ 0is small and may safely be set to zero, and thus W1,A= 0. Under these circumstances the cross section due to the transverse part o f the current operator is given by (31) and (33). In figure 5 we show the ratio between the cross sections for the transverse electric part ( Te) and the one for the Coulomb part ( MC), as a function of the average pionium excitation energy ω0. Note that the ratio is given in per cent. The solid lines correspond to total cross sections for coher ent scattering of pionium in various initial s-states as indicated in the figure. The dashed lines correspo nd to the target-inclusive process (with ∆ 0= 0). At the typical average pionium excitation energy ω0=Ebindthe transverse cross section σTfor coherent scattering amounts to 0.4% of the Coulomb part in the ground state, decreasing rapidly for initially excited states of the pionium. Finally figure 6 compares various models for F00,AandSinc,Aover the range of k values relevant for our calculations of cross sections in th e context of experiment DIRAC. The solid lines correspond to our calculations using (39) fo r the coherent form factor and (45) for the incoherent scattering function, respectiv ely. We use relativistic Dirac orbitals in both cases. The squares represent tabulated val ues from two compilations of state-of-the-art Hartree-Fock calculations with vario us corrections (configuration interaction, relativistic effects, a.s.o.). Note that whil e the tabulated results for F00 correspond to relativistic calculations [24], the tables f orSincin [25] contain only non- relativistic results. The log-log representation in the le ft frame serves merely to displayCoherent and incoherent atomic scattering: Formalism and a pplication 20 10-210-1100101102103 k [a.u.]10-210-1100101102 Moliere Thomas-Fermi / Moliere No-correlation / Moliere Tabulated values DHFS calculationF00Sinc 10-210-1100101102 k [a.u.]0510152025 Moliere Thomas-Fermi / Moliere No-correlation / Moliere Tabulated values DHFS calculationF00 Sinc Figure 6. Electronic part F00of the coherent atomic form factor and incoherent scattering function Sincfor Ti ( Z= 22). The asymptotic behavior is more easily seen from the log-log diagram on the left. The range of relevance f or the cross section calculations is 0 .1≤k≤100 a.u. For an explanation of the different models, see text. the asymptotic behavior of the simple models discussed in se ction 5. Analyzing the dashed line corresponding to (52) we note that the incorrect asymptotic fall-off for F00 at large kposes no difficulties. In the right panel we see that between 5 ≤k≤10 a.u. the Thomas-Fermi-Moli` ere model misses features of the ato mic shell structure, but the resulting deviation from (39) is insignificant. Much more pr oblematic are the crude approximations for Sinc. While the no-correlation limit (54), using (52) and shown w ith the dotted line, increases with k2at small k(as it should), it dramatically overestimates the scattering function in the most relevant range between 0 .1 and 100 a.u. On the other hand, the Thomas-Fermi model for incoherent scattering as d eveloped by Heisenberg [20], with modifications [21, App. B] for the use of Moli` ere’ s approximation, is quite successful at k≥5 a.u., but it fails completely at smaller k. 7. Conclusions We have reviewed the formalism for incoherent atomic scatte ring. The basic expressions for the atomic form factors and scattering functions have th en been evaluated in the framework of Dirac-Hartree-Fock-Slater theory, i.e., using numerically determined electron orbitals. For comparison, both the form factor for coherent scattering, as well as the incoherent scattering function have been derived in sim ple analytical models based on the Thomas-Fermi model of the atom. Applying these differe nt descriptions we performed detailed numerical studies in the context of pion ium scattering incoherently off the electrons of various target atoms. Due to the much larg er reduced mass of the pionium system ( µΠ=mπ/2≈136.566me), the length and momentum scales of the pionium and the target atom are very different. An investigation of the relevant momentum transfer q⊥revealed that the cross sections are dominated by the contributions from the region between the Thomas- Fermi momentum kTF=Z1/3/aBohrfor the atom, and the momentum scale of theCoherent and incoherent atomic scattering: Formalism and a pplication 21 pionium at kΠ=µΠ/aBohr. Under these circumstances the simple models for incoheren t scattering discussed in section 5 prove not sufficiently accu rate for our application in the context of pionium–atom scattering with a required accu racy of 1% or better. Whereas the analytical models discussed in section 5 provid e sufficiently accurate coherent form factors, the incoherent contribution really requires the more accurate treatment developed in section 4, despite its lesser import ance (as compared to coherent scattering). Only an explicit DHFS calculation can provide satisfactory scattering functions. From our detailed discussion of the q⊥-dependence of the integrand for the cross section we also conclude that the loosely bound outer shell e lectrons dominate the target- inelastic cross sections. Their contribution to the integr and dσ/dq⊥covers a much larger range of q⊥values than the one corresponding to inner shell electrons. Following this argument further, free electrons would show a behavior rath er similar to the one of the outer shells, their contribution to the cross section would stretch even further down to smaller q⊥. However, the cross section hardly depends on this modificat ion at very small q⊥. Our calculation therefore applies equally well to quasi-f ree electrons and to electrons in the conduction band. Thus solid state effects and chemistr y need not be considered explicitly as they prove irrelevant for calculations perta ining to experiment DIRAC. The same conclusion had been inferred in our earlier study [7] ba sed on an analysis of the relevant impact parameter range. Recalling that target-in elastic scattering constitutes merely a correction of order 1 /Zof the atomic part, variations on the order of a few per cent in the incoherent scattering cross sections are insign ificant. These findings are confirmed in our analysis of the dependence of the cross sections on the average excitation energies for pionium and atom. Our calculation of incoherent scattering contributions resolved accordin g to individual target electron shells demonstrates that the average excitation energy for the atom may safely be set to zero, ∆ 0= 0. For the pionium, on the other hand, a non-vanishing excit ation energy in the amount of the binding energy is needed when calculatin g total cross sections in the closure approximation. Earlier investigations on pionium–atom interaction [11, 7 ] invoked properties of the hydrogen-like pionium system to obtain crude estimates for the magnitude of the magnetic terms so far neglected in this interaction. Fro m the non-relativistic wavefunctions for the pionium, its internal velocity is of t he order of v/c≈α/2. Thus magnetic terms are believed to be small of this order. We esta blished a much better justified estimate for the magnetic terms (actually for the t ransverse part of the current) in the long-wavelength limit. Our investigation showed tha t this limit applies very well for the pionium, whereas it does not apply for the atom. Howev er, the transverse current does not contribute in the elastic case on the atom side, and f or the inelastic case it is suppressed even more than the corresponding term on the pi onium side because the relevant atomic excitation energies (of the outer shells) a re smaller than those of the pionium.Coherent and incoherent atomic scattering: Formalism and a pplication 22 Acknowledgments We would like to thank R.D. Viollier, J. Schacher, L. Nemenov , and L. Afanasyev for stimulating discussions on the subject of this article. Spe cial thanks are due to Zlatko Halabuka whose efficient and accurate integration routine pr oved instrumental in the calculation of atomic form factors and scattering function s. References [1] McGuire J H, Stolterfoht N and Simony P R 1981 Phys. Rev. A2497 [2] Anholt R 1985 Phys. Rev. A313579 [3] Wheeler J A and Lamb W E Jr 1939 Phys. Rev. 55858 [4] Sørensen A H 1998 Phys. Rev. A582895 [5] Voitkiv A B, Gr¨ un N and Scheid W 1999 Phys. Lett. A260240 [6] Nemenov L L et al1995Lifetime measurement of π+π−-atoms to test low-energy QCD predictions CERN/SPSLC 95–1, SPSLC/P 284 [7] Halabuka Z, Heim T A, Hencken K, Trautmann D and Viollier R D 1999 Nucl. Phys. B55486 [8] de Forest T and Walecka J D 1966 Adv. Phys. 151 [9] Halzen F and Martin A D 1984 Quarks and Leptons (New York: John Wiley) [10] Greiner W and Sch¨ afer A 1995 Quantum Chromodynamics (Berlin: Springer) [11] Afanasyev L G et al1994Phys. Lett. B338478 [12] Walecka J D 1983 ANL-83-50 (Argonne National Laboratory; unpublished) [13] Hencken K, Trautmann D and Baur G 1995 Z. Phys. C68473 [14] Blatt J M and Weisskopf V F 1952 Theoretical Nuclear Physics (New York: John Wiley) [15] Slater J C 1960 Quantum theory of atomic structure (New York: McGraw-Hill) Chapters 13ff [16] Baur G, Hencken K and Trautmann D 1998 J. Phys. G: Nucl. Part. Phys. 241657 [17] Caso C et al (Particle Data Group) 1998 Eur. Phys. J. C31 [18] Salvat F, Mart´ ınez J D, Mayol R and Parellada J 1987 Phys. Rev. A36467 [19] Moli` ere G 1947 Z. Naturforsch. 2a133 [20] Heisenberg W 1931 Phys. Zeitschr. 32737 [21] Tsai Y S 1974 Rev. Mod. Phys. 46815 [22] Salvat F, Fern´ andez-Varea J M and Williamson W Jr 1995 Comput. Phys. Commun. 90151 [23] Halabuka Z, private communication [24] Hubbell J H and Øverbø I 1979 J. Phys. Chem. Ref. Data 869 [25] Hubbell J H, Veigele Wm J, Briggs E A, Brown R T, Cromer D T a nd Howerton R J 1975 J. Phys. Chem. Ref. Data 4471

G. Van Hooydonk CB00-02 short, version 2: Gauge symmetry.... 05/04/00 1 of 13Gauge symmetry, left-right asymmetry and atom-antiatom systems: Coulomb's law as a universal molecular function. G. Van Hooydonk, Department of Library Sciences and Department of Physical & Inorganic Chemistry, Ghent University, Rozier 9, B-9000 Ghent, Belgium (Email: guido.vanhooydonk@rug.ac.be) We prove that Coulomb's 1/R-law is the universal function, needed for scaling in molecular spectroscopy. To obtain this result we introduce intra-atomic charge inversion. This generates an algebraic switch in the Hamiltonian of 4- particle systems when going from atom-atom XX to atom-antiatom X X systems. This switch is a consequence of atomic left-right asymmetry (handedness, chirality). For X X systems, this parity operator can reduce the 10 term Hamiltonian to 1 or 2 terms. The reduced Hamiltonian reproduces and scales potential-energy curves (PECs) of normal bonds. For 9 bonds (HH, HM and MM, M alkali metal) observed levels and turning points of XX systems coincide with those calculated for X X systems. Hydrogen-antihydrogen reactions, feasible in the near future, will produce a normal HH molecule. If true, this would solve the problem about the existence of antimatter. Nature prefers charge anti- symmetry in neutral bound 2-particle systems. In neutral bound 4-particle systems dipole anti-symmetry is preferred, which implies that charges in two interacting neutral particles are not assigned according to convention. 1. Introduction. In the 2nd quarter of the 20th century there was a parallel development in physics and chemistry. With parity violation discovered around 1950 a break occurred. This led to the new physics, the Standard Model (and beyond), whereas in chemistry mainly computational procedures for many particle systems were refined [1a]. Still, physics and chemistry remain united in solving 4-particle systems, important in the advent of hydrogen-antihydrogen reactions. H 2 is the simplest stable 4-particle system with unit charges, the ultimate test case for elementary particle theories. Quantum mechanics even describes H 2 exactly [1a]. But interest remains in classical and semi-classical approximations for various reasons, not only because of the complexity of quantum mechanical procedures for large systems but also because of the study of chaos in quantum systems and the correspondence principle. In this context, Bohr's molecular bonding models [1b] were reanalyzed [1c]. In both quantum mechanical and classical methods, the 4-particle Hamiltonian is rather complex [1a ,1c] and the way in which charges are distributed is critical for system- stability deriving from Coulomb forces [1c, 1d] . For instance, the computation of 2-electron interactions is the bottleneck of quantum chemistry (the Coulomb problem [1e]). To avoid computational problems, Coulomb-attenuated HFS calculations have been proposed, originating from a suggestion to cut off the long range branch of a Coulomb potential [1f]. The present work focuses on unconventional if not chaotic charge distributions in 4 particle systems, without altering Coulomb forces. If leptons have mass m a, mb, nucleons M 1, M2, the 10 term Hamiltonian is H= ½m ava2+½m bvb2+½M 1v12+½M 2v22 -e2/R1a-e2/R1b -e2/R2a -e2/R2b +e2/Rab +e2/R12 (1)The standard premise in chemistry [1a] that leptons have negative, nucleons positive charges secures that nucleon-lepton interactions account for bonding as in Heitler-London (H-L) theory [2]. However, (1) as it stands does not give a hint about two basic issues: (i) singlet-triplet splitting observed invariantly, and (ii) shape/scale invariance of potential-energy curves (PECs). Triplet PECs follow a repulsive Coulomb law, singlet PECs show left-right asymmetry at the minimum but the function is unknown. Both problems are connected but only (i) was solved satisfactorily in H-L theory [2]. Their solution derives from symmetry effects of both spin and wave function (Pauli's fermion anti-symmetry) in a wave mechanical framework. The wave equation with (1) must be solved first and atomic energies are then subtracted to get bond details (PECs). The complex H-L solution for (i) prohibits a simple one for (ii), shape/scale invariance of PECs. For singlet states, this can only be accounted for by means of a universal function, still to be found and, according to Tellinghuisen [3], this is the Holy Grail of Molecular Spectroscopy. PEC invariance is reflected in part [3,4] in a simple behavior of spectroscopic constants ωe, Be, ωexe and αe. Hamiltonian (1) can not lead to shape/scale invariant PECs, unless it is reduced to a function with just R 12 as a variable. For singlet states, empirical 1/R-potentials can indeed account for many PECs [4] but the R 12-term in (1) is repulsive, not attractive. In addition, the best asymptote for scaling is Coulomb's asymptote e2/Re [4], not the atomic dissociation limit D e. This is contrary to convention, since D e is the standard scaling factor in molecular spectroscopy (the Sutherland parameter). To solve the wave equation for (1) assumptions must be made about the molecular wave function. H- L theory uses only the VB part ψVB of the complete Hund-Mulliken MO wave functionG. Van Hooydonk CB00-02 short, version 2: Gauge symmetry.... 05/04/00 2 of 13ψMO = ψVB + ψION although the ionic function, apparently, is equally important. Here, we try to solve problem (i) and (ii) starting from (1) directly without relying on external symmetry effects. A solution derives from ψION, not from ψVB, the H-L approximation for ψMO. 2. Theory. a. Algebraic switch in the 10 term Hamiltonian. Interaction energies referring to asymptote e2/Re and D e are instructive. Let X consist of lepton- nucleon pair m a, M1, Y of mb, M2. Pairs are charge conjugated but charge symmetry is broken by the large particle mass difference (m/M=1/1836 for H). With the non-Coulomb asymptote D e, (1) leads to V(R)=H XY=H-(HX+HY)= -e2/R1b-e2/R2a+e2/Rab+e2/R12 (2) or 4 terms to cope with a pair of two neutral atoms . The ionic Coulomb asymptote gives 3 terms V'(R)=H X+Y- =H-(HX++HY-)=-e2/R1a-e2/R1b+e2/R12 (3) for a pair of two charge conjugated ions . Here, charge symmetry is not broken by mass difference, which is small (order 2/1836 for H). In both (2) and (3), the internuclear term remains repulsive. At least here a switch in sign is needed for singlet states to obtain a function starting off as an attraction at the asymptote with variable R = R 12. Switching signs o f Coulomb terms corresponds with switching fermion chiralities, a result of charge conjugation in combination with the particle hole transformation [5]. Out of the 6 Coulomb terms in (1), 4 switch sign when antiatom X (m+M-) replaces atom X (m-M+). Charge invariance secures that such a switch leaves the (intra-) atomic energy invariant. Asymptotes for all 4 states XY, X Y, XY and XY are identical. This has an important consequence when a switch1, a parity operator p is introduced in (1) Hp=(½m ava2+½m bvb2+½M 1v12+½M 2v22-e2/R1a-e2/R2b) +(-1)p(-e2/R1b-e2/R2a+e2/Rab+e2/R12) (4a). Terms between the first pair of brackets reflect intra- atomic charge invariance. For the remaining 4 terms, p=0 gives the classical system XX or XX, p=1 gives XX or X X. If asymptotes are really charge invariant, p explains splitting and solves problem (i), since Hp = H± = H0 ±V(R) (4b) The first 6 terms in (4a) belong to charge invariant asymptote H 0, the remaining 4 are interactions V(R) V(R)XX = V(R) XX = -V(R) XX V(R)XX = V(R) XX (4c) Two pairs of degenerate states appear, one being 1 This switch can be visualized by 2 hand-held permanent magnets (atomic EDMs). A parallel (p=0) or anti-parallel (p=1) alignment of magnets reflects mirror symmetry and is felt when the distances between aligned magnets vary.dipole symmetric ( ↑↑or↓↓), the other anti-symmetric (↑↓or↓↑) just like with spins. The approach differs from H-L theory but Pauli-matrices apply to both, as spin and dipole symmetries are similar. Spin ( ±½) and charge ( ± 1) operators only differ by a factor 2. With non-Coulomb asymptote D e, intra-atomic charge inversion p = 1 gives, instead of (2) V(R)=H XY=H-(HX+HY)= +e2/R1b+e2/R2a-e2/Rab-e2/R12 (5a) With the ionic asymptote, (3) tra nsforms in V'(R)=H X-Y+=H- (H X-+HY+)= -e2/R1a+e2/R1b -e2/R12 (5b) where Y+ is a composite 3 particle antianion. Both (5a) and (5b) give attraction for the R 12-term. A solution with an algebraic switch is generic and independent of the system's unknown geometry. We will not reconsider classical Bohr models [1b, 1c]. Intra-atomic charge inversion leads to 4 states with different dipole alignments, useful for classical analysis at long range (related spin states have only minor energetic consequences). First, (2) and (5a) approach a charge invariant asymptote D e symmetrically, conforming to (4b). If (2) reaches D e from the repulsive, (5a) reaches it from the attractive side or vice versa. With atomic radius d= ½Rab(e)=½R e, dipole-dipole interactions (the magnet metaphor1) give V(R)XX = V(R) XX = +¼(e2/Re)(Re/R)3... V(R)XX = V(R) XX = -¼(e2/Re)(Re/R)3... (6a) varying as R-3. Splitting is twice as large. Result (6a) applies to long range only. With respect to D e, this naive Coulomb treatment gives repulsion for dipole symmetric states and attraction for dipole anti- symmetric states. Approximating chiral effects by rotating one dipole by 180° gives the opposite result in first order2 but this is not exactly the same as a mirror symmetry effect with charge inversion (p=1). Second, (3) and (5b) behave similarly. Ionic models have the advantage of their simple geometry. Long and short-range interactions obey Coulomb's law. Equating ion attraction with (3) gives -e2/R12=- e2/R1a-e2/R1b+e2/R12 for p=0. At short-range, e2/R12= e2/R1a, if e2/R1a=e2/R1b. This is wrong by a factor 2. For p=1 (5b) gives -e2/R12=-e2/R1a+e2/R1b-e2/R12 or e2/R12 ≈ e2/R12, the correct ionic Coulomb attraction for all R-values even close to R e. a |d R =R 12 2-------------------------------------1 |d b Scheme a 2 Dipole rotation by π gives Vp(R)=- ¼(-1)p(e2/Re)(Re/R)3. In 2d order, mirror symmetry based states in (6a) are lower by a term in (e2/Re)(Re/R)5. Linear dipole alignments always give zero energy in first order.G. Van Hooydonk CB00-02 short, version 2: Gauge symmetry.... 05/04/00 3 of 13We can refine this with ionic models (as in Scheme a) where R ab=2d is perpendicular to R. Without polarization, we have for p=0 and p=1 V'(R)X+X- = V'(R) X+X- = -(e2/R)(1-(d/R)2) V'(R)X+X- = V'(R) X+X- = -e2/R (6b). With p=0, lepton-nucleon polarization is hampered by inter-lepton repulsion. For p=1, lepton-nucleon interactions cancels exactly. Although ionic states are always attractive by definition, a small p- dependence is observed. For R ab parallel to R, we obtain respectively V'(R) X+X- = V'(R) X+X- = -e2/R V'(R)X+X- = V'(R) X+X- = -(e2/R)(1+2d/R) (6c) In ionic cases (6b, 6c), p-effects are subtle. Yet, a charge inverted ionic state is always low. These are easily verifiable consequences of a parity operator in Hamiltonian (4a). The covalent case is ambiguous but only in first order2 (see also below). Charge inversion solves problem (i) generically but for problem (ii) additional information is needed. From both long and short-range interactions in ionic models (6b, 6c), the solution about stability is unambiguous. Unfortunately, it is impossible to get an extremum. The most critical issue is a minimum for the lowest charge inverted ionic X+X- or X+X-- state in (6c). Whether or not a minimum exists is uncertain. BOA (Born-Oppenheimer Approximation) may apply but this is not evident. Comparing V'(R), dV(R)/ dR is largest for (6c). Unlike charge inverted states V(R )XX3, the p=1 state in (6c) will intersect asymptote D e at large R and reach the extremum, if it exists, before any other state. Extrapolating long- range attraction to short range, gives the p=1 case in (6c) as most stable state. But for the simple case HH, the character of H H is still an open question. Since data on H--- H reactions could be available soon a solution is needed. But even sophisticated quantum mechanical methods are controversial about H H. b. Controversy about H H. Modern quantumchemical methods for 4-particle systems have ab initio status [1a] and reach spectroscopic accuracy. Then it is surprising there is a controversy about H H. The Richard group [6-8] suggests H H is unbound, which is confirmed by Monte Carlo simulations [9]. Abdel-Raouf and Ladik [10] claim H H is bound. Computational efforts are great and both methods seem reliable. The origin of the controversy lies in the effect p in (4a). Perhaps, something is wrong with the premises of quantum chemistry (charges, wave functions, correlation4), 3 This state will deviate from D e at much slower rate (6a).4 With a James-Coolidge procedure for correlation 1/R ab. Some of these important computational difficulties with electron correlation are discussed in Ref. [1a, 1e, 1f].despite its successes [1a]. We showed above th at p=0 and p=1 solutions for (4a) reach the same asymptote D e from different sides. If HH is bound, H H is unbound as in the first thesis [6-9]. A different spatial lepton configuration can invert this result but only in first order2: maybe this minor difference is a classical explanation for the origin of wave mechanical ambiguity with H H. Even at long range, a charge inverted atomic state is always below a dipole rotated normal state. In the cold atom region, R e/R-values equal to 1/20 and 1/10 respectively give small energy differences of only <0.01 and <0.4 cm-1 if C = 110,000cm-1 in favor of charge inverted p=1 states. Also all ionic states lead to enhanced attraction for H H, supporting Abdel- Raouf and Ladik's thesis [10]. Conventional HH is less stable than its charge inverted version H H, with H and H related by mirror or left-right asymmetry. This is not a spin and orbital symmetry but a dipole symmetry effect, completely absent as such in conventional H-L theory. These are all classical easily verifiable results deriving from long range behavior and dipole-dipole Coulomb interactions. The extrapolation to short range is critical. As for problem (ii), the extremum, if any, for X X and in particular for H H must be found. c. Reducing the 10 term 4-particle p=1 Hamiltonian to a Kratzer and a Coulomb potential. For p=1, lepton-nucleon interactions in (4a) can cancel exactly for an unknown geometry. In this ad hoc hypothesis and independent of asymptote choice , any p=1 solution for (4a) invariantly gives Hp=1=(½m ava2+½m bvb2-e2/Rab) +(½M 1v12+½M 2v22-e2/R12) (7). This is either positronium-protonium or hydrogen- antihydrogen, depending on the value of the reduced mass (central force character, BOA or geometry of subsystems). In the limit, a mass-less 4-unit charge system results. Without the term in R 12, (7) applies to anion/helium like 3-particle systems. With electronic wave functions, the R 12-term may be disregarded first and added at the end of the procedure. Instead of sophisticated quantum mechanical methods which lead to a controversy for H H [6-10], we rely on easy to check solutions for (7) to solve problem (ii) and to finalize the stability problem about X X. Two solutions exist, which must be confronted with observation. a. First, assume (7) is a generic result and that leptons can be redistributed over nucleons in such a way that two unspecified quasi-central force systems emerge (BOA). In practice, this configuration is restricted to R e-domain in VB schemes. If R ≈ Re, then R ≈ Rab also and a first solution for (7) can be Hp=1 ≈ 2(½ µv2 -e2/R) (8)G. Van Hooydonk CB00-02 short, version 2: Gauge symmetry.... 05/04/00 4 of 13a Bohr-like model [1b] with charge inversion . Here, µ is a reduced mass and the asymptote is situated between zero and the total well depth. Eqn. (8) is a reduction from 10 to 2 terms in Hamiltonian (1). In contrast with long-range forces above, (8) is valid at short range, the R e-domain, where the 4 lepton- nucleon interactions can cancel exactly5. Then, one can say something about the dynamics of the system only by assuming the lepton's angular momentum is constant. Bohr's quantum condition secures that µv2 ~ B/R2 (9) where B is a constant. Its value depends on R e. With asymptote A, a Kratzer potential [11] is the result Hp=1 = -A + 2(B/R2 -e2/R) (10) This potential has a classical minimum when the derivative with respect to R vanishes or when B = ½e2Re (11). Result (11) leads to the eigenvalue Hp=1(Re) = -A -e2/Re (12) a Coulomb asymptote, anchored at A between zero and the total well depth. Kratzer's potential (10) was proposed already in 1920, after Bohr's atom theory but before Schrödinger's wave mechanics. It is itself a generalized Bohr atom formula but it is also very useful for molecules [4, 12] in another generalized form due to Varshni [13]. Kratzer-Varshni's potential scales molecular spectroscopic constants efficiently [4]. It accounts for PECs only up to De[14], which is crosses at the critical distance since e2/Re >> D e. The usefulness of (10) sh ows in oscillator form W(R) = (e2/Re)(1- R e/R) 2 = Ck2(13) Here k is the Kratzer variable6 k = 1 - R e/R (14) and C is the ionic Coulomb asymptote. Form (13) appears naturally for (7), though no assumption was made about asymptotes. In fact, equations (9) and (10) are confined to the R e-domain (see above). This first solution for X X systems may require ad hoc hypotheses about an unknown geometry for the system but an acceptable oscillator model and asymptote are obtained. This result is consistent with spectroscopic data [4, 14-18] for diatomic bonds XX, i.e. for atom-atom systems in H-L sense. However, solution (13) can only apply for a bound singlet state (p=1) and there is no information about the triplet state. Nevertheless, it throws another light on the controversy about the stability of H H: it seems this is not only the more stable version of HH but it should 5 Configurations as in ionic schemes or in a classical Watt regulator are possible. The latter appear in Bohr's first molecular models [1b] (see also [1c]).6 Dunham's widely used potential based upon variable k/(1-k) can not converge. Function (13) will always reach the Coulomb asymptote C [4]. Expansions of variable k near Re lead to continued fractions also important for fractal/chaotic behavior [1d, 18].also have a minimum. If true, H-L theory is only a complex way for introducing an atom-antiatom switch in (1) as in (4a) to describe what we now call a chemical bond. Confronting (13) with observed PECs is done below but preliminary studies show this conclusion may be unavoidable [4, 14-18]. b. A second solution derives from a 3-particle anion/helium like subsystem. The internuclear term is treated separately as in ionic central force systems, see (6b). Without polarization, the 4 lepton-nucleon interactions now always cancel for all R , not only in the R e-domain as with solution a. This is the greatest advantage of ionic over covalent models but also the reason why ionic models are described as 'naive'. Ionic bonding is 200-years old (Davy and Berzelius [4]). Their naive7 scheme uses just one Coulomb attraction to describe a complex 4-particle system, which seems like an oversimplification indeed. With X- and Y+ defined above, (7) leads to Hp=1 = -A' -e2/R = -IE Y - EA Y -e2/R (15). IEY and EA Y are respectively, ionization energy and electron affinity of Y, both atomic not molecular constants, not varying with R. A' = -IE Y -EA Y (16a). is the eigenvalue of the 3-particle ion/He system. By definition, a naive picture with one Coulomb interaction (15) gives the same Coulomb asymptote as in (12) at a more specified intercept A' in the total well depth. Unlike (10), (15) can never lead to a minimum nor a description of a repulsive triplet state. Whereas (10) gave a minimum and led to an acceptable oscillator form (13), (15) does neither, since no dynamics is involved. Taking derivatives of (15) with respect to R does not lead to an extremum. A minimum, if any, can only be generic and, if so, it must be hidden in Coulomb's law itself (see below) as must be the information about splitting towards a triplet state. If (7) were really positronium-protonium, the energy would be Hp=1 = -½IE H - e2/R with an asymptote of about 54800 cm-1. This is of the correct order of magnitude but only applicable to H. Also the problem with the minimum remains. In (6b), the small perturbation term -e2/R1a+e2/R1b (≈ 0) (16b) was neglected. This is important to find an extremum and oscillator behavior for (15). If so, problem (ii) is also solved with the naive ionic approximation. d. Gauge symmetry and the generic minimum. Coulomb's law for 2 charges (equal masses) is 7 In fact, establishment immediately rejected these naive ionic models after H-L theory was available.G. Van Hooydonk CB00-02 short, version 2: Gauge symmetry.... 05/04/00 5 of 13V±(R) = ±e2/R = (-1) t e2/R (17) where t is a parity operator, representing an algebraic symmetry for attraction and repulsion. With charge invariance, (17) led to (4c) and a degeneracy of states with symmetries governed by atomic dipoles. Coulomb's law (17) is not complete, since potentials, acting upon a unit charge, are not gauge invariant. When a 2-particle system is bound, V ±(Re) = ±e2/Re = ±C. The sign of C determines the position on the axis, where (17) will be anchored. This choice of the sign for C is arbitrary and fixed by convention only. C-symmetry is represented with parity operator g C± = ±C = (-1) g C (18) With g in (18) different from t in (17), degeneracy like in (4c) for dipole interactions is removed. Two- dimensional scaling of axes y (C) and x (R) by means of scaling factors R e for R and |e2/Re| for |C| is possible. Instead of (17), 4 non-degenerate states are generated generically (i.e. without convention) W(R) = C ± + V±(R) = (-1) g C (1+(-1) t-g Re/R) w(m) = W(R)/C = (-1) g (1+(-1) t-g Re/R) (19) The two pairs have symmetries t-g = 0 and t-g = 1, whereby gauge and interactions have the same or opposite signs. In any pair, one of the two states is charge symmetric (++ or --), the other charge anti- symmetric (+- or -+). The behavior of the states in (19) at the tree level is shown in Fig. 18,9. Two states belong to a positive (C > 0), two to a negative world (C < 0). The worlds are isospectral, as algebraic symmetry applies. Only one world can be allowed by convention but a t-g=1 state starting in one world will extend into the other. Unlike (17), the 2 non-degenerate attractive states (+- and --) with t-g =1 always cross by definition, t-g=0 states never cross. Symmetry must be broken at crossing point Re/R=1. With a total gap of 2C, a generic minimum C=e2/Re is obtained. Gauge symmetry applied to Coulomb schemes produces a generic minimum, independent of convention and dynamics . The virial is always obeyed. Fig. 1 applies only to fermions and a relation with bonding between 2 neutral atoms (bosons) is not evident. The two attractive t-g=1 states with different signs for asymptote and interaction, are C - e2/R (20a) -C + e2/R (20b) and originate in the positive (20a) or negative world (20b) (see Fig. 1), although conventionally (20b) would be repulsive (--). Fig. 1 results solely from algebraic R n-laws and conventions about gauges and 8 Extending Fig. 1 results to a second much larger gap C 1 >> C, crossing at smaller R' e=e2/C1 will result in a fine structure at R'e. The opposite case also applies.9 The positive-negative world distinction is only formal and is easily removed by adding a large constant gauge +C0 >> C such as the absolute mass equivalent (m+M) c2.charges. PECs deriving from n=-1 have all the characteristics of observed singlet XX PECs, except for curvatures [18]. But states w ith the same t - g = 1 symmetry can not cross and perturbation is required at R e. Sum and difference of (20a) and (20b) are 0 and 2(C - e2/R). With constant perturbation P, the perturbed Coulomb state in the positive world is W(R) = C (( k2 + p2) ½ - p) (21) Here p=P/C is a reduced perturbation (not the parity operator in (4a)). This can be subtracted to obtain an oscillator presentation with zero energy at Re. At the tree level, the effect of a small constant perturbation p2=0,1 (p ≈0.33, see Eqn. (24) below) is shown in Fig. 2. For comparison, a Kratzer potential (13) is added but this is shifted upward with an amount p to make minima coincide. Although for (13) and (21) k-dependencies may be very different, the PECs are not. Both are consistent with shape and scale invariance of observed atom-atom singlet PECs [18]. Both exhibit the correct left-right asymmetry at the minimum. Nevertheless, both PECs in Fig. 2 are derived from atom-antiatom Hamiltonian (4a) with p=1. Both closed form analytical X X potentials refer to Coulomb asymptote e2/Re, so important for scaling XX PECs and constants [4, 18]. But (intra-atomic) charge invariance secured that charge inversion leaves atomic energies invariant when going from X to X. Fig. 1 shows that this is not so: gauge symmetry overrules charge invariance. By allowing for a (virtual) negative world, degeneracy (4c) is removed in (19), as illustrated in Fig. 1. This gives the generic minimum in (21) for Coulomb's law in the positive world (Fig. 2). But the net result is that a charge invariant asymptote in the positive world is restored, since the origin of the repulsive left branch of a PEC is to be found in attraction in the (virtual) negative world. Gauge symmetry leads to an extremum for 2- particle systems. For 4-particle systems, the problem is more complicated, see (1) instead of (17) and (19). The perfect symmetry in Fig. 1 and 2 applies to 2 unit-charges for which it is difficult to imagine that the perturbation needed in (21) is at work, if self- perturbation is excluded. Examples are positronium and protonium. Coulomb attraction is used in full to describe the system, so it may not be used once more as a perturbation. Describing mass symmetrical 2- particle systems with gauge symmetry seems useless (annihilation, see above). Positronium is stable10 with energy -½IE H on account of dynamics and quantum behavior. But in a more complex neutral N-particle system the perturbation needed in (21) can be present. The neutrality condition leads to N=4 (chemical bonds), 10 Positronium (and protonium) states have short lifetimes.G. Van Hooydonk CB00-02 short, version 2: Gauge symmetry.... 05/04/00 6 of 13where the single Coulomb interaction for a 2-particle system is replaced by 6 as in (1). But for ionic p=1 systems a single Coulomb attraction (15) emerges which would allow us to apply a gauge symmetry based 2-particle scheme like (19) to chemical 4- particle systems. The only condition is that these are partitioned in mass asymmetrical ionic subsystems (a composite 3-particle ion and its charge conjugated non-composite antiion). In this way, the perturbation needed to break symmetry in (21) but lacking in mass symmetrical 2-particle systems, appears naturally for N=4 as in (16b). All this applies to so- called naive ionic bonding models. Theoretically, intra-atomic charge inversion can be important for chemical bonding. The removal of the degeneracy in 2-particle systems (19) applies to 4-particle systems. In Fig. 1, the arrows represent the dipole-dipole interactions (similar to a spin notation) for 4 the states. All what has been said above about (virtual) PECs for 2-particle systems can become a reality for 4-particle systems and their PECs11. Anion-cation systems obey similar schemes [17,18] but the difference between normal and charge inverted anion-cation pairs are subtle (see above). Solution b. starts from long range, see (6b, 6c), (15) and is rather vague about the minimum, where 'a' perturbation should occur. Solution a. is specific about the minimum but, nevertheless, refers to the same asymptote as solution b. Therefore it is quite tempting to confront both solutions. Whether or not (15) is an alternative for bonding now depends on the analytical form of function P(R). e. Perturbed Coulomb potential. Classically, dipole-dipole interactions use atom polarizabilities and functions of general form R-n with n>>1 and/or sums of similar terms with higher n, depending on the degree of approximation (see above and Ref. [18]). Instead of applying this classical analytical solution, we shortcut the circuit and confront both solutions, as explained above. This gives the possibility to amend Kratzer's result (13) to find out whether it can be improved or not. Both give the same asymptote e2/Re. Equating (13) with (21) will provide us with information about P(R). If C K is any Kratzer and C C any Coulomb asymptote, we obtain CKk2 = CC ((k2 + p2) ½ - p) (22a) 11 If the present scheme is validated by experiment (see below), nature would assign charges to interacting neutral particles in a way that, with respect to convention [1a], may seem chaotic in simple quantum systems (see Fig. 8 in Ref [1c] and see also [1d]). In this work, 'charge-chaos' is restricted to just a choice between atom X and charge- inverted antiatom X, a mirror effect.With C K = CC, this leads to P(R) or p(k) p = ½ (1-k2) (22b). This is trivial and points to the corresponding virial, as it should. Using this dependence in equation (22a) again gives CKk2 = CC ((k2 + p2(1-k2)2) ½ - p) (23) Instead of Coulomb PEC (21), the result is now W(R)/C = (k2 + (0,366025(1-k2))2) ½ - 0,366025 (24) If C = e2/Re, PEC (24) solves the problems with the one term solution b. (15): minimum and an oscillator form are obtained. PEC (24) is similar to but always below the Kratzer PEC (13) if both start from the same asymptote (see Fig. 2 where p = 0,33 was used, close to 0,366). Numerically, p = 0,366025 = (31/2- 1)/2 (a more detailed derivation is in Ref. [18]). PEC (24) is an hybrid function of Kratzer solution (13) and gauge symmetry based Coulomb function (21). PECs reaching the atomic dissociation limit D e instead of C = e2/Re are available with PEC = ½(W(R)+D e)-½((W(R)-D e) 2+V'(R)) ½(25) where W(R) can either be (13) or (24). Perturbation V'(R)12 in (25) is only needed to avoid crossing at the critical distance, since C >> D e (see above and [4]). Both Coulomb-based results a-b, varying with k or k2 for atom-antiatom systems, point to empirical 1/R-potentials, so valuable for interpreting PECs for atom-atom systems, chemical bonds in H-L theory. Both can solve problem (ii) in the Introduction. The H-L scheme can not account for the well behaving13 empirical 1/R potentials [18], which is exactly why the search for universal 1/R-functions has been going strong for many decades [3,4,18]. If H-L theory were really complete, scaling PECs analytically would not have to be a problem [3,18] but, in practice, the scaling issue (ii) remains unsolved. This proves once more that H-L theory is not complete or at least too complicated. Both solutions a-b for (7) show that H- L theory is a cumbersome way to introduce a parity operator in (1) as in (4a). Unfortunately [1c ,1d, 18], all connections with classical/semi-classical results like (13) and (15) are lost in H-L theory. Strictly spo ken, gauge symmetry would even get a solution for ionic models without charge inversion. But the connection with (7), generic singlet-triplet splitting and solution a. would not have been found. These are essential to arrive finally at (24) and we showed above what exactly the differences are between classical and charge inverted ionic models. Generic gauge symmetry smoothly transforms fermion behavior (splitting at large R as in Fig. 1) into boson behavior (atoms in equilibrium around R e as in Fig. 2) without the super-potentials of SUSY. If 12 We will use V'(R) = 0 throughout.13 Traditionally, a complete theory explains exactly why empirical relations can behave well. In this respect, H-L theory fails; this raises questions about its completeness.G. Van Hooydonk CB00-02 short, version 2: Gauge symmetry.... 05/04/00 7 of 13PECs (24) are better than (13), left-right asymmetry must be the rule in nature. f. Ionic systems and wave functions. Ionic wave functions conform to the framework MO. Heitler-London-Pauling-Slater VB theory uses wave functions of general form ψVB = u1au2b + u1bu2a LCAOs where AO is atomic wave function u. Hund-Mulliken-Roothaan MO theory uses ψMO = (u1a + u2a)(u1b + u2b) = (u1au2b + u1bu2a) + (u 1au1b + u2au2b) = ψVB + ψION an equal mixture of VB and ionic structures: in MO theory ψION is as important as ψVB. Ionic functions stand for the ionic configurations above. ψVB and ψMO are valuable molecular wave functions [1a]. Solutions with ψION instead of ψVB (H-L theory) can therefore not be neglected. The difference with VB and MO theory is that for ψION gauge symmetry and atom-antiatom switches have to be introduced. These were effects not yet considered in bonding (see [17] for an early suggestion). A generic theoretical extension of ionic Coulomb theory for diatomic molecules is that the principle of charge alternation will be important in the case of polyatomic molecules, which is as observed [17,18]. Finally, partitioning a 4-particle bond into a pair of charge conjugated but mass asymmetrical ions avoids pair annihilation (mass ratio 1836/1838 for H). Mass-differences of 2/1836 are so small that charge symmetry in ion-antiion pairs is not broken. g. Zero molecular parameter potentials. Using the 3 molecular parameters R e, De and k e to test the predictions with atom-antiatom XX PECs (13) and (24) can even be avoided. Instead of using e2/Re (Re is a molecular constant), we extract this asymptote just from atomic data by choosing half the well depth of a bond as a substitute. This is equal to IEX+IEY+De ≈ IEX+IEY since D e/(IEX+IEY) <<1. Now, the atomic ionization energy IE X acts as a molecular asymptote for bonds XX. For bonds XY, ½ (IE X+ IE Y) is a first approximation, conform to standard practices for determining atomic radii rX. This leads to ionic Coulomb asymptotes since C= e2/Re=e2/2rX=IEX. Errors generated by this procedure will not be too large (order 10 %). With asymptotes deriving from atomic data only, both (13) and (24) become zero molecular parameter functions, whereas it is generally accepted [4,13,19] that universal molecular functions must at least use the 3 molecular parameters above. Just for (25), D e would be needed, but only away from the minimum. Calculating molecular PECs from atomic data is areal challenge. The approach has ab initio status, as soon as it applies to 3 simple bonds H 2, Li2 and LiH. 3. Results and Discussion. PECs (RKR/IPA) for 9 bonds are analyzed: H 2 [20], LiH[21], KH[22], Li 2[23,24], KLi[25], NaCs [26], Rb 2[27], RbCs[28] and Cs 2[29]. Valence-state ionization energies are taken from NIST tables. (a) H2 (Fig. 3). The agreement is best for the generic Coulomb equation (24), although the Kratzer result (13) is still acceptable. Deviations are found at the repulsive side but the trend is as observed. For the attractive branch, the agreement is good even at long range (the important cold atom region [18] near De). This asymptote intersects generic curve (13) and (24). For this region, we used (25), which consumes one molecular parameter D e, but only away from the minimum (about 20 % of a total PEC [18]). This first principle's molecular PEC for H 2 derives from H H bonding using atomic data only. It is closer to the observed H 2 PEC than that calculated in H-L theory, the origin of quantum chemistry. In the context of particle/antiparticle theory, Dirac's frequently cited 1929-remark, quoted recently by Pople [1a], about solving chemical problems by quantum mechanics is of interest. Dirac, the inventor of the algebraic switch we used in (4a), assumed at that time that H-L theory for 4 particles (fermions) was correct and complete. Despite the controversy in quantum mechanics about the bound or unbound character of H H, we can not but say that the reaction between H and H will give a normal H 2 molecule14 in the H-L sense. (b) Li2 (Fig. 4). Results for the Cs 2-PEC reaching De are included. For both, agreement is better than for H 2 at both branches. For Li 2 absolute deviation for 30 turning points is 2,5% for Kratzer and only 0,6% for perturbed Coulomb potential [18], showing that PEC (24) is again better than Kratzer's (13). (c) Li2, LiH, KH, KLi, NaCs, Rb 2, RbCs and Cs 2 (Fig. 5). Here, observed level energies are plotted against theoretical ones obtained with zero molecular parameter functions. Differences are shown. Only the KH repulsive branch deviates. The slope is close to 1 and goodness of fit is high. Average deviation for 310 turning points is 9.42 %: Cs 2 11.3, RbCs 10.8, Rb 2 9.0, NaCs 12.2, KLi 9.3, KH 7.5 and LiH 6.3. These errors include long-range situations where applicable but calculated15 with V'(R)=0 in (25). As 14 Although H and H are isospectral in first order, small differences can show in second order. If H 2 consists of H H and HH, this information can be hidden in atomic spectra. Well-known fine structures in molecular spectra deriving from lepton and nucleon spin need not be discussed here.15 Part of the errors is due the use of atomic ionization energy as molecular asymptote and to the neglect of (6a).G. Van Hooydonk CB00-02 short, version 2: Gauge symmetry.... 05/04/00 8 of 13above Coulomb PECs are better than Kratzer's. Fig. 5 also illustrates the solution for problem (ii). We can easily scale PECs. Scale/shape invariance of 8 observed XX PECs is quantitatively accounted for with an X X-scheme [18], an unprecedented result. (d) LiH (Fig. 5). This simple bond is intermediate between H 2 and Li 2. It has the largest D e-value of the 8 bonds. Both slope and goodness of fit in Fig. 5 indicate that observed data for LiH perfectly fit in an LiH/LiH bond scheme. When the two variables k2 in (13) and generic k gen, deriving from (24) are plotted versus level energies [18], a linear fit for k gen again gives the best results. Generic ionic asymptote e2/Re is reproduced within 0,086 % of experiment. 40 calculated turning points are within 0,54 % of experiment, an average deviation of 0,012 Å. Larger deviations are found at the attractive branch. For the left branch deviation is only 0,0047 Å, very close to spectroscopic accuracy [18]. In general, left branches are easier to reproduce [3,18,30,31] than right branches where long-range forces interfere. For KLi, deviation is even less than 0,001 Å [18]. In review [18] other examples are given. Atom- antiatom theory applied to 500 level energies and turning points for 13 bonds has a CL of 98-99 [18]. Atom-antiatom bonding theory is an overlooked atom-atom bonding theory. This is illustrated by 3 test cases H 2, LiH and Li 2 or, should we say H H, LiLi and LiH. Permutational symmetry requires that X2 is not X X but consists of a hybrid of states XX and X X. The simplicity of all analytical results and the classical concept (the magnet metaphor) contrast with complex H-L theory. PECs obey closed form formulae, in agreement with empirical evidence, an unprecedented result also. H-L theory eventually prevented finding earlier that bonding is due to symmetry effects residing in charges within atoms. If an exchange mechanism can explain bonding classically, charge inversion in one atom is a realistic choice. Mirror symmetry, left-right asymmetry or chiral dipole behavior lead to anti- symmetry at large. This is proven by the fact that perturbed Coulomb function (24), not obeying the virial exactly, is even better than Kratzer's. An algebraic switch in (4a) may remove the bottleneck in quantum chemistry (see Introduction). Instead of attenuating Coulomb forces, switching their signs is a better option. In fact, gauge symmetry provides with a first principles recipe to split a single Coulomb law in two parts: a short and a long range one, the transition point being situated at R e. It is not necessary to invent a computational 'trick' to reduce computer time [1e, 1f]. Disregarding either the short or long range branch of a Coulomb law is perfectly admissible in a gauge symmetry based approach. The generic PECs have an attractive Coulomb long range part -e2/R that reaches to the minimum. Here, it isreplaced by a repulsive Coulomb part +e2/R. Whether or not the unconventional distribution of charges between 4 particles is a form of 'quantum chaos'11 is not clear. In any case, our approach nicely fits in the persistent revival of classical and semi- classical approximations for small systems described well by quantum mechanics (because of chaos, the correspondence principle or both [1c, 1d]). It is remarkable that even Coulomb's original law with strictly unit charges accounts well for PECs of various bonds. It is not even necessary to allow for chemical effects on charges (an otherwise legitimate parameterization procedure). This makes Coulomb's law the really universal parameter-free molecular potential, so badly needed for scaling in molecular spectroscopy [3,4,18]. 4. Further evidence. Atom-antiatom schemes are also consistent with a number of other observations. 1. Exceptions to orbital-symmetry based Woodward- Hoffman rules [32] in organic reactivity: conrotatory ring-closure of cis-butadiene is an example [17], 2. Dunitz-Seiler observations [33] about the absence of valence electron-density in the bonding region between nuclei, until today the only experimental indication that H-L schemes need revision [17, 33], 3. Occurrence of 5-fold symmetry in alloys [34] and the role of Euclid's golden number [35], indicating chaotic/fractal behavior in small quantum systems, 4. Absence of isotope effects in high-T C supercon- ductors [36] and charge inversion in CuO [37], 5. Meaning and role of Cooper-pairs in general and aromatic systems in particular [38]: attraction makes lepton appearances as charge conjugated pairs their natural morphology; it is no longer difficult to explain why Cooper pairs can exist; and finally 6. The hole-concept (see above). As outlined before [17], the interaction matrix for neutral particles X and X is | X X -------|------------------ X| XX XX X| XX XX This matrix is easily extended towards aggregates of neutral atoms X. If M is a neutral diatomic molecule X2, M may replace X in this matrix. If C is a (linear) aggregate of atoms X n, a neutral chain, C may replace X in this matrix and so on. A consequence of Coulomb's law and this matrix for many particle systems is that charge alternation is important. We refer to, but not only to, arrays of hydrogen bridges like OH...X responsible for the coupling or breaking of two linear DNA-chains [17].G. Van Hooydonk CB00-02 short, version 2: Gauge symmetry.... 05/04/00 9 of 135. Conclusion. Unitary symmetry, one algebraic switch in the 10 term Hamiltonian generically accounts for splitting and for scale/shape invariant PECs. Atom-antiatom bonding explains classically and quantitatively atom- atom bonding. Theory is in agreement with universal scaling [4,18], a problem in molecular spectroscopy [3,4,18,19]. The classical intuitive 200-year-old ideas of Davy and Berzelius about (ionic) bonding are still valid today if gauge symmetry and charge inversion is considered. Coulomb's law and gauge symmetry justify the 'trick' invented to remove the bottleneck in quantum chemistry [1e, 1f]. The capacity of a Coulomb asymptote to scale spectroscopic constants is impressive [4] and shows that X X bonding schemes have universal character, deriving from first principles. Coulomb's law with unit charges is a truly universal molecular function. Gauge symmetry for 1/R-laws provides with a simple recipe for a transition from fermion (splitting) to boson (oscillator) behavior. Hydrogen-antihyd rogen reactions, feasible in the near future [39], will probably produce normal H 2. This reaction is crucial for 20th century physics on discrete symmetries. X X systems are bound and stable. Wave mechanics is still controversial on H H stability, which remains difficult to understand if the H2-PEC can be calculated with great accuracy [40]. No-extremum theories for H H [41] are probably incorrect [35] because of gauge symmetry. Atomic symmetries like handedness, chirality or left-right asymmetry must be reconsidered at atomic and at molecular energy levels to look for higher order effects of gauge symmetry and charge inversion, not discussed8,14. Many experiments are under way in atomic physics because of implicationsfor the Standard Model (and beyond) and in molecular physics the study of cold atoms is important (references are given in [18]). But even cosmology, baryogenesis [42] and super-unification are at stake if antimatter is really present in nature [17,18,35]. The perfect (algebraic) equilibrium in the X X pair production process in chemistry gives exactly equal amounts of matter and antimatter, if atoms stand for matter and antiatoms for antimatter. We have now found that molecular singlet and triplet PECs are witnesses for the presence of antiatoms in bonds (matter). A dipole- symmetry conjugated charge-neutral and mass- positive particle pair (atom/antiatom) does not (have to) annihilate, although its 6 Coulomb terms refer to pairs of charged particles and antiparticles fermions in Dirac sense. Since a Coulomb function is even better than Kratzer's, this conclusion is unavoidable. The gap e2/Re covered by chemical interactions is ½α2mc2, where α is the fine structure constant. This is ½(1/18,800)(1/1,836) or 0.2 ppm of the gap m Hc2 covered by H-annihilation. Chemistry may be just a fine structure effect8,9 but the underlying mechanism is universal and applicable to larger or smaller gaps if the scheme is scale invariant [18]. Extensions to other fields are given elsewhere [18]. Like all living material, man is also made up from composite neutral particle pairs. 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arXiv:physics/0003006v1 [physics.data-an] 2 Mar 2000On mixing times for stratified walks on the d-cube Nancy Lopes Garcia Universidade Estadual de Campinas Campinas, Brasil and Jos´ e Luis Palacios Universidad Sim´ on Bol´ ıvar Caracas, Venezuela. February 2, 2008 Abstract Using the electric and coupling approaches, we derive a seri es of results concerning the mixing times for the stratified random walk on the d-cube, inspired in the results of Chung and Graham (1997) Stratified random walks on the n-cube. Random Structures and Algorithms , 11,199-222. Key Words: effective resistance, coupling, birth and death c hains 1991 Mathematics Subject Classification. Primary: 60J15; secondary: 60C05. 11 Introduction. The stratified random walk (SRW) on the d-cube Qdis the Markov chain whose state space is the set of vertices of the d-cube and whose transition probabilities are defined thus: Given a set of non-zero probabilities p= (p0,p1,... ,p d−1), from any vertex with k1’s, the process moves either to any neighboring vertex with k+ 1 1’s with probabilitypk d; or to any neighboring vertex with k−1 1’s with probabilitypk−1 d; or to itself with the remaining probability. The simple random walk on the d-cube corresponds to the choice pk= 1 for all k. Vaguely speaking, the mixing time of a Markov chain is the tim e it takes the chain to have its distribution close to the stationary distribution unde r some measure of closeness. Chung and Graham studied the SRW on the d-cube in [5], mainly with algebraic methods, and found bounds for the mixing times under total variation and relati ve pointwise distances. Here we use non-algebraic methods, the electric and coupling appro aches, in order to study the same SRW and get exact results for maximal commute times and bound s for cover times and mixing times under total variation distance. We take advantage of t he fact that there seems to be some inequality or another linking hitting times, commute times , cover times and any definition of mixing time with any other under any measure of closeness (se e Aldous and Fill [2] and Lov´ asz and Winkler [8]). 2 The electric approach On a connected undirected graph G= (V,E) such that the edge between vertices iandjis given a resistance rij(or equivalently, a conductance Cij= 1/rij), we can define the random walk on Gas the Markov chain X={X(n)}n≥0that from its current vertex vjumps to the neighboring vertex wwith probability pvw=Cvw/C(v), where C(v) =/summationtext w:w∼vCvw, and w∼v means that wis a neighbor of v. There may be a conductance Czzfrom a vertex zto itself, giving rise to a transition probability from zto itself. Some notation: EaTbandEaCdenote the expected value, starting from the vertex a, of respectively, the hitting time Tbof the vertex 2band the cover time C, i. e., the number of jumps needed to visit all the states in V;Rabis the effective resistance, as computed by means of Ohm’s law, betw een vertices aandb. A Markov chain is reversible if πiP(i,j) =πjP(j,i) for all i,j, where {π.}is the stationary distribution and P(·,·) are the transition probabilities. Such a reversible Marko v chain can be described as a random walk on a graph if we define conductances thus: Cij=πiP(i,j). (2.1) We will be interested in finding a closed form expression for t he commute time E0Td+EdT0 between the origin, denoted by 0, and its opposite vertex, de noted by d. Notice first that the transition matrix for X={X(n),n≥0}, the SRW on the d-cube, is doubly stochastic and therefore its stationary distribu tion is uniform. If we now collapse all vertices in the cube with the same number of 1’s into a sing le vertex, and we look at the SRW on this collapsed graph, we obtain a new reversible Markov chain S={S(n),n≥0}, a birth-and-death chain in fact, on the state space {0,1,... d}, with transition probabilities P(k,k+ 1) =d−k dpk, (2.2) P(k,k−1) =k dpk−1, (2.3) P(k,k) = 1 −P(k,k+ 1)−P(k,k−1). (2.4) It is plain to see that the stationary distribution of this ne w chain is the Binomial with parameters dand1 2. It is also clear that the commute time between vertices 0 and dis the same for both XandS. For the latter we use the electric machinery described abov e, namely, we think of a linear electric circuit from 0 to dwith conductances given by (2.1) for 0 ≤i≤d, j=i−1,i,i+ 1, and where πi=/parenleftbiggd i/parenrightbigg1 2d. It is well known (at least since Chandra et al. proved it in [4] ) that EaTb+EaTb=Rab/summationdisplay zC(z), (2.5) where Rabis the effective resistance between vertices aandb. 3If this formula is applied to a reversible chain whose conduc tances are given as in (2.1), then it is clear that C(z) =πz and therefore the summation in (2.5) equals 1. We get then thi s compact formula for the commute time: EaTb+EaTb=Rab, (2.6) where the effective resistance is computed with the individu al resistors having resistances rij=1 Cij=1 πiP(i,j). In our particular case of the collapsed chain, because it is a linear circuit, the effective resistance R0dequals the sum of all the individual resistances ri,i+1, so that (2.6) yields E0Td+EdT0=R0d= 2dd−1/summationdisplay k=01 pk/parenleftbigd−1 k/parenrightbig. (2.7) Because of the particular nature of the chain under consider ation, it is clear that E0Td+EdT0 equals the maximal commute time ( τ∗in the terminology of Aldous [2]) between any two vertices. (i) For simple random walk, formula (2.7) is simplified by tak ing all pk= 1. This particular formula was obtained in [8] with a more direct argument, and i t was argued there that d−1/summationdisplay k=01/parenleftbigd−1 k/parenrightbig= 2 + o(1). An application of Matthews’ result (see [9]), linking maxim um and minimum expected hitting times with expected cover times, yields immediately that th e expected cover time is EvC= Θ(|V|log|V|),which is the asymptotic value of the lower bound for cover tim es of walks on a graph G= (V,E) (see [6]). Thus we could say this SRW is a “rapidly covered” w alk. (ii) The so-called Aldous cube (see [5]) corresponds to the c hoice pk=k d−1. This walk takes place in the “punctured cube” that excludes the origin. Form ula (2.7) thus, must exclude k= 0 in this case, for which we still get a closed-form expression for the commute time between vertex 4d, all of whose coordinates are 1, and vertex swhich consists of the collapse of all vertices with a single 1: EsTd+EdTs= 2dd−1/summationdisplay k=11/parenleftbigd−2 k−1/parenrightbig. (2.8) The same argument used in (i) tells us that the summation in (2 .8) equals 2 + o(1) and, once again, Matthews’ result tells us that the walk on the Aldous c ube has a cover time of order |V|log|V|. (iii) The choice pk=1 (d−1 k)would be in the terminology of [5] a “slow walk”: the commute time is seen to be exactly equal to |V|log2|V|and thus the expected cover time is O(|V|log2|V|). In general, the SRW will be rapidly covered if and only if d−1/summationdisplay k=01 pk/parenleftbigd−1 k/parenrightbig=c+o(1), for some constant c. Remark. A formula as compact as (2.7) could be easily obtaine d through the commute time formula (2.6). It does not seem that it could be obtained that easily, by just adding the individual hitting times EiTi+1. (A procedure that is done, for instance, in [5], [10], [11], and in the next section). 3 The coupling approach In order to asess the rate of convergence of the SRW on the cube Qdto the uniform stationary distribution π, we will bound the mixing time τdefined as τ= min {t:d(t′)≤1 2e,for all t′> t}, where d(t) = max x∈Qd/bardblPx(X(t) =·)−π(·)/bardbl, and/bardblθ1−θ2/bardblis the variation distance between probability distributio nsθ1andθ2, one of whose alternative definitions is (see Aldous and Fill [2]), chapte r 2): /bardblθ1−θ2/bardbl= min P(V1/ne}ationslash=V2), 5where the minimum is taken over random pairs ( V1,V2) such that Vmhas distribution θm,m= 1,2. The bound for the mixing time is achieved using a coupling arg ument that goes as follows: let{X(t),t≥0}and{Y(t),t≥0}be two versions of the SRW on Qdsuch that X(0) =xand Y(0)∼π. Then /bardblPx(X(t) =·)−π(·)/bardbl ≤P(X(t)/ne}ationslash=Y(t)). (3.1) A coupling between the processes XandYis a bivariate process such that its marginals have the distributions of the original processes and such th at once the bivariate process enters the diagonal, it stays there forever. If we denote by Tx= inf{t;X(t) =Y(t)} the coupling time, i. e., the hitting time of the diagonal, th en (3.1) translates as /bardblPx(X(t) =·)−π(·)/bardbl ≤P(Tx> t), (3.2) and therefore, d(t)≤max x∈QdP(Tx> t). (3.3) If we can find a coupling such that ETx=O(f(d)), for all x∈Qdand for a certain function f of the dimension d, then we will also have τ=O(f(d)). Indeed, if we take t= 2ef(d), then (3.3) and Markov’s inequality imply that d(t)≤1/2eand the definition of τimplies τ=O(f(d)). We will split TxasTx=T1 x+T2 x, where T1 xis a coupling time for the birth-and-death process S, andT2 xis another coupling time for the whole process X, once the bivariate Sprocess enters the diagonal, and we will bound the values of ET1 xandET2 x. More formally, for any x,y∈Qddefine s(x) =d/summationdisplay i=1xi (3.4) d(x,y) =d/summationdisplay i=1|xi−yi|. (3.5) 6Define also for the birth-and-death process S(t) =s(X(t)) its own mixing time: τ(S)= inf{t;dS(t)≤1 2e}, where dS(t) = max i/bardblPi(S(t) =·)−πS(·)/bardbl, and πSis the stationary distribution of S. Notice thats(Y(0))∼πSsinceY(0)∼π. Now we will prove that τ(S)=O(fS(d)),for a certain function fSof the expected hitting times of the “central states”, and that this bound implies an analogous bound for ET1 x. Indeed, as shown by Aldous [1], we can bound τ(S)by a more convenient stopping time τ(S)≤K2τ(3) 1 (3.6) where τ(3) 1= min µmax iminUiEiUiand the innermost minimum is taken over stopping times Uisuch that Pi(S(Ui)∈ ·) =µ(·). In particular, τ(3) 1≤min bmax imin Ub iEiUb i (3.7) ≤min bmax iEiTb (3.8) = max( E0Td/2,EdTd/2) (3.9) where the innermost minimum in (3.7) is taken over stopping t imesUb isuch that Pi(S(Ui) = b) = 1. Expression (3.9) follows from (3.8) since we are dealin g with birth and death chains. Therefore, combining (3.6) and (3.9) we have τ(S)≤K2max(E0Td/2,EdTd/2) :=fS(d). (3.10) In general, a coupling implies an inequality like (3.2). How ever, the inequality becomes an equality for a certain maximal (non-Markovian) coupling, d escribed by Griffeath [7]. Let T1 xbe the coupling time for the maximal coupling between s(X(t)) and s(Y(t)) such that /bardblPx(S(X(t)) =·)−πS(·)/bardbl=P(T1 x> t). Then dS(t) = max x∈Qd/bardblPx(S(X(t)) =·)−πS(·)/bardbl= max x∈QdP(T1 x> t). 7By the definition of τ(S)it is clear that P(T1 x> τ(S))≤1 2e. Moreover, by the “submultiplica- tivity” property (see Aldous and Fill [2], chapter 2) d(s+t)≤2d(s)d(t), s,t ≥0, (3.11) we have that P(T1 x> kτ(S))≤1 2ek, k≥1. (3.12) Thus ET1 x=∞/summationdisplay k=1E(T1 x1((k−1)τ(S)< T1 x≤kτ(S))) ≤τ(S)∞/summationdisplay k=1kP((k−1)τ(S)< T1 x≤kτ(S)) ≤τ(S)∞/summationdisplay k=1kP((k−1)τ(S)< T1 x) ≤τ(S)/parenleftBigg 1 +∞/summationdisplay k=2k1 2ek−1/parenrightBigg . Since the series in the right hand side converges we have ET1 x=O(fS(d)). Once the bivariate Sprocess hits the diagonal D={(x,y)∈Qd×Qd;d/summationdisplay i=1xi=d/summationdisplay i=1yi}, (3.13) we devise one obvious coupling that forces the bivariate Xprocess to stay in Dand such that the distance defined in (3.5) between the marginal processes doe s not decrease. In words: we select one coordinate at random; if the marginal processes coincid e in that coordinate, we allow them to evolve together; otherwise we select another coordinate in order to force two new coincidences. Formally, for each ( X(t),Y(t))∈D, letI1,I2andI3be the partition of {0,1,... ,d }such that I1={i;Xi(t) =Yi(t)} I2={i;Xi(t) = 0,Yi(t) = 1} I3={i;Xi(t) = 1,Yi(t) = 0} 8Given ( X(t),Y(t))∈D, choose iu.a.r. from {0,1,... ,d }. (a) If i∈I1; 1. IfXi(t) = 1 then make Xi(t+1) = Yi(t+1) = 0 with probability ps(X(t))−1; otherwise Xi(t+ 1) = Yi(t+ 1) = 1. 2. IfXi(t) = 0 then make Xi(t+ 1) = Yi(t+ 1) = 1 with probability ps(X(t)); otherwise Xi(t+ 1) = Yi(t+ 1) = 0. (b) If i∈I2; 1. Select j∈I3u.a.r.; 2. Make Xi(t+ 1) = Yj(t+ 1) = 1 with probability ps(X(t)); otherwise Xi(t+ 1) = Yj(t+ 1) = 0. (c) If i∈I3; 1. Select j∈I2u.a.r.; 2. Make Xi(t+ 1) = Yj(t+ 1) = 0 with probability ps(X(t))−1; otherwise Xi(t+ 1) = Yj(t+ 1) = 1. Then, it is easy to check that ( X(t+ 1),Y(t+ 1)) ∈Dandd(X(t+ 1),Y(t+ 1)) ≤ d(X(t),Y(t)). Moreover, noticing that |I2|=|I3|=d(X(t),Y(t))/2, we have P(d(X(t+ 1),Y(t+ 1)) = d(X(t),Y(t))−2|X(t),Y(t)) =d(X(t),Y(t)) 2d(ps(X(t))+ps(X(t))−1) (3.14) P(d(X(t+1),Y(t+1)) = d(X(t),Y(t))|X(t),Y(t)) = 1 −d(X(t),Y(t)) 2d(ps(X(t))+ps(X(t))−1). (3.15) In this case, it is straightforward to compute m(i,s(X(t))) = i−E[d(X(t+ 1),Y(t+ 1)) |d(X(t),Y(t)) =i,X(t),Y(t)] =i d(ps(X(t))+ps(X(t))−1). (3.16) LetT2 xbe the coupling time for the second coupling just described. That is, let T2 x= inf{t > T1 x:d(X(t),Y(t)) = 0 }. Then, as a consequence of the optional sampling theorem for martingales we have the following comparison lemma ( cf.Aldous and Fill [2], Chapter 2). 9Lemma 3.17 E[T2 x|d(X(T1 x),Y(T1 x)) =L,(X(T1 x),Y(T1 x))∈D,s(X(T1 x)) =s]≤L/summationdisplay i=1d i(ps+ps−1)(3.18) for all s= 0,1,... ,d . Proof. Define ( X′(t),Y′(t)) = (X(t+T1 x),Y(t+T1 x) for all t≥0. Define Z(t) =d(X′(t),Y′(t)) andFt=σ(X′(t),Y′(t)). Then, it follows from (3.16) that m(i,s) =i−E[Z(1)|Z(0) = i,s(X′(0)) = s]. (3.19) Also, for all s∈ {1,... ,d −1}, 0< m(1,s)≤m(2,s)≤...≤m(d,s). Fix s∈ {1,... ,d −1} and write h(i) =i/summationdisplay j=11 m(i,s)(3.20) and extend hby linear interpolation for all real 0 ≤x≤d. Then his concave and for all i≥1 E[h(Z(1))|Z(0) = i,s(X′(0)) = s]≤h(i−m(i,s)) ≤h(i)−m(i,s)h′(i) =h(i)−1, where the first inequality follows from the concavity of handh′is the first derivative of h. Now, defining ˜hsuch that h(i) = 1 +/summationdisplay jP[h(Z(1))|Z(0) = i,s(X′(0)) = s]h(j) +˜h(i) (3.21) and M(t) =t+h(Z(t)) +t−1/summationdisplay u=0˜h(Z(u)) (3.22) we have that Mis anFt-martingale and applying the optional sampling theorem to t he stopping timeT0= inf{t;Z(t) = 0}we have E[M(T0)|Z(0) = i,s(X′(0)) = s] =E[M(0)|Z(0) = i,s(X′(0)) = s] =h(i). (3.23) 10Noticing that M(T0)≥T0andT0=T2 x, we obtain the desired result • Since s(X(t)) is distributed as πS, we can write: E[T2 x|d(X(T1 x),Y(T1 x)) =L,(X(T1 x),Y(T1 x))∈D]≤d/summationdisplay s=0πS(s)L/summationdisplay i=1d i(ps+ps−1):=g(d). (3.24) Putting the pieces together, we have found a coupling time Txfor the whole process such that ETx≤fS(d) +g(d). The task now is to find explicit bounds for fS(d) and g(d) for particular workable cases. To avoid unnecessary complications, we will assume d= 2m, and compute only the hitting times for the Sprocess of the type E0Tm. Hitting times in birth-and-death processes assume the following closed-form (see [11] for an electrical deriv ation): EkTk+1=1 πkP(k,k+ 1)k/summationdisplay i=0πi,0≤k≤d−1, and in our case this expression turns into EkTk+1=/summationtextk i=0/parenleftbigd i/parenrightbig /parenleftbigd−1 k/parenrightbig pk. Therefore E0Tm=m−1/summationdisplay k=0/summationtextk i=0/parenleftbig2m i/parenrightbig /parenleftbig2m−1 k/parenrightbig pk. (3.25) (i) In case all pk= 1, we have the simple random walk on the cube, and it turns out there is an even more compact expression of (3.25), namely: E0Tm=m−1/summationdisplay k=0/summationtextk i=0/parenleftbig2m i/parenrightbig /parenleftbig2m−1 k/parenrightbig=mm−1/summationdisplay k=01 2k+ 1, (3.26) as was proved in [3], and the right hand side of (3.26) equals m[H(2m)−1 2H(m)], where H(n) = 1+1 2+· · ·+1 n, allowing us to conclude immediately that in this case E0Tm=E0Td/2≈ d 4logd+d 4log 2. 11Also, we have E[T2 x|d(X(T1 x),Y(T1 x)) =L,(X(T1 x),Y(T1 x))∈D]≤d 2pL/summationdisplay i=11 i≈d 2pO(logL). (3.27) Thus in this case both fS(d) and g(d), and a fortiori ETxandτ, areO(dlogd). (ii) For the Aldous cube, pk=k d−1, and (3.25) becomes (recall this cube excludes the origin): E1Tm=m−1/summationdisplay k=1/summationtextk i=0/parenleftbig2m i/parenrightbig /parenleftbig2m−2 k−1/parenrightbig=m−2/summationdisplay k=0/summationtextk i=0/parenleftbig2m i/parenrightbig /parenleftbig2m−2 k/parenrightbig+m−2/summationdisplay k=0/parenleftbig2m k+1/parenrightbig /parenleftbig2m−2 k/parenrightbig. (3.28) After some algebra, it can be shown that the second summand in (3.28) equals (2 m− 1)[H(2m−1)−1 m], and the first summand can be bounded by twice the expression in (3.26), on account of the fact that/parenleftbigg2m−1 k/parenrightbigg ≤2/parenleftbigg2m−2 k/parenrightbigg ,for 0 ≤k≤m−1. Therefore, we can write E1Td/2≤3 2dlogd+ smaller terms , thus improving by a factor of1 2the computation of the same hitting time in [5]. Also, we have E[T2 x|d(X(T1 x),Y(T1 x)) =L,(X(T1 x),Y(T1 x))∈D,s(X(T1 x)) =s]≤L/summationdisplay i=1d(d−1) i(2s−1)(3.29) Thus, in this case E[T2 x|d(X(T1 x),Y(T1 x)) =d,(X(T1 x),Y(T1 x))∈D] ≤d/summationdisplay s=1πS(s)d/summationdisplay i=1d(d−1) i(2s−1) ≤Φ(−√ d/3)d/summationdisplay i=1d(d−1) i+ (1−Φ(−√ d/3))d/summationdisplay i=1d(d−1) i(2d/3−1) ≈e−d/9d(d−1)log d+ (1−e−d/9)dlogd. (3.30) And so τ=O(dlogd) also in this case. 12(iii) Slower walks. Consider the case when the probability pkgrows exponentially in k, more specifically pk=/parenleftBigk+ 1 n+ 1/parenrightBigα (3.31) withα >1. In this case, it seems that (3.25) is useless to get a closed expression for E0Td/2. However, Graham and Chung [5] provide the following bound EiTd/2≤c0(α)dα,for all d≥d0(α),0≤i≤d (3.32) where c0(α) and d0(α) are constants depending only on α. Moreover, (3.24) becomes g(d) =d/summationdisplay s=0πS(s)d/summationdisplay i=1d(d+ 1)α i((s+ 1)α+sα) =d(d+ 1)αd/summationdisplay s=0πS(s) (s+ 1)α+sαd/summationdisplay i=11 i ≈d(d+ 1)αlogdd/summationdisplay s=0πS(s) (s+ 1)α+sα ≤d(d+ 1)αlogdd/summationdisplay s=0πS(s) (s+ 1)α =d(d+ 1)αlogdE/bracketleftBig1 (1 +X)α/bracketrightBig , (3.33) where Xis a Binomial ( d,1 2) random variable. Jensen’s inequality and the same argumen t that lead to (3.30) show that E[(1 +X)−α]∼O(d−α) and (3.33) can be bound by O(dlogd). This fact together with (3.32) allows us to conclude that τ=O(dα) in this case, thus improving on the rate of the mixing time provided in [5] by a factor of log d. Acknowledgments. This paper was initiated when both authors were visiting sch olars at the Statistics Department of the University of California at Be rkeley. The authors wish to thank their hosts, especially Dr. David Aldous with whom they shar ed many fruitful discussions. This work was partially supported by FAPESP 99/00260-3. References 13[1] Aldous, D.J. (1982) Some inqualities for reversible Mar kov chains, Journal of the London Mathematical Society ,2, 25:564–576. [2] Aldous, D. J. and Fill, J. (2000) Reversible Markov chain s and random walks on graphs, book draft. [3] Blom, G. (1989) Mean transition times for the Ehrenfest u rn,Advances in Applied Proba- bility,21, 479-480. [4] Chandra, A. K., P. Raghavan, W. L. Ruzzo, R. Smolensky and P. Tiwari (1989), The electrical resistance of a graph captures its commute and co ver times, in Proceedings of the Twenty First Annual ACM Symposium on Theory of Computing , Seattle, Washington, pp. 574-586. [5] Chung, F. R. K. and Graham, R. L. (1997) Stratified random w alks on the n-cube, Random Structures and Algorithms ,11, 199-222. [6] Feige, U. (1995)) A tight lower bound on the cover time for random walks on graphs, Random Structures and Algorithms ,6, 433-438. [7] Griffeath, D. (1975) A maximal coupling for Markov chains ,Z. Wahrscheinlichkeitstheorie verw. Gebiete ,31, 95-106. [8] Lov´ asz, L. and Winkler, P. (1998) Mixing times, in DIMACS Series in Discrete Mathemat- ics and Theoretical Computer Science ,41, American Mathematical Society. [9] Matthews, P. C. (1988) Covering problems for Markov chai ns,Annals of Probability ,16, 1215-1228. [10] Palacios, J. L. (1994) Another look at the Ehrenfest urn via electric networks, Advances in Applied Probability ,26, 820-824. [11] Palacios, J. L. and Tetali, P. (1996) A note on expected h itting times for birth and death chains, Statistics & Probability Letters ,30, 119-125. 14

arXiv:physics/0003007v1 [physics.chem-ph] 2 Mar 2000Performance of Discrete Heat Engines and Heat Pumps in Finit e Time. Tova Feldmann and Ronnie Kosloff Department of Physical Chemistry the Hebrew University, Je rusalem 91904, Israel Abstract The performance in finite time of a discrete heat engine with i nternal friction is analyzed. The working fluid of the engine is composed of an e nsemble of noninteracting two level systems. External work is appli ed by changing the external field and thus the internal energy levels. The fr iction induces a minimal cycle time. The power output of the engine is optimi zed with respect to time allocation between the contact time with the hot and cold baths as well as the adiabats. The engine’s performance is al so optimized with respect to the external fields. By reversing the cycle of operation a heat pump is constructed. The performance of the engine as a h eat pump is also optimized. By varying the time allocation between the a diabats and the contact time with the reservoir a universal behavior can be i dentified. The optimal performance of the engine when the cold bath is appro aching absolute zero is studied. It is found that the optimal cooling rate con verges linearly to zero when the temperature approaches absolute zero. I. INTRODUCTION Analysis of heat engines has been a major source of thermodyn amic insight. The second law of thermodynamics resulted from Carnot’s study of the re versible heat engine [1]. Study of the endo-reversible Newtonian engine [2] began the field o f finite time thermodynamics 1[3–6]. Analysis of a virtual heat engine by Szilard led to the connection between thermo- dynamics and information theory [7,8]. Recently this conne ction has been extended to the regime of quantum computation [9]. Quantum models of heat engines show a remarkable similarity to engines obeying macro- scopic dynamics. The Carnot efficiency is a well established l imit for the efficiency of lasers as well as other quantum engines [10–14]. Moreover, even the irreversible operation of quan- tum engines with finite power output has many similarities to macroscopic endo-reversible engines [15–19]. It is this line of thought that serves as a motivation for a det ailed analysis of a discrete four stroke quantum engine. In a previous study [20], the sam e model served to find the limits of the finite time performance of such an engine but wit h the emphasis on power optimization. In that study the working medium was composed of discrete level systems with the dynamics governed by a master equation. The purpose was to gain insight into the optimal engine’s performance with respect to time alloc ation when external parameters such as: the applied fields, the bath temperatures and the rel axation rates were fixed. The present analysis emphasizes the reverse operation of th e heat engine as a heat pump. For an adequate description of this mode of operation inner f riction has to be a consideration. Without it the model is deficient with respect to optimizing t he cooling power. Another addition is the optimization of the external fields. This is a common practice when cold temperatures are approached. With the addition of these two attributes, the four stroke quantum model is analyzed both as a heat engine and as a refrig erator. Inner friction is found to have a profound influence on perfor mance of the refrigerator. A direct consequence of the friction is a lower bound on the cy cle time. This lower bound excludes the non-realistic global optimization solutions found for frictionless cases [20] where the cooling power can be optimized beyond bounds. This obser vation, has led to the sug- gestion of replacing the optimization of the cooling power b y the optimization of the cooling efficiency per unit time [21–24]. Including friction is there fore essential for more realistic models of heat engines and refrigerators with the natural op timization goal becomes either 2the power output or the cooling power. The source of friction is not considered explicitly in the present model. Physically friction is the result of no n-adiabatic phenomena which are the result of the rapid change in the energy level structu re of the system. For example friction can be caused by the missalignement of the external fields with the internal polariza- tion of the working medium. For a more explicit description o f the friction the interactions between the individual particles composing the working flui d have to be considered. The present model is a microscopic analogue of the Ericsson refr igeration cycle [25] where the working fluid consists of magnetic salts. The advantage of th e microscopic model is that the use of the phenomenological heat transfer laws can be avo ided [16]. The results of the present model are compared to a recent analysis of macroscop ic chillers [27]. In that study, a universal modeling was demonstrated. It is found that the d iscrete quantum version of heat pumps has behavior similar to that of macroscopic chill ers. There is a growing interest in the topic of cooling atoms and m olecules to temperatures very close to absolute zero [28]. Most of the analysis of the c ooling schemes employed are based on quantum dynamical models. New insight can be gained by employing a thermo- dynamic perspective. In particular the temperatures achie ved are so low that the third law of thermodynamics has to be considered. The discrete lev el heat pump can serve as a model to study the third law limitations. The finite time per spective of the third law is a statement on the asymptotic rate of cooling as the absolute temperature is approached. These restrictions are imposed on the optimal cooling rate. The behavior of the optimal cooling rate as the absolute temperature is approached is a t hird law upper bound on the cooling rate. The main finding of this paper is that the optima l cooling rate converges to zero linearly with temperature, and the entropy production reaches a constant when the cold bath temperature approaches absolute zero. 3II. BASIC ASSUMPTIONS AND FORMAL BACKGROUND FOR THE HEAT ENGINE AND THE HEAT PUMP Heat engines and heat pumps are characterized by three attri butes: the working medium, the cycle of operation, and the dynamics which govern the cyc le. Heat baths by definition are large enough so that their temperatures is constant duri ng the cycle of operation. The heat engine and the heat pump are constructed from the same co mponents and differ only by their cycle of operation. A. The Working Medium The working medium consists of an ideal ensemble of many non- interacting discrete level systems. Specifically, the analysis is carried out on two-le vel systems (TLS) but an ensemble of harmonic oscillators [20] would lead to equivalent resul ts. The TLS systems are envisioned as spin-1/2 systems. The lack of spin-spin interactions enables the description of the energy exchange between the w orking medium and the sur- roundings in terms of a single TLS. The state of the system is t hen defined by the average occupation probabilities P+andP−corresponding to the energies1 2ωand−1 2ω, where ωis the energy gap between the two levels. The average energy per spin is given by E=P+·/parenleftbigg1 2ω/parenrightbigg +P−·/parenleftbigg −1 2ω/parenrightbigg (2.1) The polarization, S, is defined by S=1 2(P+−P−), (2.2) and thus the energy can be written as E=ωS. Energy change of the working medium can occur either by population transfer from one level to the oth er (changing S) or by changing the energy gap between the two levels (changing ω). Hence dE=Sdω+ωdS . (2.3) 4Population transfer is the microscopic realization of heat exchange. The energy change due to external field variation is associated with work. Eq. (2.3 ) is therefore the first law of thermodynamics: DW ≡ Sdω ;DQ ≡ ωdS . (2.4) Finally, for TLS the internal temperature, T′, is always defined via the relation S=−1 2tanh/parenleftbiggω 2kBT′/parenrightbigg . (2.5) Note that the polarization Sis negative as long as the temperature is positive. B. The Cycle of Operation 1. Heat engines cycle The cycle of operation is analyzed in terms of the polarizati on and frequency ( S, ω). A schematic display is shown in Fig.(1) for a constant total cy cle time, τ. The present engine is an irreversible four stroke engine [20] resembling the St irling cycle, with the addition of internal friction. The direction of motion along the cycle i s chosen such that net positive work is produced. 5FIGURES /patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10 /patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11/patchar11 Sω ABC C’ Dωb ωa S3 S1 S2 S4 TcTh FE τbτhτaτa τcmin/patchar31/patchar31/patchar31/patchar31/patchar31/patchar31/patchar31/patchar31/patchar31τc .. SceqSheq C1 D1 FIG. 1. The heat engine with friction in the ω,Splane. This the hot bath temperature. τhis the time allocation when in contact with the hot bath. Tcandτcrepresent the temperature and time allocation for the cold bath. τarepresents the time allocation for compression (field chang e fromωbtoωa) and τbfor expansion. The area A,B, C1, D1is the positive work done by the system, while the areas C, C1, S1, S3, and D1, D, S 4, S2represent the negative work done by the system. The four branches of the engine will be now briefly described. On the first branch, A→B, the working medium is coupled to the hot bath of temper- ature Thfor period τh, while the energy gap is kept fixed at the value ωb. The conditions are such that the internal temperature of the medium is lower thanTh. In this branch, the polarization is changing from the initial polarization S2to the polarization S1. The inequality to be fulfilled is therefore: S1<−1 2tanh/parenleftbiggωb 2kBTh/parenrightbigg . (2.6) Since ωis kept fixed, no work is done and the only energy transfer is th e heat ωb(S1−S2) absorbed by the working medium. In the second branch, B→Cthe working medium is decoupled from the hot bath for a period τa, and the energy gap is varied linearly in time, from ωbtoωa. In this branch work is done to overcome the inner friction which develops heat, cau sing the polarization to increase fromS1toS3(Cf. Fig. 1). The change of the internal temperature is the re sult of two opposite contributes. First lowering the energy gap leads t o a lower inner temperature for 6constant polarization S. Second increase in polarization due to friction, leads to a n increase of the inner temperature for fixed ω. The inner temperature T′at point C might therefore be lower or higher than the initial temperature at point B. The third branch C→D, is similar to the first. The working medium is now coupled to a cold bath at temperature Tcfor time τc. The polarization changes on this branch from S3 to the polarization S4. For the cycle to close, S4should be lower than S2. At the end of the cycle the internal temperature of the working medium should be higher than the cold bath temperature, T′> Tc, leading to: S4>−1 2tanh/parenleftbiggωa 2kBTc/parenrightbigg . (2.7) Since S4< S1(Fig. 1), it follows from Eq. ( 2.6) and Eq.( 2.7), that: /parenleftbiggωa Tc/parenrightbigg >/parenleftbiggωb Th/parenrightbigg (2.8) Inequality (2.8) is equivalent to the Carnot efficiency bound , from Eq. (2.8) one gets: 1−/parenleftbiggωa ωb/parenrightbigg <1−/parenleftbiggTc Th/parenrightbigg =ηCarnot (2.9) The present model is a quantum analogue of the Stirling engin e which also has Carnot’s efficiency as an upper bound. The polarization Schanges uni-directionally along the ’adiabats’ due to the i ncrease of the excited level population as a result of the heat develope d in the working fluid when work is done against friction, irrespective of the direction of t he field change. The fourth branch D→A, closes the cycle and is similar to the second. The working medium is decoupled from the cold bath. In a period τbthe energy gap is changing back to its original value, ωb. The polarization increases from S4to the original value S2. 7TABLES TABLE I. Work and heat exchange along the branches of the heat engine with friction branch work+[work against friction] heat A→B 0 ωb(S1−S2) B→C(ωa−ωb)(S1+σ2/(2τa)) + [ σ2(ωa+ωb)/(2τa) ] 0 C→D 0 ωa((S2−S1)−σ2(1/τa+ 1/τb)) D→A(ωb−ωa)(S2−σ2/(2τb)) + [ σ2(ωa+ωb)/(2τb) ] 0 2. Refrigerator cycle The purpose of a heat pump is to remove heat from the cold reser voir by employing external work. The cycles of operation in the ( S, ω) plane is schematically shown in Fig. 2, /patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10/patchar10 /patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6 Sω Cωb ωa S2eq S2 S1 TcThD Eτb τc τa. S1eq .AF B τh 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/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6/patchar6 Sω Cωb ωa S2 S1 TcThD Eτa τc τb. .AB τhS3 S2eq S1eqS4/patchar31/patchar31/patchar31/patchar31/patchar31/patchar31/patchar31/patchar31/patchar31F /patchar31/patchar31/patchar31/patchar31/patchar31/patchar31/patchar31/patchar31/patchar31/patchar31/patchar31/patchar31τhminA1B1 FIG. 2. The cycle of operation of the heat pump. Left: without friction. Seq 1is the hot bath equilibrium polarization. Seq 2is the cold bath equilibrium polarization. The area enclose d by D,C,S 2S1is the heat absorbed form the cold bath. The area enclosed by D CBA is the work done on the system. Right: with friction. The area enclosed by D,C ,S2,S1is the heat absorbed form the cold bath. The work on the system is the area defined by the rect angles B B1S2S3and B1A1D C and A1A S4S1. 8The cycle of operation resembles the Ericsson refrigeratio n cycle [25]. The differences are in the dynamics of the microscopic working fluid which are described in subsection IIC. The work and heat transfer for the heat pump is summarized in T able II. The four branches for the heat pump become: In the first branch, D→C, the working medium is coupled to the cold bath of tem- perature Tcfor time τc, while the energy gap is kept fixed at the value ωa. The conditions are such that the internal temperature of the medium is lower thanTcduring τc. Along this branch, the polarization changes from the initial pola rization S1to the polarization S2. Since ωis kept fixed, no work is done and the only energy transfer is th e heat ωa(S2−S1) absorbed by the working medium. On this branch: S2<−1 2tanh/parenleftbiggωa 2kBTc/parenrightbigg . (2.10) In the second branch, C→Bthe working medium is decoupled from the cold bath, and the energy gap is varied. In the frictionless case the polari zation S2is constant (Left of Fig. 2). The only energy exchange is the work done on the system ( Ta ble II). When friction is added the polarization is changing from S2toS3in a period τa. The energy gap changes fromωatoωb(Right of Fig. 2), according to a linear law. In addition to wo rk, heat is developing as a result of the inner friction ( Table II). The third branch B→A, is similar to the first. The working medium is coupled to the hot bath at temperature Th, for time τh, keeping the energy gap ωbfixed. In this branch the polarization changes from S2toS1in the frictionless case, and from S3toS4when friction is added. The constraint is that the internal temperature of the working medium should be higher than the hot bath temperature during the time τh,T′> Th, leading to the inequality (Fig. 2), S1> S 4>−1 2tanh/parenleftbiggωb 2kBTh/parenrightbigg . (2.11) therefore S2> S1. From Eqs. (2.10) and (2.11), the condition for the interrel ation between the bath temperatures and the field values becomes: 9/parenleftbiggωa Tc/parenrightbigg </parenleftbiggωb Th/parenrightbigg (2.12) which is just the opposite inequality of the heat engine, (Eq . 2.8). In the heat pump work is doneonthe working fluid and since no useful work is done Carnot’s bou nd is not violated. The fourth branch A→D, closes the cycle and is similar to the second. The working medium is decoupled from the cold bath, and the energy gap cha nges back, during a period τbto its original value, ωb. The results are summarized in Table II. TABLE II. Work and heat exchange along the branches of the hea t pump without/with friction. branch frictionless work/work+[work against friction] heat D→C 0 ωa(S2−S1) C→B(ωb−ωa)S2 /parenleftbigωb−ωa)(S2+σ2/(2τa)/parenrightbig+ [σ2(ωa+ωb)/(2τa)] 0 B→A 0ωb(S1−S2) ωb/parenleftbig(S1−S2)−σ2(1/τa+ 1/τb)/parenrightbig A→D(ωa−ωb)S1 (ωa−ωb)/parenleftbigS1−σ2/(2τb)/parenrightbig+ [σ2(ωa+ωb)/(2τb)] 0 10C. Dynamics of the working medium The dynamics of the system along the heat exchange branches i s represented by changes in the level population of the two-level-system. This is a re duced description in which the dynamical response of the bath is cast in kinetic terms [1 8]. Since the dynamics has been described previously [20] only a brief summary of the ma in points is presented here, emphasizing the differences in the energy exchanges on the ’a diabats’. 1. The dynamics of the heat exchange branches The dynamics of the population at the two levels, P+andP−, are described via a master equation   dP+ dt=−k↓P++k↑P− dP− dt=k↓P+−k↑P−, (2.13) where k↓andk↑are the transition rates from the upper to the lower level and vice versa. The explicit form of these coefficients depend on the nature of the bath and the system bath coupling interactions. The thermodynamics partition betw een system and bath is consistent with a weak coupling assumption [18]. Temperature enters th rough detailed balance. The equation of motion for the polarization Sobtained from Eq. (2.13) becomes: dS dt=−Γ(S−Seq) (2.14) where Γ = k↓+k↑ (2.15) and Seq=−1 2k↓−k↑ k↓+k↑=−1 2tanh/parenleftbiggω 2kBT/parenrightbigg (2.16) where Seqis the corresponding equilibrium polarization. It should b e noticed that in a TLS there is a one to one correspondence between temperature and polarization thus internal temperature is well defined even for non-equilibrium situat ions. 11The general solution of Eq (2.14) is, S(t) =Seq+ (S(0)−Seq)e−Γt. (2.17) where S(0) is the polarization at the beginning of the branch . From Eqs. (2.14) and (2.16) the rate of heat change becomes: ˙Q=ω˙S (2.18) See also [16]. For convenience, new time variables are defined: x=e−Γcτc, y=e−Γhτh (2.19) These expressions represent a nonlinear mapping of the time allocated to the hot and cold branches by the heat conductivity Γ. As a result, the time all ocation and the heat conduc- tivity parameter become dependent on each other. Figure 1 and 2 show that the friction induces an asymmetry bet ween the time allocated to the hot and cold branches since more heat has to be dissipat ed on the cold branch. 2. The dynamics on the ’adiabats’ The external field ωand its rate of change ˙ ωare control parameters of the engine. For simplicity it is assumed that the field changes linearly with time: ω(t) = ˙ ωt+ω(0) (2.20) Rapid change in the field causes non-adiabatic behavior whic h to lowest order is propor- tional to the rate of change ˙ ω. In this context non-adiabatic is understood in its quantum mechanical meaning. Any realistic assumption beyond the id eal non-interacting TLS will lead to such non-adiabatic behavior. It is therefore assume d that the phenomena can be described by a friction coefficient σwhich forces a constant speed polarization change ˙S: 12˙S=/parenleftbiggσ t′/parenrightbigg2 (2.21) where t′is the time allocated to the corresponding ’adiabat’. There fore, the polarization as a function of time becomes: S(t) = S(0) +/parenleftbiggσ t′/parenrightbigg2 t (2.22) where t≥0 ,t≤t′. A modeling assumption of internally dissipative friction , similar to Eq.(2.21), was also made by Gordon and Huleihil ( [26]). Fric tion does not operate on the heat-exchange branches, there is no nonadiabtic effect sinc e the fields ωaandωbare constant in time. The irreversibilities on those branches are due to t he transition rates (Γ) of the master equation. From Fig.(1), Eq. (2.4), and Eq. (2.22) the polarization, fo r theB→Cbranch of the heat engine becomes: SC=S3=S1+/parenleftBiggσ2 τa/parenrightBigg . (2.23) The work done on this branch is: WBC=/integraldisplayτa 0DW=/integraldisplayτa 0S˙ωdt= (ωa−ωb)/parenleftBigg S1+1 2/parenleftBiggσ2 τa/parenrightBigg/parenrightBigg (2.24) The heat generated on this branch in the working fluid, which i s the work against the friction, becomes: QBC=/integraldisplayτa 0DQ=/integraldisplayτa 0ω˙Sdt=σ2(ωa+ωb) 2τa(2.25) This work is dependent on the friction coefficient and inverse ly on the time allocated to the ’adiabats’. The computation for the other branches of the he at engine and heat pump are similar. 3. Explicit expressions for the polarizations imposed by th e closing of the cycle. By forcing the cycle to close, the four corners of the cycle ob served in Fig. 1 are linked. Applying Eq. (2.17) leads to the equations: 13S1=S2y+Seq h(1−y) S3=S1+σ2 τa S4=S3x+Seq c(1−x) S2=S4+σ2 τb(2.26) The solutions for S1,S2andS1−S2are S1=Seq c+∆Seq(1−y) +σ2yG(x) (1−xy)=Seq h−∆Seqy(1−x)−σ2yG(x) (1−xy) S2=Seq c+∆Seqx(1−y) +σ2G(x) (1−xy)=Seq h−∆Seq(1−x)−σ2G(x) (1−xy)(2.27) and S1−S2= (∆Seq)F(x, y)−σ2(1−y)G(x) (1−xy)(2.28) where F(x, y) =(1−x)(1−y) (1−xy),∆Seq= (Seq h−Seq c), G(x) = ( x/τa+ 1/τb) The constraint that the cycle must close leads to conditions on the polarizations S1andS2 and on the minimum cycle time τc,min. Eqs. (2.27) shows that both S1andS2are bounded bySeq handSeq c. The minimum cycle time is obtained when the polarizations c oincide with the hot bath polarization: S1=S2=Seq h. In this case, τh=0, and from Eqs. (2.19) and (2.28) the minimum time allocation on the cold bath τc,minis computed, xmax=(Seq h−Seq c)−σ2/τb (Seq h−Seq c) +σ2/τa(2.29) or τc,min =−1/Γclg(Seq h−Seq c)−σ2/τb (Seq h−Seq c) +σ2/τa(2.30) From this expression for τc,minthe lower bound for the overall cycle time, is obtained (The left of Fig. 3) : τ≥τmin=τc,min+τa+τb (2.31) 14When the minimum cycle time Eq. (2.30) diverges, the cycle ca nnot be closed. This condi- tion imposes an upper bound on the friction coefficient σ σ≤σup=/radicalBig τb(Seq h−Seq c). (2.32) or τb> τb,min=σ2 (Seq h−Seq c). (2.33) Closing of the cycle imposes similar constraints on the mini mal cycle time under friction for the heat pump. The value of the polarization difference S2−S1using the notation of Fig. 2 becomes: S2−S1= (Seq 2−Seq 1)F(x, y)−σ2(1−x)(y/τa+ 1/τb) (1−xy)(2.34) The minimum cycle time is calculated in the limit when τc=0, leading to S2=S1=Seq 2. From Eqs. (2.19) and (2.34) the minimum time allocation on the hot branch τh,minis computed: ymax=(Seq 2−Seq 1)−σ2/τb (Seq 2−Seq 1) +σ2/τa(2.35) τh,min =−1/Γhlg(Seq 2−Seq 1)−σ2/τb (Seq 2−Seq 1) +σ2/τa, (2.36) where Seq 2is point F and Seq 1is point E on Fig. 2. Using τh,minthe lower bound for the overall cycle time, is computed τ≥τmin=τh,min+τa+τb (2.37) Closing the cycle imposes a minimum cycle time for both the he at engine and the heat pump, which is a monotonically increasing function of the frictio n coefficient σ. The divergence of τminimposes a maximum value for the friction coefficient σ. D. Finite Time Analysis 151. Quantities to be Optimized. The primary variable to be optimized is the power of the heat e ngine and the heat-flow extracted from the cold reservoir of the heat pump. For a pres et cycle time, optimization of the power is equivalent to optimization of the total work, while optimization of heat flow is equivalent to the optimization of the heat absorbed. The e ntropy production will also be analyzed. (1)The total work done on the environment per cycle of the Heat En gine. The total work of the engine, is the sum of the work on each bran ch: Cf. (Table I and Fig. 1): Wcyle1=/contintegraldisplay DW=−(WAB+WBC+WCD+WDA) (2.38) which becomes: Wcyle1= (ωb−ωa)(S1−S2)−σ2ωa(1/τa+ 1/τb) (2.39) The negative sign is due to the convention of positive Wwhen work is done on the system. Analyzing Eq. (2.39), the work is partitioned into three pos itive and negative areas. The positive area (left rotation) Wp= (ωb−ωa)(S1−S2) (2.40) is defined by the points A, B, C1, D1in Fig. 1. The two negative areas (right rotation) Wn=σ2ωa(1/τa) + σ2ωa(1/τb) (2.41) are defined by the points C, C1, S1, S3andD1, D, S 4, S2in Fig. 1. The cycle which achieves the minimum cycle time τ=τc,min, produces zero positive work Wp= 0. The corners A and B coincide at E, and C1coincides with D1. The negative work of Eq. (2.41), is defined by the corners C, D, S 4, S3and is ’cut’ by the Seq hline (Cf. the right of Fig. 4). The cycle has negative total work, meaning that wo rk is done onthe working fluid against friction. When τincreases beyond τc,min,S1diverts from S2, becoming lower 16thanSeq h(Cf.Eq. (2.27)). At a certain point, the work done against fr iction is exactly balanced by the useful work of the engine. The minimum time in which this balance is achieved is designated τ0. Its value which can be deduced from Eq. (2.39) is worked out i n appendix B. The minimum cycle time τminis compared to τ0, the minimum time needed to obtain positive power shown in the right of Fig. 3 as a function of the friction σ. Both functions increase with friction, but τ0diverges at a much lower friction parameter. Above this friction parameter no useful work can be obtained from the en gine. The divergence of τmin corresponds to a larger friction value where the cycle canno t be closed. 0.00 0.01 0.02 0.03 0.04 σ01234τminσup 0.000 0.005 0.010 0.015 σ0.00.51.01.52.0ττ0 τmin FIG. 3. Left: Minimal cycle time τminas a function of the inner friction parameter σfor the heat engine. The vertical line represents the upper-bound o fσ. Dimensionless units are used in which kb= 1 and ¯ h= 1. The parameters used are: ωa= 1794, ωb= 4238, Tc= 500, Th= 2500, Γc= 1 and Γ h= 2. Right: Comparison between τminandτ0, the minimum cycle time for power production. When the total time allocation is sufficient, i.e. τ > τ 0, work is done on the environment, andS1starts to increase. For long cycle times S1will approach Seq h, while S2will approach Seq c. The constant negative area will become negligible in compa rison to the positive area ( Fig. 5). To study the influence of friction on the work output the polar ization difference from 17Eq. (2.28) S 1-S2is inserted into the work expression Eq. (2.39), leading to: Wcyle1= (ωb−ωa)(Seq h−Seq c)F(x, y)− W σ1 (2.42) where Wσ1=σ2/parenleftBiggωb(1−y)(x/τa+ 1/τb) 1−xy+ωa(1−x)(1/τa+y/τb) 1−xy/parenrightBigg (2.43) Wσ1is the additional ’cost’ due to friction and is always positi ve. The emergence of positive power Pis shown in Fig. 4. For a fixed cycle time the optimization of work is equivalent to the optimization of po wer. The first two cycles have a cycle time shorter than τ0, and therefore do not produce useful work. For cycle 3, τ > τ 0and positive work is obtained when the time allocation on the cold bath is sufficient τc≥0.08. 0 0.1 0.2 0.3 τc−1000100Power3 1,2 0.0 1000 2000 3000 4000 5000ω−0.41−0.39−0.37−0.35SSheq A1B1 A2B2 A3B3C1 D1 C2 D2 D3C3 FIG. 4. Left: Power as a function of the time allocation on the cold branch corresponding to the friction coefficient σ= 0.005 with changing cycle times. The cycle time values are: for curve 1,τ=τmin=0.059, for curve 2, τ= 0.1 (the first two plots overlap) and for curve 3, τ= 0.5. Other parameters are the same as in figure 3. The dashed horizo ntal line is the line of zero power. Right: The cycles corresponding to the power plots. Negativ e work is in blue and positive work is in red. Note that for cycles 1 and 2, the total area is negative and, therefore, the power output is negative. For longer total cycle times, the ratio between the negative area to the positive area decreases as can be seen in Fig. 5. 18The position of the cycles in the S,ωcoordinates relative to Seq handSeq cchanges as a function of the cycle time. Insight to the origin of the behav ior of the ’moving’ cycles is presented in Fig. 11 of Appendix A. 0 1 2 3 τc−20020406080100Power45 6 1000.0 2000.0 3000.0 4000.0 5000.0 ω−0.50−0.45−0.40−0.35S A4B4 A5B5 A6B6Sheq SceqC6 C5 C4 D4 D5 D6 FIG. 5. Left: Power as a function of the time allocation on the cold branch corresponding to the friction coefficient σ= 0.005 with changing cycle times. The cycle time values are: for curve 4,τ= 1, for curve 5, τ= 2 and for curve 6, τ= 5. The dashed horizontal line is the line of zero power. Right: The cycles corresponding to the power plots. A ll the constant parameters are as in Fig. 4 The calculation of the total work done on the working fluid per cycle,Won cycle3for the heat pump is described in appendix D. See also ( Cf. Table (II) and Cf. Fig. 2). (2)The heat-flow( QF) The heat-flow, QF, extracted from the cold reservoir is: QF=ωa(S2−S1)/τ (2.44) Due to the dependence of QFonly on S2-S1, the cycle is similar to the cycle of the heat engine. (3)The entropy production ( ∆Su). The entropy production of the universe, ∆ Su, is concentrated on the boundaries with 19the baths since, for a closed cycle, the entropy of the workin g fluid is constant. The compu- tational details for both the heat engine and the heat pump ar e shown in appendix C. The entropy production and the power have a reciprocal relation (See Fig. 12). For example, the entropy production increases with σ, while the power decreases. (4)Efficiency. The efficiency of the heat engine is the ratio of useful work to t he heat extracted from the hot bath. ηH.E.=Wcycle Qabsorbed=ηfricles H.E. −/parenleftBiggσ2ωa(1/τa+ 1/τb) ωb(S1−S2)/parenrightBigg (2.45) where ηfricles H.E. = (1 −ωa/ωb) When the cycle time approaches its minimum τ→τmin,the efficiency diverges: ηH.E.−→ − ∞ . The efficiency becomes positive only when τ≥τ0. Using Eq. (2.45) a bound for the efficiency is obtained: 0< η H.E.≤ηfricles H.E. −Tc Th/parenleftBiggσ2(1/τa+ 1/τb) (S1−S2)/parenrightBigg (2.46) The cooling efficiency of the refrigerator will be: ηRf=QDC Won cycle=ωa(S2−S1) ((ωb−ωa)(S2−S1) +σ2ωb(1/τa+ 1/τb))(2.47) or: 1 ηRf+ 1 =1 COP+ 1 =ωb ωa/parenleftBigg 1 +σ2(1/τa+ 1/τb) (S2−S1)/parenrightBigg >Th Tc/parenleftBigg 1 +σ2(1/τa+ 1/τb) (S2−S1)/parenrightBigg , (2.48) leading to the expression for the efficiency: ηRf=ωa ωb1 ηfricles H.E. +σ2(1/τa+ 1/τb) S2−S1<Tc Th1 ηfricles H.E. +σ2(1/τa+ 1/τb) S2−S1(2.49) For both the heat engine and the heat pump, the efficiency is exp licitly dependent on time allocation, cycle time, and bath temperatures. 202. Optimization The performance of both the heat engine and the heat pump can b e optimized with respect to: •(a) The overall time period τof the cycle, and its allocation between the hot and cold branches. •(b) The overall optimal time allocation between all branche s. (This optimization is performed only for the heat pump.) •(c) The external fields, ( ωa,ωb). (a)Optimization with respect to time allocation. The optimization of time allocation is carried out with the c onstant fields ωaandωb. The Lagrangian for the work output becomes: L(x, y, λ ) =Wcycle+λ/parenleftbigg τ+1 Γcln(x) +1 Γhln(y)−τa−τb/parenrightbigg , (2.50) where λis a Lagrange multiplier. Equating the partial derivatives ofL(x, y, λ ) with respect toxandyto zero, the following condition for the optimal time alloca tion becomes: Γcx((1−y)2(Seq h−Seq c) +σ2(1−y)(1/τa+y/τb)) = Γhy((1−x)2(Seq h−Seq c)−σ2(1−x)(x/τa+ 1/τb))(2.51) When σ= 0, the previous frictionless result is retrieved. (Optimi zing the entropy pro- duction ∆ Suleads to an identical time allocation to Eq. (2.51)). Eq. (2.51) can also be written in the following way: Γcx((1−y)(1−y xmax)) = Γ hy((1−x)(xmax−x)) (2.52) where xmaxwas defined in Eq. (2.29). The result is dependent on the time a llocations of the ’adiabats’, through the dependence of xmax. For the special case when Γ c= Γ h, the relation between the time allocations in contact with the hot and cold baths becomes: 21x=xmaxy (2.53) For the frictionless case, this result coincides with the fo rmer frictionless one x=y, meaning that equal time is allocated to contact with the cold and hot r eservoirs. When friction is added this symmetry is broken, Eq. (2.53), to compensate for the additional heat generated by friction the time allocated to the cold branch, becomes la rger than the time on the hot branch. The Lagrangian for the heat-flow, QF, extracted from the cold reservoir is defined in parallel to the Lagrangian for the total work. Substituting Γhfor Γ c,xforyand vice versa, alsoymaxforxmax, where ymaxwas defined in Eq. (2.35), one gets the optimal time allocatio n for the heat pump. Optimization of power with respect to time allocation as a fu nction of the cycle time, τ for different friction coefficients is shown in Fig. 6 (Left), t ogether with the corresponding heat-flow (Middle) and the corresponding entropy productio n (Right). The left part shows that in the frictionless case the power obtains its maximum a t zero cycle time with the value consistent with Eq. (2.58). When friction is introduced, th e maximum power decreases and is shifted to longer cycle times. The figure also shows, that f or short times the work done by the system is negative, and as the friction coefficient σincreases, the boundary between positive and negative power shifts to longer cycle times. In the Middle of Fig. 6, the heat- flow corresponding to the optimal power on the left is shown. T he shapes of the power and heat flow curves are similar. The heat-flow values are always p ositive and larger than the corresponding power values. The entropy production (Right ) shows that unlike the power curves the friction changes significantly the shape of the cu rves. The entropy production rate for the case with friction sharply decreases. The paral lel graphs for the heat pump are similar. 220.0 5.0 10.0 τ−120.0−80.0−40.00.040.080.0120.0Optimal Power1 2 3 4 5 6 0.0 5.0 10.0 τ0.050.0100.0150.0200.0QF1 2 3 4 5 6 0.0 5.0 τ0.000.100.200.300.40Entropy production( ΔSu) 1234567 FIG. 6. Left: The optimal power with respect to time allocati on as a function of τ, for different values of friction. Middle: The corresponding heat-flow( QF). The parameter values for both Left and Middle are: for plot 1, τa=τb=σ= 0. For all the other plots τa=τb= 0.01 The σvalues for the curves from plot 2 to 6 are: 0.002, 0.005, 0.007, 0.0135, 0 .02 respectively. Right: The entropy production rate corresponding to the optimal power on the le ft part of the figure. The additional curve is curve ’1’, which corresponds to σ= 0, and τa=τb= 0.01 . The parameter values for the other plots are : for plot 2, τa=τb=σ= 0, for all the other plots τa=τb= 0.01. The σfor the curves from plot 3 to plot 7 are: 0.002, 0.005, 0.007 0.0135, 0 .02 accordingly. (b) Time allocation optimization between all branches of th e refrigerator Further optimization of the performance of the heat pump is p ossible by relaxing the assumption of constant time on the ’adiabats’. First the tim e allocation between the two ’adiabats’ is optimized, when τa+τb=δ, where δis a constant. Finally the time allocation between the ’adiabats’ and the heat exchange branches, is op timized. These results are compared to the recent analysis of Gordon et. all. [27]. From Eqs. (2.44) and (2.34) with constant time allocations a long the heat exchange branches one gets for the cooling power: QF=A0−A1(y τa+1 (δ−τa)), (2.54) where A0andA1are constant functions of the parameters of the system. And o nδ, a double inequality is imposed τ > δ > the larger of [ ( τ−τh,min);τb,min], see Eq. (2.33). The optimal τadepends only on yand on δ. The optimal value of τa,optbecomes: τa,opt =δ−y+√y 1−y(2.55) 23Further optimization by changing the the value of δ, changes the cycle time τ. This optimization step is done by numerical iteration. Typicall y the sum of the final optimal values of τaandτbis about twice their value before, and their ratio is about 0. 7 of the value which was chosen initially. The next step is to study the time allocation between the ’adi abats’ and the heat exchange branches when all other controls of the heat pump have optima l values. These controls include also the external fields of optimization which are de scribed later. For comparison with Gordon et. all. [27], the results of opti mization are plotted in the 1/QF,1/ηplane for a fixed cycle time τ. The following example demonstrates the method followed: First an optimal starting value for QFwas found which determines the time allocation control parameters, τc= 0.44221, τh= 0.31779, τa= 0.0084, τb= 0.0116 with a total cycle time of τ= 0.78. Under such conditions QF,max= 2.9158 (1 /QF,max= 0.34296). Changing the time allocation between the ’adiabats’ and the heat exchange branches changes the balance between optimal cooling power and efficie ncy. Denoting the sum τc+τh byτch, the ratio τh/τcbyrhc, the sum τa+τbbyτab, the ratio τa/τbbyrab, time is transfered fromτchby small steps to τab, while keeping the the ratios rhcandrabconstant. For each step the corresponding 1 /QFand 1/η, are calculated as in Fig. 7. The relation between the reciprocal efficiency and the reciprocal cooling power sh ows the tradeoff between losses due to friction and losses due to heat transfer. Following th e curve in Fig. 7, starting from pointAwhere the cooling power is optimal, resources represented b y time allocation are transferred from the heat exchange branches to the ’adiabat s’, reducing the friction losses. At point Ban optimum is reached for the efficiency. This point has been fo und by Gordon et. al. to be the universal operating choice for commercial c hillers. Point Brepresents the optimal compromise between maximum efficiency and cooling po wer. 240 0.1 0.2 0.3 0.4 arbitrary time scale05101/QF (1) ; 1/ η(2) 12 0 2 4 6 1/QF11.611.81212.21/ηA(0.343,12.202) B(0.6322,11.6379)0.33 0.35 0.37 1/QF11.512.012.513.01/η A FIG. 7. The relation between efficiency and cooling power for t he heat pump. The parameters are: The constant optimal cycle time, τ, =0.78; T c=51.49, T h=257.45, ωa=47.699, ωb=600, Γ c=1, Γh=2,σ=0.005 Left: Comparison between 1 /QF(plot 1) and 1 /η(plot 2) as a function of the allocated time transfer from the heat exchange branches to t he ’adiabats’. Zero time is the optimal heat-flow time allocation. Right: The Universal plot for the heat pump. The starting optimal point in the plane of (1 /QF, 1/η), was (0.34296, 12.202), while the maximum efficiency point B is (0.6322,11.6379) and time allocation ( τc,τh,τa,τb)= (0.22721,0.16328,0.1636,0.2259). The insert shows the neighborhood of point A. PointAis located at the maximum cooling power. If more time is alloc ated to the heat exchange branches both 1 /QFand 1/ηwill continue to increase as seen in the insert of Fig. 7. (c)Optimization with respect to the fields. The values of the fields ωaandωbare control parameters of the engine. In a spin system these fields are equivalent to the value of the externa l magnetic field applied on the system. They directly influence the energy spacing of the TLS . The work function Wcycle, or equivalently the power ( P) is optimized with respect to the fields, subject to the Carno t constraint: ωa Tc≥ωb Th(2.56) 25Optimal power is obtained by equating independently to zero the partial derivatives of Wcycle, or of P=Wcycle/τby varying ωaandωb. In addition the optimal solutions have to fulfill the inequality constraints in Eq. (2.56). As a resu lt two transcendental equations inωaandωbare obtained which are solved numerically. The two equations are: (1−y xmax) (ωb−ωa)(∆Seq+σ2/τa) cosh2/parenleftBig ωa 2kBTc/parenrightBig =1−y (4kBTc) (xmax−x) (ωb−ωa)(∆Seq+σ2/τa) cosh2/parenleftBig ωb 2kBTh/parenrightBig =1−x (4kBTh)(2.57) Where ∆ Seq= Seq h- Seq cas defined in Eq. (2.28). Examining Eq. (2.57), and fixing the friction σ, it is found that ∆ Seqis an extensive function of order zero (intensive ) with respect to the quartet of variables ωa, Tc, ωb, Th. This means that scaling these parameters simultaneously will not change ∆ Seq. Also x max, and cosh2/parenleftBig ω 2kBT/parenrightBig are extensive (order zero). The work function however, is extensive with order on e (Eqs. (2.42) and (2.43)). This property will be exploited in paragraph III. The optimization of power with respect to the fields is shown i n Fig. 8 for the frictionless engine, as a function of the fields with fixed time allocation. A global maximum can be identified. 1760 1800 1840419442344274 1760 1800 1840ωaωb P FIG. 8. Power for the frictionless engine as a function of the fields ( ωa,ωb) for constant bath temperatures, and constant time allocations. The maximum p ower is achieved at ωa= 1794 and ωb= 4239 where the bath temperatures are Tc= 500 ,Th= 2500. 26The heat pump optimization of QFwith respect to the fields is different and therefore will be presented in Section III. The analysis for the optimization with respect to the fields f or the entropy production ∆Su, is presented in appendix C. The optimal solution without fr iction( σ= 0) leads to ∆Su min= 0. When σ/negationslash= 0, the minimum value of ∆ Suis different from zero, and is achieved on the boundary of the region. E. Global Optimization of the Heat Engine Global optimization of the power means searching for the opt imimum with respect to the control parameters cycle time, time allocation and the fi elds. An iterative procedure is used. The procedure is initiated by setting the optimal time alloc ation from the corresponding Lagrangian, with σ= 0. The power becomes a product of two functions, one dependi ng only on time the other only on the fields, and therefore, the fields c an be changed independently of time. The optimal fields for the above time allocation are the n sought. For the frictionless case, the overall time on the adiabats tends to zero. The opti mal field values become independent of time. The value P= 107 .501 is the short time limit in accordance with the equation: P −→ (ωb−ωa)(Seq h−Seq c)(ΓcΓh)/(/radicalBig Γc+/radicalBig Γh)2(2.58) These fields are inserted into the expression with friction σ/negationslash= 0, and the new optimal times and fields are computed. The iteration converges after two to three steps, as indicated by Table III for σ= 0.005. Notice that the location of the optimum is not very sensi tive to the friction parameter. 27TABLE III. Global optimization of power. The notation Pmax(ωa,ωb) stands for fixed time allocations, and the notation Pmax(τa,τb) stands for fixed field values. The other parameters are: Tc= 500, Th= 2500, τa=τb=0.01. Γ c= 1 and Γ h= 2 σ τ Pmax(ωa,ωb) Pmax(τc,τh) ωa ωb τc/τ 0.005 2 84.46 1794 4239 0.5999 0.005 1.367 87.18 1794 4239 0.5891 0.005 1.367 88.68 1719.1 4036.31 0.5891 0.005 1.347 87.47 1719.1 4036.31 0.58856 0.005 1.347 88.704 1718.16 4033.67 0.58856 In Table IV, the extensive properties Eq. (2.57) are examine d for k=2 and k=10 with respect to Table III. The temperature values will change to T c= 1000, T h= 5000 for k=2 and Tc= 5000, T h= 25000 for k=10 . The results verify the analysis. TABLE IV. Global optimization of power. By multiplying the f our values, T c, Th,ωa,ωbby k and searching first for optimal time allocation, then multip lying only the temperature values by k and searching for the optimal fields. All the notations and ot her parameters as in Table III. σ k τ Pmax(ωa,ωb)Pmax(τc,τh) ωa ωb τc/τ 0.005 2 1.367 174.9 3438.2 8072.6 0.58852 0.005 2 1.367 174.9 3436.7 8070.3 0.58852 0.005 2 1.347 174.94 3436.7 8070.3 0.58856 0.005 2 1.347 179.8 3437.7 8069.4 0.58856 0.005 10 1.347 887.04 17181.6 40336.7 0.58856 28III. ASYMPTOTIC PROPERTIES OF THE HEAT PUMP WHEN THE COLD BATH TEMPERATURE APPROACHES ABSOLUTE ZERO. The goal is to obtain an asymptotic upper bound on the cooling power when the heat pump is operating close to absolute zero temperature. This r equires optimizing the perfor- mance of the heat pump with respect to all control parameters . A. Optimization of the heat-flow QFwith respect to the fields and to the cooling power upper bound. The heat-flow, QFextracted from the cold reservoir now becomes the subject of interest: QF=ωa(S2−S1)/τ (3.1) or from Eq. (2.34), QF= (ωa/τ)/parenleftBigg (Seq 2−Seq 1)F(x, y)−σ2(1−x)(y/τa+ 1/τb) (1−xy)/parenrightBigg (3.2) No global maximum for the QFwith respect to the fields is found. The derivative of QF with respect to ωbbecomes: ∂QF ∂ωb=F(x, y)ωa τ1 4kBThcosh2ωb 2kBTh≥0 (3.3) leading to the result that QFis monotonic in ωb. Under such conditions, ωbis set, and the optimum with respect to ωais sought for. The derivative of QFwith respect to ωabecomes: ∂QF ∂ωa= (Seq 2−Seq 1)−σ2 (1−y)(y/τa+ 1/τb)−ωa1 4kBTccosh2ωa 2kBTc= 0 (3.4) Introducing from Eq. (3.4) the optimal value of ( Seq 2−Seq 1)−σ2 (1−y)(y/τa+ 1/τb), into Eq. (3.2), leads to the optimal cooling rate: Qoptimum F =F(x, y)ω2 a τ1 4kBTccosh2ωa 2kBTc=F(x, y) 4kBτ/parenleftbiggωa Tc/parenrightbigg2Tc cosh2ωa 2kBTc(3.5) 29Due to its extensivity, the ratioωa Tcbecomes a constant, while both ωaand T ccan approach zero. From Eq.(3.5), an upper-bound for the cooling rate QFis obtained: Qoptimum F ≤F(x, y) 4kBτ/parenleftbiggωa Tc/parenrightbigg2 Tc. (3.6) From Eqs. (3.6), when T capproaches zero, the cooling rate vanishes, at least linear ly with temperature. This is a third law statement which shows that a bsolute zero cannot be reached since the rate of cooling vanishes as absolute zero is approa ched. B. The asymptotic relation between the internal and externa l temperature on the cold branch When the bath temperature tends to zero, the internal workin g fluid temperature has to follow. This becomes a linear relationship between T′andTcasTctends to zero. Calculating the polarization at the end of the contact with t he cold bath S2: S2=Seq 2−(Seq 2−Seq 1)x(1−y)−σ2x(1/τb+y/τa) (1−xy)(3.7) Assuming the relation Th=ρ TcasTctends to zero, the exponents can be expanded to the first order to give: S2=Tc ωa1−xymax+ (ωa/ωb)ρx(ymax−y) (1−xy)+ 1/2−x(σ2/τa)(ymax−y) (1−xy)(3.8) Also,S2defines the internal temperature T′through the relation: S2=−1 2tanh/parenleftBig ωa 2kBT′/parenrightBig . Expanding the hyperbolic tangent, one gets: T′=Tc1−xymax+ρ(ωa/ωb)x(ymax−y) (1−xy)−xωa(σ2/τa)(ymax−y) (1−xy)(3.9) proving that TcandT′both tend asymptotically to zero. It should be noted that the term independent of Tcdepends on ωa, which also tends to zero as Tctends to zero ( (Eq. 3.6). Eq. (3.6) also shows that Qoptimum F *Tcis a quadratic function of ωa, Cf. Fig. 9. 30Eq. (3.6) represents an upper-bound to the rate of cooling. I n order to determine how closely this limit be approached, a strategy of cooling must be devised, which re-optimizes the cooling power during the changing conditions when T capproaches zero. C. Optimal cooling strategy The goal is to follow an optimal cooling strategy, which expl oits the properties of the equations and achieves the upper-bound for the rate of cooli ng,QF. The properties of the equations employed are; •i: The derivative with respect to ωaofQF( Eq. (3.4)), is extensive of order zero in the ’quartet’ ( ωa, ωb, Tc, Th). •ii: For∂QF ∂ωathe extensivity holds also for the ’doublets’ ( ωb, Th) or (ωa, Tc). Scaling these variables by the same number, leaves Eq. (3.4) equal to zero, and the value of Qoptimum F does not change. •iii: In spite of QFbeing monotonic in ωb,Qoptimum F is independent of ωb(and of T h), therefore QFsaturates as ωbis increased. From property (i) it follows, that once an optimal ’quartet’ (ωa, ωb, Tc, Th) is created, it is possible to cool optimally with a set of quartets, which ar e scaled by a decreasing set rn<1, lim n→∞rn= 0. For this set the limit of the ratioωa Tcis a non zero constant. Therefore in Eq. (3.5) ωaand T Care optimal leading to: Qoptimum F =F(x, y) 4kBτoptimal/parenleftBiggωa,optimal Tc,optimal/parenrightBigg2Tc,optimal cosh2ωa,optimal 2kBTc,optimal(3.10) In general, the hot bath temperature is constant, and the pro perty (ii) is used to scale back the value of the optimal T hto the bath temperature. As a result, the optimal high field is also scaled. Property (iii) will be exploited by changing only ωbin the optimal quartets and checking for saturation. See Fig. 13 and the dashed curves of Fig. 9. Su mmarizing, for every ’quartet’ 31the upper-bound in Eq. (3.6) can be reached. The details of th e cooling strategy can be found in Appendix E 0 5 10 15 Tc0.00.51.01.5QF 0 5 10 15 ωa02468101214QF * Tc FIG. 9. Left, (solid line): The optimal heat-flow Qoptimum Ffor the heat pump as a function of T c. The fixed parameter values of the pair ( ωband T h) are, Th= 64.359,51.49,42.9082,30.03555,23.599,17.153,4.291,0.8582,0.1717,0.01717, and accordingly ωb= 150 ,120,100,70,55,40,10,2,0.4,0.04. The other constant parameter values are: σ= 0.005, Γc= 1, Γ h= 2. (dashed line): fixing ωb= 3000 for every point. The other parameters are the same as the solid line. Right: The optimal heat-flows multipl ied by the corresponding T cas a function of ωa. The optimal time is constant for the chosen parameters; τoptimal = 0.885 for the solid curves, and 0.825 for the dashed curves. Fig. 10 shows that the cooling strategy ( Tables V and VI ) can a pproach the upper bound leading to a linear relation of the optimal cooling pow er with temperature. With respect to the fields the optimal strategy leads to a decrease of the field ωawhich is in contact with the cold bath. This causes the internal tempera ture of the TLS T’ to be lower than the cold bath temperature T c. On the hot side the optimal solution requires as large an energy separation as possible ωa→ ∞ but this effect saturates. 320.000 0.001 0.002 0.003 Tc0.000000.000020.000040.000060.00008QF 0 10000 20000 30000 1/Tc0.000.010.020.030.04ΔSu FIG. 10. Left: The optimal cooling rate as a function of the co ld bath temperature T c, com- pared with the upper-bound for the cooling rate. Right: The e ntropy production during cooling shown for the case with friction (upper line, circles) and wi thout friction (lower line, squares). The common parameters for all three figures are: τa=τb= 0.01,σ= 0.005, Γ c= 1, Γ h= 2. The linear relation of the cooling rate with T cleads to a constant asymptotic entropy production as can be seen in the right of Fig. 10 ( Cf. Appendix C). IV. CONCLUSION The detailed study of the four stroke discrete heat engine wi th internal friction serves as a source of insight on the performance of refrigerators at temperatures which are very close to absolute zero. The next step is to find out if the behav ior of the specific heat pump described in the study can be generalized. A comparison with other systems studied indicates that the conclusions drawn from the model are gene ric. As a heat engine the model shows the generic behavior of maximum power as a function of c ontrol parameters found in finite time thermodynamics [3–6]. This is despite the fact that the heat transfer laws in the microscopic model of the working fluid are different fro m the macroscopic laws such as the Newtonian heat transfer law [16]. When operated as a he at pump with friction, the present model shows the universal behavior observed for com mercial chillers [27] caused by 33a tradeoff between allocating resources to the ’adiabats’ or to the heat exchange branches. Another question is whether the linear scaling of the optima l cooling power at low cold bath temperatures is a universal phenomenon. For low temper atures the results of the present model can be extended to a working fluid consisting of an ensemble of harmonic oscillators or any N-level systems. This is because at the li mit of absolute zero only the two lowest energy levels are relevant. When examined, other mod els with different operating cycles show an identical behavior. For example the continuo us model of a quantum heat engine [18] based on reversing the operation of a laser shows this linear scaling phenomena. Another example is the Ericsson refrigeration cycle Cf. Eq. (23) in the study of Chen et al [25] which shows the same asymptotic linear relationship. A point of concern is the dependence of the heat transfer laws on temperature when absolute zero is approached. The kinetic parameters k↓andk↑represent an individual coupling of the two-level-system to the bath. Considering c oefficients derived from gas phase collisions they settle to a constant asymptotic value as the temperature is lowered [29]. The reason is that the slow approach velocity is compen sated by the increase in the thermal De-Broglie wavelength. There has been an ongoing interest in the meaning of the third law of thermodynamics [30–36]. The issue at stake has been: is the third law an indep endent postulate or it is a consequence of the second law and the vanishing of the heat ca pacity. This study presents a dynamical interpretation of the third law. The absolute te mperature cannot be reached because the maximum rate of cooling vanishes linearly at lea st with temperature. ACKNOWLEDGMENTS This research was supported by the US Navy under contract num ber N00014-91-J-1498. The authors want to thank Jeff Gordon for his continuous help, discussions and willingness to clarify many fine points. T.F. thanks Sylvio May for his hel p. 34APPENDIX A: ANALYSIS FOR THE ’MOVING’ CYCLES. Insight into the origin of the behavior of the ’moving’ cycle s is seen in Fig. 11, where the polarizations S 1, S2are shown as monotonically decreasing functions of the time allocation on the cold bath. However, the envelope of S 1for maximal power, namely for maximal S 1-S2 is worth noticing. It is a decreasing function for short cycl e times, achieves a minimum atτ0, and starts to increase for τ > τ 0. Thus it is responsible for shifting the cycles to smaller polarization for short cycle times, and for the chan ge of that trend for larger cycle times. The envelope of S 2for maximal S 1-S2is also a monotonically decreasing function of τc, or equivalently of τ, supporting the increase of S 1-S2. The figure also shows, that for a short time allocation both S 1and S 2are close to the equilibrium polarization Seq h, When not enough time is allocated on the hot bath both the polarizatio ns S 1and S 2approach Seq c. 0.0 0.5 1.0 1.5 2.0 τc−0.48−0.43−0.38SSheq Sceq FIG. 11. Comparison between the polarizations S 1and S 2as a function of τc, for ten different τvalues, 0.06, 0.08, 0.1, 0.25, 0.5, 0.75, 1 1.25, 1.5 and 2. Th e solid curves are S 1while the dashed curves are S 2. Superimposed are the values of S 1and S 2for the maximal S 1- S2. 35APPENDIX B: THE COMPUTATION OF τ0 The computation of τ0Eq. (2.39) is not sufficient since it gives only the relation be - tween the times spent on the cold and hot branches for zero wor k. The natural addi- tional requirement is to seek for the optimal allocations, τc,0andτh,0using Eq. (2.52): τ0=τc,0+τh,0+τa+τb Denoting by x0andy0the corresponding x and y values defined in Eq. (2.19), the following two equations for x 0and y 0are obtained: y0=(xmax−x0)−R (xmax−x0)−Rx0(B1) and Γcx0((1−y0)(1−y0xmax)) = Γ hy0((1−x0)(xmax−x0)) (B2) Where R is defined as: R=σ2ωa(1/τa+ 1/τb) (ωb−ωa) (Seq h−Seq c+σ2/τa)(B3) and x maxwas defined in Eq. (2.29) as: xmax=(Seq h−Seq c)−σ2/τb (Seq h−Seq c) +σ2/τa(B4) The quadratic equation to be solved for x 0is, AA x2 0+BB x 0+CC= 0 (B5) Where AA = Γ h(1 + R) BB = - (Γ h((1 + R) (x max-R) + x max) + Γ c(1 + R - xmax)) and CC = Γ h(xmax-R) x max APPENDIX C: ENTROPY PRODUCTION. (1)Heat Engine. ∆Su cyle1=−(QAB/Th+QCD/Tc) (C1) 36Or from Table (I) ∆Su cyle1= (ωa/Tc−ωb/Th)(S1−S2) +σ2ωa Tc(1/τa+ 1/τb) (C2) The entropy production results are shown in Fig. 12. The left figure shows ∆ Suwith increasing friction. The middle figure shows the correspond ing cycles, while the right figure shows the corresponding power values. 0.00 0.02 0.04 0.06 0.08 0.10 τc0.00.10.20.30.4ΔSu 010012345 1000 2000 3000 4000 5000 ω−0.41−0.39−0.37−0.35S A00B00C00 D00D01C01A01B01A1B1A2B2A3B3A4B4A5B5C5 D5 C4 D4 C3 D3C2 D2C1D1Sheq 0.00 0.02 0.04 0.06 0.08 0.10 τc−200−1000100200P 543210100 FIG. 12. Left: The entropy production of the heat engine as a f unction of the time spent on the cold branch for the fixed values of ωa= 1794, ωb= 4238, Tc= 500, Th= 2500, Γ c= 1, Γ h= 2 and τ= 0.1. Middle: The corresponding cycles. Right: The correspond ing powers. Seven cases are shown. Case ’00’ is the frictionless case, when the times spe nt on the ’adiabats’, are zero, case ’01’ is the frictionless case, when the times spent on the ’adiaba ts’,τaandτbare different from zero and equal: 0.01. The other five cases are with increasing frictio n, when also τa=τb= 0.01, whereas the different friction coefficients σare: for plot 1: σ= 0.003, for plot 2: σ= 0.004, for plot 3: σ= 0.005, for plot 4: σ= 0.006 and for plot 5: σ= 0.007. The reciprocal behavior of the entropy production and the po wer is clear from Fig. 12. One also observes, that for the given cycle time the ’free’ ti me for the cycles with increasing σbecomes more restricted. This follows from the dependence o fτc,minonσ. See also Fig.( 3) Introducing Eq. (2.28) into Eq. (C2). The entropy productio n becomes, ∆Su cyle1= (ωa/Tc−ωb/Th)(Seq h−Seq c)F(x, y) + ∆ Su σ1 (C3) where 37∆Su σ1=σ21 (1−xy)/parenleftbiggωa Tc(1−x)(1/τa+y/τb) +ωb Th(1−y)(x/τa+ 1/τb)/parenrightbigg (C4) Notice, that ∆ Su σ1is always positive. For σ= 0 Eq. (C4) reduces to the frictionless results [20]. (2)Heat Pump The entropy production for the heat pump becomes: ∆Su ref=/parenleftbiggωb Th−ωa Tc/parenrightbigg (S2−S1) +σ2ωb Th(1/τa+ 1/τb) =/parenleftbiggωb Th−ωa Tc/parenrightbigg (Seq 2−Seq 1)·F(x, y) + σ2F(x, y)/braceleftBiggωb Th1 1−x(1 τa+x τb) +ωa Tc1 1−y(1 τb+y τa)/bracerightBigg (C5) The asymptotic entropy production as T ctends to zero can be calculated leading to ∆Su ref=F(x, y) [(ωb/(ρωa))(1−ρ(ωa/ωb))2+ σ2/parenleftBiggωb ρTc1 (1−x)(1/τa+x/τb) +ωa Tc1 (1−y)(1/τb+y/τa)/parenrightBigg ] (C6) Since Th=ρTc, the r.h.s. of Eq. (C6) tends to a constant, for each term depe nds on the constant ratios ( ωb/Th), (ωa/Tc) or on their ratio. This result is demonstrated on the right side of Fig. 10. The optimization with respect to time allocation has the sam e result as for the heat engine. Therefore, only optimization with respect to the fie lds are presented; Equating to zero the derivatives with respect to x an y of the e ntropy production, one gets two similar equation to the total work derivatives: (1−y xmax) (ωa/Tc−ωbTh)(∆Seq+σ2/τa) cosh2/parenleftbiggωa 2kBTc/parenrightbigg +1−y (4kBTc)≥0 (C7) (xmax−x) (ωa/Tc−ωbTh)(∆Seq+σ2/τa) cosh2/parenleftbiggωb 2kBTh/parenrightbigg +1−x (4kBTh)≥0 (C8) Where ∆ Seq, isSeq h-Seq c. Eqs. (C7) and (C8) show that the entropy production is a monot onic function in the allowed range, namely, for 38ωa Tc>ωb Th. (C9) To conclude the entropy production has a minimum value: ∆ Su min, will be ∆Su min=ωa Tcσ2(1/τa+ 1/τb) (C10) obtained on the boundary of the range. APPENDIX D: THE TOTAL WORK DONE ON THE SYSTEM FOR THE HEAT PUMP The total work done on the system becomes, Won cyle3= (ωb−ωa)(S2−S1) +σ2ωb(1/τa+ 1/τb) (D1) or Won cyle3= (ωb−ωa)(Seq 2−Seq 1)F(x, y) +Wσ3 (D2) where Wσ3=σ2 (1−xy)(ωb(1−y)(1/τa+x/τb) + ωa(1−x)(y/τa+ 1/τb)) =σ2F(x, y)/braceleftBiggωb 1−x(1 τa+x τb) +ωa 1−y(1 τb+y τa)/bracerightBigg (D3) Eq. (D1) can be interpreted as the work done on the working flui d see (Cf. Fig. 2), as the sum of three positive areas, ( ωb−ωa)(S2−S1),σ2ωb(1/τa) and σ2ωb(1/τb) with the corresponding corners, D,C,B1,A1, B,B1,S2,S3and A1,A,S 4,S1. APPENDIX E: THE OPTIMAL COOLING STRATEGY CLOSE TO THE ABSOLUTE ZERO TEMPERATURE The first step in the cooling strategy is to create the first opt imal quartet; •(0) The systems external parameters σ,τa,τb, Γcand Γ hare set. 39•(1) A decreasing set of ωbis chosen. •(2) A constant ratio ( ρ) for T h/Tc, is chosen which is the ratio of the initial bath temperatures. •(3) For the above chosen values, the optimal values of ωa, Tc,τ, and its optimal allocations between the branches to give maximal Q Fare found for each ωbin the set in (1) , by solving numerically the following additional equ ation to Eq. (3.4), with the condition that T h=ρ·Tc: ∂QF ∂Tc=F(x, y) 4τ k B/parenleftbiggωa Tc/parenrightbigg2 1 cosh2ωa 2kBTc−ωb ρωacosh2 ωb 2ρ kBTc = 0 (E1) The above strategy causes the decrease of Thtogether with the Tc. Nevertheless according to (ii) above, the doublet ωbandThcan be rescaled to increase Thback to its original value. The solid curves of Fig. 9 are optimal in the in the above descr ibed sense. Increasing only the value of ωbin the optimal quartet according to point (iii), leads to lar ger values of the cooling rate, but eventually the increase of QFwill slow down and saturate. See Fig. 13 and the dashed curves of Fig. 9. Fig. 13 represents the saturation phenomenon on ωb. Three points from Fig. 9 are chosen, and all parameters are fixed, except ωb, which is allowed to increase. 400 200 400 600 800 1000 ωb0.00.51.01.5QF 0.82 0.84 0.86 0.88 0.90 τmax0.00.51.01.5QF FIG. 13. Left: The optimal heat-flow for the heat pump as a func tion of ωb, showing the saturation phenomenon. The fixed parameter values are: f or triangles; Th= 64.5725, Tc= 12 .9145, ωa= 11 .9233, starting with ωb= 150, for squares; Th= 42 .90815, Tc= 8.58168, ωa= 7.94986, starting with ωb= 100, for circles; Th= 23.59905, Tc= 4.71981, ωa= 4.37247, starting with ωb= 55. The common parameters for all three figures are: τa=τb= 0.01,σ= 0.005, Γ c= 1, Γ h= 2. Right: The optimal heat-flows as a function of τmax, the time at which the optimum is achieved. The fixed paramete r values are the same, as on the left. We note, that the optimal time is becoming constant only at saturation. In order to approach the upper-bound for QFin Eq. (3.6), a decreasing set of ωa/ Tcis created, achieved in an optimal way: First step: After having an optimal ’quartet’, T cand T h, are fixed. Then, by lowering ωb, one finds the corresponding optimal ωavalues. This procedure is checked globally, by also iterating the time allocations. The results of a typica l example are shown in Table V. Second step: Using again the property of extensivity, the co oling will be achieved by multiplying the rows of Table V by a decreasing sequence, e.g . by 2−nfor the n-th row. Table VI describes the cooling strategy, checking also the n on-divergence of the entropy production both for the frictionless case and the case with f riction. The results are also summarized in Fig. 10. 41Table V demonstrates, that the procedure shifts down to the C arnot bound. The ratio R =ωb Th/ωa Tc. was computed showing only small changes. TABLE V. First step. Starting from an optimal quartet, the pr ocedure creates for a given decreasing set of ωb-s, a decreasing set of of ωa-s for fixed bath temperatures. Tc Th ωb ωoptimal a ωoptimal a /Tc ωb/Th R QF 0.0025 50 60 1.370(-3) 0.5392 1.2 2.226 5.812(-5) 0.0025 50 55 1.273(-3) 0.5090 1.1 2.161 5.013(-5) 0.0025 50 50 1.164(-3) 0.4653 1 2.149 4.241(-5) 0.0025 50 45 1.051(-3) 0.4205 0.9 2.140 3.50549(-5) 0.0025 50 40 9.320(-4) 0.3728 0.8 2.146 2.826(-5) 0.0025 50 35 8.250(-4) 0.3300 0.7 2.121 2.182(-5) 0.0025 50 30 6.985(-4) 0.2794 0.6 2.147 1.613(-5) TABLE VI. A procedure to get an optimal set of pairs of ωa, Tcwhere their ratio tends to zero. T h= 50 for every cold bath temperature, T c. The index ’fl’ stands for the frictionless case, andQup Fdenotes the upper-bound for QF. Tc ωb ωoptimal a ∆Su∆Su,flQF Qup F 0.0025 60 1.370(-3) 0.0333353 0.0285 5.812(-5) 6.084(-5) 0.00125 55 6.365(-4) 0.0285075 0.02406 2.457(-5) 2.626(-5) 0.000625 50 2.91(-4) 0.0242662 0.02021 1.039(-5) 1.098(-5) 0.0003125 45 1.3138(-4) 0.020338 0.01669 4.2939(-6) 4.467(-6) 0.00015625 40 5.825(-5) 0.016788 0.01354 1.7239(-6) 1.759(-6) 0.0000781 35 2.578(-5) 0.013210 0.010357 6.6772(-7) 6.888(-7) 0.0000391 30 1.0914(-5) 0.010356 0.007915 2.4673(-7) 2.468(-7) 42REFERENCES [1] S. Carnot, R´2 sur la Puissance Motrice du Feu et sur les Machines propres ` a D´ evelopper cette Puissance (Bachelier, Paris, 1824). [2] F.L. Curzon and B. Ahlborn, Am. J. Phys. 43, 22 (1975) [3] P.Salamon, B. Andresen and R.S. Berry Phys. Rev. A 15, 2094 (1977) [4] P. Salamon, A. Nitzan, B. 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arXiv:physics/0003008v1 [physics.class-ph] 2 Mar 2000Electromagnetic interaction between two uniformly moving charged particles: a geometrical derivation using Minkowski diagrams C˘ alin Galeriu Physics Department, Clark University, Worcester, MA, 0161 0, USA Abstract This paper presents an intuitive, geometrical derivation o f the relativistic addition of velocities, and of the electromagnetic interac tion between two uni- formly moving charged particles, based on 2 spatial + 1 tempo ral dimensional Minkowski diagrams. We calculate the relativistic additio n of velocities by projecting the world-line of the particle on the spatio-tem poral planes of the reference frames considered. We calculate the real compone nt of the electro- magnetic 4-force, in the proper reference frame of the sourc e particle, starting from the Coulomb force generated by a charged particle at res t. We then obtain the imaginary component of the 4-force, in the same re ference frame, from the requirement that the 4-force be orthogonal to the 4- velocity. The 4-force is then projected on a real 3 dimensional space to get the Lorentz force. I. INTRODUCTION Special Relativity, as presented in today’s textbooks, is a complex mathematical theory. The 1 spatial + 1 temporal dimensional Minkowski diagrams , w hich initially introduce the Lorentz transformation, the time dilatation and the len gth contraction, are soon put aside in favor of an approach based on differential calculus a nd linear algebra. One gets 1little intuitive understanding of the law of relativistic a ddition of velocities, and of the fact that ”magnetism is a kind of ’second-order’ effect arisi ng from relativistic changes in the electric fields of moving charges” [1]. However, by int roducing only slightly more elaborate Minkowski diagrams, and using geometrical deriv ations, one can get back the intuitive understanding, to the great delight of the physic ist who still believes in the spirit of Descartes’ philosophy. II. REAL PLANE AND COMPLEX PLANE: SAME TRIGONOMETRY The complex plane, like the real plane, is a two-dimensional (2D) vector space. The scalar products for the real plane (1) and for the complex pla ne (2) are defined as follows ˆx·ˆx= 1 ˆy·ˆy= 1 ˆx·ˆy=ˆy·ˆx= 0 (1) ˆx·ˆx= 1 ˆi·ˆi=−1 ˆx·ˆi=ˆi·ˆx= 0 (2) where ˆx,ˆyand respectively ˆx,ˆiare the basis vectors. Since both planes have a scalar product, one can talk about or thogonal vectors (their scalar product is zero) and about the magnitude of a vector (t he square root of the scalar product of a vector with itself). This allows us to define the c ircle (the geometric locus of the points equally spaced from a given point), the angle in ra dians (the length, between two points of a circle of radius one, measured along the circumfe rence), and the trigonometric functions sine and cosine (the magnitudes of the projection s of a radius one vector on the two coordinate axes). From these definitions it follows that, fo r both the real and the complex planes, one has the relations: sin2(α) + cos2(α) = 1 (3) [d dαsin(α)]2+ [d dαcos(α)]2= 1. (4) 2From (3)-(4) one can get the derivatives of the trigonometri c functions, the Taylor series expansions of sine and cosine, and then all the well known tri gonometric relations. The only detail we have to keep in mind is that the angle αin the real plane is a real number, while in the complex plane it is a purely imaginary number, du e to the non-positive definite scalar product used in the last case. There are a few more rele vant differences, which we can best point out if we represent the complex plane as an Eucl idean plane. Two vectors in the complex plane are orthogonal if they make the same angl e with the bisecting line of the first quadrant. The circle in the complex plane looks like a hyperbola [2]. Not any line passing through the origin intersects the right (or left) br anch of the hyperbola. This means that there are pairs of lines passing through the origin to wh ich we cannot assign an angle. However, for the triangles we will be working with, the ratio of segments behaves as if it were an angle, of negative value. The true angle is obtained b y symmetry with respect to the first bisecting line, as the pair αand−αindicates in Figure 1. III. RELATIVISTIC ADDITION OF VELOCITIES Consider a reference frame K’ which is moving with a velocity V=Vˆxrelative to another one K, and a particle moving with a velocity v′=v′ xˆx′+v′ yˆy′+v′ zˆz′in the reference frame K’. The reference frames are chosen such that their origins a nd the particle coincide at the space-time point O, as shown in Figure 1. Notice that ˆy=ˆy′andˆz=ˆz′, because Vhas a component only in the x direction. The Oz axis is not plotted, but is similar to the Oy axis. The question is: What is the velocity v=vxˆx+vyˆy+vzˆzof the particle in the reference frame K? The world-line OP of the particle is projected on the complex planes (x,O,ict), (y,O,ict), (z,O,ict), (x’,O,ict’), (y’,O,ict’), (z’,O,ict’), and th e resulting angles from the respective projections give the components of the velocity of the parti cle in the two reference frames considered. For the situation considered the planes (x,O,i ct) and (x’,O,ict’) coincide. It is seen from Figure 1 that 3tan(−α) =EF OE=V ictan(α) =−V ic(5) tan(−β) =DC OD=v′ x ictan(β) =−v′ x ic(6) tan(−γ) =DA OD=v′ y ictan(γ) =−v′ y ic(7) tan(−δ) =EB OE=vy ictan(δ) =−vy ic(8) tan(−θ) =EC OE=vx ictan(θ) =−vx ic. (9) In order to express vxandvyas functions of V, v′ xandv′ ywe need to express δandθas functions of α,βandγ. In the plane (x,O,ict) of the Lorentz boost the addition of ve locities is based on the addition of angles [3] θ=α+β (10) tan(θ) = tan( α+β) =tan(α) + tan( β) 1−tan(α) tan(β). (11) From (11), by substitution of the tangents (5)-(9), it follo ws that vx=V+v′ x 1 +V v′x/c2. (12) Two rectangles, APCD and BPCE, result from the projection pr ocess. It is evident that CP OC=EB OC=EB OEOE OC= tan( −δ) cos(−θ) (13) CP OC=DA OC=DA ODOD OC= tan( −γ) cos(−β). (14) From (13)-(14) it follows that tan(δ) =cos(β) tan(γ) cos(α+β)=tan(γ) cos(α)[1−tan(α) tan(β)]. (15) By substitutions of the tangents (5)-(8) and of cos( α) = [1 + tan2(α)]−1/2we get vy=v′ y(1−V2/c2)1/2 1 +V v′x/c2. (16) A similar expression is obtained for the vzcomponent. 4IV. ELECTROMAGNETIC INTERACTION BETWEEN TWO UNIFORMLY MOVING CHARGED PARTICLES Consider two charged particles (with charges Q1andQ2) at some arbitrary positions, moving with arbitrary, but uniform, velocities. We orient o ur 3D reference frame in such a way that the first particle (which generates the field) is init ially at the origin, moving along the Ox axis with velocity V=Vˆx, and the vector R=Rcos(θ)ˆx+Rsin(θ)ˆyconnecting the two particles is in the (x,O,y) plane. The angle between Rand the Ox axis is θ. The second particle (subject to the electromagnetic field gener ated by the first one) is moving with velocity v=vxˆx+vyˆy+vzˆz. A section through the (x,O,y) plane can be seen in Figure 2. The first particle is at point O and the second one is a t point A. A. Analytical calculation of the Lorentz force The electric field (in Gaussian units) generated by the first p article at the position of the second particle is [4,5] E=Q1R R3(1−V2 c2)[1−V2 c2sin2(θ)]−3/2. (17) The magnetic field generated by the first particle is H=1 cV×E. (18) The Lorentz force acting on the second particle is F=Q2E+Q2 cv×H. (19) From (17)-(19) the Cartesian components of the force [6] are obtained Fx=Q1Q2 R2(1−V2 c2)[1−V2 c2sin2(θ)]−3/2[cos(θ) + sin( θ)vyV c2] (20) Fy=Q1Q2 R2(1−V2 c2)[1−V2 c2sin2(θ)]−3/2sin(θ)(1−vxV c2) (21) Fz= 0. (22) 5B. Geometrical derivation of the Lorentz force The force components (20)-(22) can be obtained in a more grap hical way, if we start with the Coulomb force generated by a charged particle at res t. One key assumption or experimental fact is that in a frame where all the source char ges producing an electric field Eare at rest, the force on a charge qis given by F=qEindependent of the velocity of the charge in that frame [7]. The reference frame K’ in which the s ource particle is at rest is moving with velocity Vrelative to the original frame K. In the reference frame K the particle at A is observed to inter act with the particle at O. The distance between particles is R, the length of the segment OA. In the reference frame K’ the particle at A is observed to inte ract with the particle at B, where the segment BAis a position vector R′parallel to the plane (x’,O,y’). The following construction gives the position of point B: the segment AEis parallel to Oy and intersects the Ox axis at E, whereas the segment EBis parallel to Ox’ and intersects the world-line COat B.BDprojects the point B on the Ox axis at D. Relative to K’, the particle at B exerts a radial Coulomb forc e on the particle at A. This force (in Gaussian units) is F′=Q1Q2 R′3R′(23) where R′=R′[cos(θ′)ˆx′+ sin( θ′)ˆy′]. The key point in getting the force Fin the reference frame K is to notice that the force, in any reference frame considered, is given by the projectio n on the real 3D space of that frame of the 4-force F(which is a Minkowski-space vector), that is F=Freal+Fimag=γ(v)F+ˆiγ(v)P c(24) F=F′ real+F′ imag=γ(v′)F′+ˆi′γ(v′)P′ c(25) where γ(v) = (1 −v2/c2)−1/2andP=F·v. 6We will obtain the 4-force Ffrom its real and imaginary components ( F′ realandF′ imag) in the reference frame K’, then we will decompose the same 4-for ce into its real and imaginary components ( FrealandFimag) in the reference frame K. The Lorentz force we are looking for is just F=Freal/γ(v). From (23)-(25) it follows that F′ real=γ(v′)Q1Q2 R′3R′. (26) To get the imaginary component F′ imagwe use the orthogonality between the 4-force and the 4-velocity, F·V= 0, where the 4-velocity is V=γ(v′)v′+ˆi′γ(v′)c. The orthogonality condition leads to γ2(v′)Q1Q2 R′2R′·v′ R′+F′ imag·ˆi′γ(v′)c= 0 (27) F′ imag=ˆi′γ(v′)Q1Q2 R′2v′ rad c(28) where the radial component of the velocity is v′ rad=R′·v′ R′=v′ xcos(θ′) +v′ ysin(θ′). (29) The components of the force Fin the reference frame K are given by the projection of the 4-force Fon the 3D real space of K. An easy way to do this is to notice that we can decompose F′ real(which has the direction of the segment BA) and F′ imag(which has the direction of the segment BO) into sums of 4-vectors, each of the 4-vectors being paralle l to one of the axes of the reference frame K: rBA=rBD+rDE+rEA (30) rBO=rBD+rDO (31) Because these expansions do not involve any component along the Oz axis, this simply means that Fz= 0. The projections of the 4-force on the Ox and Oy axes are 7γ(v)Fx=F′ realDE BA+F′ imagDO BO(32) γ(v)Fy=F′ realEA BA. (33) The lengths of the various segments needed above are as follo ws: AO=R (34) EA=AOsin(θ) =Rsin(θ) (35) OE=AOcos(θ) =Rcos(θ) (36) BE=OEcos(α) =Rcos(θ) cos(α) (37) DE=BEcos(α) =Rcos(θ) cos2(α) (38) AB= (AE2+BE2)1/2=Rcos(α)[1 + tan2(α) sin2(θ)]1/2=R′(39) We also notice that DO/BO = sin( −α). The force components in (32)-(33) become Fx=γ(v′) γ(v)Q1Q2 R2cos(θ) cos(α)[1 + tan2(α) sin2(θ)]3/2 +iγ(v′) γ(v)Q1Q2 R2v′ rad csin(−α) cos2(α)[1 + tan2(α) sin2(θ)](40) Fy=γ(v′) γ(v)Q1Q2 R2sin(θ) cos3(α)[1 + tan2(α) sin2(θ)]3/2. (41) We can also calculate sin(θ′) =EA AB=sin(θ) cos(α)[1 + tan2(α) sin2(θ)]1/2(42) cos(θ′) =BE AB=cos(θ) [1 + tan2(α) sin2(θ)]1/2. (43) 8If the velocity of the particle at A has the components vx, vy, vz, as measured in the reference frame K, and K is moving with the velocity V′=−Vˆx′relative to K’, then the particle will have the following components of the velocity (compare with equations (12) and (16)) in the reference frame K’ v′ x=vx−V 1−V vx/c2v′ y=vy(1−V2/c2)1/2 1−V vx/c2v′ z=vz(1−V2/c2)1/2 1−V vx/c2. (44) With these components we find that γ(v′) =γ(v)(1−V vx/c2) (1−V2/c2)1/2(45) and the radial velocity (29) becomes v′ rad=(vx−V) cos(α) cos(θ) +vy(1−V2/c2)1/2sin(θ) (1−V vx/c2) cos(α)[1 + tan2(α) sin2(θ)]1/2. (46) Substituting γ(v′) and v′ radin (40)-(41), and also using the fact that sin( α) =i(V/c)γ(V), cos(α) =γ(V) and tan( α) =iV/c, we finally obtain the components in (20)-(21). V. CONCLUSIONS We have presented a geometrical calculation of the relativi stic addition of velocities, and of the electromagnetic interaction between two uniformly m oving charged particles. The geometrical approach used is an elegant, more intuitive and alternative way of obtaining these important results of Special Relativity. We hope our w ork will usefully complement other pedagogical efforts [8–10] centered on Minkowski spac e diagrams. 9REFERENCES [1] Purcell E M, Electricity and Magnetism (New York: McGraw-Hill, 1985) p 238 [2] Minkowski H, Space and Time , reprinted in Einstein A, Lorentz H A, Weyl H, Minkowski H,The Principle of Relativity (New-York: Dover, 1952) [3] Einstein A, The Meaning of Relativity (Princeton: Princeton University Press, 1988) p 37 [4] Landau L and Lifchitz E, Th´ eorie du Champ (Moscow: Mir, 1966) p 119 [5] Donnelly R and Ziolkowski R W, Electromagnetic field gene rated by a moving point charge: A fields-only approach, Am. J. Phys. 62(10), October 1994, p 916 [6] Rosser W G V, Classical Electromagnetism via Relativity (New York: Plenum Press, 1968) p 32 [7] Jackson J D, Classical Electrodynamics (New York: Wiley, 1975) p 580 [8] Mermin D, An introduction to space-time diagrams, Am. J. Phys.65(6), June 1997, p 476 [9] Mermin D, Space-time intervals as light rectangles, Am. J. Phys. 66(12), December 1998, p 1077 [10] Saletan E J, Minkowski diagrams in momentum space, Am. J . Phys. 65(8), August 1997, p 799 10FIGURES FIG. 1. Relativistic addition of velocities. The world-lin e OP is projected on various spa- tio-temporal planes. OA is the projection on (y’,O,ict’), O B is the projection on (y,O,ict) and OC is the projection on (x,O,ict). The planes (x,O,ict) and (x’ ,O,ict’) coincide. FIG. 2. Electromagnetic interaction between two uniformly moving charged particles. CO is the world-line of the source particle, and AG is the world- line of the test particle. In the proper reference frame of the source particle there is a Coul omb force directed along the BA radial direction. 11/G32 /G5C /G5C /GA9/G5B /G5B/GA9 /G4C /G46 /G57/G4C /G46 /G57 /GA9 /G33/G26 /G27 /G24 /G25/G28−α−β −δ−γ /G29/G4C/G4A/G58 /G55 /G48/G03 /G14 /G0F/G03/G26 /G44 /G4F/G4C /G51/G03 /G2A/G44/G4F /G48 /G55 /G4C /G58 /G0F/G03 /G28 /G4F /G48 /G46/G57 /G55/G52 /G50 /G44 /G4A /G51 /G48/G57 /G4C/G46/G03/G4C /G51/G57 /G48 /G55/G44/G46 /G57/G4C /G52 /G51/G29 α −θ/G32/G28 /G27/G4C /G46 /G57 /GA9 /G29 /GA9/G55 /G48 /G44 /G4F/G29 /GA9/G4C /G50 /G44 /G4Aθ /GA9θ /G25 /G24 /G29/G4C/G4A/G58 /G55 /G48/G03 /G15 /G0F/G03/G26 /G44 /G4F/G4C /G51/G03 /G2A/G44/G4F /G48 /G55 /G4C /G58 /G0F/G03 /G28 /G4F /G48 /G46/G57 /G55/G52 /G50 /G44 /G4A /G51 /G48/G57 /G4C/G46/G03/G4C /G51/G57 /G48 /G55/G44/G46 /G57/G4C /G52 /G51/G35α /G2A/G5B /G4C /G46 /G57/G5B/GA9 /G5C/G5C /GA9/G35 /GA9α /G10α /G26

arXiv:physics/0003009v1 [physics.ao-ph] 2 Mar 20001 On the effect of small-scale oceanic variability on topography-generated currents A.´Alvarez SACLANT Undersea Research Centre, La Spezia, Italy E. Hern´ andez-Garc´ ıa and J. Tintor´ e Instituto Mediterr´ aneo de Estudios Avanzados, CSIC-Univ ersitat de les Illes Balears, Palma de Mallorca, Spain Short title: EFFECTS OF SMALL SCALE VARIABILITY ON OCEANIC CURRENTS2 Abstract. Small-scale oceanic motions, in combination with bottom to pography, induce mean large-scale along-isobaths flows. The direction of these me an flows is usually found to be anticyclonic (cyclonic) over bumps (depressions). Here we employ a quasigeostrophic model to show that the current direction of these topographi cally induced large-scale flows can be reversed by the small-scale variability. This re sult addresses the existence of a new bulk effect from the small-scale activity that could h ave strong consequences on the circulation of the world’s ocean.3 Introduction Small-scale ocean motions have an important effect on oceani c flows several orders of magnitude larger than them. The best-known bulk effect of s mall-scale processes is a substantial contribution to the transport of heat, salt, mo mentum, and passive tracers in all parts in the world’s oceans. This effect is usually includ ed in ocean circulation models by modifying the transport and mixing properties of the fluid from their molecular values to larger ones, giving rise to eddy-diffusion approaches of i ncreasing sophistication and predictive power [ Neelin and Marotzke, 1994]. The transport processes parametrized by these effective changes of the diffusive fluid properties ha ve been shown to control important aspects of the Earth’s climate [ Danabasoglu et al., 1994]. Beyond eddy diffusion approaches, physical effects of small- scale activity are still poorly understood. For this reason, the nature and variabil ity of small-scale oceanic motions have been exhaustively examined in different oceano graphic contexts [ Wunsch and Stammer, 1995]. Given the nature of small-scale activity – disordere d, fluctuating and turbulent – a contribution to diffusion and dispersion eff ects is obvious on physical grounds. But a more coherent influence of processes occurrin g at small-scales on large scales motions is unexpected unless some oceanograph ic factor is able to get a significant mean component out of the fluctuating behavior. B ottom topography is one of such factors breaking the symmetry of the fluctuation stat istics, and thus provides a dynamical link for energy transfer from the small to the larg e scale. Evidence has been accumulated in the last decade showing tha t mean flows4 following the topographic contours are often found in the vi cinity of topographic features of different scales [ Klein and Siedler, 1989;Pollard et al., 1991;de Madron and Weatherly, 1994;Brink, 1995]. These topographically generated currents have been shown to influence both local and global aspects of the Earth’ s climate [ Dewar, 1998]. For example, large-scale motions related to topographic an omalies have been found in the North and South Atlantic playing a major role in determin ing regional circulation and climatic characteristics [ Lozier et al., 1995;Saunders and King, 1995]. Coriolis force, topography and fluctuations have been point ed out as the main ingredients to generate these along-isobaths coherent mot ions [ Alvarez et al., 1997; Alvarez and Tintor´ e, 1998; Alvarez et al., 1998]. Briefly, Coriolis force links topography to the dynamics of ocean vorticity. Thus changes in ocean depth provide a symmetry breaking factor distinguishing according to thei r vorticity content otherwise isotropic mesoscale fluctuations. The result is that the mea n effect of small-scale fluctuations does not average to zero yielding the existence of mean flows. Finally, the topographic structure determines the circulation pattern s of the originated currents. On the basis of present knowledge, anticyclonic (cyclonic) tendencies are expected over bumps (depressions) for generated mean flows over topog raphy. However, Alvarez et al. [1998] pointed out that these circulations tendencies coul d be strongly dependent on the properties of the small-scale variability. They theo retically addressed the possibility that the above mentioned circulation pattern c ould be even reversed (cyclonic (anticyclonic) circulations over bumps (depressions)) by the action of the small scales. The same effect is predicted when considering bottom frictio n instead of viscosity as the5 damping mechanism [ Alvarez et al., 1999]. The present Letter attempts to elucidate, by means of computer simulations, if the direction of these m ean flows is sensitive to the statistical characteristics of the small-scale, as it was a rgued on theoretical grounds. Model and results To explore in detail the possible relations between large an d small scales in the presence of topography an ideal ocean represented by a singl e layer of fluid subjected to quasigeostrophic dynamics will be considered. Baroclin ic effects which in real oceans give rise to meso-small scale activity are modeled he re by an explicit stochastic forcing with prescribed statistical properties [ Williams, 1978]. This term might also be considered as representing any high frequency wind forci ng components and other processes below the resolution considered in the numerical model. This random forcing, in combination with viscous dissipation, will bring the oce an model to a statistically steady state. While highly idealized, the simplifying mode ling assumptions above arise from our interest in isolating just the essential processes by which small-scale variability leads to topography-generated currents. Within our approximations, the full ocean dynamics can be de scribed by [ Pedlosky, 1987]: ∂∇2ψ ∂t+/bracketleftBig ψ,∇2ψ+h/bracketrightBig =ν∇4ψ+F , (1) The ocean dynamics described by Eq. (1) is an f-plane quasige ostrophic model6 whereψ(x,t) is the streamfunction, F(x,t) is the above mentioned stochastic vorticity input,νis the viscosity parameter and h=f∆H/H 0, withfthe Coriolis parameter, H0the mean depth, and ∆ H(x) the local topographic height over the mean depth. The Poisson bracket or Jacobian is defined as [A,B] =∂A ∂x∂B ∂y−∂B ∂x∂A ∂y. (2) A set of numerical simulations has been carried out to determ ine the dependence of the large scale pattern circulation on the structure and v ariability of the small-scales. The description of the numerical model and different paramet ers employed in the simulations are summarized in Appendix A. A randomly genera ted bottom topography is used in all the cases. As a way of changing in a continuous ma nner the statistical properties of the forcing F(x,t) we assume it to be a Gaussian stochastic process of zero mean, white in time, and spatial spectrum given by S(k)∝k−y, wherekis the wavenumber. A positive exponent yrepresents relative-vorticity fluctuations more dominant at the large scales, whereas negative yrepresents fluctuations dominant at the smaller scales. The distribution of fluctuation variance am ong the scales can thus be controlled by varying y. The spectrum of the energy input corresponding to the above stochastic vorticity forcing is also white in time, with a wa venumber dependence given by the relation E(k) =S(k)k−2. We have started first considering a situation where the small-scale variability is described by S(k)∝k0. This power-law has been observed for vorticity forcing induced by winds in the Pacific ocean [ Freilich and Chelton, 1985]. The model has been integrated until a stationary state is ach ieved. Figure 1b shows the7 mean currents obtained from this specific simulation. In the mean state the currents do not average to zero, despite the isotropy of the fluctuations and dissipation. Instead the final mean state is characterized by the existence of large-s cale mean currents strongly correlated with bottom topography. The spatial correlatio n coefficient between the streamfunction and the bottom topography is for this case 0 .85. This positive spatial correlation implies the existence of mean anticyclonic (cy clonic) circulations over bumps (depressions). As a next step, we have modeled the action of s mall-scales as a noisy process with a correlation described by the power-law k4.8. This spectrum describes a situation where the small-scale variability is more energ etic than the one induced by the previous k0power-law. The response of the system is drastically change d by this small-scale activity. As shown in Figure 1c the mean sta te of the ocean displays a pattern of circulation practically uncorrelated with bott om topography. Specifically, the spatial correlation coefficient is 0 .091. Increasing again the exponent of the power law toS(k)∝k6, we obtain the generation of mean currents anticorrelated w ith bottom topography, as it can be observed from Figure 1d. The spatial correlation coefficient is −0.77 in this case of high small-scale activity, indicating the existence of mean cyclonic (anticyclonic) motions over bumps (depressions). Note tha t Figures 1c and d display topography-generated currents much weaker than those obta ined for the k0power-law case, Figure 1b. This feature comes from the scale-selectiv e character of the viscosity, more efficient at small scales where the forcing energy is most localized in the k4.8and k6cases. Besides this effect, it should also be mentioned that f orcing with a k0spectral power-law directly provides more energy input to the large- scale components than the8 k4.8andk6forcings. Additional numerical simulations, for different initial conditions and noise and topography realizations, consistently confir m the results of the simulations presented in Fig. 1, that is the sensibility of the large-sca le circulations not only to the particular structure of the underlying topography but also to the characteristics of the small-scale variability of the environment. In particular , as the small-scale content in the vorticity forcing is augmented with respect to the large -scale one, mean currents are always seen to reverse direction. Conclusion On the basis of the property of potential-vorticity conserv ation, anticyclonic (cyclonic) motions are traditionally expected over topogr aphic bumps (depressions) [Pedlosky, 1987]. If potential vorticity is not preserved because of th e presence of some kind of forcing mechanism, then different circulation p atterns can be generated. Small-scale activity constitutes a systematic and persist ent forcing of the circulation in the whole ocean. Due to the relatively small and fast space and time scales that characterize this variability, the physical characterist ics of this forcing are usually described in terms of their statistical properties [ Williams, 1978]. In other words, small-scale activity can be considered as a fluctuating back ground in which the large-scale motions are embedded. The relevance of the role played by this fluctuating environment in modifying the transport and viscous propert ies of the large scales is widely recognized. Beyond these diffusive and viscous effect of the small-scale activity, the numerical results presented in this Letter show that in t he presence of bottom9 topography, statistical details of the variability of the s mall-scales can induce different large-scale oceanic circulations. The strength of this effe ct will be affected by the spectral characteristics of the topographic and forcing fie lds as well as by the real baroclinic nature of the ocean. Preliminary computer simul ations indicate that the strength of the mean currents increases when the topography contains more proportion of large-scale features. This effect and the dependence of th e current direction on the forcing spectral exponent, presented in Fig. 1, nicely vali dates the theoretical results in [Alvarez et al., 1998]. Extension of the theoretical methods to the baroclin ic case is in progress. However, a complete analysis of the influence of different forcings and topography shapes can only be addressed numerically. The sh ape of the topography was already shown to play a fundamental role in the energy tra nsfer between different scales in baroclinic quasigeostrophic turbulence [ Treguier and Hua, 1988]. Finally, the results shown in this Letter stress the need for a better obse rvational characterization of the space and time variability of oceans at small-scales in o rder to achieve a complete understanding of the large-scale ocean circulation. Appendix A: Numerical model description Numerical simulations of Eq. (1) have been conducted in a par ameter regime of geophysical interest. A value of f= 10−4s−1was chosen as appropriate for the Coriolis effect at mean latitudes on Earth and ν= 200m2s−1for the viscosity, a value usual for the eddy viscosity in ocean models. We use the numerical sche me developed in Cummins [1992] on a grid of 64 ×64 points. The distance between grid points corresponds to 2 010 km, so that the total system size is L= 1280 km. The algorithm, based on Arakawa finite differences and the leap-frog algorithm, keeps the val ue of energy and enstrophy constant when it is run in the inviscid and unforced case. The consistent way of introducing the stochastic term into the leap-frog scheme c an be found in Alvarez et al. [1997]. The amplitude of the forcing has been chosen in order to obtain final velocities of several centimeters per second. The topographic field is r andomly generated from a isotropic spectrum containing, with equal amplitude and ra ndom phases, all the Fourier modes corresponding to scales between 80 km and 300 km. The mo del was run for 5×105time steps (corresponding to 206 years) after a statistical ly stationary state was reached. The streamfunction is then averaged during this la st interval of time. Acknowledgments. Financial support from CICYT (AMB95-0901-C02-01-CP and MA R98-0840), and from the MAST program MATTER MAS3-CT96-0051 (EC) is greatly ackn owledged. Comments of two anonymous referees are also greatly appreciated.11 References Alvarez, A., E. Hern´ andez, and J. Tintor´ e, Noise-sustain ed currents in quasigeostrophic turbulence over topography, Physica A,247, 312-326, 1997. Alvarez, A., and J. Tintor´ e, Topographic stress: Importan ce and parameterization, Ocean modeling and parameterization, edited by E. P. Chassignet and J. Verron, Kluwer Academic Publishers, Netherlands, 1998. Alvarez, A., E. Hern´ andez, and J. Tintor´ e, Noise rectifica tion in quasigeostrophic forced turbulence, Phys. Rev. E,58, 7279-7282, 1998. Alvarez, A., E. Hern´ andez, and J. Tintor´ e, Noise-induced flow in quasigeostrophic turbulence with bottom friction, Phys. Lett. A,261, , 179-182, 1999. Brink, K. H., Tidal and lower frequency currents above Fiebe rling Guyot, J. Geophys. Res.,100, 10817-10832, 1995. Cummins, P. F., Inertial gyres in decaying and forced geostr ophic turbulence, J. Mar. Res., 50,545-566, 1992. Danabasoglu, G., J. C. MacWilliams, and P. R. Gent, The role o f mesoscale tracer transports in the global ocean circulation, Science, 264, 1123-1126, 1994. de Madron, D. X., and G. Weatherly, Circulation, transport a nd bottom boundary layers of the deep currents in the Brazil Basin, J. Mar. Res., 52, 583-638, 1994. Dewar, W. K., Topography and barotropic transport control b y bottom friction, J. Mar. Res., 56,295-328, 1998. Freilich, M. H., and D. B. Chelton, Wavenumber spectra of Pac ific winds measured by the Seasat scatterometer, J. Phys. Oceanogr., 16, 741-757, 1985.12 Klein, B., and G. Siedler, On the origin of the Azores current ,J. Geophys. Res., 94, 6159-6168, 1989. Lozier, M., W. Owens, and R. Curry, The climatology of the Nor th Atlantic, Prog. Oceanogr., 36,1-44, 1995. Neelin, D. J., and J. Marotzke, Representing ocean eddies in climate models, Science, 264, 1099-1100, 1994. Pedlosky, J. Geophysical Fluid Dynamics, 212 pp., Springer-Verlag, New York, 1987. Pollard, R., M. Griffiths, S. Cunningham, J. Read, F. Perez, an d A. Rios, A study of the formation, circulation, and ventilation of Eastern North A tlantic Central Water, Prog. Oceanogr., 37, 167-192, 1991. Saunders, P., and B. King, Bottom currents derived from a shi pborne ADCP on WOCE cruise A11 in the South Atlantic, J. Phys. Oceanogr., 25, 329-347, 1995. Treguier, A. M., and B. L. Hua, Influence of bottom topography on stratified quasi-geostrophic turbulence in the Ocean, Geophys. Astrophys. Fluid Dynamics, 43, 265-305, 1988. Williams, G. P., Planetary circulations:1. Barotropic rep resentation of Jovian and terrestrial turbulence, J. Atmos. Sci., 35, 1399-1426, 1978. Wunsch, C., and D. Stammer, The global frequency-wavenumbe r spectrum of oceanic variability estimated from TOPEX/POSEIDON altimetric mea surements, J. Geophys. Res., 100, 24895-24910, 1995. A. Alvarez, SACLANT Undersea Research Centre, 19138 San Bar tolomeo, La Spezia, Italy. (e-mail: alvarez@saclantc.nato.int)13 E. Hern´ andez-Garc´ ıa and J. Tintor´ e, Instituto Mediterr ´ aneo de Estudios Avanzados, CSIC-Universitat de les Illes Balears, 07071 Palma Mallorc a, Spain. (e-mail: emilio@imedea.uib.es; dfsjts0@ps.uib.es) Received September 01, 1999; revised November 23, 1999; acc epted January 19, 2000.14 Figure 1. a) Random bottom topography. Maximum and minimum topograph y heights are 500mand−599m, respectively, over an average depth of 5000 m. b) Computed mean streamfunction ψ(x,t) inm2s−1for the case when the small-scale variability is described by a spectral power law k0. Bottom topography levels have been superimposed (black lines) as reference over the streamfunction field. The stron g correlations between the streamfunction and topography are clear from this figure. c) Same as b) but with power spectra law k4.8. In this case the flow remains practically uncorrelated with the underlying topography (black lines). d) For k6the flow is anticorrelated with the topography.This figure "figpaper.gif" is available in "gif" format from: http://arXiv.org/ps/physics/0003009v1

arXiv:physics/0003010v1 [physics.ao-ph] 3 Mar 20001 To appear in the Journal of Geophysical Research , 2000. A non-linear optimal estimation inverse method for radio oc cultation measurements of temperature, humidity and surface pressur e Paul I. Palmer1, J. J. Barnett Department of Physics, Clarendon Laboratory, Oxford, Unit ed Kingdom J. R. Eyre, S. B. Healy Satellite Applications Division, United Kingdom Meteorol ogical Office, Bracknell, United Kingdom Abstract An optimal estimation inverse method is presented which can be used to retrieve simultaneously vertical profiles of temperature and specific humidity, in ad dition to surface pressure, from satellite-to-satellite radio occultation observations o f the Earth’s atmosphere. The method is a non-linear, maximum a posteriori technique which can accommodate most aspects of the real radio occultation problem and is found to be stable and to con verge rapidly in most cases. The optimal estimation inverse method has two distinct advanta ges over the analytic inverse method in that it accounts for some of the effects of horizontal gradi ents and is able to retrieve optimally temperature and humidity simultaneously from the observat ions. It is also able to account for observation noise and other sources of error. Combined, the se advantages ensure a realistic retrieval of atmospheric quantities. A complete error analysis emerges naturally from the optima l estimation theory, allowing a full characterisation of the solution. Using this analysis a qua lity control scheme is implemented which allows anomalous retrieval conditions to be recognis ed and removed, thus preventing gross retrieval errors. The inverse method presented in this paper has been implemen ted for bending angle measurements derived from GPS/MET radio occultation obser vations of the Earth. Preliminary results from simulated data suggest that these observation s have the potential to improve NWP model analyses significantly throughout their vertical ran ge.2 1. Introduction Radio occultation (RO) experiments have played a prominent role in the NASA programme for so- lar system exploration for more than two decades and have contributed to studies of the atmosphere of Mars [ Fjeldbo and Eshleman, 1968], Venus [ Fjeldbo and Kliore, 1971], Jupiter [ Kliore et al, 1975], Saturn [Lindal et al, 1985], Uranus [ Lindal et al, 1987], and Neptune [ Lindal 1992]. This method of radio occulta- tion uses a receiver on Earth and a satellite occulted by a planetary atmosphere (which may occur from either a fly-by or by a satellite orbit of the planet). Suitably accurate atmospheric RO measurements of the Earth’s atmosphere became possible with the ad- vent of the Global Positioning System (GPS), but it was not until the late 1980s −early 1990s that the po- tential of RO using the GPS was widely appreciated (e.g.Gurvich and Krasil’nikova, [1990]). The radio occultation method used to sound the Earth’s atmosphere is different from that used by the planetary experiments, in that both the receiver and the transmitters are orbiting the planet. Data from the prototype GPS space-borne re- ceiver, GPS/MET, launched in April 1995, confirmed the potential of obtaining accurate, global observa- tions of the Earth’s atmosphere from the radio occul- tation technique. Temperature comparisons between early results from the GPS/MET receiver and collo- cated radiosondes and numerical weather prediction (NWP) model analyses showed good agreement (e.g. Kursinski et al, [1996]; Ware et al, [1996]). The analytic method of inverting radio occulta- tion measurements to obtain meteorological param- eters (i.e. the method used to sound other planetary atmospheres) involves the use of an integral trans- form, using the assumption of a horizontally homo- geneous atmosphere, to obtain a profile of refractiv- ity (as a function of geometric height) [ Fjeldbo and Eshleman, 1968]. The hydrostatic relation is used to obtain pressure and temperature from refractiv- ity via density. For the Earth’s atmosphere, where a reasonable prior knowledge of horizontal gradients is available, the analytic inversion does not represent the most suitable method since inadequate modelling of such gradients can cause large retrieval errors (e.g. Ahmad and Tyler, [1998]). Eyre, [1994] addresses this issue and suggests a sta- tistically optimal retrieval approach, using variational methods, to enable the direct assimilation of bending angle or refractivity ( Healy and Eyre, [1999] investi-gate the latter quantity). Zou et al, [1995] also looked at the impact of atmospheric radio refractivity mea- surements using a 4-D variational data assimilation approach. Their results showed that the measure- ments were effective in recovering the vertical pro- files of water vapour, and found that the accuracy of the derived water vapour field was significantly bet- ter than that obtained through the analytic retrieval technique. The assimilation of these measurements were also shown to provide useful temperature infor- mation. There have also been several numerical ex- periments which have assessed simulated GPS/MET refractivity measurements to predict cyclonic distur- bances (e.g. Kuo et al, [1998]), and have concluded that these measurements are likely to have a signif- icant impact on short-range operational NWP, with the caveat that the number of GPS receivers will have to be increased before the full potential impact of this measurement could be realised. In this paper we utilise a non-linear optimal esti- mation technique which is implemented and validated using an ensemble of simulated retrieval scenarios, us- ing the bending angle quantity as the ‘observation’. Section 2 outlines the details of the RO method- ology for the Earth’s atmosphere necessary to derive the bending angle quantity, recalls the analytic in- verse method and discusses the impetus for pursuing an alternative inverse method. In section 3 we out- line the theory for the non-linear optimal estimation inverse method and give details of its implementation for GPS RO observations. Section 4 is devoted to details of the validation of the optimal estimation in- verse method with reference to GPS RO observations. The sensitivity of the inverse model assumptions is also investigated, and we conclude the paper with a discussion of the results obtained. 2. Radio occultation measurements of Earth’s atmosphere Kursinski et al, [1997] give a detailed description of the method used to measure the RO atmospheric observables; the following section gives a summary of the theory, assuming no external encryption of the signals. The GPS satellites transmit on two L-band radio frequencies1. Assuming a continuous link between the receiver and transmitter, when the receiver passes be- hind the atmosphere with respect to a GPS transmit- 1Namely, L1: 1.57542 GHz and L2: 1.2272 GHz.3 ter the signal travels through the atmosphere and is refracted in response to variations of refractive index along its path. This refraction causes the ray to travel over a longer path than it would in the absence of the atmosphere, in accordance with Fermat’s principle of least time, which subsequently causes an atmospheric time delay in the received signal. The Doppler shift of the signal is calculated from the additional atmospheric delay (the derivative of the phase delay). Using the geometry and notation of figure 1, the Doppler shift fdof the carrier frequency f0measured by the receiver is given by: fd=f0/bracketleftbigg(vT·nT+vR·nR) c/bracketrightbigg (1) wherevT,Rare transmitter and receiver velocity vec- tors,nT,Rare path direction vectors of the transmit- ter and receiver, and cis the velocity of light. There are also relativistic terms which need to be considered in equation 1 (due to different grav- itational potentials and higher order corrections for spacecraft velocity) but these can be eliminated based on knowledge of orbital geometry and the Earth’s gravity field [ Kursinski et al, 1997]. Note that the rel- ative positions and velocities of the two satellites can be calculated very accurately using available tracking data, which is independent of radio occultation data. By specifying radial and tangential components of the velocity of satellite iin the plane coinciding with the ray trajectory as vr iandvt i, and taking into ac- count Snell’s law, the angles φRandφTcan be calcu- lated from the following relations: fd=f0c−1(vr RcosφR+vr TcosφT+ vt RsinφR−vt TsinφT) (2) The cumulative effect of the atmosphere on the ray path can be expressed in terms of the total refractive bending angle, ε, as a function of the impact param- eter,a. The impact parameter may be defined as the perpendicular distance between the local curva- ture of the Earth at the tangent point of the ray and the asymptotic straight line followed by the ray as it approaches the atmosphere. From Bouguer’s rule [ Born and Wolf, 1993], and the geometry defined by figure 1, ε(a) can be calcu- lated thus rRsinφR=rTsinφT=a (3) ε=φR+φT+θ−π (4)It is this rule that introduces the assumption of spher- ical symmetry ( nrsinφ=a, where ais a constant along the ray path), i.e. a horizontally homogeneous atmosphere. Departures from this assumption can introduce significant errors if not properly accounted for. These errors have been studied by Ahmad and Tyler, [1999] and Healy, [1999], but are not addressed in the study presented here. Measurements of the time delay become possible for neutral atmospheric sounding when the GPS sig- nal begins to transect the mesosphere at an altitude of about 85 km; at this altitude the atmospheric phase delay is about 1 mm (3 ×10−12s) which can be ob- served by the LEO GPS receiver [ Ware et al, 1996]. Further information about the measurement charac- teristics may be obtained from Kursinski et al, [1997]. 2.1. The analytic inverse method Using an Abel integral transform (equation 5), or making use of a similar integral transform when applying Fresnel diffraction theory [ Mortensen and Høeg, 1998], these bending angle measurements can be inverted to obtain a profile of refractivity. For com- pleteness sake, the ‘forward’ Abel integral transform (equation 6) is presented alongside the ‘inverse’ Abel integral transform: lnn(x) =1 π/integraldisplay+∞ xε(a)(a2−x2)−1/2da(5) ε(a) =−2a/integraldisplay+∞ a/parenleftbigg∂lnn(x) ∂x/parenrightbigg (x2−a2)−1/2dx,(6) where nis the refractive index and xis the refractive radius (i.e. x=rn). Geometric height levels, z, can be obtained from the refractive index profile, as a function of impact parameter, and the local radius of curvature, Rc, thus z=x n−Rc. (7) Because refractive index is near unity, refractivity Nis used to describe the refractive medium which is given by N= (n−1)×106. Refractivity is affected primarily by air density (dependent on pressure and temperature) and water vapour density, thus the measurement contains infor- mation about both. Equation 8 describes this rela- tionship,4 N=c1Pa T+c2PW T2(8) where Pais the total atmospheric pressure (hPa), PW is the partial pressure of water vapour, Tis the tem- perature (K), and c1andc2represent constants of proportionality, whose values are 77 .6 (KhPa−1) and 3.73×105(K2hPa−1), respectively. The form of the dry and moist terms in equation 8 is from Smith and Weintraub, [1953]. Refractivity is also affected by charged particles in the ionosphere and the scattering by water droplets suspended in the atmosphere. The first-order iono- spheric contribution to refractivity can be removed by combining the two GPS signals (described by Vorob’ev and Krasil’nikova, [1997]), leaving higher- order terms, and the scattering contribution is found to be negligible compared to the contribution due to air and water vapour density [ Kursinski et al, 1997]. There is no measurement information to allow the separation of the effects of temperature and water vapour, and therefore these quantities can be re- trieved only using prior information. If the abso- lute humidity is judged to be small (e.g. in the cold- est regions of the troposphere and stratosphere, with temperatures less than 250 K), it may be neglected, and density calculated from refractivity. The hydro- static relation can be used to calculate values of pres- sure, and hence temperature. However, if humidity is judged to be significant, then an iterative process may be used to calculate temperature/humidity if an a priori profile of humidity/temperature is used. This prior information can be taken from various sources such as collocated NWP model output. The inability of this inverse method to account for horizontal refractivity inhomogeneities, and the sub- optimal way this method retrieves temperature and humidity with prior values, represents two key disad- vantages of this method, and form part of the impetus to develop a new inverse method. The hydrostatic relation, used to compute values of pressure on the retrieved height levels, requires an assumed pressure value at a particular geometric height level. A variety of methods have been imple- mented to tackle this initial value problem. Kursin- ski et al, [1996] assumed a temperature of 260 K at 50 km. This method has the problem that if the as- sumed temperature is inconsistent with the measure- ments an error is introduced, which decreases expo- nentially with depth. More elaborate methods (e.g. Rocken et al, [1997] and Steiner et al, [1999]) initialisethe GPS/MET retrieval at some high altitude (e.g. 100 km) using climate model data, and combine the measurements and model data to minimise downward propagation of errors. In principle, the method pre- sented in this paper also combines the model data and the observations but achieves this is an optimal way. The method presented also has the advantage of a straightforward error characterisation. Since GPS RO observations sometimes reach near- surface altitudes (i.e. less than 1 km from the sur- face), surface pressure is also retrieved using the op- timal estimation inverse method. 3. The optimal estimation inverse method The method outlined here is also known as one- dimensional variational data analysis. The main advantage of this method is that it pro- vides simultaneous estimates of temperature and hu- midity profiles that are statistically optimal, given prior estimates from an NWP model (together with their error covariances). It also provides a framework for assessing the error characteristics of the estimates. In this study only the 1-d problem has been stud- ied. However, the errors introduced by the neglect of the horizontal gradients have been estimated and al- lowed for as part of the error budget (see section 3.2 “Forward modelling errors”). 3.1. Theory A brief description of the theory used in optimal estimation in presented here; a more detailed descrip- tion may be found in Rodgers, [1976] and Rodgers, [1990]. For brevity, the observation noise, the error associ- ated with any forward modelling parameters and the forward model error (which includes the representa- tiveness error [ Lorenc, 1986] ) will be accounted for in one vector which will be denoted by ǫand its ensem- ble characteristics described by the covariance matrix E. The rationale behind optimal estimation is to min- imise a cost functional J(x) (or to solve ∇xJ(x)=0), which measures the degree of fit of estimates of the atmospheric state to the measurements and to some prior information, and possibly to some other physical or dynamical constraints. In this case J(x) is given by5 J(x) = ( yo−y(x))TE−1(yo−y(x)) + (x−xb)TC−1(x−xb) (9) where xbandxrepresent the background and up- dated state vectors, respectively; yoandy(x) repre- sent the observation vector and the estimated obser- vation vector calculated from the state vector ( Eyre, [1994]), respectively; and Crepresents the back- ground error covariance matrix. There are a number of methods available to min- imiseJ(x): the scheme described here uses the Leven- berg-Marquardt iterative method [e.g. Press et al, 1992]: xi+1=xb+ ((1 + γ)C−1+KTE−1K)−1 [(KTE−1(yo−y(xi))) + (γC−1+KTE−1K)(xi−xb)] (10) where Kis∇xiy(xi),γis a non-dimensional weight- ing factor2, and all other variables are as before. Using the optimal estimation theory it is possible to obtain an error covariance for the retrieved prod- ucts. Indeed, it can be argued that the retrieved prod- ucts are of limited value without an estimate of their uncertainty. The solution error covariance ˆSis given approximately (i.e. at the linear limit) by ˆS= (C−1+KTE−1K)−1. (11) The solution error covariance can then be com- pared with the prior error covariance to ascertain how the retrieval has improved upon the prior knowledge of the atmospheric state. 3.2. Implementation of the optimal estimation inverse model This subsection describes in detail the components of equation 9. The background state vector and its uncer- tainty covariance matrix In this case the back- ground knowledge of the atmosphere state xbwas obtained from short-range forecasts provided by the UKMO unified model [ Cullen, 1993]. The model from which the data are derived had 19 levels, which were expressed on hybrid-sigma pressure coordinates 2for increasing values of γthis minimisation method degen- erates into the method of steepest descent.(surface −10 hPa). The global model had a resolution of 0.833◦(180◦/217) latitude and 1.25◦(360◦/288) longitude. The data used are 6-hour forecasts which have been interpolated to occultation event positions, using the mean latitude and longitude of each occultation3. These 19 levels are linearly interpolated (in ln- pressure) onto the state vector levels used for TOVS retrievals [ Eyre, 1990]. CIRA climatology [ CIRA, 1986] is assumed above the UKMO model account- ing for the latitudinal and seasonal variation of the profile. This climatology provides a reasonable prior and first guess information in the upper stratosphere. For the forecast error covariance matrix C(de- scribed in Eyre, [1989]), lower atmospheric values (surface −50 hPa) were generated from radiosonde −forecast difference statistics and upper stratosphere values were found by regression from the levels provided [Eyre, 1989]. The radio occultation retrieval uses 40 tempera- ture elements, 15 ln(specific humidity) elements4and a surface pressure element from this forecast error co- variance matrix. The temperature and ln(specific hu- midity) inter-quantity covariance values have been set to zero. These inter-quantity covariances are not well known and assuming zero covariance between them is more conservative than an erroneous covariance. The surface pressure element is uncorrelated with both temperature and ln(specific humidity). Since CIRA climatology is used to form the a pri- ori(and the first-guess) it is necessary to consider the errors that may be attached to such information. In general, if the standard deviation values from the diagonal elements of the forecast error covariance ma- trix are smaller than the uncertainties assumed for the climatology, then the climatological errors are used at the levels in the upper atmosphere described by the CIRA climatology (off-diagonal elements remain the same): at latitude θ, for|θ| ≥45◦σ=15 K (winter) andσ=5 K (summer); and for |θ| ≤45◦σ=5 K. The values for the diagonal elements of the UKMO forecast error covariance matrix are shown by figure 2. As expected for temperature, the lower atmosphere 3The mean latitude and longitude of an occultation corre- sponds typically to altitudes in the lower stratosphere/up per troposphere. 4Specific humidity is expressed as the natural logarithm of specific humidity since forecast errors in this quantity are more constant than in specific humidity.6 forecast errors are reasonably small (of the order of 1.5 K) and increase as a function of altitude. The actual values have been developed over recent years at the UKMO and reflect the average error in six-hour forecasts. The observation vector and its error covari- ance matrix The observation vector yocontains bending angle measurements as a function of impact parameter (section 2). In practice, atmospheric phase delay measurements from the GPS/MET receiver are low-pass filtered to reduce noise. The cut-off frequency of the filter is tuned to pass phase variations corresponding to verti- cal scales of 2 to 3 km in the stratosphere and approx- imately 200 m in the lower troposphere [ Rocken et al, 1997]. From the phase observable, Doppler shifts (and subsequently bending angle profiles) for the two GPS signals are computed. First-order ionospheric effects are removed from the data by combining the two sig- nals to form a single corrected profile [ Vorob’ev and Krasil’nikova, 1994]. Typically, after filtering, there are 100 −200 neutral atmosphere bending angle mea- surements, which span a vertical range of approxi- mately 0.5 −60 km. In addition to the observation error covariance ma- trix consisting of observation noise estimates, errors from the forward modelling and forward model pa- rameters are considered. Observation noise Observation errors are cre- ated by the hardware of the measurement system and by the pre-processing of the observations. The ob- servation noise estimates are taken from the work described in Luntama, [1997]. They include ther- mal noise, residual errors from the ionospheric cor- rection, local multipath (distortions when the trans- mitted signal is reflected from a surface near the sig- nal propagation path), orbit determination accuracy, and clock instabilities of low-Earth-orbit receiver and GPS satellites and ground stations. These observation error estimates are based on phase noise levels during the measurement or esti- mated from other noise sources during a radio occul- tation event. The effect from satellite clock errors and from the selective availability military encryp- tion process were found to be negligible by assum- ing a differencing decryption technique (involving the differencing of the several sets of signals, using the satellites and the ground stations) [ Kursinski et al, 1997]. The only exception is the residual error from the ionospheric correction which was obtained from refractivity error estimates published in Kursinski etal,[1997] and mapped to bending angle space using a forward model [ Luntama, 1997]. The observation error estimates5used in the op- timal estimation technique are shown by figure 3. These estimates represent normal atmospheric condi- tions with a relatively small multipath error (3 mm) and normal ionospheric conditions. Panel (a) shows that there are a number of noise contributions of com- parable size in the lower atmosphere. At altitudes above 30 km the residual ionospheric correction er- ror begins to dominate the total bending angle error curve (panel (b)). These noise estimates are assumed to be fully in- dependent, i.e. their inter-level (and inter-quantity) covariance is zero. However, the bending angle mea- surements do contain a small, local correlation be- tween successive levels due to filtering of the phase measurements. Because this correlation is small, the diagonal form is a good approximation to the full ma- trix [J. P. Luntama: personal communication]. Real observations from the GPS/MET data used have been found to be noisier than theoretical esti- mates [ Luntama, 1997]. Bending angle fluctuations in the upper stratosphere (of the order of 10−5radi- ans) are present and are thought to be due to residual errors from the LEO satellite clock calibration in the differencing decryption technique (see “Observation noise”) [ Syndergaard, 1999]. As such, a suitable error is attached to reflect the upper atmosphere measure- ments. Forward modelling errors In this work the forward model used to map from state space to obser- vation space is described in Eyre, [1994] but applied to an atmosphere approximated as spherical symmet- ric about the given profile at the tangent point. Es- sentially the geophysical parameters are converted to refractivity as a function of height, and subsequently impact parameter using the local radius of curvature. The resulting profile is mapped into observation space using the ‘forward’ Abel integral transform (equation 6). The two main forward modelling errors are due to the assumption of a horizontally homogeneous atmo- sphere and a representativeness error. Estimates for the first of these errors are obtained using a version of the forward model which can ac- 5These estimates have been computed by a nominal bend- ing angle profile defined using an exponential curve with a scale height of approximately 7 km [J. P. Luntama: personal communication].7 count for horizontal inhomogeneities in the plane of the ray path (described in Eyre, [1994]) and com- paring observation vectors with the version of the forward model which assumes a horizontal homoge- neous atmosphere [ Palmer, [1998]]. Mid-latitude two- dimensional NWP fields (0 −360◦) were used to sim- ulate typical horizontal gradients in temperature and humidity. By considering small sections of the field at a time (typical of the horizontal resolution of ra- dio occultation measurements which is of the order of 300 km), the complete field was traversed. Comput- ing the ensemble mean from the difference between the two versions of the forward models allowed a rea- sonable estimate of the forward modelling error in- curred by the spherical symmetry assumption to be computed. It is noted that this error estimate does not represent the true error in observation space since the forward model does not simulate the full error characterisation. Both Ahmad and Tyler, [1999] and Healy, [1999] consider bending angle errors from hori- zontal gradients for specific cases. However there is no published material that quantifies this error statisti- cally. Simulation with a full 3-d raytracer through the UKMO mesoscale model fields suggest that the errors are approximately 3% for ray paths near the surface, which is consistent with the value used in this work. An error arises from representing an intrinsically high resolution problem with a crude resolution. This type of error is often called a representativeness error and describes the error from the inability of NWP model vertical grids to represent small-scale atmo- spheric structure, which are evident in GPS/MET RO measurements [ Kursinski et al, [1997]]. The method used to estimate this quantity is described by Healy, [1998], and is found to be of the order of 2% of the bending angle measurement in the troposphere and upper stratosphere, decreasing slightly in the middle stratosphere. This variation in the error is associated with the temperature variations in these region. The forward model used is based on geometric optics therefore does not account for atmospheric diffraction. However, Kursinski et al, [1997] have shown that the geometric optics assumption is suc- cessful in describing propagation characteristics above a certain diffraction limit, and diffraction is therefore not considered here. Forward model parameter errors Uncertain- ties associated with the physical constants used to model the physical system also cause modelling er- rors. The major physical constants used in the for- ward model are the refractivity coefficients ( c1andc2in equation 8) and the local radius of curvature. The uncertainty of the refractivity coefficients do not represent the error associated with their values but the uncertainty of the measured quantity; for this reason the information will not be included in the to- tal observation error budget since it will result in a bias in the retrieval6. The local radius of curvature assumed for zero altitude is a parameter that is used to compute the bending angle observations from at- mospheric phase delay, therefore any error associated with this parameter will be present in the bending an- gle observations. This parameter is also used to com- pute geometric height levels from impact parameter levels. The uncertainty of this value is estimated to be approximately 100 metres [ Kursinski et al, [1997]]. Total observation error The total observation error covariance Eis constructed by adding the co- variance matrices from observation noise, forward modelling and forward model parameters. The diagonal terms have also been constrained not to fall below a minimum value to account for the noisy upper stratosphere measurements. Figure 3 shows how the standard deviation values of the principal diagonal from each error contribute to the observation error covariance matrix. The dominant source of error for the majority of the vertical range considered is the forward modelling error, i.e. representativeness error and horizontal in- homogeneity error, with the upper stratospheric noise limit providing the second largest contribution to the total error. In the upper stratosphere the total bend- ing angle reverts to the upper level minimum noise used. At near-surface altitudes, the forward model parameter error, i.e. local radius of curvature, is sig- nificant. The overall effect from the local radius of curvature decreases exponentially due to the hydro- static relation. 3.3. Convergence and quality control The method used to judge convergence relies upon values of the cost function, i.e. if the relative change is smaller than a specified value (0.5%) then the solution is determined to have converged. This method alone is found to be a good indicator of convergence in this case. For the work presented in this paper, the maxi- mum number of iterations considered is 10; if the so- 6Optimal estimation theory assumes that all the errors are unbiased.8 lution has not converged (determined by the method presented above within 10 iterations) then the calcu- lation is halted and a numerical ‘flag’ set. If, after convergence has been determined, the J(x) value is greater than the χ2value given the number of degrees of freedom at a set confidence level (in this case 99.9%) then a numerical ‘flag’ is set. Retrievals with flags set are omitted from any statistics. Furthermore, to ensure the solution computed at each iteration is physical, the ln(specific humidity) el- ements are checked for super-saturation and corrected if necessary. 4. Results In this section the performance of the optimal esti- mation retrieval scheme is examined using simulated profiles and realistic error estimates. For each simu- lated profile, a ‘true’ profile is established by taking one of a set of profiles of UKMO model analyses from which to compute the ‘true’ observation vector. The associated background atmospheric profile is calcu- lated by perturbing the ‘true’ profile thus xb=xt+n/summationdisplay i=1ǫiλ1/2 iPi (12) where the superscript tdenotes the ‘truth’, λiandPi are the ith eigenvalues and eigenvectors of the fore- cast error covariance matrix and ǫirepresents the ith number drawn from a normal distribution of random numbers. Observation noise is modelled and superimposed onto the observation vector using the method anal- ogous to equation 12, utilising the eigenvector and eigenvalues from the total observation error covari- ance matrix. 4.1. Ensemble of numerical simulations Using the method described by equation 12, sim- ulated observations and realistic background profiles were produced. These were used as inputs to the in- version scheme to obtain retrieved profiles which were subsequently compared with the ‘true’ profiles to as- sess the impact of the observations on the background information. The observation level values (i.e. impact parame- ter, local radius of curvature and geographical posi- tion) have been taken from data during ‘prime-times’7 7Periods of time when the received signals are free from1 and 2 [ Rocken et al, 1997], in an effort to simulate realistic retrieval scenarios. The latitudinal and longi- tudinal distribution of these occultation events (deter- mined by GPS sampling) are varied, thus providing a mixed ensemble of polar, tropical and mid-latitudinal occultation events. Five hundred profiles with random temperature, humidity and surface pressure conditions have been tested, and successful retrievals (i.e. which pass qual- ity control) were obtained in all but eight cases. In most cases convergence is obtained within three or four iterations. In general, the J(x) values at conver- gence were comparable to the number of degrees of freedom considered, as expected [ Marks and Rodgers, 1993]. This suggests that although the χ2quantity is only strictly valid for linear problems it can be used reliably as a quality control for the retrieval. The eight cases which do not pass the specified quality control have been examined. They are found to be cases in which the cost function has been min- imised successfully but the converged value is too large compared with the χ2distribution. These spu- rious converged profiles represent artificial outliers, which are generated when the increment described by equation 12 is large enough to make the inverse prob- lem grossly non-linear. A small number of profiles are expected to have this problem due to the normal dis- tribution of random numbers used in the method of simulating atmospheric profiles. The profiles which failed the quality control have not been included in the statistics shown. For each successful retrieval, the retrieval error and the background error (first-guess error) have been cal- culated, and the mean and standard deviation values of these data have been computed. The standard de- viation values represent the errors ascribed to each element of the solution and background state vector, and can be compared directly with the square root values of the principal diagonal of the background er- ror covariance matrix assumed in the retrieval. The ratio of the retrieval error estimates to the forecast er- ror estimates is related to the amount of information the measurements supply to the NWP system. An improvement vector is defined which indicates how the retrieval has improved the knowledge of the background state throughout the atmospheric profile, and will be used to complement the r.m.s. statistics presented. The improvement vector ηis given by anti-spoofing military encryption.9 ηj= 100 × 1−/parenleftBiggˆSj,j Cj,j/parenrightBigg1 2  (13) where jis the matrix and improvement vector element index, and all other variables are as before. The results from the ensemble of simulated re- trievals are summarised by figure 4. The upper panels ((a) and (b)) show the computed r.m.s. errors from the the simulated retrievals. The forecast errors re- semble those shown by figure 2 as expected, modified slightly by the modelled observation noise. The temperature improvement vector suggests that optimal estimation considerably improves upon the prior knowledge of the atmospheric temperature, from the lower-troposphere to the mid-stratosphere. The temperature improvement vector declines in the up- per stratosphere partly because of the upper noise limit used, which is comparable to forecast errors in observation space, and partly because it represents the lower limit of the observation vectors used in the ensemble of simulations. The temperature improve- ment vector declines in the lower-troposphere where humidity becomes more significant, and both quan- tities are retrieved simultaneously, thus the emphasis is shifted from temperature to ln(specific humidity). It is clear from the plot that there is a gradual de- cline in the temperature retrieval quality from 300 to 1000 hPa as the humidity retrieval quality improves, where the background uncertainty information is be- ing used to resolve the temperature-specific humidity ambiguity in the refractivity. Both the temperature and humidity knowledge de- cline near the surface because the majority of occulta- tion events presented here terminate typically above one kilometre. The results shown by figure 4 confirm that the method is suitable for the purpose of non-linear op- timal estimation using RO measurements. Together, the theoretical r.m.s. errors and the computed im- provement vectors suggest that there are improve- ments in the prior knowledge of the atmospheric state from near-surface to the upper stratosphere. In par- ticular, the level of surface pressure improvement sug- gests that the RO observations can improve the prior knowledge of the surface pressure. 4.2. Solution error characterisation Using the optimal estimation inverse theory out- lined in section 3 an error analysis can be obtainedwhich allows a full characterisation of the solution (for a detailed account see Rodgers, [1990]). The error as- sociated with the solution vector can be split into its constituent parts, namely error from the background error estimates, forward modelling, forward model pa- rameters and observation noise. A mid-latitude re- trieval8(which spans 0.7 −60 km) has been used as an example to illustrate the method (figure 5). Panels (a) and (b) show that the a priori pro- vides almost all the information to the temperature and ln(specific humidity) retrieval above 10 hPa and 500 hPa, respectively. Panels (c) and (d) show the observation noise con- tribution to the retrieval error which is comparatively small. Panels (e) and (f) show the forward modelling con- tribution to the retrieval error. The structure of this contribution is very similar to that of the observation noise, but has a larger associated error. In general, forward modelling represents the second largest con- tribution to retrieval error. The temperature solution error contributions shown by panels (c) and (e) peak at 3 hPa, at the point where the stratospheric noise limit contribution to the total observation error budget peaks (figure 3). The ob- servation contribution to the solution error begins to increase above about 0.3 hPa. Above this pressure level the a priori increases more rapidly, and so the observation is given more weight, resulting in a larger contribution to the solution error. Panels (g) and (h) show that forward model pa- rameter error represents the smallest contribution to the total retrieval error. This contribution to the tem- perature and ln(specific humidity) solutions peak near 1000 hPa due to the local radius of curvature error. The contribution to the surface pressure solution rep- resents a small fraction of the total retrieval error. Figure 5 indicates that the dominant contributions to the solution error are from the a priori and forward modelling, suggesting that particular efforts should be made to improve their accuracies. It should be noted that the solution error depends on the assumptions made about the uncertainty statistics. 4.3. Quality of surface pressure retrievals It is found that the quality of the surface pres- sure retrievals is dependent on the vertical extent of 8Henceforth will be referred to as the example occultation profile.10 the occultation, i.e. how closely it approaches the sur- face. To illustrate this point, the example occultation profile is used. By systematically removing observa- tions near the surface and re-retrieving temperature, ln(specific humidity) and surface pressure, it can be shown how the vertical range of the occultation is im- portant to the quality of the surface pressure retrieval (figure 6). Panels (a), (b) and (c) show the temperature, ln(specific humidity) and surface pressure improve- ment vectors get smaller in the lower atmosphere with increasing values for the lowest geometric height level, as expected. Panel (d) shows that the retrieval which includes all the observations provides enough informa- tion to retrieve accurately the true value for surface pressure; as the number of near-surface observations decreases the retrieval becomes smaller and smaller, approaching the prior value. It is interesting to note that the surface pressure information does not decrease as quickly as expected. Indeed there is still a considerable amount of surface pressure information towards the upper troposphere. This variation in surface pressure information is due to the link between height and pressure through the hydrostatic relation. It can be concluded from this experiment that us- ing optimal estimation, radio occultation measure- ments of the Earth possess surface pressure informa- tion even if the occultation has been completed in the mid-troposphere (e.g. due to atmospheric multipath interrupting the transmitted signal). This is impor- tant to appreciate when validating surface pressure retrievals using real data. 4.4. Sensitivity to inverse model statistics In this section we present the results from a sensi- tivity study in which the statistics used to compute the optimal estimate are changed in order to investi- gate the retrieval sensitivity to such alterations. The three statistics that are altered are the observation noise, the forward modelling error and the forecast error since these represent the largest contributions to the solution error as shown by figure 5. Altering the observation noise may simulate the possible changes in observation noise sources, e.g. im- proved high-order ionospheric or poor quality clocks aboard the GPS receivers. Changing the forward modelling error is a crude method of simulating the possibility of assimilating intrinsically high resolution GPS radio occultationmeasurements with lower or higher resolution NWP model fields, and/or an occultation through a frontal system or a relatively horizontally inhomogeneous at- mosphere. Altering the prior error covariance matrix repre- sents the effect of changing the prior knowledge of the atmosphere. Increasing these error estimates can represent the extent of the knowledge of the atmo- sphere in the southern hemisphere where other at- mospheric information is sparse. Decreasing this er- ror may represent a more realistic model dynam- ics/climatology and/or greater confidence in other similar atmospheric observations used to initialise the model. For this particular test the variance values are changed, whilst retaining the existing correlations. Figure 7 shows the improvement vectors from the retrieval sensitivities outlined above. Improvement vectors are used to present the results of this study because it is the relative improvement on the prior es- timate of the atmospheric state that we are interested in. Panels (a) and (b) shows that doubling or halv- ing the observation noise has little effect on the over- all improvement, reflecting the contribution from this error source to the total observation error covariance matrix (figure 3). The changes due to ln(specific hu- midity) are very slight. The surface pressure improve- ment changes typically by a few percent. Panels (c) and (d) correspond to increasing or de- creasing the forward modelling error (by 50%). This error source provides the largest contribution to the total observation error, and as such has a large influ- ence on the degree of improvement. The difference in the temperature improvement is of the order of 15% in the upper troposphere and lower stratosphere, above which other error sources are more important, and below which (the lower troposphere) the empha- sis is shifted from the temperature retrieval to the ln(specific humidity) and surface pressure retrieval. The improvement response for ln(specific humidity) is less pronounced than for temperature peaking at ±8%. The surface pressure improvement variation is approximately ±15%. Panels (e) and (f) shows that increasing or de- creasing the standard deviation of the prior error in- creases or decreases the improvement of temperature, ln(specific humidity) throughout the vertical range of the observations as expected. Decreasing the a pri- orierror means better background knowledge, con- sequently the weighting of the a priori /observation is increased/decreased. This corresponds to a small11 improvement relative to the background atmospheric knowledge. Positive and negative temperature im- provement differences are of the order of 20% through- out the range described by the observation vector; the ln(specific humidity) improvement is of the order of ±10%; and the surface pressure improvement is of the order of ±10%. This sensitivity study has looked at some of the ex- treme case scenarios in which the statistics assumed for the optimal estimation inverse method have been changed. It has been shown that the retrieval method is most sensitive to the background errors and forward modelling errors; the latter being related to errors due to the spherical symmetry assumption. The results from increasing the background error estimates are es- pecially important to note since they represent a real possibility when dealing with any reasonable measure- ments in the data sparse southern hemisphere. 5. Conclusions In this paper we have demonstrated a prototype optimal estimation inverse method for GPS radio oc- cultation observations. The method is a non-linear, maximum a posteri- oritechnique which can accommodate most aspects of the real radio occultation problem. In particular, it is able to account for some of the error incurred from assuming local spherical symmetry which is not pos- sible using the analytic inverse method. The optimal estimation technique handles the temperature −water vapour ambiguity in a more rigorous way, rather than the sub −optimal manner inherent with the analytic inverse method. The optimal estimation inverse method is used here as an iterative method but is found to be stable and to converge rapidly in most cases. The value of the cost function at each iteration can be used reliably to judge convergence and as an indicator of sensi- ble results, allowing anomalous retrieval conditions to be recognised and omitted, thus preventing gross retrieval errors. The method is shown to be suitable for retrieving values for surface pressure. Hence, this method of utilising radio occultation observations of the Earth’s atmosphere has the potential to improve both atmo- spheric and oceanographic models, which may lead to improved predictions of the weather and climate. The quality of the surface pressure retrieval is shown to depend on the vertical extent of the occultation, i.e. higher quality retrievals are attainable with oc-cultations that reach low altitudes. It should be noted that the background statistics assumed in the paper represent global statistics. How- ever, for purposes of demonstrating a prototype re- trieval scheme for radio occultation observations they are found to be adequate. It has been shown that the retrieval accuracies, and hence the weight which should be given to the data in the subsequent model analysis, are sensitive to both forecast error uncer- tainties and values used to describe the forward mod- elling error. Acknowledgments. The authors would like to thank the Jet Propulsion Laboratory for providing the GPS/MET radio occultation dataset, J. P. Luntama for the bending angle error estimates, and C. D. Rodgers, A. Dudhia and H. Roscoe for comments on earlier drafts. References B. Ahmad and G. L. Tyler, Systematic errors in atmo- spheric profiles obtained from Abelian inversion of ra- dio occultation data: Effects of large-scale horizontal gradients, J. Geophys. Res., 104 , 3971-3992, 1999. M. Born and E. Wolf, Principles of Optics, Pergamon Press, 1993. Cospar International Reference Atmosphere: 1986. Part II: Middle Atmosphere Models, Adv. Space Res., 10, 1990. M. P. Cullen, The unified forecast/climate model, Met. Mag.112, 1449-1451, 1993. J. R. Eyre, Inversion of cloudy satellite sounding radiance s by nonlinear optimal estimation. I: Theory and simu- lation for TOVS, Q. J. R. Meteorol. Soc., 116 , 401-434, 1989. J. R. 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Rodgers, A Retrieval Method for Atmospheric Composition from Limb Emission Mea- surements, J. Geophys. Res., 98 , D8, 14939-14953, 1993. M. D. Mortensen and P. Høeg, Inversion of GPS occulta- tion measurements using diffraction theory, Geophys. Res. Lett., 25 , 13, 2441-2444, 1998. P. I. Palmer, Analysis of atmospheric temperature and humidity from radio occultation measurements, D.Phil. thesis, Oxford University, 1998. R. A. Phinney and D. L. Anderson, On the Radio Occul- tation Method for Studying Planetary Atmospheres, J. Geophys. Res., 73, 1819-1827, 1968. W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes in FORTRAN: The art of scientific computing, Cambridge Univ. Press, 1992.C. Rocken, R. Anthes, M. Exner, D. Hunt, S. Sokolovskiy, R. Ware, M. Gorbunov, W. Shreiner, D. Feng, B. Her- man, Y.-H. Kuo and X. Zou, Analysis and validation of GPS/MET data in the neutral atmosphere, J. Geo- phys. Res., 102 , D25, 29849-29866, 1997. C. D. Rodgers, Retrieval of Atmospheric Temperature and Composition from Remote Measurements of Thermal Radiation, Rev. Geophys., 14 , 609-624, 1976. C. D. Rodgers, Characterization and Error Analysis of Profiles Retrieved from Remote Sounding Measure- ments, J. Geophys. Res., 95 , D5, 5587-5595, 1990. E. K. Smith and S. Weintraub, The constants in the equa- tion for atmospheric refractive index at radio frequen- cies,Proc. of the I.R.E., 41, 1053-1037, 1953. A. K. Steiner, G. Kirchengast and H. P. Ladreiter, Inver- sion error analysis, and validation of GPS/MET occul- tation data, Ann. Geophysicae, 17, 122-138, 1999. S. Syndergaard, Retrieval analysis and methodologies in atmospheric limb sounding using the GNSS radio oc- cultation technique, Ph.D. thesis, DMI Scientific Re- port, 99-6, Danish Meteorological Institute, Copen- hagen, 1999. R. Ware, M. Exner, D. Feng, M. Gorbunov, K. Hardy, W. Melbourne, C. Rocken, W. Schreiner, S. Sokolovskiy, F. Solheim, X. Zou, R. Anthes, S. Businger and K. Tren- berth, GPS Sounding of the Atmosphere from Low- Earth Orbit: Preliminary Results, Bull. Am. Meteorol. Soc.,19-40, 1996. V. V. Vorob’ev and T. G. Krasil’nikova, Estimation of the Accuracy of the Atmospheric Refractive Index Recov- ery from Doppler Shift Measurements at Frequencies Used in the NAVSTAR System, USSR Physics of the Atmosphere and Ocean (Engl. Transl.), 29, 5, 602-609, 1994. X. Zou, Y. H. Kuo and Y. R. Guo, Assimilation of At- mospheric Radio Refractivity Using a Nonhydrostatic Adjoint Model, Mon. Weather Rev., 123 , 2229-2249, 1995. Dr. Paul I. Palmer, Division of Engineering & Ap- plied Science, Pierce Hall, Harvard University, Cam- bridge, MA 02138 (e-mail: pip@io.harvard.edu) Received August, 1999; revised January, 2000; accepted February, 2000. 1Now at Division of Engineering & Applied Science, Pierce Hall, Harvard University, Cambridge, MA 02138 This preprint was prepared with AGU’s L ATEX macros v5.01. File gpsinverse formatted February 2, 2008.13 /0/0/0/0/1/1/1/1 vrφφn Earthvrn TR R RR TTε T transmitterGPS receiverLow-Earth orbit θ Centre of local curvaturepointRay tangent ra Figure 1. Defining the radio occultation geometry used to obtain bendi ng angle information from the time delay caused by the Earth’s atmosphere. A tangential sphere is sup erimposed on to the oblate Earth (exaggerated) to emphasise the position of the local radius of curvature at th e ray periapsis. The dashed lines indicate motion. (a) 02468101214 Temperature uncertainty [K]1000.0100.010.01.00.1Pressure [hPa](b) 0.00.20.40.60.81.0 ln(specific humidity) uncertainty 300 400 500 600 700 800 900 1000P0uncertainty 2.5hPa Figure 2. The square-root values of the principal diagonal of the UKMO forecast error covariance matrix. Panels (a) and (b) show the assumed uncertainty for prior temperatu re and ln(specific humidity), respectively, with the assumed prior uncertainty for the surface pressure element inset of panel (b).14 (a) 10-710-610-510-4 Bending angle error [radians]010203040506070Altitude [km]Thermal noise Multipath noise(3mm) Satellite velocity error Residual ionosphericerror Total measurementnoise 1000.00100.0010.001.000.10 Approximate pressure [hPa](b) 020406080100 Contribution to total bending angle error [%]010203040506070Altitude [km] 1000.00100.0010.001.000.10 Approximate pressure [hPa] (c) 10-810-610-410-2100 Bending angle error [radians]010203040506070Altitude [km]Measurement noise Forward modellingerror Forward modelparameter error Upper stratospherenoise value Total bending angleerror 1000.00100.0010.001.000.10 Approximate pressure [hPa](d) 020406080100 Proportion of total error [%]010203040506070Altitude [km] 1000.00100.0010.001.000.10 Approximate pressure [hPa] Figure 3. Measurement noise budget and total measurement error budge t for the RO optimal estimation inverse method [after Luntama, [1997]]. The error estimates shown i n panel (a) represent normal atmospheric conditions, i.e. small multipath error (3 mm) and normal ionospheric con ditions. Panel (b) shows the percentage contributions from the different error sources to the total bending angle er ror. Panel (c) shows typical standard deviation values from the principal diagonal from each contribution to the to tal measurement error covariance matrix; and panel (d) shows the individual measurement error sources as a perc entage proportion of the total error.15 (a) 02468101214 Temperature [K]1000.0100.010.01.00.1Pressure [hPa](b) 0.00.20.40.60.81.0 ln(specific humidity)1000900800700600500400300Pressure [hPa]P0 error [hPa]: (background) (solution) 2.5 2.2 (c) 020406080100 Temperature [%]1000.0100.010.01.00.1Pressure [hPa](d) 020406080100 ln(specific humidity) [%]1000900800700600500400300Pressure [hPa]P0 [%]: 11.8 Figure 4. Theoretical error estimates from the ensemble of simulated retrievals. The upper panels show the theoretical r.m.s errors, where the solid and dotted lines r epresent the retrieval and background errors, respectivel y. Panel (a) shows the errors from the temperature solution and the panel (b) shows the errors from the ln(specific humidity) solution with the surface pressure solution erro r inset. The lower panel show the corresponding improve- ment vector. Panels (c) and (d) show the temperature and ln(s pecific humidity) elements with the surface pressure improvement inset of panel (d).16 Figure 5. Solution error characterisation of the optimal estimation inverse method for the example occultation profile. Left panels of a pair show the contribution to the tem perature retrieval error and the right panels show the error contributions to the ln(specific humidity) retrie val error. The surface pressure retrieval error is inset of the left panels. Panels (a) and (b) show the a priori contribution to the retrieval error, where the dashed shown represent the standard deviation values of the principal di agonal of the a priori error covariance matrix; (c) and (d) the measurement noise contribution; (e) and (f) the forward modelling contribution; and (g) and (h) the forward model parameter contribution. The total solution error is s uperimposed on all panels (dotted line) to provide an indication as to which contributions are prominent.17 (a) 020406080100 Temperature improvement [%]1000.0100.010.01.00.1Pressure [hPa]Lowest geometric height [km]: 0.7 2.9 5.7 7.6(b) 020406080100 ln(specific humidity) improvement [%]1000900800700600500400300Pressure [hPa] (c) 0246810 Lowest geometric height level [km]05101520253035Surface pressure improvement [%](d) 0246810 Lowest geometric height level [km]-4-202468Surface pressure error [hPa] Figure 6. Investigating the relationship between the quality of surf ace pressure retrievals and the lowest geometric height level of the occultation using the example occultati on profile. Panel (a), (b) and (c) show the temperature, ln(specific humidity) and surface pressure improvement vec tors; and panel (d) shows the difference (with errors bars) between the truth and the retrieval. The broken line in panel (d) represents the difference between the ‘true’ and the assumed value for the surface pressure.18 (a) 020406080100 Temperature [%]1000.0100.010.01.00.1Pressure [hPa]Measurement noise(b) 020406080100 ln(specific humidity) [%]300 400 500 600 700 800 9001000Pressure [hPa]P0 improvement: 30.24% 30.48% 30.54% (c) 020406080100 Temperature [%]1000.0100.010.01.00.1Pressure [hPa]Forward modelling error(d) 020406080100 ln(specific humidity) [%]300 400 500 600 700 800 9001000Pressure [hPa]P0 improvement: 16.33% 30.48% 54.38% (e) 020406080100 Temperature [%]1000.0100.010.01.00.1Pressure [hPa]a priori error(f) 020406080100 ln(specific humidity) [%]300 400 500 600 700 800 9001000Pressure [hPa]P0 improvement: 40.86% 30.48% 15.67% Figure 7. Improvement vectors from the retrieval sensitivity studie s using the example occultation profile. In general, left panels show the temperature improvement vect or elements and the right panels show the ln(specific humidity) improvement vector elements, with the surface pr essure improvement vector element inset. For panels (a)−(f), dashed curves represent a reduction in the quantity alt ered, dotted curves represent an increase in the quan- tity altered and solid curves represent the control case. Pa nels (a) and (b) show the results from doubling/halving the measurement noise; panels (c) and (d) show the results fr om increasing and decreasing the representative error by±50%; and panels (e) and (f) show the results from increasing a nd decreasing the principal diagonal standard deviation errors from the a priori error covariance matrix by ±50%.

arXiv:physics/0003011v1 [physics.acc-ph] 4 Mar 2000The Superconducting TESLA Cavities Dedicated to the memory of Bjørn H. Wiik B. Aune1, R. Bandelmann4, D. Bloess2, B. Bonin1,11, A. Bosotti7, M. Champion5, C. Crawford5, G. Deppe4, B. Dwersteg4, D.A. Edwards4,5, H.T Edwards4,5, M. Ferrario6, M. Fouaidy9, P.-D. Gall4, A. Gamp4, A. G¨ ossel4, J. Graber4,12, D. Hubert4, M. H¨ uning4, M. Juillard1, T. Junquera9, H. Kaiser4, G. Kreps4, M. Kuchnir5, R. Lange4, M. Leenen4, M. Liepe4, L. Lilje4, A. Matheisen4, W.-D. M¨ oller4, A. Mosnier1, H. Padamsee3, C. Pagani7, M. Pekeler4,10, H.-B. Peters4, O. Peters4, D. Proch4, K. Rehlich4, D. Reschke4, H. Safa1, T. Schilcher4,13, P. Schm¨ user8, J. Sekutowicz4, S. Simrock4, W. Singer4, M. Tigner3, D. Trines4, K. Twarowski4, G. Weichert4, J. Weisend4,14, J. Wojtkiewicz4, S. Wolff4and K. Zapfe4 1CEA Saclay, Gif-sur-Yvette, France 2CERN, Geneve, Switzerland 3Cornell University, Ithaca, New York, USA 4Deutsches Elektronen Synchrotron, Hamburg, Germany 5FNAL, Batavia, Illinois, USA 6INFN Frascati, Frascati, Italy 7INFN Milano, Milano, Italy 8II. Institut f¨ ur Experimentalphysik, Universit¨ at Hambu rg, Germany 9I.P.N. Orsay, Orsay, France 10Now at ACCEL Instruments GmbH, Bergisch-Gladbach, Germany 11Now at CEA/IPSN, Fontenay, France 12Now at Center for Advanced Biotechnology, Boston Universit y, USA 13Now at Paul Scherrer Institut, Villigen, Switzerland 14Now at SLAC, Stanford, California, USA Abstract The conceptional design of the proposed linear electron-po sitron collider TESLA is based on 9-cell 1.3 GHz superconducting niobium cavities wi th an accelerating gradient ofEacc≥25 MV/m at a quality factor Q0≥5·109. The design goal for the cavities of the TESLA Test Facility (TTF) linac was set to the more modera te value of Eacc≥15 MV/m. In a first series of 27 industrially produced TTF caviti es the average gradient at Q0= 5·109was measured to be 20 .1±6.2 MV/m, excluding a few cavities suffering from serious fabrication or material defects. In the second production of 24 TTF cavities additional quality control measures were introduced, in pa rticular an eddy-current scan to eliminate niobium sheets with foreign material inclusions and stringent prescriptions for carrying out the electron-beam welds. The average gradient of these cavities at Q0= 5·109 amounts to 25 .0±3.2 MV/m with the exception of one cavity suffering from a weld de fect. Hence only a moderate improvement in production and prepara tion techniques will be needed to meet the ambitious TESLA goal with an adequate safe ty margin. In this paper we present a detailed description of the design, fabricatio n and preparation of the TESLA Test Facility cavities and their associated components and report on cavity performance in test cryostats and with electron beam in the TTF linac. The ongoing R&D towards higher gradients is briefly addressed. 11 Introduction In the past 30 years electron-positron collisions have play ed a central role in the discovery and detailed investigation of new elementary particles and the ir interactions. The highly successful Standard Model of the unified electromagnetic and weak inter actions and Quantum Chromo- dynamics, the quantum field theory of quark-gluon interacti ons, are to a large extent based on the precise data collected at electron-positron collide rs. Important questions still remain to be answered, in particular the origin of the masses of field quanta and particles – within the Standard Model explained in terms of the so-called Higgs mechanism – and the existence or nonexistence of supersymmetric particles which appear t o be a necessary ingredient of any quantum field theory attempting to unify all four forces know n in Nature: the gravitational, weak, electromagnetic and strong forces. There is general a greement in the high energy physics community that in addition to the Large Hadron Collider unde r construction at CERN a lepton collider will be needed to address these fundamental issues . Electron-positron interactions in the center-of-mass (cm ) energy range from 200 GeV to more than a TeV can no longer be realized in a circular machine like LEP since the E4depen- dence of the synchrotron radiation loss would lead to prohib itive operating costs. Instead the linear collider concept must be employed. This new principl e was successfully demonstrated with the Stanford Linear Collider SLC providing a cm energy of mor e than 90 GeV. Worldwide there are different design options towards the next generation of l inear colliders in the several 100 GeV to TeV regime. Two main routes are followed: colliders eq uipped with normal-conducting (nc) cavities (NLC, JLC, VLEPP and CLIC) or with superconduc ting (sc) cavities (TESLA). The normal-conducting designs are based on high frequency s tructures (6 to 30 GHz) while the superconducting TESLA collider employs the comparativ ely low frequency of 1.3 GHz. The high conversion efficiency from primary electric power to bea m power (about 20%) in combi- nation with the small beam emittance growth in low-frequenc y accelerating structures makes the superconducting option an ideal choice for a high-lumin osity collider. The first international TESLA workshop was held in 1990 [1]. A t that time superconducting rf cavities in particle accelerators were usually operated in the 5 MV/m regime. Such low gradi- ents, together with the high cost of cryogenic equipment, wo uld have made a superconducting linear electron-positron collider totally non-competiti ve with the normal-conducting colliders proposed in the USA and Japan. The TESLA collaboration, form ally established in 1994 with the aim of developing a 500 GeV center-of-mass energy superc onducting linear collider, set out with the ambitious goal of increasing the cost effectiveness of the superconducting option by more than an order of magnitude: firstly, by raising the accel erating gradient by a factor of five from 5 to 25 MV/m, and secondly, by reducing the cost per un it length of the linac by using economical cavity production methods and a greatly si mplified cryostat design. Impor- tant progress has been achieved in both directions; in parti cular the gradient of 25 MV/m is essentially in hand, as will be shown below. To allow for a gra dual improvement in the course of the cavity R&D program, a more moderate goal of 15 MV/m was s et for the TESLA Test Facility (TTF) linac [2]. The TESLA cavities are quite similar in their layout to the 5- cell 1.5 GHz cavities of the elec- tron accelerator CEBAF in Newport News (Virginia, USA) whic h were made by an industrial company [3]. These cavities exceeded the specified gradient of 5 MV/m considerably and hence offered the potential for further improvement. While the CEB AF cavity fabrication methods were adopted for TTF without major modifications, important new steps were introduced in the cavity preparation: 2•chemical removal of a thicker surface layer •a 1400◦C annealing with titanium getter to improve the Nb heat condu ctivity and to homogenize the material •rinsing with ultrapure water at high pressure (100 bar) to re move surface contaminants •destruction of field emitters by a technique called High Powe r Processing (HPP). The application of these techniques, combined with extreme ly careful handling of the cavities in a clean-room environment, has led to a significant increas e in accelerating field. The TESLA Test Facility (TTF) has been set up at DESY to provid e the necessary infra- structure for the chemical and thermal treatment, clean-ro om assembly and testing of indus- trially produced multicell cavities. In addition a 500 MeV e lectron linac is being built as a test bed for the performance of the sc accelerating structures wi th an electron beam of high bunch charge. At present more than 30 institutes from Armenia, P.R . China, Finland, France, Ger- many, Italy, Poland, Russia and USA participate in the TESLA Collaboration and contribute to TTF. The low frequency of 1.3 GHz permits the acceleration of long trains of particle bunches with very low emittance making a superconducting linac an id eal driver of a free electron laser (FEL) in the vacuum ultraviolet and X-ray regimes. For this r eason the TTF linac has recently been equipped with undulator magnets, and its energy will be upgraded to 1 GeV in the coming years to provide an FEL user facility in the nanometer wavele ngth range. An X-ray FEL facility with wavelengths below 1 ˚A is an integral part of the TESLA collider project [4]. The present paper is organized as follows. Sect. 2 is devoted to the basics of rf super- conductivity and the properties and limitations of superco nducting (sc) cavities for particle acceleration. The design of the TESLA cavities and the auxil iary equipment is presented in Sect. 3. The fabrication and preparation steps of the caviti es are described in Sect. 4. The test results obtained on all TTF cavities are presented in Se ct. 5, together with a discussion of errors and limitations encountered during cavity produc tion at industry, and the quality control measures taken. The rf control and the cavity perfor mance with electron beam in the TTF linac are described in Sect. 6. A summary and outlook is gi ven in Sect. 7 where also the ongoing research towards higher gradients is shortly addre ssed. 2 Basics of RF Superconductivity and Properties of Su- perconducting Cavities for Particle Acceleration 2.1 Basic principles of rf superconductivity and choice of s upercon- ductor The existing large scale applications of superconductors i n accelerators are twofold, in magnets and in accelerating cavities. While there are some common re quirements like the demand for as high a critical temperature as possible1there are also characteristic differences. In magnets operated with a dc or a low-frequency ac current the so-calle d “hard” superconductors are 1The High- Tcceramic superconductors have not yet found widespread appl ication in magnets mainly due to technical difficulties in cable production and coil windin g. Cavities with High- Tcsputter coatings on copper have shown much inferior performance in comparison to niobi um cavities. 3needed featuring high upper critical magnetic fields (15–20 T) and strong flux pinning in order to obtain high critical current density; such properties ca n only be achieved using alloys like niobium-titanium or niobium-tin. In microwave applicatio ns the limitation of the superconduc- tor is not given by the upper critical field but rather by the so -called “superheating field” which is well below 1 T for all known superconductors. Moreover, st rong flux pinning appears undesir- able in microwave cavities as it is coupled with hysteretic l osses. Hence a “soft” superconductor must be used and pure niobium is still the best candidate alth ough its critical temperature is only 9.2 K and the superheating field about 240 mT. Niobium-ti n (Nb 3Sn) looks more favorable at first sight since it has a higher critical temperature of 18 K and a superheating field of 400 mT; however, the gradients achieved in Nb 3Sn coated single-cell copper cavities were below 15 MV/m, probably due to grain boundary effects in the Nb 3Sn layer [5]. For these reasons the TESLA collaboration decided to use niobium as the supercond ucting material, as in all other large scale installations of sc cavities. Here two alternat ives exist: the cavities are fabricated from solid niobium sheets or a thin niobium layer is sputtere d onto the inner surface of a copper cavity. Both approaches have been successfully applied, th e former at Cornell (CESR), KEK (TRISTAN), DESY (PETRA, HERA), Darmstadt (SDALINAC), Jeffe rson Lab (CEBAF) and other laboratories, the latter in particular at CERN in the e lectron-positron storage ring LEP. From the test results on existing cavities the solid-niobiu m approach promised higher acceler- ating gradients, hence it was adopted as the baseline for the TTF cavity R&D program. 2.1.1 Surface resistance In contrast to the dc case superconductors are not free from e nergy dissipation in microwave fields. The reason is that the radio frequency (rf) magnetic fi eld penetrates a thin surface layer and induces oscillations of the electrons which are not boun d in Cooper pairs. The number of these “free electrons” drops exponentially with temperatu re. According to the Bardeen-Cooper- Schrieffer (BCS) theory of superconductivity the surface re sistance in the range T < T c/2 is given by the expression RBCS∝ω2 Texp(−1.76Tc/T) (1) where f=ω/2πis the microwave frequency. In the two-fluid model of superco nductors one can derive a refined expression for the surface resistance [6, 7] RBCS=C Tω2σnΛ3exp(−1.76Tc/T). (2) HereCis a constant, σnthe normal-state conductivity of the material and Λ an effect ive penetration depth, given by Λ =λL/radicalbig 1 +ξ0/ℓ . λLis the London penetration depth, ξ0the coherence length and ℓthe mean free path of the unpaired electrons. The fact that σnis proportional to the mean free path ℓleads to the surprising conclusion that the surface resistance does not assume its minimum value when the superconductor is as pure as possible ( ℓ≫ξ0) but rather in the range ℓ≈ξ0. For niobium the BCS surface resistance at 1.3 GHz amounts to about 800 nΩ at 4. 2 K and drops to 15 nΩ at 2 K; see Fig. 1. The exponential temperature dependence is th e reason why operation at 1.8–2 K is essential for achieving high accelerating gradients in combination with very high quality factors. Superfluid helium is an excellent coolant owing to i ts high heat conductivity. 42 3 4 5 6 7RS [nΩ] Tc/TRres = 3 nΩ1000 100 10 1RBCS Figure 1: The surface resistance of a 9-cell TESLA cavity plo tted as a function of Tc/T. The residual resistance of 3 nΩ corresponds to a quality factor Q0= 1011. In addition to the BCS term there is a residual resistance Rrescaused by impurities, frozen- in magnetic flux or lattice distortions. This term is tempera ture independent and amounts to a few nΩ for very pure niobium but may readily increase if the s urface is contaminated. 2.1.2 Heat conduction in niobium The heat produced at the inner cavity surface has to be guided through the cavity wall to the superfluid helium bath. Two quantities characterize the hea t flow: the thermal conductivity of the bulk niobium and the temperature drop at the niobium-h elium interface caused by the Kapitza resistance. For niobium with a residual resisti vity ratio2RRR = 500 the two contributions to the temperature rise at the inner cavity su rface are about equal. The thermal conductivity of niobium at cryogenic temperatures scales a pproximately with the RRR, a rule of thumb being λ(4.2K)≈0.25·RRR [W/(m·K)]. However, λis strongly temperature dependent and drops by about an orde r of magnitude when lowering the temperature to 2 K, as shown in Fig. 2. Impurities influence the RRR and the thermal conductivity of niobium. Bulk niobium is contaminated by interstitial (mostly hydrogen, carbon, ni trogen, oxygen) and metallic impu- rities (mostly tantalum). The resulting RRR can be calculated by summing the individual contributions [10] RRR =/parenleftBigg/summationdisplay ifi/ri/parenrightBigg−1 (3) 2RRR is defined as the ratio of the resistivities at room temperatu re and at liquid helium temperature. The low temperature resistivity is either measured just above Tcor at 4.2 K, applying a magnetic field to assure the normal state. 51101001000 2 4 6 8 10 30λ[ W / m · K ] T [K]RRR = 380RRR = 1070 Figure 2: Measured heat conductivity in niobium as a functio n of temperature [8]. Continu- ous curves: parametrization by B. Bonin, using the RRR and the average grain size as input parameters [9]. These data do not show an enhancement at 2 K (t he so-called “phonon peak”) which was observed in some earlier experiments [5]. Table 1: Resistivity coefficients of common impurities in Nb [ 10] Impurity atom iN O C H Ta riin 104wt. ppm 0.44 0.58 0.47 0.36 111 where the fidenote the fractional contents of impurity i(measured in wt. ppm) and the rithe corresponding resistivity coefficients which are listed in T able 1. A good thermal conductivity is the main motivation for using high purity niobium with RRR≈300 as the material for cavity production. The RRR may be further improved by post-purification of the entire cavity (see Sect. 5). The Kapitza conductance depends on temperature and surface conditions. For pure niobium in contact with superfluid helium at 2 K it amounts to about 600 0W/(m2K) [11]. 2.1.3 Influence of magnetic fields Superheating field . Superconductivity breaks down when the rf magnetic field ex ceeds the critical field of the superconductor. In the high frequency case the so -called “superheating field” is relevant which for niobium is about 20% higher than the therm odynamical critical field of 200mT [12, 13]. Trapped magnetic flux . Niobium is in principle a soft type II superconductor witho ut flux pinning. In practice, however, weak magnetic dc fields are no t expelled upon cooldown but remain trapped in the niobium. Each flux line contains a norma l-conducting core whose area is roughly πξ2 0. The coherence length ξ0amounts to 40 nm in Nb. Trapped magnetic dc flux 6therefore results in a surface resistance [6] Rmag= (Bext/2Bc2)Rn (4) where Bextis the externally applied field, Bc2the upper critical field and Rnthe surface resis- tance in the normal state3. At 1.3 GHz the surface resistance caused by trapped flux amou nts to 3.5 nΩ /µT for niobium. Cavities which are not shielded from the Earth ’s magnetic field are therefore limited to Q0values below 109. 2.2 Advantages and limitations of superconducting cavitie s The fundamental advantage of superconducting cavities is t he extremely low surface resistance of about 10 nΩ at 2 K. The typical quality factors of normal con ducting cavities are 104–105 while for sc cavities they may exceed 1010, thereby reducing the rf losses by 5 to 6 orders of magnitude. In spite of the low efficiency of refrigeration t here are considerable savings in primary electric power. Only a tiny fraction of the incident rf power is dissipated in the cavity walls, the lion’s share is either transferred to the beam or r eflected into a load. The physical limitation of a sc resonator is given by the requ irement that the rf magnetic field at the inner surface has to stay below the superheating field o f the superconductor (200–240 mT for niobium). For the TESLA cavities this implies a maximu m accelerating field of 50– 60 MV/m. In principle the quality factor should stay roughly constant when approaching this fundamental superconductor limit but in practice the “exci tation curve” Q0=Q0(Eacc) ends at considerably lower values, often accompanied with a strong decrease of Q0towards the highest gradient reached in the cavity. The main reasons for the perf ormance degradation are excessive heating at impurities on the inner surface, field emission of electrons and multipacting4. 2.2.1 Thermal instability and field emission One basic limitation of the maximum field in a superconductin g cavity is thermal instability. Temperature mapping at the outer cavity wall usually reveal s that the heating by rf losses is not uniform over the whole surface but that certain spots exh ibit larger temperature rises, of- ten beyond the critical temperature of the superconductor. Hence the cavity becomes partially normal-conducting, associated with strongly enhanced pow er dissipation. Because of the expo- nential increase of surface resistance with temperature th is may result in a run-away effect and eventually a quench of the entire cavity. Analytical models as well as numerical simulations are available to describe such an avalanche effect. Input parame ters are the thermal conductivity of the superconductor, the size and resistance of the normal conducting spot and the Kapitza resistance. The tolerable defect size depends on the RRR of the material and the desired field level. As a typical number, the diameter of a normal-conduct ing spot must exceed 50 µm to be able to initiate a thermal instability at 25 MV/m for RRR > 200 . There have been many attempts to identify defects which were localized by temperature mapping. Examples of defects are drying spots, fibers from ti ssues, foreign material inclusions, weld splatter and cracks in the welds. There are two obvious a nd successful methods for reducing the danger of thermal instability: 3Benvenuti et al. [14] attribute the magnetic surface resistance in niobium s putter layers to flux flow. 4“Multipacting” is a commonly used abbreviation for “multip le impacting” and designates the resonant multiplication of field emitted electrons which gain energy in the rf electromagnetic field and impact on the cavity surface where they induce secondary electron emissi on. 7Figure 3: Superconducting 1.3 GHz 9-cell cavity for the TESL A Test Facility. •avoid defects by preparing and cleaning the cavity surface w ith extreme care; •increase the thermal conductivity of the superconductor. Considerable progress has been achieved in both aspects ove r the last ten years. Field emission of electrons from sharp tips is the most sever e limitation in high-gradient superconducting cavities. In field-emission loaded caviti es the quality factor drops exponentially above a certain threshold, and X-rays are observed. The field emission current density is given by the Fowler-Nordheim equation [15]: jFE=c1E2.5 locexp/parenleftbigg −c2 βEloc/parenrightbigg (5) where Elocis the local electric field, βa so-called field enhancement factor and c1,c2are constants. There is experimental evidence that small parti cles on the cavity surface (e.g. dust) act as field emitters. Therefore perfect cleaning, for examp le by high-pressure water rinsing, is the most effective remedy against field emission. By applying this technique it has been possible to raise the threshold for field emission in multicell caviti es from about 10 MV/m to more than 20 MV/m in the past few years. The topics of thermal instability and field emission are disc ussed at much greater detail in the book by Padamsee, Knobloch and Hayes [16]. 3 Design of the TESLA Cavities 3.1 Overview The TTF cavity is a 9-cell standing wave structure of about 1 m length whose lowest TM mode resonates at 1300 MHz. A photograph is shown in Fig. 3. The cav ity is made from solid niobium and is cooled by superfluid helium at 2 K. Each 9-cell cavity is equipped with its own titanium helium t ank, a tuning system driven by a stepping motor, a coaxial rf power coupler capable of tra nsmitting more than 200 kW, a pickup probe and two higher order mode couplers. To reduce th e cost for cryogenic installations, eight cavities and a superconducting quadrupole are mounte d in a common vacuum vessel and constitute the so-called cryomodule of the TTF linac, shown in Fig. 4. Within the module the cavity beam pipes are joined by stainless steel bellows and fl anges with metallic gaskets. The cavities are attached to a rigid 300 mm diameter helium suppl y tube which provides positional accuracy of the cavity axes of better than 0.5 mm. Invar rods e nsure that the distance between adjacent cavities remains constant during cooldown. Radia tion shields at 5 K and 60 K together with 30 layers of superinsulation limit the static heat load on the 2 K level to less than 3 W for the 12 m long module. 3.2 Layout of the TESLA cavities 8Figure 4: Cryogenic module of the TESLA Test Facility linac c omprising eight 9-cell cavities and a superconducting quadrupole. 3.2.1 Choice of frequency The losses in a microwave cavity are proportional to the prod uct of conductor area and surface resistance. For a given length of a multicell resonator, the area scales with 1 /fwhile the surface resistance of a superconducting cavity scales with f2forRBCS≫Rresand is independent of f forRBCS≪Rres. At an operating temperature T= 2 K the BCS term dominates above 3 GHz and hence the losses grow linearly with frequency whereas fo r frequencies below 300 MHz the residual resistance dominates and the losses grow with 1 /f. To minimize the dissipation in the cavity wall one should therefore select fin the range 300 MHz to 3GHz. Cavities in the 350 to 500 MHz regime are in use in electron-po sitron storage rings. Their large size is advantageous to suppress wake field effects and h igher order mode losses. However, for a linac of several 10 km length the niobium and cryostat co sts for these bulky cavities would be prohibitive, hence a higher frequency has to be chos en. Considering material costs f= 3GHz might appear the optimum but there are compelling argu ments for choosing about half this frequency. •The wake fields losses scale with the second to third power of t he frequency ( W/bardbl∝f2, W⊥∝f3). Beam emittance growth and beam-induced cryogenic losses are therefore much higher at 3 GHz. •Thef2dependence of the BCS resistance sets an upper limit5of about 30MV/m at 3GHz, hence choosing this frequency would definitely preclude a po ssible upgrade of TESLA to 35–40MV/m [17]. The choice for 1.3GHz was motivated by the availability of hi gh power klystrons. 3.2.2 Cavity geometry A multicell resonator is advantageous for maximizing the ac tive acceleration length in a linac of a given size. With increasing number of cells per cavity, how ever, difficulties arise from trapped modes, uneven field distribution in the cells and too high pow er requirements on the input coupler. Extrapolating from the experience with 4-cell and 5-cell cavities a 9-cell structure appeared manageable. A side view of the TTF cavity with the be am tube sections and the coupler ports is given in Fig. 5. The design of the cell shape was guided by the following consi derations: •a spherical contour near the equator with low sensitivity fo r multipacting, •minimization of electric and magnetic fields at the cavity wa ll to reduce the danger of field emission and thermal breakdown, •a large iris radius to reduce wake field effects. 5See Fig. 11.22 in [16]. 91061 mm 1276 mm115.4 mmpick up flange HOM coupler flange (rotated by 65Ê)HOM coupler flange power coupler flange Figure 5: Side view of the 9-cell TTF cavity with the ports for the main power coupler and two higher-order mode (HOM) couplers. The shape of the cell was optimized using the code URMEL [18]. The resonator is operated in the πmode with 180◦phase difference between adjacent cells. The longitudinal d imensions are determined by the condition that the electric field has to be inverted in the time a rela- tivistic particle needs to travel from one cell to the next. T he separation between two irises is therefore c/(2f). The iris radius Ririsinfluences the cell-to-cell coupling6kcell, the excitation of higher order modes and other important cavity parameters , such as the ratio of the peak electric (magnetic) field at the cavity wall to the accelerat ing field and the ratio ( R/Q) of shunt impedance to quality factor. For the TESLA Test Facility cav itiesRiris= 35 mm was chosen, leading to kcell= 1.87% and Epeak/Eacc= 2. The most important parameters are listed in Table 2. The contour of a half-cell is shown in Fig. 6. It is composed of a circular arc around the equator region and an elliptical section near the iris. The d imensions are listed in Table 3. The half-cells at the end of the 9-cell resonator need a slightly different shape to ensure equal field amplitudes in all 9 cells. In addition there is a slight asymm etry between left and right end cell which prevents trapping of higher-order modes (see Sect. 3. 5). 3.2.3 Lorentz-force detuning and cavity stiffening The electromagnetic field exerts a Lorentz force on the curre nts induced in a thin surface layer. The resulting pressure acting on the cavity wall p=1 4(µ0H2−ε0E2) (6) leads to a deformation of the cells in the µm range and a change ∆ Vof their volume. The consequence is a frequency shift according to Slater’s rule ∆f f0=1 4W/integraldisplay ∆V(ε0E2−µ0H2)dV . (7) Here W=1 4/integraldisplay V(ε0E2+µ0H2)dV (8) 6The coupling coefficient is related to the frequencies of the c oupled modes in the 9-cell resonator by the formula fn=f0//radicalbig 1 + 2kcellcos(nπ/9) where f0is the resonant frequency of a single cell and 1 ≤n≤9. 10Table 2: TTF cavity design parameters.a type of accelerating structure standing wave accelerating mode TM010,πmode fundamental frequency 1300 MHz design gradient Eacc 25 MV/m quality factor Q0 >5·109 active length L 1.038 m number of cells 9 cell-to-cell coupling 1.87 % iris diameter 70 mm geometry factor 270 Ω R/Q 518 Ω Epeak/Eacc 2.0 Bpeak/Eacc 4.26 mT/(MV/m) tuning range ±300 kHz ∆f/∆L 315 kHz/mm Lorentz force detuning at 25 MV/m ≈600 Hz Qextof input coupler 3·106 cavity bandwidth at Qext= 3·106430 Hz RF pulse duration 1330µs repetition rate 5 Hz fill time 530µs beam acceleration time 800µs RF power peak/average 208 kW/1.4 kW number of HOM couplers 2 cavity longitudinal loss factor k /bardblforσz= 0.7 mm 10.2 V/pC cavity transversal loss factor k ⊥forσz= 0.7 mm 15.1 V/pC/m parasitic modes with the highest impedance : type TM011 π/9 ( R/Q)/ frequency 80 Ω/2454 MHz 2π/9 ( R/Q)/ frequency 67 Ω/2443 MHz bellows longitudinal loss factor k /bardblforσz= 0.7 mm 1.54 V/pC bellows transversal loss factor k ⊥forσz= 0.7 mm 1.97 V/pC/m aFollowing common usage in ac circuits and the convention ado pted in the Handbook of Accelerator Physics and Engineering [19], page 523, we define the shunt impedance by the relation R=V2/(2P), where Pis the dissipated power and Vthe peak voltage in the equivalent parallel LCR circuit. Not e that another definition is common, which has also been used in the TESLA Conceptual Desi gn Report: R=V2/P, leading to a factor of 2 larger shunt impedance. 11ab RirisRequatorRc length cavity axis Figure 6: Contour of a half cell. Table 3: Half-cell shape parameters (all dimensions in mm). cavity shape parameter midcup endcup 1 endcup 2 equator radius Requat . 103.3 103.3 103.3 iris radius Riris 35 39 39 radius Rarcof circular arc 42.0 40.3 42 horizontal half axis a 12 10 9 vertical half axis b 19 13.5 12.8 length l 57.7 56.0 57.0 is the stored energy and f0the resonant frequency of the unperturbed cavity. The compu ted frequency shift at 25MV/m amounts to 900Hz for an unstiffened cavity of 2.5mm wall thick- ness. The bandwidth of the cavity equipped with the main powe r coupler ( Qext= 3·106) is about 430Hz, hence a reinforcement of the cavity is needed. N iobium stiffening rings are welded in between adjacent cells as shown in Fig. 7. They reduce the f requency shift to about 500Hz for a 1.3ms long rf pulse7, see Fig. 26. The deformation of the stiffened cell is negligible near the i ris where the electric field is large, but remains nearly the same as in the unstiffened cell near the equator where the magnetic field dominates. The deformation in this region can only be reduce d by increasing the wall thickness. 3.2.4 Magnetic Shielding As shown in Sect. 2.1.3 the ambient magnetic field must be shie lded to a level of about a µT to reduce the magnetic surface resistance to a few nΩ. This is accomplished with a two-stage passive shielding, provided by the conventional steel vacu um vessel of the cryomodule and a high-permeability cylinder around each cavity. To remove t he remanence from the steel vessel 7Part of this shift is due to an elastic deformation of the tuni ng mechanism. 12reference flangeconical head plate stiffening ring Figure 7: End section of a cavity with stiffening ring, conica l head plate for welding into the helium tank and reference flange for alignment. the usual demagnetization technique is applied. The result ing attenuation of the ambient field is found to be better than expected from a cylinder without an y remanence. The explanation is that the procedure does not really demagnetize the steel b ut rather remagnetizes it in such a way that the axial component of the ambient field is counterac ted. This interpretation (see also ref. [20]) becomes obvious if the cylinder is turned by 180◦: in that case the axial field measured inside the steel cylinder is almost twice as large as the ambi ent longitudinal field component, see Fig. 8a. The shielding cylinders of the cavities are made from Cryope rm8which retains a high per- meability of more than 10000 when cooled to liquid helium tem perature. Figure 8b shows the measured horizontal, vertical and axial components inside a cryoperm shield at room temper- ature, which was exposed to the Earth’s field. The combined ac tion of remagnetized vacuum vessel and cryoperm shield is more than adequate to reduce th e ambient field to the level of some µT. An exception are the end cells of the first and the last cavit y near the end of the cryomodule where the vessel is not effective in attenuating l ongitudinal fields. Here an active field compensation by means of Helmholtz coils could reduce t he fringe field at the last cavity to a harmless level. 3.3 Helium vessel and tuning system The helium tank contains the superfluid helium needed for coo ling and serves at the same time as a mechanical support of the cavity and as a part of the t uning mechanism. The tank is made from titanium whose differential thermal contractio n relative to niobium is 20 times smaller than for stainless steel. Cooldown produces a stres s of only 3 MPa in a cavity that was stress-free at room temperature. Titanium has the addit ional advantage that it can be directly electron-beam welded to niobium while stainless s teel-niobium joints would require an 8Cryoperm is made by Vacuumschmelze Hanau, Germany. 13Figure 8: Shielding of the Earth’s magnetic field. (a) Shield ing of axial component by the steel vacuum vessel of the cryomodule. Shown is also the arrangeme nt of the cavity string in the vessel. (b) Shielding of the axial, horizontal and vertical field components by the cryoperm cylinder surrounding the cavity (measured without vacuum v essel). 14intermediate metal layer. The assembly of cavity and helium tank proceeds in the follow ing sequence: a titanium bellows is electron-beam (EB) welded to the conical Nb head p late at one side of the cavity, a titanium ring is EB welded to the conical Nb head plate at othe r side (see Fig. 7). The cavity is then inserted into the tank and the bellows as well as the ti tanium ring are TIG welded to the Ti vessel. The tuning system consists of a stepping motor with a gear box and a double lever arm. The moving parts operate at 2K in vacuum. The tuning range is a bout±1mm, corresponding to a frequency range of ±300kHz. The resolution is 1Hz. The tuning system is adjusted in such a way that after cooldown the cavity is always under compress ive force to avoid a backlash if the force changes from pushing to pulling. 3.4 Main Power Coupler Design requirements A critical component of a superconducting cavity is the powe r input coupler. For TTF several coaxial couplers have been developed [21], consisting of a “ cold part” which is mounted on the cavity in the clean room and closed by a ceramic window, an d a “warm part” which is assembled after installation of the cavity in the cryomodul e. The warm section contains the transition from waveguide to coaxial line. This part is evac uated and sealed against the air-filled wave guide by a second ceramic window. The elaborate two-win dow solution was chosen to get optimum protection of the cavity against contamination dur ing mounting in the cryomodule and against window fracture during linac operation. The couplers must allow for some longitudinal motion9inside the 12m long cryomodule when the cavities are cooled down from room temperature to 2K . For this reason bellows in the inner and outer conductors of the coaxial line are needed . Since the coupler connects the room-temperature waveguide with the 2K cavity, a compromis e must be found between a low thermal conductivity and a high electrical conductivity. T his is achieved by several thermal intercepts and by using stainless steel pipes or bellows wit h a thin copper plating (10–20 µm) at the radio frequency surface. The design heat loads of 6W at 70K, 0.5W at 4K and 0.06W at 2K have been undercut in practice. Electrical properties An instantaneous power of 210kW has to be transmitted to prov ide a gradient of 25MV/m for an 800 µs long beam pulse of 8 mA. The filling time of the cavity amounts to 530 µs and the decay time, after the beam pulse is over, to an additional 500 µs. At the beginning of the filling, most of the rf wave is reflected leading to voltage enhancemen ts by a factor of 2. The external quality factor of the coupler is Qext= 3·106at 25 MV/m. By moving the inner conductor of the coaxial line, Qextcan be varied in the range 1 ·106– 9·106to allow not only for different beam loading conditions but also to facilitate an in-situ hi gh power processing of the cavities. This feature has proved extremely useful on several occasio ns to eliminate field emitters that entered the cavities at the last assembly stage. 9The motion of the coupler ports is up to 15 mm in the first cryomo dules but has been reduced to about 1 mm in the most recent cryostat design by fixing the distance b etween neighboring cavities with invar rods. 15Figure 9: Simplified view of the power input coupler version A . Input coupler A The coupler version A is shown in Fig. 9. It has a conical ceram ic window at 70 K and a commercial planar waveguide window at room temperature. A conical shape was chosen for the cold ceramic window to obta in broad-band impedance matching. The Hewlett-Packard High Frequency Structure Si mulator program HFSS was used to model the window and to optimize the shape of the tapered in ner conductor. The reflected power is below 1%. The ceramic window is made from Al 2O3with a purity of 99.5%. OFHC copper rings are brazed to the ceramic using Au/Cu (35%/65%) braze alloy. The inner con- ductors on each side of the ceramic are electron-beam welded , the outer conductors are TIG welded. The ceramic is coated on both sides with a 10 nm titani um nitride layer to reduce multipacting. The waveguide-to-coaxial transition is realized using a cy lindrical knob as the impedance- transforming device and a planar waveguide window. Matchin g posts are required on the air side of the window for impedance matching at 1.3 GHz. Input couplers B, C Coupler version B uses also a planar wave guide window and a do or-knob transition from the wave guide to the coaxial line, but a cylindrical ceramic win dow at 70 K without direct view of the beam. Owing to a shortage in commercial wave guide wind ows a third type, C, was developed using a cylindrical window also at the wave guide - coaxial transition. It features a 60 mm diameter coaxial line with reduced sensitivity to mul tipacting and the possibility to apply a dc potential to the center conductor. In case of the LE P couplers [22] a dc bias has proved very beneficial in suppressing multipacting. Simila r observations were made at DESY. All couplers needed some conditioning but have then perform ed according to specification. 3.5 Higher order modes The intense electron bunches excite eigenmodes of higher fr equency in the resonator which must be damped to avoid multibunch instabilities and beam br eakup. This is accomplished by extracting the stored energy via higher-order mode (HOM) couplers mounted on the beam pipe sections of the nine-cell resonator. A problem arises f rom “trapped modes” which are concentrated in the center cells and have a low field amplitud e in the end cells. An example is the TE121mode. By an asymmetric shaping of the end half cells one can en hance the field amplitude of the TE 121mode in one end cell while preserving the “field flatness” of th e fundamental mode and also the good coupling of the HOM couplers to the untrappe d modes TE 111, TM 110and TM011. The effects of asymmetric end cell tuning are sketched in Fig . 10. The two polarization states of dipole modes would in princip le require two orthogonal HOM couplers at each side of the cavity. In a string of cavities, h owever, this complexity can be avoided since the task of the “orthogonal” HOM coupler can be taken ov er by the HOM coupler of the neighboring cavity. The viability of this idea was verified i n measurements. 16left end cell right end cellTM010, TE111, TM110 TE121 TM011TM011 TE121 Figure 10: Effect of asymmetric end cell shaping on various mo des. The main accelerating mode TM010and the higher modes TE 111and TM 110are not affected while TM 011is enhanced in the left end cell, TE 121in the right end cell. Using HOM couplers at both ends, all hig her order modes can be extracted. HOM coupler design The HOM couplers are mounted at both ends of the cavity with a n early perpendicular ori- entation10to ensure damping of dipole modes of either polarization. A 1 .3 GHz notch filter is incorporated to prevent energy extraction from the accel erating mode. Two types of HOM couplers have been developed and tested, one mounted on a flan ge, the other welded to the cavity. The demountable HOM coupler is shown in Fig. 11a. An antenna l oop couples mainly to the magnetic field for TE modes and to the electric field for TM m odes. The pickup antenna is capacitively coupled to an external load. The 1.3 GHz notch fi lter is formed by the inductance of the loop and the capacity at the 1.9 mm wide gap between loop and wall. A niobium bellows permits tuning of the filter without opening the cavity vacuu m. The antenna is thermally connected to the 2K helium bath. In a cw (continuous wave) tes t at an accelerating field of 21MV/m the antenna reached a maximum temperature of 4K, whic h is totally uncritical. The welded version of the HOM coupler is shown in Fig. 11b. It r esembles the couplers used in the 500MHz HERA cavities which have been operating for sev eral years without quenches. The good cooling of the superconducting inner conductor by t wo stubs makes the design insen- sitive to γradiation and electron bombardment. Both HOM couplers permit tuning of the fundamental mode reje ction filter when mounted on the cavity. It is possible to achieve a Qextof more than 1011thereby limiting power extraction to less than 50 mW at 25 MV/m. 4 Cavity Fabrication and Preparation 4.1 Cavity fabrication 4.1.1 Niobium properties The 9-cell resonators are made from 2.8 mm thick sheet niobiu m by deep drawing of half cells, followed by trimming and electron beam welding. Niobium of h igh purity is needed. Tantalum with a typical concentration of 500 ppm is the most important metallic impurity. Among the interstitially dissolved impurities oxygen is dominant du e to the high affinity of Nb for O 2 10The angle between the two HOM couplers is not 90◦but 115◦to provide also damping of quadrupole modes. 17(a) (b)capacitive coupling output capacitor of notch filter output Figure 11: The higher-order-mode couplers: (a) demountabl e HOM coupler, (b) welded HOM coupler. 18Table 4: Technical specification for niobium used in TTF cavi ties Impurity content in ppm (wt) Mechanical Properties Ta≤500 H≤2 Residual resistivity ratio RRR ≥300 W≤70 N≤10 grain size ≈50µm Ti≤50 O≤10 yield strength >50 MPa Fe≤30 C≤10 tensile strength >100 MPa Mo≤50 elongation at break 30 % Ni≤30 Vickers hardness HV 10 ≤50 above 200◦C. Interstitial atoms act as scattering centers for the unpa ired electrons and reduce theRRR and the thermal conductivity, see Sect. 2.1. The niobium ing ot is highly purified by several remelting steps in a high vacuum electron beam furna ce. This procedure reduces the interstitial oxygen, nitrogen and carbon contamination to a few ppm. The niobium specification for the TTF cavities is listed in Table 4. After forging and sheet rolling, the 2.8 mm thick Nb sheets ar e degreased, a 5 µm surface layer is removed by etching and then the sheets are annealed f or 1–2 hours at 700–800◦C in a vacuum oven at a pressure of 10−5– 10−6mbar to achieve full recrystallization and a uniform grain size of about 50 µm. 4.1.2 Deep drawing and electron-beam welding Half-cells are produced by deep-drawing. The dies are usual ly made from a high yield strength aluminum alloy. To achieve the small curvature required at t he iris an additional step of forming, e.g. coining, may be needed. The half-cells are machined at t he iris and the equator. At the iris the half cell is cut to the specified length (allowing for weld shrinkage) while at the equator an extra length of 1 mm is left to retain the possibility of a pr ecise length trimming of the dumb-bell after frequency measurement (see below). The acc uracy of the shape is controlled by sandwiching the half-cell between two metal plates and me asuring the resonance frequency. The half-cells are thoroughly cleaned by ultrasonic degrea sing, 20 µm chemical etching and ultra-pure water rinsing. Two half-cells are then joined at the iris with an electron-beam (EB) weld to form a “dumb-bell”. The EB welding is usually done fro m the inside to ensure a smooth weld seam at the location of the highest electric field in the r esonator. Since niobium is a strong getter material for oxygen it is important to carry out the EB welds in a sufficiently good vacuum. Tests have shown that RRR = 300 niobium is not degraded by welding at a pressure of less than 5 ·10−5mbar. The next step is the welding of the stiffening ring. Here the we ld shrinkage may lead to a slight distortion of the cell shape which needs to be correct ed. Afterwards, frequency measure- ments are made on the dumb-bells to determine the correct amo unt of trimming at the equators. After proper cleaning by a 30 µm etching the dumb-bells are visually inspected. Defects an d foreign material imprints from previous fabrication steps are removed by grinding. After the inspection and proper cleaning (a few µm etching followed by ultra-clean water rinsing and clean room drying), eight dumb-bells and two beam-pipe sect ions with attached end half-cells are stacked in a precise fixture to carry out the equator welds which are done from the outside. The weld parameters are chosen to achieve full penetration. A reliable method for obtaining a smooth weld seam of a few mm width at the inner surface is to ra ster a slightly defocused beam in an elliptic pattern and to apply 50 % of beam power duri ng the first weld pass and 19100 % of beam power in the second pass. 4.2 Cavity treatment Experience has shown that a damage layer in the order of 100 µm has to be removed from the inner cavity surface to obtain good rf performance in the sup erconducting state. The standard method applied at DESY and many other laboratories is called Buffered Chemical Polishing (BCP), using an acid mixture of HF (48 %), HNO 3(65 %) and H 3PO4(85 %) in the ratio 1:1:2 (at CEBAF the ratio was 1:1:1). The preparation steps adopte d at DESY for the industrially produced TTF cavities are as follows. A layer of 80 µm is removed by BCP from the inner surface, 30 µm from the outer surface11. The cavities are rinsed with ultra-clean water and dried in a class 100 clean room. The next step is a two-hour ann ealing at 800◦C in an Ultra High Vacuum (UHV) oven which serves to remove dissolved hydr ogen from the niobium and relieves mechanical stress in the material. In the initial p hase of the TTF program many cavities were tested after this step, applying a 20 µm BCP and ultra-clean water rinsing before mounting in the cryostat and cooldown. Presently, the cavities are rinsed with clean water after th e 800◦C treatment and then immediately transferred to another UHV oven in which they ar e heated to 1350–1400◦C. At this temperature, all dissolved gases diffuse out of the mate rial and the RRR increases by about a factor of 2 to values around 500. To capture the oxygen coming out of the niobium and to prevent oxidation by the residual gas in the oven (pres sure<10−7mbar) a thin titanium layer is evaporated on the inner and outer cavity surface, Ti being a stronger getter than Nb. The high-temperature treatment with Ti getter is often call ed post-purification. The titanium layer is removed afterwards by a 80 µm BCP of the inner surface. A BCP of about 30 µm is applied at the outer surface since the Kapitza resistance of titanium-coated niobium immersed in superfluid helium is about a factor of 2 larger than that of p ure niobium [11]. After final heat treatment and BCP the cavities are mechanically tuned to adj ust the resonance frequency to the design value and to obtain equal field amplitudes in all 9 c ells. This is followed by a slight BCP, three steps of high-pressure water rinsing (100 bar) an d drying in a class 10 clean room. As a last step, the rf test is performed in a superfluid helium b ath cryostat. A severe drawback of the post-purification is the considerab le grain growth accompanied with a softening of the niobium. Postpurified-treated cavit ies are quite vulnerable to plastic deformation and have to be handled with great care. 5 Results on Cavity Performance and Quality Control Measures 5.1 Overview Figure 12 shows the “excitation curve” of the best 9-cell res onator measured so far; plotted is the quality factor12Q0as a function of the accelerating electric field Eacc. An almost constant and high value of 2 ·1010is observed up to 25 MV/m. 11These numbers are determined by weighing the cavity before a nd after etching and represent therefore the average over the whole surface. Frequency measurements ind icate that more material is etched away at the iris than at the equator. 12The quality factor is defined as Q0=f/∆fwhere fis the resonance frequency and ∆ fthe full width at half height of the resonance curve of the “unloaded” cavity. 200 5 10 15 20 25 30 35 E [MV/m]Q acc0 81091010101110 Figure 12: Excitation curve of the best TESLA 9-cell cavity m easured up to date. The cavity was cooled by superfluid helium of 2 K. The importance of various cavity treatment steps for arrivi ng at such a good performance are illustrated in the next figure. A strong degradation is us ually observed if a foreign particle is sticking on the cavity surface, leading either to field emiss ion of electrons or to local overheating in the rf field. At Cornell University an in situ method for destroying field emitters was invented [23], called “high power processing” (HPP), which in many ca ses can improve the high-field capability, see Fig. 13a. Removal of field-emitting particl es by high-pressure water rinsing, a technique developed at CERN [24], may dramatically improve the excitation curve (Fig. 13b). The beneficial effect of a 1400◦C heat treatment, first tried out at Cornell [25] and Saclay [2 6], is seen in Fig. 13c. Finally, an incomplete removal of the tit anium surface layer in the BCP following the 1400◦C heat treatment may strongly limit the attainable gradient . Here additional BCP is of advantage (Fig. 13d). 5.2 Results from the first series of TTF cavities After the successful test of two prototype nine-cell resona tors a total of 27 cavities, equipped with main power and HOM coupler flanges, were ordered at four E uropean companies. These cavities were foreseen for installation in the TTF linac wit h an expected gradient of at least 15 MV/m at Q0>3·109. However, in the specification given to the companies no guar anteed gradient was required. According to the test results obtain ed at TTF these resonators can be classified into four categories: (1) 16 cavities without any known material and fabrication d efects, or with minor defects which could be repaired, (2) 3 cavities with serious material defects, 2110910101011 0 5 10 15 20 25 30before HPP after HPPQ0 Eacc [MV/m](a) 10910101011 0 5 10 15 20 25 30before HPR after HPRQ0 Eacc [MV/m](b) 10910101011 0 5 10 15 20 25 30first test second testQ0 Eacc [MV/m](d) 10910101011 0 5 10 15 20 25 30Q0 Eacc [MV/m]no HT, RRR 400 HT 800 C, RRR 400 HT 1400 C, RRR 770quenchquench no quench limited by amplifierx-ray starts(c) Figure 13: Improvement in cavity performance due to various treatments: (a) high power pro- cessing, (b) high pressure water rinsing, (c) successive ap plication of 800◦C and 1400◦C heat treatment, (d) removal of surface defects or titanium in gra in boundaries by additional BCP. (3) 6 cavities with imperfect equator welds, (4) 2 cavities with serious fabrication defects (not fully p enetrated electron beam welds or with holes burnt during welding; these were rejected). One cavity has not yet been tested. The test results for the cavities of class (1) in a vertical ba th cryostat with superfluid helium cooling at 2 K are summarized in Fig. 14. It is seen that the TTF design goal of 15 MV/m is clearly exceeded. Nine of the resonators fulfill even the mor e stringent specification of TESLA (Eacc≥25 MV/m at Q0≥5·109). The excitation curves of the class 2 cavities (Fig. 15) are ch aracterized by sudden drops in quality factor with increasing field and rather low maximu m gradients. Temperature map- ping revealed spots of excessive heating at isolated spots w hich were far away from the EB welds. An example is shown in Fig. 16a. The defective cell was cut from the resonator and sub- jected to further investigation [27]. An eddy-current scan , performed at the Bundesanstalt f¨ ur Materialforschung (BAM) in Berlin, showed a pronounced sig nal at the defect location. With X-ray radiography, also carried out at BAM, a dark spot with a size of 0.2–0.3 mm was seen (Fig. 16b) indicating an inclusion of foreign material with a higher nuclear charge than nio- bium. Neutron absorption measurements at the Forschungsze ntrum GKSS in Geesthacht gave no signal, indicating that the neutron absorption coefficien t of the unknown contamination was similar to that of Nb. The identification of the foreign in clusion was finally accomplished using X-ray fluorescence (XAFS) at the Hamburger Synchrotro nstrahlungslabor HASYLAB 2210910101011 0 5 10 15 20 25 30C3 C18 C21 C28Q0 Eacc [MV/m] (a) 01234 0 5 10 15 20 25 30 35 40number of cavities Eacc [MV/m] (b) Figure 14: (a) Excitation curves of the best 9-cell resonato r of each of the four manufacturers. (b) Distribution of maximum gradients for the resonators of class 1, requiring a quality factor Q0≥5·109. 2310910101011 0 5 10 15 20Q0 Eacc [MV/m] Figure 15: Excitation curves of three cavities with serious material defects (class 2). Cavity C5 (/trianglesolid) exhibited a jump in quality factor. (a) (b) Figure 16: (a) Temperature map of cell 5 of cavity C6 showing e xcessive heating at a localized spot. (b) Positive print of an X-ray radiograph showing the “ hot spot” as a dark point. at DESY. Fluorescence was observed at photon energies corre sponding to the characteristic X-ray lines of tantalum L 1= 11.682 keV, L 2= 11.136 keV and L 3= 9.881 keV. The SYRFA (synchrotron radiation fluorescence analysis) method feat ures sufficient sensitivity to perform a scan of the tantalum contents in the niobium by looking at th e lines Ta-K α1= 57.532 keV, Ta-K α2= 56.277 keV and Ta-K β1= 65.223 keV. The average Ta content in the bulk Nb was about 200 ppm but rose to 2000 ppm in the spot region. The RRR dropped correspondingly from 330 to about 60. The six cavities in class 3 were produced by one company and ex hibited premature quenches at gradients of 10–14 MV/m and a slope in the Q(E) curve (Fig. 17). Two of the resonators were investigated in greater detail [28]. Temperature mapp ing revealed strong heating at several spots on the equator weld (Fig. 18b). The temperature rise as a function of the surface magnetic field is plotted in Fig. 18c for one sensor position above the w eld and three positions on the weld. In the first case a growth proportional to B2is observed as expected for a constant surface resistance. On the weld, however, a much stronger rise is see n ranging from B5toB8. This is clear evidence for a contamination of the weld seam. Once the reason for the reduced performance of the cavities i n class 3 had been identified a new 9-cell resonator was manufactured by the same company a pplying careful preparation steps of the weld region: a 2 µm chemical etching not more than 8 hours in advance of the EB welding, rinsing with ultrapure water and drying in a clean r oom. A rastered electron beam 2410910101011 0 5 10 15 20 25 30Q0 Eacc [MV/m] Figure 17: Excitation curves of six cavities with imperfect equator welds (class 3). Also shown is a resonator ( /squaresolid) made later by the same company, following stringent cleani ng procedures at the equator welds. was used for welding with 50% penetration in the first weld lay er and 100% in the second. The new cavity indeed showed excellent performance and achieve d 24.5 MV/m, see Fig. 17. The same applies for later cavities made by this company. The average gradient of the cavities without serious materi al or fabrication defects amounts to 20.1±6.2 MV/m at Q0= 5·109where the error represents the rms of the distribution. 5.3 Diagnostic methods and quality control The deficiencies encountered in the first series production o f TESLA cavities have initiated the development of diagnostic methods and quality control proc edures. Electron microscopy Scanning electron microscopy with energy-dispersive X-ra y analysis (EDX) is used to identify foreign elements on the surface. Only a depth of about 1 µm can be penetrated, so one has to remove layer by layer to determine the diffusion depth of tita nium or other elements. Alter- natively one can cut the material and scan the cut region. The titanium layer applied in the high temperature treatment has been found to extend to a dept h of about 10 µm in the bulk niobium. The sensitivity of the EDX method is rather limited ; a Ti fraction below 0.5 % is undetectable. Auger electron spectroscopy offers higher se nsitivity and using this method tita- nium migration at grain boundaries has been found to a depth o f 50–100 µm. Hence this large thickness must be removed from the rf surface by BCP after pos t-purification with Ti getter. The detrimental effect of insufficient titanium removal has al ready been shown in Fig. 13d. The microscopic methods are restricted to small samples and cannot be used to study entire cavities. 25boardcavity surface equator weld 24 x1.2 cm (a) 0.250 0.000 24 0246810121416182022 ΔT [K] (b) ΔT [K] 0.020 0.000 ~B2Board 14 above the weldΔT [K] 0.040 0.000 ~B5Board 22 on the weld ΔT [K] 0.070 0.000 ~B8Board 1 on the weldΔT [K] 0.250 0.000 ~B2quench location Board 24 on the weld (c) Figure 18: (a) Location of temperature sensors to determine heating at the equator weld. (b) Temperature map of the equator region from cell 5 of cavit y C9 just below the quench. (c) Temperature rise at various locations as a function of Bnwithnbetween 2 and 8. 26(a) (b) Figure 19: (a) Schematic view of the xy eddy-current scannin g system. (b) Photo of the new rotating scanning system. X-ray fluorescence The narrow-band X-ray beams at HASYLAB permit element ident ification via fluorescence analysis. In principle the existing apparatus allows the sc anning of a whole niobium sheet such as used for producing a half-cell, however the procedure wou ld be far too time-consuming. Eddy-current scan A practical device for the quality control of all niobium she ets going into cavity production is a high-resolution eddy-current system developed by the Bun desanstalt f¨ ur Materialforschung (BAM) in Berlin. The apparatus is shown in Fig. 19. The freque ncy used is 100 kHz corre- sponding to a penetration depth of 0.5 mm in niobium at room te mperature. The maximum scanning speed is 1 m/s. The scanning probe containing the in ducing and receiving coils floats on an air pillow to avoid friction. The machined base plate co ntains holes for evacuating the space between this plate and the Nb sheet. The atmospheric pr essure is sufficient to flatten the 265 x 265 mm2niobium sheets to within 0.1 mm which is important for a high s ensitivity scan. The performance of the apparatus was tested with a Nb test she et containing implanted tan- talum deposits of 0.2 to 1 mm diameter. The scanned picture (F ig. 20a) demonstrates that Ta clusters are clearly visible. Using this eddy-current appa ratus the tantalum inclusion in cavity C6 was easily detectable. In the meantime an improved eddy-current scanning device ha s been designed and built at BAM which operates similar to a turn table and allows for much higher scanning speeds and better sensitivity since the accelerations of the probe hea d occuring in xy scans are avoided. A two frequency principle is applied in the new system. Scanni ng with high frequency (about 1 MHz) allows detection of surface irregularities while the l ow frequency test (about 150 kHz) is sensitive to bulk inclusions. The high and low frequency sig nals are picked up simultaneously. Very high sensitivity is achieved by signal subtraction. Neutron activation analysis The eddy-current scan allows the detection of foreign mater ials in the niobium but is not suitable for identification. Neutron activation analysis p ermits a non-destructive determination of the contaminants provided they have radioactive isotope s with a sufficiently long half life. Experiments were carried out at the research reactor BER II o f the Hahn Meitner Institut in Berlin. The niobium sheets are exposed to a thermal neutron fl ux of 109cm−2s−1for some 5 hours. The radioactive isotope94Nb has a half life of 6.2 min while182Ta has a much longer half life of 115 days. Two weeks after the irradiation the94Nb activity has dropped to such a low level that tantalum fractions in the ppm range can be ide ntified. Figure 20b shows the implanted tantalum clusters in the specially prepared N b plate with great clarity. Also the uniformly dissolved Ta is visible and the inferred concentr ation of 200 ppm is in agreement with the chemical analysis. The activation analysis is far too ti me consuming for series checks but can be quite useful in identifying special contaminations f ound with the eddy-current system. 27(a) (b) Figure 20: (a) Eddy-current scan of a specially prepared Nb s heet with Ta implantations at 5 locations. (b) Neutron activation analysis of the same shee t. Ten Nb sheets from the regular production were investigated without showing any evidence for tantalum clusters. 5.4 Present status of TTF cavities 5.4.1 Improvements in cavity production For the second series 25 cavities have been ordered at four fir ms. The second production differed from the first one in three main aspects. (1) Stricter quality control of niobium The niobium sheets for the second series were all eddy-curre nt scanned to eliminate material with tantalum or other foreign inclusions before the deep dr awing of half cells. From 715 sheets 637 were found free of defects, 63 showed grinding marks or im prints from rolling and 15 exhibited large signals which in most cases were due to small iron chips. No further Nb sheets with tantalum inclusions were found. Most of the rejected sh eets will be recoverable by applying some chemical etching. The iron inclusions were caused by me chanical wear of the rolls used for sheet rolling. In the meantime new rolls have been instal led. The eddy-current check has turned out to be an important quality control not only for the cavity manufacturer but also for the supplier of the niobium sheets. (2) Weld preparation Stringent requirements were imposed on the electron-beam w elding procedure to prevent the degraded performance at the equator welds encountered in th e first series. After mechanical trimming the weld regions were requested to be cleaned by a sl ight chemical etching, ultrapure water rinsing and clean-room drying not more than 8 hours in a dvance of the EB welding. The success of these two additional quality control measure s has been convincing: no foreign material inclusions nor weld contaminations were found in t he cavities tested so far. (3) Replacement of Nb flanges by NbTi flanges In the first cavity series the flanges at the beam pipes and the c oupler ports were made by rolling over the 2 mm thick niobium pipes. The sealing agains t the stainless steel counter flanges was provided by Helicoflex gaskets. This simple design appea red satisfactory in a number of prototype cavities but proved quite unreliable in the serie s production, mainly due to a softening of the niobium during the 1400◦C heat treatment. Most of the nine-cell cavities had to be flan ged more than once to become leak tight in superfluid helium. This caused not only time delays but also severe problems with contamination and field emissi on. Therefore an alternative flange design was needed [29]. The material was selected to be EB-we ldable to niobium and to possess a surface hardness equivalent to that of standard UHV flange m aterial (stainless steel 316 LN/ DIN 4429). Niobium-titanium conforms to these requirement s at a reasonable cost. Contrary 28051015202530 C29C30C31C32C33C34C35C36C37C38C39C40C41C42C43C44C45C46C47C49Eacc [MV/m] C28bd bd bdbd bd bd bdbd bd bdpw pwpwpw pw pwpw pw pwpwpw TESLA goal Figure 21: Test results of the second cavity series (cross-h atched bars); plotted is the highest gradient achieved in the first rf test of each cavity at Q0≥5·109. Cavities with poor initial performance were subjected to an additional BCP and high pre ssure water rinsing and tested again (open bars). Field limitation by amplifier power (pw) o r thermal breakdown (bd) is indicated for the best gradient. to pure niobium the alloy NbTi (ratio 45/55 in wt %) shows no so ftening after the 1400◦C heat treatment and only a moderate crystal growth. O-ring type al uminum gaskets provide reliable seals in superfluid helium and are easier to clean than Helico flex gaskets. During cavity etching the sealing surface must be protected from the acid. 5.4.2 Test results in vertical cryostat All new cavities were subjected to the standard treatment de scribed in Sect. 4.2, including the post-purification with titanium getter at 1400◦C. Twenty resonators have been tested up to date. Only one rf test was performed for each resonator in the first round. If some limitation was found the cavity was put aside for further treatment. The results of the first test sequence are summarized in Fig. 21. It is seen that 8 cavities reach or e xceed the TESLA specification ofEacc≥25 MV/m with a quality factor above 5 ·109. Eight cavities are in the range of 18 to 23 MV/m while four cavities show a much lower performance. In cavity C43 a hole was burnt during equator welding which was repaired by welding i n a niobium plug; the cavity quenched at 13 MV/m at exactly this position. It is rather unl ikely that C43 can be recovered by repeating the repair. Therefore, in future cavity produc tion repaired holes in EB welds will no longer be acceptable. The cavities C32, C34 and C42 showed very strong field emission in the first test. They have been improved in the meantime by addi tional BCP and high pressure water rinsing, see Fig. 21. Excluding the defective cavity C 43, the average gradient is 25 .0±3.2 MV/m at Q0= 5·109. 29010203040 0 10 20 30 40Eacc [MV/m] horizontal test Eacc [MV/m] vertical test Figure 22: Comparison of vertical and horizontal test resul ts. The average accelerating field achieved in the vertical test with cw excitation is 22.3 MV/m , in the horizontal test with pulsed excitation 22.5 MV/m. Most of these cavities are from the sec ond production. 5.4.3 Tests with main power coupler in horizontal cryostat After the successful test in the vertical bath cryostat the c avities are welded into their liquid helium container and equipped with the main power coupler. T he external Qis typically 2 ·106, while in the vertical test an input antenna with an external Qof more than 1011is used. Four cavities of the first production series and thirteen of the se cond series have been tested together with their main power coupler in a horizontal cryostat. The a ccelerating fields achieved in the vertical and the horizontal test are quite similar as shown i n Fig. 22. In a few cases reduced performance was seen due to field emission while several cavi ties improved their field capability due to the fact that with the main power coupler pulsed operat ion is possible instead of the cw operation in the vertical cryostat. These results indica te that the good performance of the cavities can indeed be preserved after assembly of the liqui d helium container and the power coupler provided extreme care is taken to avoid foreign part icles from entering the cavity during these assembly steps. 5.4.4 Cavity improvement by heat treatment The beneficial effect of the 800◦C and 1400◦C heat treatments has been shown in Fig. 13c. Ten of the 9-cell cavities have been tested after the intermedia te 800◦C step yielding an average gradient of 20.7 MV/m. The 1400◦C treatment with titanium getter raised the average gradien t to 24.4 MV/m. It should be noted that part of the 3.7 MV/m impro vement may be due to the additional etching of about 80 µm. An interesting correlation is obtained by plotting the maximum gradient as a function of the measured RRR of the cavity, see Fig. 23. This figure clearly indicates that a higher heat conductivity leads to h igher accelerating fields, at least if 300100 200 300 400 500 600 700 800051015202530 RRREacc [MV/m] annealed at 800C annealed with Ti at 1400C Figure 23: Maximum gradient as a function of RRR. the standard BCP treatment is applied to prepare the cavity s urface. 6 RF Control System and Performance of the Cavities with Electron Beam 6.1 General demands on the rf control system The requirements on the stability of the accelerating field i n a superconducting acceleration structure are comparable to those in a normal-conducting ca vity. However the nature and mag- nitude of the perturbations to be controlled are rather diffe rent. Superconducting cavities pos- sess a very narrow bandwidth and are therefore highly suscep tible to mechanical perturbations. Significant phase and amplitude errors are induced by the res ulting frequency variations. Per- turbations can be excited by mechanical vibrations (microp honics), changes in helium pressure and level, or Lorentz forces. Slow changes in frequency, on t he time scale of minutes or longer, are corrected by a frequency tuner, while faster changes are counteracted by an amplitude and phase modulation of the incident rf power. The demands on amplitude and phase stability of the TESLA Tes t Facility cavities are driven by the maximum tolerable energy spread in the TTF lina c. The design goal is a relative spread of σE/E= 2·10−3implying a gradient and phase stability in the order of 1 ·10−3and 1.6◦, respectively. For cost reasons up to 32 cavities will be pow ered by a single klystron. Hence it is not possible to control individual cavities but only th e vector sum of the field vectors in these 32 cavities. One constraint to be observed is that the rf power needed for c ontrol should be minimized. The rf control system must also be robust against variations of system parameters such as beam loading and klystron gain. The pulsed structure of the rf power and the beam at TTF, shown in Fig. 24, puts demanding 31time[µs] 500 1000 1500 20000Accelerating voltage cavity detuningincident power cavity phasebeam currentFill time Flat topamplitude [a.u] Figure 24: Pulse structure of TTF cavity operation. Shown ar e: accelerating voltage with 500µs filling time and 800 µs flat top, incident power, beam current, cavity phase and cav ity detuning. requirements on the rf control system. Amplitude and phase c ontrol is obviously needed during the flat-top of 800 µs when the beam is accelerated, but it is equally desirable to control the field during cavity filling to ensure proper beam injection co nditions. Field control is aggravated by the transients induced by the subpicosecond electron bun ches which have a repetition rate of 1 to 9 MHz. For a detailed discussion of the basic principles of rf syste ms used in superconducting electron linacs and their operational performance, refer to [30] and [31]. 6.2 Sources of field perturbations There are two basic mechanisms which influence the magnitude and phase of the accelerating field in a superconducting cavity: •variations in klystron power or beam loading (bunch charge) •modulation of the cavity resonance frequency. Perturbations of the accelerating field through time-varyi ng field excitations are dominated by changes in beam loading. One must distinguish between tra nsients caused by the pulsed structure of the beam current and stochastic fluctuations of the bunch charge. The transients caused by the regular bunch train in the TTF linac (800 picose cond bunches of 8 nC each, spaced by 1 µs) are in the order of 1% per 10 µs; the typical bunch charge fluctuations of 10% induce field fluctuations of about 1%. In both cases the effect o f the fast source fluctuations on the cavity field is diminuished by the long time constant of th e cavity13. Mechanical changes of the shape and eigenfrequency of the ca vities caused by microphonics are a source of amplitude and phase jitter which has bothered superconducting accelerator technology throughout its development. In the TTF cavities the sensitivity of the resonance frequency to a longitudinal deformation is about 300 Hz/ µm. Heavy machinery can transmit 13The cavity with power coupler is adjusted to an external Qof 3·106at 25 MV/m, corresponding to a time constant of about 700 µs. The consequence is a low-pass filter characteristic. 3217:30 18:00 18:30−30−25−20−15−10−5 cavity C3 time [h]detuning [Hz] −20 −10 0 10 20010203040 f-f [Hz]no. of measurementscavity C3 σ = 4 HzΔf 0 (a) (b) Figure 25: Fluctuations of the cavity resonance frequency. (a) Slow drifts caused by helium pres- sure variations. The sensitivity is 10 Hz/mbar. (b) Random v ariations of resonance frequency after correction for the slow drift. vibrations through the ground, the support and the cryostat to the cavity. Vacuum pumps can interact with the cavity through the beam tubes, and the c ompressors and pumps of the refrigerator may generate mechanical vibrations which tra vel along the He transfer line into the cryostat. Also helium pressure variations lead to changes i n resonance frequency as shown in Fig. 25a. The rms frequency spread due to microphonics, meas ured in 16 cavities, is 9 .5±5.3 Hz and thus surprisingly small for a superconducting cavity sy stem (see Fig. 25b). At high accelerating gradients the Lorentz forces become a s evere perturbation. The cor- responding frequency shift is proportional to the square of the accelerating field according to ∆f=−K·E2 accwithK≈1 Hz/(MV/m)2. Figure 26a shows a cw measurement of the resonance curve with a strong distortion caused by Lorentz forces. In c w operation the frequency shift can be easily corrected for by mechanical tuning. In the pulsed m ode employed at the TTF linac this is not possible since the mechanical tuner is far too slo w. Hence a time-dependent detuning is unavoidable. In order to keep the deviation from the nomin al resonance frequency within acceptable limits the cavities are predetuned before fillin g. The measured dynamic detuning of cavity C39 during the 1.3 ms long rf pulse is shown in Fig. 26 b for accelerating fields of 15 to 30 MV/m. Choosing a predetuning of +300 Hz, the eigenfrequ ency at 25 MV/m changes dynamically from +100 Hz to −120 Hz during the 800 µs duration of the beam pulse. In steady state (cw) operation at a gradient of 25 MV/m and a be am current of 8 mA a klystron power of 210 kW is required per nine-cell cavity. In pulsed mode ≈15 % additional rf power is needed to maintain a constant accelerating gradien t in the presence of cavity detuning. The frequency changes from microphonics and helium pressur e fluctuations lead to comparable extra power requirements. The klystron should be operated 1 0 % below saturation to guarantee sufficient gain in the feedback loop. 6.3 RF control design considerations The amplitude and phase errors from Lorentz force detuning, beam transients and microphonics are in the order of 5% and 20◦, respectively. These errors must be suppressed by one to two orders of magnitude. Fortunately, the dominant errors are r epetitive (Lorentz forces and beam transients) and can be largely eliminated by means of a feedf orward compensation. It should be noted, however, that bunch-to-bunch fluctuations of the b eam current cannot be suppressed by the rf control system since the gain bandwidth product is l imited to about 1 MHz due to the 33−80 −60 −40 −20 0 20 4001234567 Δf = - K · Eacc2gradient [MV/m] Δf [Hz]K = 0.9Hz (MV/m)2 0 500 1000 1500 2000−300−200−1000100200300 time [µs]detuning [Hz] fill: 500 µs flat: 800 µs15 MV/m 20 MV/m 25 MV/m 30 MV/m (a) (b) Figure 26: (a) Influence of Lorentz forces on the shape of the r esonance curve of a sc cavity in cw operation. The left part of the curve was mapped out by appr oaching the resonance from below, the right part by coming from above. (b) Dynamical det uning of cavity C39 during the TESLA pulse. In pulsed operation the resonance is approache d from above. low-pass characteristics of the cavity, the bandwidth limi tations of electronics and klystrons, and cable delay. Fast amplitude and phase control can be accomplished by modu lation of the incident rf wave which is common to 32 cavities. The control of an individ ual cavity field is not possible. The layout of the TTF digital rf control system is shown in Fig . 27. The vector modulator for the incident wave is designed as a so-called “I/Q modulator” controlling real and imaginary part of the complex cavity field vector instead of amplitude a nd phase. This has the advantage that the coupling between the two feedback loops is minimize d and the problem of large phase uncertainties at small amplitude is avoided. The detectors for cavity field, incident and reflected wave ar e implemented as digital detec- tors for real and imaginary part. The rf signals are converte d to an intermediate frequency of 250 kHz and sampled at a rate of 1 MHz, which means that two subs equent data points yield the real and the imaginary part of the cavity field vectors. Th ese vectors are multiplied with 2×2 rotation matrices to correct for phase offsets and to calibr ate the gradients of the indi- vidual cavity probe signals. The vector sum is calculated an d a Kalman filter is applied which provides an optimal state (cavity field) estimate by correct ing for delays in the feedback loop and by taking stochastic sensor and process noise into accou nt. Finally the nominal set point is subtracted and a time-optimal gain matrix is applied to ca lculate the new actuator setting (theReandImcontrol inputs to the vector modulator). Adaptive feedforw ard is realized by means of a table containing the systematic variations, ther eby reducing the task of the feedback loop to control the remaining stochastic fluctuations. The f eedforward tables are continuously updated to take care of slow changes in parameters such as ave rage detuning angle, microphonic noise level and phase shift in the feedforward path. 6.4 Operational experience The major purpose of the TESLA Test Facility linac is to demon strate that all major accelerator subsystems meet the technical and operational requirement s of the TESLA collider. Currently the TTF linac is equipped with two cryomodules each containi ng 8 cavities. The cavities are 34DAC DACReImCavity 32 ......8xCavity 25klystronvector modulator master oscillator 1.3 GHz Cavity 8 ......8xCavity 1 cryomodule 4... cryomodule 1. . . .LO 1.3 GHz + 250 kHz 250 kHzADC f = 1 MHzs . . . .... vector-sumΣ( )aba-b 1 8( )aba-b 25( )aba-b 32( )aba-b DSP systemsetpoint tablegain tablefeed tableforward+ +digital low pass filter ImRe ImRe ImRe clockLO ADCLO ADCLO ADC ImRepower transmission line 1.3GHz field probe Figure 27: Schematic of the digital rf control system. routinely operated at the design gradient of TTF of 15 MV/m, p roviding a beam energy of 260 MeV. An important prerequisite of the proper functioning of the v ector-sum control of 16 to 32 cavities is an equal response of the field pickup probes in the individual cavities. A first step is to adjust the phase of the incident rf wave to the same value in all cavities by means of three- stub tuners in the wave guides. Secondly, the transients ind uced by the bunched beam are used to obtain a relative calibration of the pickup probes, both i n terms of amplitude and phase. Typical data taken at the initial start-up of a linac run are s hown in Fig. 28. Ideally the lengths of the field vectors should all be identical since the signals are induced by the same electron bunch in all cavities. The observed length differences indic ate a variation in the coupling of the pickup antenna to the beam-induced cavity field, which ha s to be corrected. The different phase angles of the field vectors are mainly due to different si gnal delays. The complex field vectors are rotated by matrix multiplication in digital sig nal processors to yield all zero phase. Moreover they are normalized to the same amplitude to correc t for the different couplings of the pickup antennas to the cavity fields. Once this calibrati on has been performed the vector sum of the 16 or 32 cavities is a meaningful measure of the tota l accelerating voltage supplied to the beam. The calibration is verified with a measurement of the beam energy in a magnetic spectrometer. The required amplitude stability of 1 ·10−3and phase stability of σφ≤1.6◦can be achieved during most of the beam pulse duration with the exception of t he beam transient induced when turning on the beam. Without feedback the transient of a 30 µs beam pulse at 8 mA would be of the order of 1 MV/m. This transient can be reduced to abou t 0.2 MV/m by turning on feedback. The effectiveness of the feedback system is limi ted by the loop delay of 5 µs and the unity-gain bandwidth of about 20 kHz. The 0.2 MV/m transi ent is repetitive with a high degree of reproducibility. Using feedforward it can be furt her suppressed by more than an order 35 0.05 0.1 0.15 0.2 0.25 30 21060 24090 270120 300150 330180 0Measured Transient [MV/m] 1 234 5 67 8expected transient Figure 28: Beam induced transients (cavity field vectors) ob tained by measuring the cavity fields with and without beam pulses. The noise in the signals c an be estimated from the erratic motion at the center of the plot. This region represents the fi rst 50 µs of the measurement before the arrival of the beam pulse. of magnitude, as shown in Fig. 29. Slow drifts are corrected f or by making the feedforward system adaptive [32]. The feedforward tables are updated on a time scale of minutes. 7 Cavities of Higher Gradients Both the TESLA collider and the X-ray FEL would profit from the development of cavities which can reach higher accelerating fields (i.e. higher part icle energies) and higher quality factors (i.e. reduced operating costs of the accelerators). The TES LA design energy of 250 GeV per beam requires a gradient of 25 MV/m in the present nine-cell c avities. The results shown in Sect. 5.4 demonstrate that TESLA could indeed be realized with a mo derate improvement in the present cavity fabrication and preparation methods. Howev er, for particle physics an energy upgrade of the collider would be of highest interest, and hen ce there is a strong motivation to push the field capability of the cavities closer to the phys ical limit of about 50 MV/m which is determined by the superheating field of niobium. Thr ee main reasons are known why the theoretical limit has not yet been attained in multicell resonators: (1) foreign material contamination in the niobium, (2) insufficient quality and cl eanliness of the inner rf surface, (3) insufficient mechanical stability of the resonators. An R&D p rogram has been initiated aiming at improvements in all three directions. Furthermore, the f easibility of seamless cavities is being investigated. 360 500 1000 1500 2000010203040506070accelerating voltage [MV] time [µs]30 µs beam only feedback (feedback gain: 70)feedback with feed forward compensation (feedback gain: 70) time [µs]400 600 800 1000 12005757.55858.55959.56060.561 30 µs beam feedback with feed forward compensation (feedback gain: 70) only feedback (feedback gain: 70)accelerating voltage [MV] Figure 29: Field regulation of the vector-sum of 8 cavities w ithout and with adaptive feedfor- ward. The lower graph shows an enlarged view of the plateau re gion. 7.1 Quality improvement of niobium Niobium for microwave resonators has to be of high purity for several reasons: (a) dissolved gases like hydrogen, oxygen and nitrogen reduce the heat con ductivity at liquid helium temper- ature and degrade the cooling of the rf surface; (b) contamin ation by foreign metals may lead to magnetic flux pinning and heat dissipation in rf fields; (c) no rmal-conducting or weakly super- conducting clusters close to the rf surface are particularl y dangerous . The Nb ingots contain about 500 ppm of finely dispersed tantalum. It appears unlike ly that the Ta clusters found in some early TTF cavities might have been caused by this “natur al” Ta content. Rather there is some suspicion that Ta grains might have dropped into the Nb m elt during the various remelt- ings of the Nb ingot in an electron-beam melting furnace beca use such furnaces are often used for Ta production as well. To avoid contamination by foreign metals a dedicated electron-beam melting furnace would appear highly desirable but seems to b e too cost-intensive in the present R&D phase of TESLA. Also more stringent requirements on the q uality of the furnace vacuum (lower pressure, absence of hydrocarbons) would improve th e Nb purity. The production steps following the EB melting (machining, forging and sheet roll ing of the ingot) may also introduce dirt. The corresponding facilities need careful inspectio n and probably some upgrading. The present TTF cavities have been made from niobium with gas con tents in the few ppm range and anRRR of 300. Ten 9-cell cavities have been measured both after 800◦C and 1400◦C firing. The average gain in gradient was about 4 MV/m. It would be high ly desirable to eliminate the tedious and costly 1400◦C heat treatment of complete cavities. One possibility migh t be to produce a niobium ingot with an RRR of more than 500. This is presently not our favored approach, mainly for cost reasons. 37For the present R&D program, the main emphasis is on the produ ction of ingots with RRR≥300, but with improved quality by starting from niobium raw m aterial with reduced foreign material content, especially tantalum well below 5 00 ppm. Stricter quality assurance during machining, forging and sheet rolling should prevent metal flakes or other foreign material from being pressed into the niobium surface deeper than a few µm. To increase the RRR from 300 to about 600, it is planned to study the technical feasibi lity14of a 1400◦C heat treatment at the dumb-bell stage (2 half cells joined by a weld at the iri s). This procedure would be preferable compared to the heat treatment of whole cavities which must be carefully supported in a Nb frame to prevent plastic deformation, while such a pre caution is not needed for dumb- bells. However, there is a strong incentive to find cavity tre atment methods which would permit elimination of the 1400◦C heat treatment altogether. According to the results obtai ned at KEK [33] electropolishing seems to offer this chance (see below) . 7.2 Improvement in cavity fabrication and preparation Once half cells or dumb-bells of high RRR have been produced it is then mandatory to perform the electron-beam welding of the cavities in a vacuum of a few times 10−6mbar in order to avoid degradation of the RRR in the welds. The EB welding machines available at industria l companies achieve vacua of only 5 ·10−5mbar and are hence inadequate for this purpose. An EB welding machine at CERN is equipped with a much better vacuum system. This EB apparatus is being used for a single-cell test program. For the future c avity improvement program a new electron-beam welding apparatus will be installed at DESY w ith a state-of-the-art electron gun, allowing computer-controlled beam manipulations, an d with an oilfree vacuum chamber fulfilling UHV standards. The industrially produced cavities undergo an elaborate tr eatment at TTF before they can be installed in the accelerator. A 150–200 µm thick damage layer is removed from the rf surface because otherwise gradients of 25 MV/m appear inacc essible. As explained in Sect. 4 the present method is Buffered Chemical Polishing (BCP) whic h leads to a rather rough surface with strong etching in the grain boundaries. An alternative method is “electropolishing” (EP) in which the material is removed in an acid mixture under curr ent flow. Sharp edges and burrs are smoothed out and a very glossy surface can be obtained. Fo r a number of years remarkable results have been obtained at KEK with electropolishing of 1 -cell niobium cavities. Recently, a collaboration between KEK and Saclay has convincingly dem onstrated that EP raises the accelerating field by more than 7 MV/m with respect to BCP. Sev eral 1-cell cavities from Saclay, which showed already good performance after the sta ndard BCP, exhibited a clear gain after the application of EP [34]. Conversely, an electropol ished cavity which had reached 37 MV/m suffered a degradation after subsequent BCP. These resu lts are a strong indication that electropolishing is the superior treatment method. CERN, DESY, KEK and Saclay started a joint R&D program with el ectropolishing of half cells and 1-cell cavities in August 1998. Recent test re sults yield gradients around 40 MV/m [35] and hence the same good performance as was achieved at KEK. The transfer of the EP technology to 9-cell resonators requires considerab le effort. It is planned to do this in collaboration with industry. Recently it has been found [36] that an essential prerequisi te for achieving gradients in the 40 MV/m regime is a baking at 100 to 150◦C for up to 48 hours while the cavity is evacuated, 14At Cornell University cavities have been successfully fabr icated from RRR = 1000 material. Likewise, the TTF cavity C19 was made from post-purified half cells and show ed good performance. 3810910101011 0 5 10 15 20 25 30Q0 Eacc [MV/m]quench Figure 30: Excitation curve of a TESLA 9-cell cavity showing a drop of quality factor without any field emission. after the final high-pressure water rinsing. In electropoli shed cavities this procedure removes the drop of quality factor towards high gradients which is of ten observed without any indication of field emission. Such a drop is usually also found in chemica lly etched cavities; see for example Fig. 30. Experiments at Saclay [37] have shown that a baking m ay improve the Q(E) curve; however, part of the Qreduction at high field may be due to local magnetic field enhan cements at the sharp grain boundaries of BCP treated cavities [38]. 7.3 Mechanical stability of the cavities The stiffening rings joining neighboring cells in the TESLA r esonator are adequate to limit Lorentz-force detuning up to accelerating fields of 25 MV/m. Beyond 25 MV/m the cavity reinforcement provided by these rings is insufficient. Hence an alternative stiffening scheme must be developed for cavities in the 35–40 MV/m regime. A pro mising approach has been taken at Orsay and Saclay. The basic idea is to reinforce a thi n-walled niobium cavity by a 2 mm thick copper layer which is plasma-sprayed onto the oute r wall. Several successful tests have been made [39]. The copper plating has a potential dange r since Nb and Cu have rather different thermal contraction. The deformation of a cavity u pon cooldown and the resulting frequency shift need investigation. Another phenomenon ha s been observed in cavities made from explosion-bonded niobium-copper sheets: when these c avities were quenched, a reduction in quality factor Q0was observed [40]. An explanation maybe trapped magnetic flu x from thermo-electric currents at the copper-niobium interface . It is unknown whether this undesirable effect happens also in copper-sprayed cavities. An alternat ive to copper spraying might be the reinforcement of a niobium cavity by depositing some sort of metallic “foam”, using the plasma or high velocity spraying technique. If the layer is porous t he superfluid helium penetrating the voids should provide ample cooling. The cavity reinforcement by plasma or high-velocity sprayi ng appears to be a promising 39approach but considerable R&D work needs to be done to decide whether this is a viable technique for the TESLA cavities. 7.4 Seamless cavities The EB welds in the present resonator design are a potential r isk. Great care has to be taken to avoid holes, craters or contamination in the welds which u sually have a detrimental effect on the high-field capability. A cavity without weld connecti ons in the regions of high electric or magnetic rf field would certainly be less vulnerable to sma ll mistakes during fabrication. For this reason the TESLA collaboration decided several years a go to investigate the feasibility of producing seamless cavities. Two routes have been followed : spinning and hydroforming. At the Legnaro National Laboratory of INFN in Italy the spinn ing technique [41] has been successfully applied to form cavities out of niobium sheets . The next step will be to produce a larger quantity of 1-cell, 3-cell and finally 9-cell caviti es from seamless Nb tubes with an RRR of 300. In the cavities spun from flat sheets a very high degree of material deformation was needed, leading to a rough inner cavity surface. Gradien ts between 25 and 32 MV/m were obtained after grinding and heavy etching (removal of more t han 500 µm) [42]. Starting from a tube the amount of deformation will be much less and a smoothe r inner surface can be expected. This R&D effort is well underway and the first resonators can be expected in early 2000. The hydroforming of cavities from seamless niobium tubes is being pursued at DESY [43]. Despite initial hydroforming difficulties, related to inhom ogeneous mechanical properties of the niobium tubes, four single cell cavities have been successf ully built so far. Three of these were tested and reached accelerating fields of 23 to 27 MV/m. In a ve ry recent test1532.5 MV/m was achieved at Q0= 2·1010. Most remarkable is the fact that the cavity was produced fro m low RRR niobium ( RRR = 100). It received a 1400◦C heat treatment raising the RRR to 300–400. The surface was prepared by grinding and 250 µm BCP. 7.5 Niobium sputtered cavities Recent investigations at CERN [44] and Saclay [45] show that single-cell copper cavities with a niobium sputter layer of about 1 µm thickness are able to reach accelerating fields beyond 20 MV/m. These results appear so promising that CERN and DESY ha ve agreed to initiate an R&D program on 1.3 GHz single cell sputtered cavities aiming at gradients in the 30 MV/m regime and quality factors above 5 ·109. High-performance sputtered cavities would certainly be of utmost interest for the TESLA project for cost reasons. Another advantage would be the suppression of Lorentz force detuning by choosing a sufficien tly thick copper wall. 7.6 The superstructure concept The present TTF cavities are equipped with one main power cou pler and two higher order mode couplers per 9-cell resonator. The length of the interc onnection between two cavities has been set to 3 λ/2 (λ= 0.23 m is the rf wavelength) in order to suppress cavity-to-cav ity coupling of the accelerating mode. A shortening of the inter connection is made possible by the “superstructure” concept, devised by J. Sekutowicz [46 ]. Four 7-cell cavities of TESLA geometry are joined by beam pipes of length ( λ/2). The pipe diameter is increased to permit 15Preparation and test of the hydroformed cavities were carri ed out by P. Kneisel at Jefferson Laboratory, Newport News, USA. 40an energy flow from one cavity to the next, hence one main power coupler is sufficient to feed the entire superstructure. One HOM coupler per short beam pi pe section provides sufficient damping of dangerous higher modes in both neighboring cavit ies. Each 7-cell cavity will be equipped with its own LHe vessel and frequency tuner. Theref ore, in the superstructure the field homogeneity tuning (equal field amplitude in all cells) and the HOM damping can be handled at the sub-unit level. The main advantages of the sup erstructure are an increase in the active acceleration length in TESLA - the design energy of 25 0 GeV per beam can be reached with a gradient of 22 MV/m - and a savings in rf components, esp ecially power couplers. A copper model of the superstructure is presently used to ver ify the theoretically predicted performance. This model allows individual cell tuning, fiel d profile adjustment, investigation of transients in selected cells, test of the HOM damping scheme and measurement of the cavity couplings to the fundamental mode coupler. Also the influenc e of mechanical tolerances is stud- ied. First results are promising [47]. A niobium superstruc ture prototype is under construction and will be tested with beam in the TTF linac beginning of 2001 . 8 Acknowledgements We are deeply indebted to the late distinguished particle an d accelerator physicist Bjørn H. Wiik who has been the driving force behind the TESLA project a nd whose determination and enthusiasm was essential for much of the progress that has be en achieved. We want to thank the many physicists, engineers and technicians in the laborato ries of the TESLA collaboration and in the involved industrial companies for their excellent wo rk and support of the superconducting cavity program. We are grateful to P. Kneisel for numerous di scussions. References [1] Proceedings of the 1st TESLA WORKSHOP, CLNS-90-1029, ed ited by H. Padamsee, Cornell University, USA (1990). [2] TESLA TEST FACILITY LINAC-Design Report, TESLA-Report 95-01, edited by D.A. 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Simrock, Proceedings EPAC 1998, Stockho lm, Sweden (1998), p. 1735. [33] K. Saito, Proceedings of the 9th Workshop on RF Supercon ductivity, Santa Fe, USA (1999) to be published. [34] E. Kako et al., Proceedings of the 9th Workshop on RF Superconductivity, S anta Fe, USA (1999) to be published. [35] L. Lilje et al., Proceedings of the 9th Workshop on RF Superconductivity, S anta Fe, USA (1999) to be published. [36] P. Kneisel, Proceedings of the 9th Workshop on RF Superc onductivity, Santa Fe, USA (1999) to be published. [37] P. Charrier et al., Proceedings EPAC 1998, Stockholm, Sweden (1998), p. 1885. [38] J. Knobloch et al., Proceedings of the 9th Workshop on RF Superconductivity, S anta Fe, USA (1999) to be published. [39] S. Bousson, Proceedings of the 9th Workshop on RF Superc onductivity, Santa Fe, USA (1999) to be published. [40] T Fujino et al., Proceedings of the 9th Workshop on RF Superconductivity, S anta Fe, USA (1999) to be published. [41] V. Palmieri, Proceedings of the 8th Workshop on RF Super conductivity, edited by V. Palmieri and A. Lombardi, Abano Terme, Italy (1997), p. 55 3 and LNL-INFN (Rep) 133/98 [42] P. Kneisel, V. Palmieri and K. Saito, Proceedings of the 9th Workshop on RF Supercon- ductivity, Santa Fe, USA (1999) to be published. [43] I. Gonin et al., Proceedings of the 9th Workshop on RF Superconductivity, S anta Fe, USA (1999) to be published. [44] C. Benvenuti et al., Proceedings of the 9th Workshop on RF Superconductivity, S anta Fe, USA (1999) to be published. 43[45] P. Bosland et al., IEEE Transactions on Applied Superconductivity, ASC’98, Vol. 9, No. 2 (1999). [46] J. Sekutowicz, M. Ferrario and Ch. Tang, Phys. Rev. ST Ac cel. Beams, Vol. 2, No. 6 (1999). [47] H. Chen et al., Proceedings of the 9th Workshop on RF Superconductivity, S anta Fe, USA (1999) to be published. 44This figure "fig1_cav1.jpeg" is available in "jpeg" format from: http://arXiv.org/ps/physics/0003011v1This figure "FIG4_schoen.gif" is available in "gif" format from: http://arXiv.org/ps/physics/0003011v1This figure "FC8.gif" is available in "gif" format from: http://arXiv.org/ps/physics/0003011v1This figure "Fig8Diagn.jpeg" is available in "jpeg" format from: http://arXiv.org/ps/physics/0003011v1This figure "Fig10Diagn.jpeg" is available in "jpeg" format from: http://arXiv.org/ps/physics/0003011v1This figure "F12a_BWPh.jpeg" is available in "jpeg" format from: http://arXiv.org/ps/physics/0003011v1This figure "F12b_OJS.jpeg" is available in "jpeg" format from: http://arXiv.org/ps/physics/0003011v1This figure "F17.gif" is available in "gif" format from: http://arXiv.org/ps/physics/0003011v1This figure "FigEDA.jpeg" is available in "jpeg" format from: http://arXiv.org/ps/physics/0003011v1

arXiv:physics/0003012v1 [physics.ed-ph] 6 Mar 2000Effective Field Theory for Pedestrians1 G.B. Tupper Institute of Theoretical Physics and Astrophysics, Department of Physics, University of Cape Town, Rondebosch 7701, South Africa. Abstract A pedagological introduction to effective field theory is pre sented. 1Invited talk given at the Millennium School on Nuclear and Pa rticle Physics, National Acceleration Centre, Faure, South Africa, 31 January to 3 February 2000Introduction If in 1975 one had asked for a brief history of hadronic physic s it would have undoubtedly gone something like this [1] : first there was a ‘classical age ’ initiated by Yuhawa’s (1935) meson hypothesis for the nuclear force and terminated ( ±1950) by invading hoards of “strange” particles and resonance. There followed a sort of ‘dark ages’ where arcane rites of dispersion relations, Regge poles and dual resonance mod els were practiced. Finally we are now in the ‘enlightened age’ of “quantum chromodynami cs” (QCD): baryons – like the proton and neutron – are composites of three “quarks ” while mesons are made of quark-antiquark pairs; these are inseparably bound by a c olour force which becomes weak at short distances, and the interaction between hadron s is a colour analogue of the van der Waal’s force between neutral atoms. Alas, some twenty five years later we still are unable to calcu late many interesting quan- tities such as the nuclear mass or nucleon-nucleon potentia l directly from QCD (albeit lattice gauge enthusiasts will tell you with the next genera tion of computers · · ·). One is left with a variety of models (bag, Skyrme, etc.) and a sort of interpolating scheme (QCD sum rules), but nothing approaching the systematics an d accuracy of quantum electrodynamics (QED). The difference is due to confinement: whereas in QED the basic entities (electrons and photons) are observable, in QCD the y (quarks and gluons) are not, rather we can only observe their hadronic composites. Still, the triumphs of QED were afforded by the realization th at one did not need to be able to calculate the electron mass to determine the effects o f the self energy of a bound electron – the Lamb shift [2]. That one could apply a modified v ersion of this and work directly with hadrons in a systematic way was first suggested by Weinberg (1979) [3] and marked the birth of a new age: the age of effective field theory w hose ramifications go far beyond hadronic physics alone. 2There are by now a number of textbook exposition [4] and revie w articles for the sophis- ticate; in this talk I will endeavour to give the novice some f eeling for what is going on using the old static model [5] as an example. Then, at the end I will return to the wider implications. A word of warning: for simplicity (mine, not yours) I will use ‘natural units’ ¯ h=c= 1 ; mass and momenta are in units of energy, and length in units of inverse energy, a useful conversion being ¯hc= 1 = 197MeV ·fm, (1) (1 fm = 10−13cm) . The Static Model Let me begin by recalling that the impetus for pre-QCD meson t heory was Yukawa’s ob- servation that in contrast to Poisson’s equation for the ele ctrostatic potential, the equation /parenleftBigg∂2 ∂t2−∆ +m2/parenrightBigg φ=gn (2) has for a static charge at the origin, n(/vector r) =δ(/vector r) φ(r) =g 4πe−mr r(3) whose range is not infinite but 1/ m. Now suppose for the moment n= 0 ; by making the the Fourier expansion φ(t,/vector r) =/integraldisplayd3k (2π3)ϕ(t,/vectork)ei/vectork·/vector r(4) one obtains for each /vectork ¨ϕ(/vectork) +ω2(/vectork)ϕ(/vectork) = 0 , ω2(/vectork) =/vectork2+m2. (5) Thus, classically one has a set of harmonic oscillators and t he dispersion relation ω(/vectork) is that for a particle of mass min relativity. Each oscillator has a “momentum” π(/vectork) = ˙ϕ(/vectork) (6) 3and the total energy is H0=/integraldisplayd3k (2π)3 1 2π2(/vectork) +ω2(/vectork) 2ϕ2(/vectork) . (7) Now each oscillator can be quantized individually, but inst ead of ϕandπits more con- venient to use aanda+ φ=1√ 2ω(a++a), π=i/radicalbiggω 2(a+−a). (8) This gives ˆH0=/integraldisplayd3k (2π)3ω(k) ˆa+(/vectork) ˆa(/vectork) (9) where we have thrown out an infinite sum at “zero point energie s”2which play no role here. The ‘ladder operators’ have non-vanishing commutato r /bracketleftBig ˆa(/vectork),ˆa+(/vectork′)/bracketrightBig = (2π)3δ(/vectork−/vectork′) (10) and the lowest energy, ground or ‘vacuum’ state |0∝angbracketrightobeys ˆa(k)|0∝angbracketright= 0 (11) so indeed it has zero energy. The state |/vectork∝angbracketright= ˆa+(/vectork)|0∝angbracketrighthas the property ˆH0|/vectork∝angbracketright=ω(/vectork)|/vectork∝angbracketright (12) so describing a particle (meson) with (3-) momentum /vectorkand energy ω(/vectork) . When the right hand side of (2) is nonzero, i.e. the nucleon is present, the oscillators are driven so the total energy (hamiltonian) is ˆH=ˆH0+ˆHI. (13) Now, H0is unmodified if the nucleon is static (in practical terms thi s means we are neglecting recoil which is a fair approximation to reality) . In writing the ‘interaction 2Specifically : EZPE=1 2/integraldisplayd3k (2π)3ω(/vectork) 4part’HIwe need to account for the fact that the light mesons (pions) a re ‘pseudoscalar’3, i.e. under ‘parity’, /vector r→ −/vector r , φ → −φwhereas for a scalar φis unchanged. Taking this together with the fact that the nucleon is spin 1/2 (occurrin g in two spin states, “up” and “down”), because the energy should not be changed by pari ty or rotations the unique choice is ˆHI=/integraldisplayd3k (2π)3 −ig/radicalBig 2ω(/vectork)K ˜/parenleftBig ˆa+(/vectork) + ˆa(/vectork)/parenrightBig  (14) where K ˜is shorthand for the 2 by 2 matrix K ˜= kz kx−iky kx+ihy −kz  (15) and for simplicity “isospin” is neglected. Note the couplin g parameter gmust have di- mension of length to compensate that of K ˜. Finally it is also worth mentioning that if one replaces the w ords nucleon and meson by electron and phonon this model bears many similarities to pr oblems in solid state physics [6]. The Self-Energy Alack, unlike ˆH0,ˆHcannot be diagonalized exactly but can be treated by time ind epen- dent perturbation theory familiar to every quantum mechani c. Taking the unperturbed state as that with no mesons and one nucleon the leading energ y shift – which is to say the nucleon mass shift because it is static – is given by4 ∆E[1]=/integraldisplayd3k (2π)3 ig K ˜√ 2ω /bracketleftbigg1 −ω/bracketrightbigg −ig K ˜√ 2ω  (16) which can be given a diagrammatic representation 3In QCD this follows from ‘spontaneously broken chiral symme try’. 4This may be compared to the usual expression E(2) n=/summationdisplay s /negationslash=nHInsHIsn En−Es. Note ∝angbracketleft/vectork|ˆHI|0∝angbracketright= −igK˜√ 2ω. 5k −ω Figure 1: Reading from right to left : the nucleon emits a meson losing e nergy ω(/vectork), remains with energy −ωfor a time and then reabsorbs the meson; it can do this for any /vectorkso we add all the intermediate states. Turning this around it is easy t o use these ‘Feynman rules’ to write down contributions corresponding to k'k k' −ω −ωk −ω −ω−ω ' −ω'−ω−ω' Figure 2: (try it!). Notice these involve more “loops”. Of course the energy shift is not a matrix but ( K ˜)2=/vectork2I and after a little work (16) leads to δM[1]=−/parenleftbiggg 2π/parenrightbigg2/integraldisplay∞ 0dk/bracketleftBigg k2−m2+m4 k2+m2/bracketrightBigg (17) where k=|/vectork|. It is painfully obvious that only the last integral converg es toπm3/2, the rest diverge! This is analogous to (even classical) electro dynamics where in the self-energy of a point change is infinite. To be honest we ought to insert a c onvergence or ‘form’ factor all the way back in HI, but then the result depends on how we choose to ‘cutoff’. 6Irrespective of details we can say that the nucleon mass Mis of the form M=ˆM+κ1m2−g2 8πm3+.... (18) where ˆM, which is what Mwould be were the meson massless, and κ1‘renormalised’ parameters hiding the strong cutoff dependence. The ellipsi s represents weakly cutoff dependent parts, higher loops, etc. The first significant thi ng about (18) is that as a function of m2, the parameter appearing in H, the unknown parameters appear in the analytic part whereas the non-analytic part is calculable. It is not hard to see why: if we tried to expand (17) in powers of m2we soon encounter integrals which diverge at the lower limit only, and these do not care how we ‘regularize ’. One reason why this is significant is that in QCD the pseudoscalar mass squared is pr oportional to the quark mass, m2∝mq; the first two terms in Mgive the Gell-Mann-Okubo relation for the barren octet and the equal splitting rule for the decuplet, t he last the correlation to these. But there is something deeper: the theory we are working with is ‘non-renormalizable’, signalled by needing κ1as a parameter in the 1-loop calculation. At 2-loops we need m ore, and ultimately to hide all our ignorance would require an infi nite number of parameters! Once more, with feeling this time, the bits which are cutoff se nsitive are analytic in m2 so M=ˆM+κ1m2+κ2m4+· · ·+ calculable . (19) Now if we replace the upper limit in (17) by Λ with Λ = 2 π/g (20) our one-loop calculation says K[1] 1= Λ−1. Generally then m= ˆm+ ¯κ1m2/Λ + ¯κ2m4/Λ3+· · ·+ calculable (21) with ¯κia pure number of order unity. 7We have arrived at the crux of why field theory is effective in th e usual sense of the word. The infinity of parameters do not contribute equally, and hig her orders are suppressed by powers of m/Λ.5Were we calculating meson-nucleon scattering the correspo nding series would be in |/vector q|/Λ and m/Λ,/vector qthe meson momentum, so this only works for energies low compared to Λ. For the case in hand, m/Λ≈mπ/mρ≈140 MeV/770 MeV and (18) is valid up to the 20% level (the same as recoil corrections). More Effective Theory In conclusion, let me stress that our modest calculation did not require that we know anything about the underlying theory, QCD. All we needed wer e the low energy degrees of freedom and their interaction. Now, quantum gravity is di scarded as a fundamental theory because it is nonrenormalizable, involving as it doe s the dimensionful newtonian coupling G=ℓ2 pℓ (22) where ℓpℓ≈10−33cm is the Planck length. As noted by Donoghue [7], however, wh atever the ultimate ‘Theory of Everything’ (GOD) quantum gravity c an be treated as an effective field theory and e.g. quantum corrections to the newtonian po tential V(r) =−Gm1m2 r 1 +β/parenleftBiggℓpℓ r/parenrightBigg2 +· · ·  (23) are calculable. βis a computable number of order unity, and the pathetic small ness of the correction is less significant than the realization that it can be done. 5Similarly, heavy particle contributions are suppressed by powers of 1/ mH. They are subsumed in ˆM and ¯κi. 8Appendix In case the reader did try and wants to check his/her work, the expressions corresponding to figure 2 are ∆E[2a]=/integraldisplayd3k (2π)3/integraldisplayd3k′ (2π)3 ig K ˜√ 2ω /bracketleftbigg1 −ω/bracketrightbigg ig K ˜′ √ 2ω′ · /bracketleftbigg1 −ω−ω′/bracketrightbigg −ig K ˜′ √ 2ω′ /bracketleftbigg1 −ω/bracketrightbigg −ig K ˜√ 2ω  ∆E[2b]=/integraldisplayd3k (2π)3/integraldisplayd3k′ (2π)3 ig K ˜′ √ 2ω′ /bracketleftbigg1 −ω′/bracketrightbigg ig K ˜√ 2ω · /bracketleftbigg1 −ω−ω′/bracketrightbigg −ig K ˜′ √ 2ω′ /bracketleftbigg1 −ω/bracketrightbigg −ig K ˜√ 2ω  These are most difficult to evaluate, but as noted in the text co ntribute only at the 20% level. 9References [1] Glimpses of pre-QCD history may be found in: R. Oppenheim er,Physics Today , November 1966, 51; G. Veneziano, ibid, September 1969, 31. [2] H.A. Bethe, Phys. Rev. 72(1947) 339. [3] S. Weinberg, Physica 96A(1979) 327. [4] H. Georgi, “Weak Interactions and Modern Particle Physi cs”, Benjamin Cumming (1984); J.F. Donoghue, E. Golowich and B.R. Holstein, “Dyna mics of the Standard Model”, Cambridge University Press (1992). [5] G.C. Wick, Rev. Mod. Phys. 27(1955) 339; also references therein. [6] H. Lipkin, “Quantum Mechanics: New Approaches to Select ed Topics”, North- Holland (1973). [7] J.F. Donoghue, Phys. Rev. Lett. 72(1994) 2996. 10

arXiv:physics/0003013v1 [physics.gen-ph] 7 Mar 2000PHYSICAL MODEL OF HADRONS: BARIONS and MESONS. PHYSICAL ESSENCE of QUARKS and GLUONS AND PHYSICAL INTERPRETATION OF THEIR PARAMETERS. Josiph Mladenov Rangelov , Institute of Solid State Physics - Bulgarian Academy of Scie nces , 72 blw Tzarigradsko chaussee , 1784 , Sofia ,Bulgaria Abstract The physical model (PhsMdl) of the hadrons is offered by means of the obvious analogy with the transparent surveyed PhsMdls of the vacuum and lept ons in our recent works. It is assumed that the vacuum is consistent by dynamides, str eamlined in junctions of some tight crystalline lattice. Every dynamide contains a n eutral pair of massless con- trary point-like (PntLk) elementary electric charges (Elm Elc Chrgs): electrino ( −) and positrino (+). By means of the existent fundamental analogy between their properties and behaviour we can understand the similarity and differenc e between them and assume that the quark parameter aroma is determined by the value of i ts size of its circular two- dimensional motion, while the quark parameter colour is det ermined by orientation of the flat of the same circular two-dimensional motion in the space . The colorless of the barions is explained by distribution of the same circular two-dimen sional motion of its elementary electric charge within three mutually perpendicular flats. Then the exchange of the colors between two quarks with different colors within some hadron c an be interpretated as some twisting of same hadron in the space. We give a new obvious phy sical interpretation of the charge values of quarks, which gives some explanation of angles of Cabibo and Weyn- berg. By some physical supposition about the structure of ch arged intermediate vector bozon Wand uncharged intermediate vector bozon Zwe have possibility to explain as the physical essence of the strong, weak and electromagneti c interactions, so the outline of all births, transformations and decays of the ElmPrts. Although up to the present nobody of scientists distinctly k nows are there some elementary micro particles (ElmMicrPrts) as a stone of the micro world a nd what the elementary micro particle (ElmMicrPrt) means, there exists an essential pos sibility for clear and obvious scientific consideration of the unusual behaviour of the quantized mic ro particles (QntMicrPrts). It is well known that the physical model (PhsMdl) presents at us as an ac tual ingradient of every good physical theory. It may be used as for an obvious visual teach ing the occurred physical processes within the investigated phenomena, so for doing with them a b igger capacity of its physical understanding and mathematical description. All scientis ts think that the ElmMicrPrt display itself as united and indivisible everywhere and always, but nobody demand for ElmMicrPrt to have no structure. About thirty years ago in the early theo ry of the ElmMicrPrt some theoretical physicists assumed that all the ElmMicrPrts ha ve point-like (PntLk)-fashion and their behaviour are no dependent on the presence of other Elm MicrPrt as neighbors. In further it was turned out that the existence of the strong interactio n between hadrons and of their own magnetic dipole moment (MgnDplMmn) is caused by existence a nd exchange of the virtual pions between them. Really, the strong interaction between nearest hadrons is provided by the 1electromagnetic interaction, which is transferred by freq uent exchanging a virtual charged pion (π±-meson). The virtual charged pions ( π±-mesons), owing of its smallest mass between all other mesons, are most probable for a frequent birth from all other virtual ElmMicrPrts, which are caused by exchanged gluon within the quantized electrom agnetic field (QntElcMgnFld). We can assume without restriction that if the fine spread (FnS pr) elementary electric charge (ElmElcChrg) of every charged lepton participates in isotr opic three-dimensional (IstThrDmn) relativistic quantized Schrodinger (RltQntSchr) self-co nsistent strong-correlated (SlfCnsStrCrr) solitary fermion vortical harmonic oscillations (SltFrmV rtHrmOscs) within its volume, then the FnSpr ElmElcChrg of every charged hadron participates succ essive in three anisotropic two- dimensional (AnIstTwoDmn) relativistic quantized (RltQn t) self-consistent strong-correlated (SlfCnsStrCrr) solitary fermion circular harmonic oscill ations (SltFrmCrcHrmOscs) within its volume. If the RltQntSchr SlfCnsStrCrr SltFrmVrtHrmOscs c an be approximately described by the orbital wave function (OrbWvFnc) of s-state ItsThrDm nNrlQnt boson harmonic oscil- lator (BsnHrmOsc), the AnIstTwoDmn RltQnt SlfCnsStrCrr Fr mCrcHrmOscs can be approxi- mately described by OrbWvFnc of p-state of AnIstThrDmn NrlQ nt boson harmonic oscillator (BsnHrmOsc). Therefore we can assume that if the averaged Fn Spr ElmElcChrg of charged lepton has one maximum, then the averaged FnSpr ElmElcChrg o f charged hadron has three maximums, which can be identified as three FnSpr ElmElcChrgs of three quarks. But in a reality their is only one FnSpr ElmElcChrg within every char ged hadron, which is determined by electric interaction only with the FnSpr ElmElcChrg of on e maximum at a dispersion of passing charged leptons. Indeed, at every dispersion of som e FnSpr ElmElcChrg of leptons from every hadron the FnSpr ElmElcChrg electromagneticall y interacts (ElcMgnIntAct) only by one maximum of FnSpr ElmElcChrg of same hadron. The existence of essential analogy between leptons and hadr ons allows us to assume that the quarks for the hadrons are analogous of the neutrinos for the leptons. Therefore the phys- ical means of the aroma of the quarks is analogous of the physi cal means of the aroma of the neutrinos, which is determined by the sizes of the vortical h armonic excitation of the neutral but fluctuating vacuum (FlcVcm). Because of that we can suppo se that if the neutrino is an isotropic three dimensional solitary vortical excitation of a spherical symmetry of the neutral vacuum, then the quark is a two-dimensional flat circular sol iton excitation of cylindrical sym- metry of the same neutral vacuum. But if all the neutrino are v ery stable excitation without elementary electric charge (ElmElcChrg) and have unlimite d time of live, all the quarks are unstable without ElmElcChgr and have limited time of live. M oreover we may assume also that if within some massive lepton (electron, muon ( µ-meson), tauon ( τ-meson)) its point-like (PntLk) ElmElcChrg takes part simultaneously in three one- dimensional very strongly corre- lated self-consistent fermion vortical harmonic oscillat ions along three axes, orientired mutually perpendicular one to another in the space, then within some m assive charged hadron its PntLk ElmElcChrg or within other massive ,,uncharged” hadron its two opportunity PntLk ElmEl- cChrgs, take part successive in three two-dimensional very strongly correlated self-consistent fermion circular harmonic oscillations along three planes , orientired mutually perpendicular one to another in the space. Therefore we may naturally assume th at three colours of the quark state correspond to three planes of motion of the PntLk ElmEl cChrg, orientired mutually per- pendicular one to another in the space. If the lepton state of its PntLk ElmElcChrg is determined by o ne spherical state, describing the self- consist motion (Zitterbewegung) of its PntLk ElmE lcChrg by one s-OrbWvFnc, then the hadron state of its PntLk ElmElcChrg is determined by thr ee flat states, describing its three dimension motion, composed by three flat cylindrical h armonic motions by three p- 2OrbWvFnc. Moreover, each p-state correspond to one quark st ate with one of three colours. Therefore the physical incomprehencible demand about colo urless of the hadrons obtains quite obvious and clear physical interpretation about the demand of the spherical symmetry of three mutual perpendicular orientated flats of two-dimensional f ermion circular harmonic oscillations in our PhsMdl. The change of the quark colour of the PntLk ElmE lcChrg flat circular harmonic motion occurs through emission or absorption of some gluon ( δ). When the PntLk ElmElcChrg of some hadron, moving within quark of some colour, absorbs s ome gluon (a quantized quasi- plane magnetic field), then this quarks changes his colour an d the hadron as whole twists the flat of the two-dimensional fermion circular harmonic oscil lations of his PntLk ElmElcChrg. In the beginning for an explanation of the existent experime ntal date some theoretical physicists suppose that there are only three quarks and the s ame number antiquarks. But for physical explanation of new experimental date which are obtained during the carrying the following experiments, the scientists had need from new thr ee quarks and the same number antiquarks. There are a common agreement that the existence of six quark (d,u,s,c,b,t) and the same number antiquark ( ˜d,˜u,˜s,˜c,˜b,˜t) is firmly determined by experimental research. Coming out from the supposed symmetry between leptons and quarks so me theoretical physicists very frequently suppose a following list of their distribution o n aroma: /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleνeνµντ e−µ−τ− u c t d s b/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle˜νe˜νµ˜ντ ˜e−˜µ−˜τ− ˜u˜c˜t ˜d˜s˜b/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(1) Indeed, we can suppose that owing of the strong localization of the PntLk ElmElcChrg of any massive hadron allow one to participate in very high freq uency oscillations within very small area of the space, what secures the obtaining its QntEl cMgnFld with very high electric and magnetic intensities and very dense electromagnetic en ergy, which same FnSpr ElmElcChrg borrows from the FlcVcm at its electromagnetic interaction with it by stochastic exchanging the virtual photons (VrtPhtns). Then it is obviously that th e existence of the QntElcMgnFld with very high electric and magnetic intensities and very de nse electromagnetic energy can secures the need probability for frequently births of virtu al charged pions pairs within the immediate neighborhood of the PntLk ElmElcChrg’s position . Therefore we mast suppose that the interacting hadrons are very quite near one to others. At these conditions we may suppose also that when some hadron (for example some proton) takes pl ace in some nucleus and has as neighbors other hadrons with other isotopic spin (neurtons ) then during the time of the motion of its PntLk ElmElcChrg within some uquark without changing its color, the hadron is visited by negative charged virtual pion ( π−-meson), emitted by neighbor neutrons. Then both the free u-quark from the proton and the u-antiquark (¯ u- quark) annihilated and negative PntLk ElmElcChrg shifts from its d-quarks to the free d-quarks of t he visited proton. After some time some gluon from FlcVcm generates another pair of u-quar k and u-antiquark with another colour from the fluctuating vacuum (FlcVcm). Then the negati ve PntLk ElmElcChrg shifts itself from d-quark of the visited proton on the new born u-an tiquark, which together with the d-quark of the old transferring negative charged pion ( π−-meson) build a new negative charged pion. In the time of such a visit the positive PntLk El mElcChrg uninterruptedly participated within the flat two-dimensional fermion circu lar harmonic oscillations within the u-quark of the visited proton. After the departure of the neg ative PntLk ElmElcChrg from the new neutron the positive PntLk ElmElcChrg shifts into th e new born u-quark, which have the colour, different from the colour of old quark, from which it left. After such an exchange 3twisted proton is ready for a new visit from another negative virtual pion ( π−- meson). In such a natural way the exchange of the negative virtual pion b etween some pair of proton and neutron causes the existence of the strong interaction. If the three different aroma of leptons correspond to three di fferent sizes of the isotropic three dimensional relativistic quantized (IstThrDmnRltQnt) Sc hrodinger self-consistent strongly cor- related (SchrSlfCnsStrCrr) solitary fermion vortical har monic oscillations (SltFrmVrtHrmOscs) of its FnSpr ElmElcChrg, composed by three SlfCnsStrCrr one -dimensional relativistic quan- tized harmonic oscillations, then the three different aroma of hadron quark state correspond to three sizes of three two-dimensional relativistic quant ized fermion circular harmonic oscilla- tions of its FnSpr ElmElcChrg. Therefore we may suppose that if within the massive lepton its averaged (over spread (OvrSpr)) ElmElcChrg has one maximum only, then within the massive hadron its averaged OvrSpr ElmElcChrg has three maximums on ly, each of which may be iden- tified by us as the ElcChrg of the quark or parton. The exchange of the FnSpr ElmElcChrg of its quark colour state occurs through absorption or emissio n of some gluon ( δ). The gluons are hypothetical particle, which exchange betwe en two quarks with different colors (space orientations) achieve the strong interactio n between them. It is turn out that they have not a ElmElcChrg and rest mass, but have a proper mec hanical moment (spin) and quantized magnetic field with a proper magnetic dipole momen t (MgnDplMmn). Therefore in magnetic interaction (MgnIntAct) between the QntMgnFlds o f the gluon and quarks ensures the turn of space orientation (change of color) of gluon. It i s turn that the impulse (mechanical momentum) sum of all gluons within some hadron is equal of the impulse sum of all quarks within same hadron. As the gluon spin is 1¯ h, therefore at its habitual virtual decay within hadron one can turn into pair virtual quark and antiquark wit h paralel spins. But such a virtual pair is no virtual meson, as more mesons with small ma ss have a spin with a zero value. Therefore such a decay of some gluon into a pair of some virtua l quark and antiquark could effectively turn upsite-down the spin of one of meson quark an d in such a way enssure the decay of this meson in first into charged intermediate vector bozon Wand in second into massive and massless leptons of equal aromas. The behavior of eight g luons can be described by eight Gell-Man’s mathrices of SU(3) space. We must note that the PntLk ElmElcChrg of the charged admeson is moving successively in two opposite orientated flats. As in positive π-meson its positive PntLk ElmElcChrg (positrino) is successively moving in an u-quark state and in a ˜d-antiquark state, which have an opposite parallel orientations, then in negative π-meson its negative PntLk ElmElcChrg (electrino) is successively moving in a d-quark state and in a ˜ u-antiquark state, which have opposite parallel orientations. In same way in the positive K-meson its positi ve massless PntLk ElmElcChrg (positrino) is successively moving in an u quark state and in a ˜s-antiquark state, which have opposite parallel orientations, and in the negative K-meso n its negative PntLk ElmElcChrg (electrino) is successively moving in a s-quark state and in a ˜u-antiquark state, which have opposite parallel orientations. However in the neutral adm esons two opposite charged PntLk ElmElcChrgs (electrino and positrino) are moving in four fla ts, which have in pair opposite parallel orientations, where are moving separately two opp osite charged PntLk ElmElcChrgs. For instance in πo-meson its massless electrino is moving in d-quark state and in ˜u-antiquark state, which both have opposite parallel orientations, and its positrino is moving in u quark state and in ˜dantiquark state, which both have opposite parallel orienta tions, mutually perpendicular orientated to the first pair flats. In such a fashion in Ko-meson its electrino is moving in a d quark state and in a ˜ u-antiquark state, which both have opposite parallel orient ations, and its positrino is moving in an u quark state and in ˜ santiquark state, which both have opposite 4parallel orientations, but the second are mutual perpendic ular oriented to the first pair flats of moving electrino. In such a fashion in ˜Ko- meson its electrino is moving in a s-quark state and in a ˜ uantiquark state, which both have opposite parallel orienta tions, and its positrino is moving in an u quark state and in a ˜d-antiquark state, which both have opposite parallel orientations, but the second are mutual perpendicular orie nted to first pair flats of moving electrino. Owing of a small difference between the compositi ons of quarks, from which are built theKo-meson and the ˜Ko-meson the some fluctuations in the FlcVcm have energy, enoug h for their conversion one meson in its antimeson. Ko(d↑,˜s↓) = (u↓, W−↑,˜s↓) = (u↓,˜c↑) = (s↑, W+↓,˜c↑) =˜Ko(s↑,˜d↓) (2) In a result of such mutual conversions are obtained short-li ved and long-lived Ko-mesons with different parameters and scheme of decay. Ko l=(Ko+˜Ko)√ 2and Ko s=(Ko−˜Ko)√ 2(3) The similar fashion of the meson construction we may find in th eDo-mesons, where its positive PntLk ElmElcChrg is successively moving in an u-qu ark state and in a ˜d-antiquark state, which both have opposite parallel orientations, and its negative PntLk ElmElcChrg is successively moving in a d-quark state and in a ˜ c-antiquark state, which both have opposite parallel orientations, but the second are mutually perpend icular to first pair flats. In such a fashion in ˜Do-meson its electrino is moving in a d-quark state and in a ˜ u-antiquark state, which both have opposite parallel orientations, and its positrin o is moving in a c quark state and in a ˜d-antiquark state, which both have opposite parallel orient ations, but the second are mutually perpendicular to first pair flats. Do(u↑,˜c↓) = (d↓, W+↑,˜c↓) = (d↓,˜s↑) = (c↑, W−↓,˜s↑) == ˜Do(c↑,˜u↓) (4) As for the radiation of the spontaneous real photon (RlPhtn) from the excitative atom it is necessary the presence of the virtual photon (VrtPhtn) fo r the creation of the electric dipole moment (ElcDplMm), so for the decay of a charged π-meson it is necessary the presence of a virtual gluon for an overturning of the spin of one of its quar ks, by which charged π-meson turns into charged virtual ρ-meson, which can immediately decay into charged intermedi ate vector boson W. At the subsequent transfer of the in a pair of massive and mas sless leptons of equal aroma the participating in the decay gluon go back in th e FlcVcm. Indeed, I think that instead of the equations of the incomprehencible decay : π+=⇒W+=⇒µ++νµ, (5) π−=⇒W−=⇒µ−+ ˜νµ, (6) π+=⇒W+=⇒e++νe, (7) π−=⇒W−=⇒e−+ ˜νe, (8) πo=⇒W++W−=⇒γ+γ , (9) 5πo=⇒W++W−=⇒γ+e++e−, (10) we must used the following equations : π++δ=⇒W+=⇒µ++νµ, (11) π−+δ=⇒W−=⇒µ−+ ˜νµ, (12) π++δ=⇒W+=⇒e++νe, (13) π−+δ=⇒W−=⇒e−+ ˜νe, (14) πo=⇒W++W−=⇒γ+γ , (15) πo=⇒W++W−=⇒γ+e++e−, (16) As it is easy to see from eqns.(15) and (16) that there is no nec essity for participating some gluon δin decay of the πo-meson. For certain of that the half-life time ( τ+ π= 2.6×10−8s) of the charged π-mesons is very different from the half-life time ( τo π= 8.3×10−17s) of the neutral π-mesons. It seems to me the existance of two very interesting facts, ha ving common physical cause.The first is concurrence of the energy of one degree of freedom in c harged lepton µ-meson and in charged admeson π-meson. Indeed, if in isotropic three dimensional solitary vortical harmonic ocsillations of FnSpr ElmElcChrg of µ-meson have three degrees of freedom and therefore 3 ¯h ω= 2m C2= 213 .2Mev. Hence the energy of one degree of freedom can be determi ned ¯h ω 2= 35.5Mev. If we take into consideration that the FnSpr ElmElcChr g ofπ-meson takes participation in two quasi-plane circular harmonic oscill ations with opportunity orientations and therefore has energy 2¯ hω= 139 .6Mev. Hence the energy of one degree of freedom can be determined¯h ω 2= 34.9Mev. As we can see by comparision of two results this coincid ence is very accurate. On this reason we can assume that the areas of their oscillations must also coincidence and therefore the OrbWvFnc of both FnSpr ElmElcChrg. May be t herefore the decay of the positive (negative) charged π-meson in 100% occurs through the positive (negative) µ-meson andµ-neutrino (antineutrino) as we can see from (5, 6, 11, 12). Th is second coincidence gives us many correct answer of the question for inner structure of the elementary micro particles (ElmMicrPrts). It is useful to remember that the existence of some analogy be tween leptons and hadrons allows us to assume that the quarks for the hadrons are analog ous of the neutrinos for the leptons. Therefore the physical means of the aroma of the qua rks is analogous of the physical means of the aroma of the neutrinos, which determines the siz es of the type excitation of the neutral but fluctuating vacuum (FlcVcm). Because of that we can suppose that if the neutrino is three dimensional spherical soliton seclusion excitation of the neutral vacuum, then the quark is the sum of two two-dimensional flat soliton excit ation of same neutral vacuum. As the vacuum is formed by neutral dynamides, which are made f rom two contrary massless electric charges ((-) electrino and (+) positrino) through their dense streamlining, then its solution seclusion oscillation secures the joint motion of the oscillating FlcVcm and the PntLk 6ElmElcChrg. However if the neutrino can accept the PntLk Elm ElcChrg for an unlimited time at a creation of the massive lepton, then the quark can accept the PntLr ElmElcChrg only for a limited time t or 2t at every its visit, after which the PntLk ElmElcChrg must leave it and go on to another quark. In this a way every PntLk ElmElcChrg in every hadron continuously and alternately changes the quarks of the hadron, in which it temporary pass into. Therefore as we could see the average distribution of the spread electr ic charge, we think that the electric charge of one type quark is1e 3and the electric charge of an other type quark is2e 3. The time of passτor 2τof the PntLk ElmElcChrg within some quark area is determined in depending on the kind (symmetric or antisymmetric) of the neutral flat vor tical oscillation of the FlcVcm. So any flat oscillation can be presented as a sum of two differen t flat oscillations, orientated in opposite directions. Therefore two kind of the quarks could be presented as symmetrical and antisymmetrical sums of two one-sided orientated protoqua rks|PK|and other-sided orientated antiprotoquark |APK|: /vextendsingle/vextendsingle/vextendsingle/vextendsingleSC ASC/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsinglecos(θo) sin( θo) −sin(θo) cos( θo)/vextendsingle/vextendsingle/vextendsingle/vextendsingle×/vextendsingle/vextendsingle/vextendsingle/vextendsinglePQ APQ/vextendsingle/vextendsingle/vextendsingle/vextendsingle (17) where the |SC|means a symmetric combination of one-side flat oscillation o f protoquark |PQ|and of one-side flat oscillation of antiprotoquark |APQ|, while the |ASC|means an antisymmertic combination of one-side flat oscillation of p rotoquark |PQ|and of one-side flat oscillation of antiprotoquark |APQ|. If we assume that θo= 30othen sin( θo) =1 2and cos(θo) =√ 3 2. Therefore in this case we can obtain from (17) the following equation : /vextendsingle/vextendsingle/vextendsingle/vextendsingleSC ASC/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle√ 3 21 2 −1 2√ 3 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle×/vextendsingle/vextendsingle/vextendsingle/vextendsinglePQ APQ/vextendsingle/vextendsingle/vextendsingle/vextendsingle (18) Hence the angle θodetermine the ratio of the participation of two oscillation s. This inter- pretations help us to understand the physical mean of number1 3and2 3. Indeed, if the |SC| means the symmetric combination of two opposite orientated oscillations then the probability to find the PntLk ElcChrg in this state is determined by the diff erence of the quadrats of its amplitudes3 4and1 4. By this natural way we have obtain the number2 4=1 2. Then, if the |ASC|means the antisymmetric combination of two opposite orient ated oscillations then the probability to find the PntLk ElcChrg in this state is determi ned by sum of the quadrats of its amplitudes3 4and1 4. By this natural way we have obtain the number4 4= 1. Consequently the number1 3and the number2 3mean that the probability of finding the PntLk ElmElcChrg in the state of the SC is1 3and the probability of finding the PntLk ElmElcChrg in the sta te of the ASC is2 3. Therefore we think that the values of the electric charges a re1e 3and2e 3. By means of this our interpretation of our PhsMdl we can obtai n the real values of the Weinberg angle and of the Cabibo angle. Indeed, in the probab ility of the finding the PntLk ElmElcChrg in the dquark state is equal of1 3and the probability of the finding same PntLk ElmElcChrg in the uquark state is equal of2 3, then it is obviously clearly that the transition probability from one kind of the quark to another kind of quar k is equal of their product. Therefore we have obtained that sin( θw)2=2 9and sin( θw) =√ 2 3. Consequently, θw= arcsin√ 2 3 orθw= 28o8,. In such a naturally easy, physically substantiated and math ematically correct way we have achieved Weinberg’s angle values, which very well coincide s with its value, determined by the experiment. It is very important that the value, ˜θw= 38o, which is determined by means of theSU(2)×SU(1) group method, is very different from the experimental val ue. 7We have explained that besides the colour the quark state has also the aroma, which dif- ference ensures perfectly different properties of the quark state. We have supposed, that three different aroma of the quark correspond to three sizes of the t wo-dimensional flat fermion har- monic oscillations of its PntLk ElmElcChrg. It turns out tha t there are possibility enough to give us clear physical interpretation and mathematical sub stantiation for obtaining the Cabibo angle value, determined by experiment. Indeed, we may assum e, that in a result of the different cover of the OrbWvFncs of different aroma of quarks, there wil l be different possibility for a transition of one aroma of quark in another aroma of quark wit h same ClcChrg. Some times this possibility is interpreted as a reflection of some mutua l influence between them. Therefore we may assume that each quark is consists from quarks of differ ent aroma but of same ElcChrg. In this supposing we may roughly write the following equatio ns : d1=docosα+sosinα;s1=socosα−dosinα; (19) u1=uocosβ+cosinβ;c1=cocosβ−uosinβ; (20) It is easy to verify that : ˜u1d1+ ˜c1s1= (˜uodo+ ˜coso) cos(α−β) + (˜uoso−˜codo) sin(α−β) ; (21) ˜u1s1−˜c1d1= (˜uoso−˜codo) cos(α−β)−(˜uodo+ ˜coso) sin(α−β) ; (22) If we present the difference ( α−β) as Cabibo angle θcthen from eqns. (21) and (22) one becomes clear that at the participation of the quarks of e qual aroma (˜ u1d1+ ˜c1s1) and different aroma (˜ u1s1−˜c1d1) in the weak interaction reaction, then Cabibo angle determ ines the decay probabilities with the hypercharge change and hyp ercharge conservation. Therefore Cabibo angle value is determined by equation sin θc=2 9andθc= arcsin2 9= 12o50,. In such a naturally, physically clear and mathematically subs tantiated way we have easy achieved Cabibo angle value 12 .84owhich very well coincides with the value 12 .7o, determined by the experiment. We assume that the proton is composed from one free positive P ntLk ElmElcChrg, which visits consecutively two states of u-quark.The free u-quark and the ˜d-antiquark are frequently visited by negative PntLk ElmElcChrg of virtual negative ch arged π−-mesons and the neutron is composed from two free opposite charged PntLk ElmElcChrg s, the positive one is oscillating in a state of u-quark and the negative one is oscillating in a state of two d-quarks. Very frequently the negative PntLk ElmElcChrg after forming the virtual negative charged point (π−-meson) by means of birthed pair of u-quark and ˜ u-antiquark owing of an absorption of some virtual gluon. The birthed by this way virtual negative pion can participate in strong interaction between very close neighbor proton and neutron by its exchanging. However I have possibility to show and you to understand by me ans of some decay relation that negative charged intermediate vectorial boson W−has spin minus ¯ h,and the positive charged intermediate vectorial boson W+have a spin ¯ h, while the neutral intermediate vectorial boson Zohave a spin zero ¯ h. Therefore the PntLk ElmElcChrg during the exchange of its self-consistent motions within one ElmMicrPrt and within o ther ElmMcrPrt transfers in a form of charged intermediate vectorial boson W. Indeed, the observing the law of total spin conservation at all decay with participating the weak inter action, when the PntLk ElmElcChrg takes the form of the charged intermediate vector boson W±, causes fulfillment of some strong selection rules from all ElmMicrPrt. The consideration of s pins of the quarks and charged intermediate vector bozon Wallow us to understand why the positive charged intermediat e 8vectorial boson W+emits only right quarks and selects only left quarks for part icipating with them in a weak interacting, while the negative charged inter mediate vectorial boson W−emits only left quarks and selects only right quarks for participa ting with them in a weak interaction. The neutral intermediate vector boson Zodon’t select the polarization of the quarks with which it participate in the weak interaction. The weak interaction performs an important role in the relat ion between hadrons and leptons as it is display in nuclear βdecay. Indeed, it is possible some a negative PntLk ElmElcCh rg (electrino) of any neutron, which takes part within the osci llations of d-quark, to change its self- consistent motion and moves on the oscillations of ˜ u-quark, which are unstable and therefore it decay in a negative charged intermediate vector boson W−andu-quark. In such a naturally way the neutron transforms itself in proton and the residual the negative charged intermediate vector boson W−after its incorporation with some new created electron neut rinoνeform the electron, while the other new created antineutrino ˜ νego a free away. When some positive PntLk ElmElcChrg (positrino) of any proton, which takes par t in the oscillations of the u- quark, changes his self-consistent motion and moves on the o scillations of the ˜d-quark, which are unstable and therefore it decay in positive charged inte rmediate vector boson W+andd- quark. In such a naturally way the proton transforms itself i n a neutron and the residual the positive charged intermediate vector boson W+after its incorporation with some new created electron antineutrino ˜ νeform the positron, while the other new created neutrino νego a free away. Therefore the described β-transitions can be written in the following form : n(u, d, d ) =p(u, u, d ) +W−=p(u, u, d ) +e−+ ˜νe (23) p(u, u, d ) =n(u, d, d ) +W+=n(u, d, d ) +e++νe (24) As within the areas of every nucleon there are two virtual opp osite charged π-mesons, at decreasing the distance between nucleons begins the creati on of some correlation between their motions, which creates some decreasing of their total energ y. At a rather decreasing of the distance between the nucleons begins some collectivizatio n of all virtual charged π-meson. In this way we understand that internucleonic forces are analo gous of interatomic Van-der- Waals force between neutral atoms. Our PhsMdls explain as the structure of hadrons and the natur e of their interaction so the existence of a possibility for joint description of a fiel d and substantial form of the matter as unity whole in the physical science, which are submitted t o an united, fundamental and invariable laws of nature. As all existent ElmMicrPrts are excitement of the vacuum the n all of them will move freely through one without any damping, that is to say without to fee l the existence of the vacuum. Moreover, the existence of some McrPrt in the vacuum twists i ts cristalline lattice.This twist of the neutral vacuum excites the gravitation field of the ElmMi crPrt’s mass, which will influence by using of some force upon mass of another ElmMicrPrt and upo n its behaviour. We can understand the physical essence of the hadrons and the ir structure and charac- teristics on these obvious representations. Our physical m odel explains as the structure of hadrons and the nature of their three type of interactions, s o the existence of the possibil- ity for a joint description of a field and substantial form of t he matter as unity whole in the physical science, which are submitted to an united, fundame ntal and invariable laws of nature. By this natural way we can see the unity of the field charged and neutral excitations within the vacuum and its substantial charged excitations, offered by modernity relativistic quantum mechanics (RltQntMch), quantum electrodynamics (QntElcD nm) and quantum theory of field (QntThrFld). 9I cherish the hope that this consideration of my physical mod el of the hadrons from my new point of view will be of great interest for all scientists . My very quality and good inter- pretation of behaviour and structure of hadrons and their in teractions with its corresponding experimentally determined values gives us the hope for corr ectness of our beautiful, simple and preposterous physical model of hadrons and fine,extraordin ary ideas,which have been inserted at its construction. References [1] H.Frauenfelder, E.M.Henley, Subatomic Physics , Prent ice-Hall, (1974) [2] K.Nishijima, Fundamental Particles, W.A.Benjamin inc . New York (1964) [3] K.Gattfried, V.F.Weisscopf, Concepts of Particle Phys ics, Clarendon Press, Oxford, (1984) [4] Gunnar Kallen, Elementary Particle Physics, Addison-W esley publisher, Comp. Mas- sachusetts, London (1966) [5] Lewis Ryder, Elementary Particles and Symmetries. Gord on and Breach Science Publisher New York (1975). [6] G.Feinberg, What is the world made of? Anchor press/Doub leday, New York (1978) [7] F.E.Close, An Introduction to Quarks and Partons, Acade mic press, London (1979) [8] S.Ogawa, S.Sawada, M.Nakagava, Composite Model of Elem entary particles, Iwanami Shoten, (1980). [9] Kerson Huang, Quarks, Leptons and Gauge Fields. World Sc ientific, New York, (1982). [10] Red.D.V.Shirkov, Small Encyclopeadia of Micro World P hysics. (in Russian) Sovietscaja encyclopeadia, Moscow (1980) [11] Harald Fritzsch, Quarks, Verlag Munchen (1983) [12] F.Halzen, A.D.Martin, Quarks and Leptons. John Wilew& Sons, New York (1984). [13] L.V.Okun, Elementary Particle Physics. (in Russian) N auka, Moscow (1988) 10

arXiv:physics/0003014v1 [physics.gen-ph] 7 Mar 2000PHYSICAL MODEL OF LEPTONS: MASSIVE ELECTRONS, MUONS, TAUONS AND THEIR MASSLESS NEUTRINOUS Josiph Mladenov Rangelov , Institute of Solid State Physics, Bulgarian Academy of Scie nces, 72 Tsarigradsko chaussee , 1784 Sofia , Bulgaria . Abstract The physical model (PhsMdl) of the leptons is offered by means of the PhsMdls of the vacuum and electron,described in our recent works. It is assumed that the vacuum is consistent by dynamides, streamlined in junctions of some t ight crystalline lattice. Every dynamide contains a neutral pair of massless point-like (Pn tLk) contrary elementary elec- tric charges (ElmElcChrgs): electrino ( −) and positrino (+). The PntLk ElmElcChrgs of the massless electrino and positrino of some dynamide in t he fluctuated vacuum may been excited or deviated by means of some energy, introduced by some photon or other micro particles (MicrPrts). The massless leptons (neutrin os) are neutral long-living soli- tary spherical vortical oscillation excitations of the unc harged fluctuating vacuum. The massive leptons are charged long-living solitary spherica l vortical excitations of its fine spread (FnSpr) elementary electric charge (ElmElcChrg). S o-called zitterbewegung is self-consistent strong-correlated vortical harmonic osc illation motion of the FnSpr ElmEl- cChrg of massive leptons. Different leptons have different se lf-consistent strong-correlated vortical harmonic oscillation motion of different sizes of t heir FnSpr ElmElcChrg, which is determined by their Kompton length λ=h m C, where mis the mass of the massive lep- tons. At mutual transition of one massive lepton into anothe r massive lepton its PntLk ElmElcChrg move up by dint of weak interaction in the form of t he charged intermediate vector meson Wfrom one self-consistent strong-correlated vortical harm onic oscillation motion of one size into another self-consistent strong-cor related vortical harmonic oscilla- tion motion of another size. Although up to the present nobody of scientists distinctly k nows are there some elementary micro particles (ElmMicrPrts) as a stone of the micro world a nd what the elementary micro particle (ElmMicrPrt) means, there exists an essential pos sibility for clear and obvious scientific consideration of the uncommon quantum behaviour and unusua l relativistic dynamical param- eters of all the relativistic quantized micro particle (Qnt MicrPrt) by means of our transparent surveyed PhsMdl. It is well known that the physical model (Ph sMdl) presents at us as an actual ingradient of every good physical theory (PhsThr). I t may be used as for an obvious visual teaching the unknown occurred physical processes wi thin the investigated phenomena, It turned out that all leptons are elementary micro particles ( ElmMicrPrt) of two kinds: a charged massive and uncharged massless. The massless leptons (neut rinos) are neutral long-living soli- tary isotropic three dimensional relativistic quantized ( SltIstThrDmnRltQnt) spherical vortex harmonic oscillation (SphrVrtNrmOsc) excitations of the n eutral fluctuated vacuum (FlcVcm) without a self-energy at a rest. The massive leptons are char ged long-living solitary isotropic 1three dimensional relativistic quantized (SltIstThrDmnR ltQnt) spherical vortical harmonic os- cillation (SphrVrtHrmOsc) motion of its fine spread (FnSpr) elementary electric charge (ElmEl- cChrg). The PhsMdl of the charged massive lepton will be pres ented by the PhsMdl of one of them, the DrEl. Therefore so-called zitterbewegung by Sc hrodinger,which is self-consistent strong-correlated vortical harmonic oscillation (VrtHrm Oscs) motion of the FnSpr ElmElcChrg of one of massive leptons, electron, is generalized for all c harged massive leptons. Therefore different charged massive leptons have different self-consi stent strong-correlated vortical har- monic oscillation motion of different sizes of their FnSpr El mElcChrg, which is determined by their Kompton length λ=h m C, where mis the mass of the massive leptons. At mutual transition of one massive lepton into another massive lepto n its PntLk ElmElcChrg move up by means of weak interaction in the form of the charged intermed iate vector meson Wfrom one self-consistent strong-correlated (VrtHrmOscs) motion o f one size into another self-consistent strong-correlated (VrtHrmOscs) motion of another size. The PhsMdl of the DrEl have offered in all my work in resent nine teen years for bring of light to the physical interpretation of physical cause of the uncommon quantum behaviour of the Schrodinger electron (SchEl) and the relativistic be haviour of the DrEl and give the thru physical interpretation and sense of all its dynamical parameters. Our PhsMdl of the DrEl explain as the physical causes for its unusual stochast ic classical dual wave-corpuscular behaviour and so give a new cleared picturesque physical int erpretation with mother wit of the physical means of its relativistic dynamical parameters. I n our transparent surveyed PhsMdl of the DrEl one will be regarded as some point like (PntLk) ElmEl cChrg, taking simultaneously part in four different motions: A) The isotropic three-dimensional relativistic quantize d (IstThrDmnRltQnt) Einstein’s stochastic (EinStch) boson harmonic shudders(BsnHrmShdr s) as a result of momentum re- coils (impulse kicks), forcing the charged QntMicrPrt at it s continuously stochastical emissions and absorptions of own high energy (HghEnr) virtual photons (StchVrtPhtns) by its PntLk ElmElcChrg. This jerky motion display almost Brownian clas sical stochastic behaviour (Brn- ClsStchBhv) with a light velocity C during a small time inter valτ1, much less then the pe- riodTof the IstThrDmnRltQnt Schrodinger fermion (SchFrm) vorti cal harmonic oscillation (VrtHrmOsc) motion and more larger then the time interval τoof the stochastically emission or absorption of the Hgh-Enr StchVrtPhtn by its PntLk ElmElc Chrg. The display of the Ist- ThrDmnRltQnt EinStch BsnHrmShds can be observed at the ligh t (RlPhtn) scattering of free DrEl. Indeed, in a consequence of the above investigation we may really consider the Ist- ThrDmnRltQnt FrthStch VrtHrmOscMtn’s ”trajectory” of the DrEl turns into cylindrically fine spread path, which has a form of the distorted figure of eig ht. The participate the FnSpr ElmElcChrg of DrEl in IstThrDmnRltQnt FrthStch VrtHrmOscM tn well spread (WllSpr) the ElmElcChrg of the SchEl. Further the behaviour of the DrEl’s FnSpr ElmElcChrg may be treated as a nearly Brownian classical stochastic behaviou r (BrnClsStchBhv) of the ClsMicr- Prt, taking part in the relativistic random trembling motio n (RltRndTrmMtn), during the time intervals τ1, much less then the period Tof the SchFrm VrtHrmOscs and more larger then the time interval τof the emission and absorption of the Hgh-Enr StchVrtPhtn, t he QntElcMgnFld of which form the RslSlfCnsVls of own RslQntElcMgmFld of the DrEl’s FnSpr ElmElcChrg. In a consequence of this the product 2 C τmay be considered as the space parth values of the DrEl’s PntLk ElmElcChrg, which is passed in the time inte rval 2 τof its absorption and emission (or scattering) of some RlPhtn. In a consequence of this we can easily understand by our felicitous PhsMdl why the classical radius of the LrEl entirely coincides with the size/radicalBig /angbracketleftξ2/angbracketrightof a spherical fine spread spot with an effective scattering su rfaceπ/angbracketleftξ2/angbracketright. Therefore 2the IstThrDmnRltQnt SchFrm VrtHrmOscMtn’s trajectory tur ns into fine spread path of a cylindrical shape with different radii. Therefore the size o f this smallest IstThrDmnRltQnt EinStch BznHrmShd’s motion of the DrEl’s PntLk ElmElcChrg c ould be determined by the Thompson total cross section. Indeed the averaged cross sec tion of the cylindrical spread path of the IstThrDmnRltQnt StchBsnHrmOscMtn can be determined by the effective total cross section of Thompson of RlPhtn’s scattering from the free DrE l’s, which has a space distribution with a spherical symmetry may be easily obtained by means of t he following simple relations, well-known by us from the classical StchMch : σ=π/angbracketleftξ2/angbracketright=3π 2/bracketleftBigg4e2 3mC2/bracketrightBigg2 =8π 3/bracketleftBigge2 mC2/bracketrightBigg2 (1) It is very important to make here, that this EinStchBznHrmSh ds have the smallest size and the fastest velocity and therefore the described smallest c ross section may be represented as a described roughly by the space distribution of the DrEl’s Fn Spt ElmElcChrg : |Υo(̺)|2=/bracketleftBigg (3 5√π)3(mC2 e2)3/bracketrightBigg exp(−̺2 κ2 o) (2) where κois parameter of the EinStch BsnHrmShds ( κo= (5/3)e2 mc2. The space distribution (2) of the DrEl’s FnSpt ElmElcChrg is described roughly by an OrbWvFnc Υ, having the following form : Υo(̺) = (√πκo)−3 2exp(−̺2 2κ2 o) (3) B) The isotropic three-dimensional relativistic quantize d (IstThrDmnRltQnt) Schrodinger fermion vortical harmonic oscillation motion (SchFrmVrtH rmOscMtn). In a consequence of such jerks of the PntLk ElmElcChrg along the IstThrDmnRltQn t EinStchHrmOscShdMtn ”trajectory” the ”trajectory” of the DrEl’s FnSpt ElmElcCh rg, participating in the IstThrDm- nRltQnt SchFrm VrtHrmOscMtn takes a strongly broken shape. Only after the correspondent averaging over the ”trajectory” of the IstThrDmnRltQnt Ein Stch BznHrmShdMtn we may ob- tain the fine spread ”trajectory” of the IstThrDmnRltQnt Sch Frm VrtHrmOscs’ one, having got the form of the distorted figure of an eight. Only such a mot ion along a spread uncom- mon ”trajectory” of the DrEl’s FnSpt ElmElcChrg could throu gh a new light over the SchEl’s well spread (WllSpr) ElmElcChrg’s space distribution and o ver the spherical symmetry of the SchEl’s WllSpr ElmElcChrg. It turns up that all relativisti c dynamical properties of the DrEl are results of the participation of its fine spread (FnSpr) El mElcChrg in the Schrodinger’s self-consistent fermion strongly correlated harmonic osc illations motion. This self-consistent strongly correlated IstThrDmnRltQnt SchFrm VrtHrmOsc’s m otion may be described mathe- matically correctly by means of the four components of its to tal wave function (TtlWvFnc) Ψ and four Dirac’s matrices; αj(γj) and β(γo). It turned out that all the massive leptons are sums of the corr esponding massless lepton (neutrino) and the FnSpr ElmElcChrg, which participates in some kind of a badly known but powerfully correlated self-consistent fermion motion, ca lled zitterbewegung. The different mas- sive leptons are distinguished between them-self by the amp litude size and frequency of its IstThrDmnRltQnt SchFrm VrtHrmOscs, called zitterbewegun g. Hence the different aroma of the leptons are different self-consistent spherical excita tions of its fine spread (FnSpr) elemen- tary electric charge (ElmElcChrg), which participates in i sotropic three dimensional relativistic 3quantized (IstThrDmnRltQnt) Schrodinger’s fermion (Schr Frm) vortical harmonic oscillations (VrtHrmOscs) of different sizes and frequency, which are det ermined by their Kompton length λ=h m C,where mis the mass of the massive leptons, at different energies, whi ch are determined by their mass m. In such a way the deviated FnSpr ElmElcChrg can creates own se lf-consistent resultant quantized electromagnetic fields (QntElcMgnFlds) by dint o f own high energy stochastic vir- tual photons (StchVrtPhtns), emitted by itself at different points of its zitterbewegung trajec- tory and in different moments in positions of the self-consis tent powerfully correlated SchrFrm VrtHrmOscs motion. Therefore the rest self-energy of the El mMicrPrt Eo=mc2is created in a results of the electromagnetic interaction (ElcMgnIntAc t) between its PntLk ElmElcChrg and MgnDplMm with the electric intensity (ElcInt) and magnetic intensity (MgnInt) of their own QntElcMgnFld. The own resultant QntElcMgnFld of the PntLk E lmElcChrg of the charged ElmMicrPrt is a result of the sum of the QntElcMgnFlds of cons tantly emitted stochastic Vrt- Phtns by same PntLk ElmElcChrg from its different positions o f the space in the zitterbewegung trajectory at different moments of a time at its self-consist ently powerfully correlated SchrFrm VrtHrmOsc motion. Therefore the different ElmMicrPrts, whi ch are different aromas of the leptons, may be considered as stable excitations of differen t energy in the uncharged fluctuating vacuum (FlcVcm). C)The isotropic three-dimensional nonrelativistic quant ized (IstThrDmnNrlQnt) Furthian stochastic boson (FrthStchBsn) circular harmonic oscilla tion motion (CrcHrmOscMtn) of the SchEl as a result of the permanent ElcIntAct of the electric i ntensity (ElcInt) of the resultant QntElcMgnFld of all the low energetic (LwEmr) StchVrtPhtns , existing within the FlcVcm and generated by dint of the VrtPhtn’s stochastic exchange b etween them. The SchEl’s motion and its unusual quantized behaviour, described in the NrlQn tMch may be easily understood by assuming it as a forced random trembling motion (RndTrmMt n) upon a stochastic joggle influence of the StchVrtPhtns scattering from some FrthQntP rt. Therefore the RndTrmMtn can be approximately described through some determining ca lculations by means of both the laws of the Maxwell ClsElcDnm and the probabilistic laws of t he classical stochastic theory (ClsStchThr). But in a principle the exact description of th e SchEl’s uncommon behaviour can be carry into a practice by means only of the laws of NrlQntMch and ClsElcDnm. Since then it is easily to understand by means of upper accoun t that if the ClsMicrPrt’s motion is going along the clear definitived smooth thin traje ctory in accordance with the Nrl- ClsMch, then the QntMicrPrt’s motion is perform in the form o f a roughly cylindrical spread path of a cylindrical shape with different radii with centers on a strongly broken line with quite unordered in its direction small straight lines of the RndTr bMtn near the classical one of any NtnClsPrt within the NrlClsMch. As a result of that we can sup pose that the unusual dualistic behaviour of QntMicrPrt can be described by dint of rj= ¯rj+δrj;pj= ¯pj+δpj; (4) It turns up that all the quantized dynamical properties of th e SchEl are results of the par- ticipation of its WllSpt ElmElcChrg in the isotropic three d imensional nonrelativistic quantized (IstThrDmnNrlQnt) Furth’s stochastic (FrthStch) boson ha rmonic oscillations(BsnHrmOscs). It is used as for a visual teaching the occurred physical proc esses within the investigated phe- nomena, so for doing them equal with the capacity of its mathe matical correct description by the mathematical apparatus of the both the quantum mechanics : t he nonrelativistic (NrlQntMch) and relativistic (RltQntMch). 4D) The classical motion of the LrEl along an well contoured sm ooth and thin trajectory realized in a consequence of some classical interaction (Cl sIntAct) of its over spread (OvrSpr) ElmElcChrg, bare mass or magnetic dipole moment (MgnDplMm) with some external classical fields (ClsFlds), described by well known laws of the Newton n onrelativistic classical mechan- ics (NrlClsMch). This motion may be finically described by vi rtue of the laws of both the NrlClsMch and the classical electrodynamics (ClsElcDnm); We must draw attention here that two massive leptons of same a roma may been distin- guished also by direction of its twirl. But if only massless l eptons have primary twirl the massive leptons may have both twirls. We must draw attention also here that the neutral spherical vortex excitations may been distinguished only b y direction of its twirl. As there are possibility for two opportunity twirls then neutrino ha s left-handed twirl and therefore the spin direction of the neutrino is antiparallel of its impuls es direction while the antineutrino has right-handed twirl and therefore the spin direction of t he antineutrino is parallel of its im- pulses direction. For knowing this it is very interesting wh y many theoretical physicists debate frequently why there are as left-handed photons so there are and right-handed photons, but no- body speaks about photons and antiphotons. However as becau se massless leptons (neurtinos) participate in the weak interactions therefore the primary twirl may been observanted in the weak interaction. Therefore many of them assert that there i s asymmetry between weak inter- action and electromagnetic interaction. Indeed if somebod y of them call the left-handed photon a photon and call the right-handed photon an antiphoton, the n both interactions, weak and electromagnetic should been semantically. In is easily to perceive that at the attentive analysis of the decay formulas of the different lepton aromas : τ+−→µ++ ˜ντ+νµ, τ+−→e++ ˜ντ+νe, µ+−→e++ ˜νµ+νe, (5) τ−−→µ−+ντ+ ˜νµ, τ−− →e−+ντ+ ˜νe, µ−−→e−+νµ+ ˜νe, (6) It is seen by means of these formulas that we are ability to wri te the follow equations: τ+=W++ ˜ντ, µ+=W++ ˜νµ, e+=W++ ˜νe, (7) τ−=W−+ντ, µ−=W−+νµ, e−=W−+νe, (8) It is easily to understand from upper that from both group dec ay formulas we are ability to assume the existence of the follow decay reactions: W+−→τ++ντ, W+− →µ++νµ, W+−→e++νe, (9) W−−→τ−+ ˜ντ, W−− →µ−+ ˜νµ, W−−→e−+ ˜νe, (10) The upper decay reaction remain us about the emmiting of a rea l photon (RlPht) by stim- ulated atom. Really as we well know the Rlpht is no found withi n stimulated atom before its radiation.Therefore we can speak only about the electrino, positrino and neutrinos.That is why we can use the following formal registration: τ+= (+) + ˜ ντ, µ+= (+) + ˜ νµ, e+= (+) + ˜ νe, (11) τ−= (−) +ντ, µ−= (−) +νµ, e−= (−) +νe (12) But the upper decay reaction have no means that the charged in termediate vector bosons W are composed from some aroma lepton and its neutrino, as in re ality the charged intermediate vector bosons Ware created only in the time of transfer beginning od the PntL k ElmElcChrg 5from one ElmPrt to other ElmPrt and are existed only during th e time interval of same transfer. We can distinctly see this from the following equation: Zo=νe+ ˜νe, Zo=νµ+ ˜νµ, Zo=ντ+ ˜ντ, (13) The equation (13) shows that the neutral intermediate vecto r boson Zocontains two opposite PntLk ElmElcChrgs (electrino and positrino). During the ti me interval of the decay of the neutral intermediate vector boson Zoboth its opposite PntLk ElmElcChrgs reconstruct own self-consistent motions and annihilate, formating one dyn amide, while its flat motions decay in two parallel neutral spherical vortical excitations: neut rino and antineutrino of same aroma. But the upper decay reaction have no means that the some massi ve charged lepton is composed from same aroma neutrino and the PntLk ElmElcChrg o f the negative charged inter- mediate vector bosons W−and every massive antilepton is sum of its antineutrino and p ositive charged intermediate vectorial boson W+. Indeed, although the decay of one lepton from an- other lepton is accompanied with leaping its negative PntLk ElmElcChrgthe in a state of the negative charged intermediate vectorial boson W−from one neutrino to another neutrino. But this decay don’t means that every massive lepton is sum of its neutrino and negative charged intermediate vectorial boson W−Really we can think that some massive lepton receives energy from fluctuating vacuum (FlcVcm) by dint of some virtual phot on (VrtPhtn) or virtual gluon (VrtGln) and therefore it generate neutrino of its aroma in t ime moment of its transition in state of the negative charged intermediate vectorial boson W−In such the way the unstable negative charged intermediate vectorial boson W−gives back of FlcVcm borrowing from it en- ergy in form of the VrtPhtn or VrtGln and after that its negati ve PntLk ElmElcChrg emits the antineutrino of same aroma, which aroma has the lepton massi ve state, which it occupy, with- out disintegrating itself of the negative charged intermed iate vectorial boson W−and massless neutrino. However there is possibility to understand by means of upper decay relation why exist lepton and antilepton numbers and weak charges, satisfying the con servation laws and why absent the electric charge from symmetry law. Indeed, if negative char ged intermediate vectorial boson W−has spin minus ¯ h, then in order to some lepton with spin a minus half ¯ h, we must add only the antineutrino with spin a half ¯ h. If we wish to make some antilepton, then to neutrino with a spin minus half ¯ hwe must accompanied with leaping the positive charged inter mediate vectorial boson W+with a spin ¯ h. The upper decay group formulas teach us that the PntLk ElmElc chrg of different leptons participates in some IstThrDmnRltQnt self-consistent and powerful correlated SchrFrnHrmOsc motion (zitterbewegung) of different sizes and at different e nergies. Therefore at the mu- tual transitions between them there give birth of pair neutr inoνland antineutrino ˜ νlof the same aroma and the PntLk ElmElcChrd pass from own neutrino (a ntineutrino) to the new birthed neutrino (antineutrino) in the form of the charged i ntermediate vectorial boson W. This is a natural way from which we can see the unity of the field neutral excitations in the FlcVcm and its substantial charged excitations, offered by m odernity relativistic quantum me- chanics (RltQntMch), quantum electrodynamics (QntElcDnm ) and quantum theory of field (QntThrFld). The electric interaction (ElcIntAct) of the r esultant QntElcMgnFld of all the StchVrtPhtns, exchanged from the FlcVcm with the ElmMicrPr t’s PntLk ElmElcChrg creates their diverse oscillations along its classical well contou red smooth and thin trajectory, spread and turned it into wide path, described by its OrbWvFnc (Ψ) wi thin the nonrelativistic quan- tum mechanics (NrlQntMch) . 6It turns out that we describe only three aroma massive and mas sless leptons. In approxi- mation of a mathematical correct substitution of the three o ne-dimensional powerful correlated fermion harmonic oscillations with three one-dimensional independent boson harmonic oscilla- tions the size of each aroma of leptons is determined by lengt h of its Kompton wave λ=h m C. As the masses of three massive lepton aromas have very big diff erent values (1 ,207,1785)meC2, then and the size of each lepton aroma must very strong differ o ne from other. Here I wish to show one very interesting coincidence. Indeed, if the total energy of each charged massive lep- tons is a sum of its self-energy of rest m.C2and of the potential energy of its PntLk ElmElcChrg in own averaged QntElcMfnFld (2 /3)e2 C¯hm.C2, then the total energy of the muon µ m µ.C2is equal of the sum of its self-energy of rest mµ.C2and of the energy of the electron me.C2, which is the potential energy of its PntLk ElmElcChrg in own averag ed QntElcMgnFld2 3e2 C¯hmµ.C2. Indeed : me/braceleftBigg 1 + (3 /2)C¯h e2/bracerightBigg ∼=mµor ( 1 + 205 .54 )∼=206.7 (14) Really by means of upper equation (14) we can assert that the t otal energy me.C2of the electron is an equal of the potential energy of the PntLk E lmElcChrg of muon in own averaged QntElcMgnFld. From research of the upper decay for mulas we can understand that the weak interaction is result of the charged (or neutral) in termediate vector bosons W(or Z) mutually interchange between leptons and others elementa ry micro particles (ElmMicrPrt). The formation of the lepton with a spin of the half of ¯ hfrom charged intermediate vector boson Wwith a spin of one ¯ hand another lepton with a spin of the half of ¯ hdetermines the choice rule of the participating the lepton in this interaction. The relation E2=p2c2+m2c4between the energy, impulse and mass of the ClsMacrPrt may be obtained through the use of the Maxwell’s equations of the classical electrodynamics (ClsEl- cDnm),as a result of the relation between the harmonic oscil lations of the impulse of a charged ClsMacrPrt and the vector-potential of its ClsElcMgnFld. T he parameters of own resultant QntElcMgnFld in the point of the moment positions of the QntM icrPrt’s PntLk ElmElcChrg and value of its rest-self energy may be determined by the age ncy of the mathematical appa- ratus of the RltQntMch, QntThrFld and QntElcDnm. The create d by this way own resultant QntElcMgnFld have zero values of the electric intensity (El cInt) of own resultant QntElcFld in the point of the moment position of a MicrPrt’s PntLk ElmEl cChrg and doubled value of the magnetic intensity (MgnInt) of own resultant QntMgnFld in this point in a respect of the MgnInt of the ClsMgnFld, created by the small spread (SmlSpr ) ElmElcChrg, participating in isotropic three dimensional relativistic classical (Is tThrDmnRltCls) Debay boson harmonic oscillations (DbBsnHrmOscs) with same the energy. The electric interaction (ElcIntAct) of the resultant QntE lcMgnFld of the StchVrtPhtns from the fluctuating vacuum (FlcVcm) with a ElmMicrPrt’s Wll Spr ElmElcChrg creates their diverse oscillations along its classical trajectory, spre ad and turned into an wide path within the nonrelativistic quantum mechanics (NrlQunMch). Such IstT hrDmnRltQnt Furthian stochastic boson circular harmonic oscillation motion (FrthStchBsnC rcHrmOscMtn) secures the existence of an additional mechanical moment (MchMm) of the QntMicrPr t, as and anomalous part of its MgnDplMm. The energy of the StchVrtPhtns exchanged betw een QntMicrPrt’s WllSpr ElmElcChrg and the FlcVcm gives a possibility of the QntMicr Prts to make tunneling through potential barriers, which are impassable in a classical way . The QntMicrPrt gathers from the FlcVcm at its IstThrMrnRltQnt BrnStchBsnCrcHrmOscMtn for potential energies of an aver- aged ElcFld of its WllSpr ElmElcChrg. Therefore the potenti al energies of this field don’t take part in equations between an inserted energy in the beginnin g of the birth of the ElmMicrPrts 7and obtained after its end. These isotropic three dimensional relativistic quantized (IstThrDmnRltQnt) Schrodinger fermion vortical harmonic oscillation (SchFrmVrtHrmOsc) motion (zitterbewegung) of the Elm- MicrPrt’s FnSpr ElmElcChrg correspond to its inner harmoni cal motion, introduced by Louis de Broglier. The Schrodinger’s zitterbewegung is some powerf ul correlation fermion self-consistent motion, who minimizes the rest self-energy of the ElmMicrPr t and secures the continuous sta- bility of the Schrodinger’s wave package (SchWvPck) in the s pace inanalogous of the Debay wave package (DbWvPck). In such a way we understand that the e nergetical advantage of the self-consistent strong correlated zitterbewegung, which minimizes the energy of ElcMgnSlfAct between the FnSpr ElmElcChrg and its own resultant QntElcMg nFld secures the stability of the SchrWvPck in the space and time. The emission and absorption of high energy StchVrtPhtns by t he PntLk ElmElcChrg of the charged ElmMicrPrt forces itself to make a isotropic thr ee-dimensional relativistic quan- tized (IstThrDmnRltQnt) Einstein stochastic (EinStch) bo son harmonic oscillation motion (BsnHrmOscMtn), which makes the smooth trajectory of the Sc hrFrm VrtHrmOscMtn a thickly and strongly broken line with shortest and very disordered s traight lines. The fine spread (Fn- Spr) of the SchrFrm VrtHrmOscMtn’s trajectory by very rapid jerk EinStch BsnHrmOscMtn may be observed at the scattering of the light (RlPhtns) on th e free Dirac electrons (DrEl). Indeed, the IstThrDmnRltQnt SchrFrm VrtHrmOscMtn’s traje ctory turns into roughly spread path of a cylindrical shape with different radii. The average d cross section of the cylindrical spread path of the IstThrDmnRltQnt StchBsn VrtHrmOscMtn ca n be determined by Thomp- son total cross section σ=8π 3/bracketleftBig e2 mC2/bracketrightBig2of the of the light (RlPhtns) at the free DrEl ,which determines the classical radius ro=2e2 mC2/radicalBig 2/3 of the LrEl. Although till now nobody knows what the ElmmICRpRT means, th ere exists a possibility for a consideration of the unusual behaviour of the quantize d micro particles (QntMicrPrts), of such as leptons and adrons by dint of an analogy with the tra nsparent surveyed PhsMdl of the DrEl. By our PhsMdl of the leptons at transition of the P ntLk ElmElkChrg from one massless leptons (its neutrino) to another massless lepton s (its neutrino) it takes form of a charged intermediate vector meson W. In such a way the weak interaction between two leptons may be realized by a transition of one PntLk ElmElcChrg from o ne neutrino (anti neutrino) to another neutrino (antineutrino) in the form of a charged int ermediate vector meson W. It seems to me the existance of two very interesting facts,ha ving common physical cause.The first is concurrence of the energy of one degree of freedom in c harged lepton µ-meson and in charged admeson π-meson. Indeed, if in isotropic three dimensional solitary vortical harmonic ocsillations of FnSpr ElmElcChrg of µ-meson have three degrees of freedom and therefore 3 ¯h ω= 2m C2= 213 .2Mev. Hence the energy of one degree of freedom can be determi ned ¯h ω 2= 35.5Mev. If we take into consideration that the FnSpr ElmElcChr g ofπ-meson takes participation in two quasi-plane circular harmonic oscill ations with opportunity orientations and therefore has energy 2¯ hω= 139 .6Mev. Hence the energy of one degree of freedom can be determined¯h ω 2= 34.9Mev. As we can see by comparision of two results this coincid ence is very accurate. On this reason we can assume that the areas o f their oscillations must also coincidence and therefore the OrbWvFnc of both FnSpr ElmElc Chrg. May be therefore the decay of the positive (negative) charged π-meson in 100% occurs through the positive (negative) µ-meson and µ-neutrino (antineutrino). This second coincidence gives u s many correct answer of the question for inner structure of the elementary micro p articles (ElmMicrPrts). 8I think that is very interesting to wtite the equations of the incomprehencible decay down: π+=⇒W+=⇒µ++νµ, π−=⇒W−=⇒µ−+ ˜νµ, (15) π+=⇒W+=⇒e++νe, π−=⇒W−=⇒e−+ ˜νe, (16) As for the radiation of the spontaneous real photon (RlPhtn) from the excitative atom it is necessary the presence of the virtual photon (VrtPhtn) fo r the creation of the electric dipole moment (ElcDplMm), so for the decay of a charged π-meson it is necessary the presence of a virtual gluon for an overturning of the spin of one of its quar ks, by which charged π-meson turns into charged virtual ρ-meson, which can immediately decay into charged intermedi ate vector boson W. At the subsequent transfer of the charged intermediate vec tor bozon Win a pair of massive and massless leptons of equal aroma the part icipating in the decay gluon go back in the FlcVcm. Therefore instead of upper decays we must used the following equations : π++δ=⇒W+=⇒µ++νµ, π−+δ=⇒W−=⇒µ−+ ˜νµ, (17) π++δ=⇒W+=⇒e++νe, π−+δ=⇒W−=⇒e−+ ˜νe, (18) For the first time in a hundred-years history of an electron, c ommon known as the smallest stable ElmMicrPrt, there exist a posibility for a considera tion of its unusual behaviour by means of a transparent PhsMdl realized in a natural way without any irreconcilable contradictions. I cherish hope that our consideration from quit new point of vi ew of my PhsMdl of all leptons by means of the PhsMdl of the electron will be of great interes for all scientists. Our PhsMdl explain as the structure of leptons and adrons and the nature of their interaction so the existence of a possibility for joint description of a field and substant ial form of the matter as unity whole in the physical science, which are submitted to an united, fu ndamental and invariable laws of nature. References [1] Rangelov J.M., University Annual (Technical Physics), 22, (2), 65, 87, (1985) ; 23, (2), 43, 61, (1986) ; 24, (2), 287, (1986); 25, (2), 89, 113, (1988). [2] Rangelov J.M.,Comptens Rendus e l’Academie Bulgarien S ciences, 39, (12), 37, (1986). [3] Rangelov J.M., Report Series of Symposium on the Foundat ions of Modern Physics, 6/8, August, 1987, Joensuu,p.95-99, FTL, 131, Turqu, Finla nd; Problems in Quantum Physics’2, Gdansk’89, 18-23, September, 1989, Gdansk, p.4 61-487, World Scientific, Sin- gapur, (1990); [4] Rangelov J.M., Abstracts Booklet of 29th Anual Conferen ce of the University of Peoples’ Friendship, Moscow 17-31 May 1993 Physical ser.; Abstracts Booklet of Symposium on the Foundations of Modern Physics, 13/16, June,(1994), Hel sinki, Finland 60-62. [5] Rangelov J.M., Abstract Booklet of B R U-2, 12-14, Septem ber, (1994), Ismir, Turkey ; Balk.Phys.Soc., 2, (2), 1974, (1994). Abstract Booklet of B R U -3, 2-5 Septembe r,(1997), Cluj-Napoca,Romania. 9[6] Rangelov J.M., LANL quant-ph 0001078; 0001079; 0001099 ; 0002018; 0002019. [7] H.Frauenfelder, E.M.Henley, Subatomic Physics , Prent ice-Hall, (1974) [8] K.Nishijima, Fundamental Particles, W.A.Benjamin inc . New York (1964) [9] K.Gattfried, V.F.Weisscopf, Concepts of Particle Phys ics, Clarendon Press, Oxford, (1984) [10] Gunnar Kallen, Elementary Particle Physics, Addison- Wesley publisher, Comp. Mas- sachusetts, London (1966) [11] Lewis Ryder, Elementary Particles and Symmetries. Gor don and Breach Science Publisher New York (1975). [12] G.Feinberg, What is the world made of? Anchor press/Dou bleday, New York (1978) [13] F.E.Close, An Introduction to Quarks and Partons, Acad emic press, London (1979) [14] S.Ogawa, S.Sawada, M.Nakagava, Composite Model of Ele mentary particles, Iwanami Shoten, (1980). [15] Kerson Huang, Quarks, Leptons and Gauge Fields. World S cientific, New York, (1982). [16] Red.D.V.Shirkov, Small Encyclopeadia of Micro World P hysics. (in Russian) Sovietscaja encyclopeadia, Moscow (1980) [17] Harald Fritzsch, Quarks, Verlag Munchen (1983) [18] F.Halzen, A.D.Martin, Quarks and Leptons. John Wilew& Sons, New York (1984). [19] L.V.Okun, Elementary Particle Physics. (in Russian) N auka, Moscow (1988) 10

arXiv:physics/0003015v1 [physics.optics] 7 Mar 2000Light propagation control by finite-size effects in photonic crystals E. Centeno, D. Felbacq LASMEA UMR CNRS 6602 Complexe des C´ ezeaux 63177 Aubi` ere Cedex France Abstract We exhibit the strong influence on light propagation of the fin ite size in photonic band-gap material. We show that light emission can be controlled by the symmetry group of the boundary of the finite device. The se results lead simply to important practical applications. Pacs: 42.70 Qs, 11.30-j, 42.82.-m, 03.50.De, 75.40.Mg Photonic crystals are expected to permit the realization of devices for integrated optics or laser microcavities [1–5]. The today technology now author izes the making of bidimensional photonic crystals in the optical range [6]. From the theoret ical point of view, there are two ways for characterizing PBG materials. The first way is th e one commonly used in solid state physics where pseudo-periodic boundary condit ions are imposed leading to Bloch waves theory. This is a powerful tool for the computation of t he band structure of the material [7]. However, this theory cannot be used when deali ng with the scattering of an electromagnetic wave by a finite device (which is the actual e xperimental situation) [8–10]. In that case, boundary effects cannot be skipped and must be ta ken into account. In this letter, we address the problem of the effect on light propagat ion of both the symmetry group of the lattice and the symmetry group of the boundary of a finit e piece of the material. The general study of the symmetry of the lattice modes has alread y been addressed in order to understand the possible degeneracies and the existence of f ull band gaps [11–15]. Here, we show that the finite size of the device strongly modifies the be haviour of the electromagnetic field, and that the symmetry of the boundary is a crucial param eter for the control of light emission from PBG devices. We deal with a bidimensional photonic crystal that is made of a finite collection of parallel dielectric rods of circular section. The rods are settled ac cording to a lattice with some symmetry group GY(Ydenotes the elementary cell) of the plane. The relative perm ittivity of the lattice is a Y-periodic function εr(x, y). The rods are contained in a domain Ω, having a symmetry group GΩ. As we deal with objects that are embebbed in an affine euclidea n space, both groups must be given as subgoups of the group of pl ane isometries O(2), in a canonical oriented basis. This is due to the fact that the use of the abstract groups does not permit to distinguish between two isomorphic realizations . Indeed, using the abstract groups and unspecified representations of degree 2, we could not, fo r instance, distinguish between two squares C1,2deduced from one another through a rotation rof angle π/4: denoting Γ 1 a realization of D4as an invariance group of C1, then Γ 2=rΓ1r−1is a representation of C2 1and therefore the equivalence class of Γ 1is not sufficient to compare the two figures and a canonical basis has to be precised. Denoting 1 Ωthe characteristic function of Ω (which is equal to 1 inside Ω and 0 elsewhere), the relative permittivity is given by εΩ(x, y) = 1+1 Ω(εr(x, y)−1). Assuming a s-polarized incident field (electric field parallel to the rods), the tota l electric field verifies the d’Alembert equation: c−2∂ttEz=ε−1 Ω∆Ez (1) Our aim is to study the invariance of this equation under the a ction of the various groups characterizing the geometry of the problem. For arbitrary c ross-sections of the fibers, we should also introduce their symmetry group (for instance th e group D4for square rods). The choice of circular rods simplifies the study in that their sym metry group O(2) contains both GYandGΩ. Note however that strong effects can be obtained by using pec uliar symmetries of the rods, including the enlargment of the band gaps [16]. L et us now denote Γ ( GY) and Γ (GΩ) the groups of operators associated to GYandGΩrespectively [17]. Both operators ∆ and∂ttcommute with Γ ( GY) and Γ ( GΩ). However, due to the function εΩ, the propagation equation is only invariant under the intersection group Γ ( GY)∩Γ (GΩ). This simple remark is a crucial point in understanding the invariant propertie s of finite crystals and it leads to extremely important effects in practical applications: it i s a clue to controlling the directions of light propagation in the structure. Indeed, due the bound ary of Ω, the degree of the global symmetry group of the device is reduced and consequently, fr om selection rules, the number of forbidden directions increases. In order to make this reasoning more explicit, we present two numerical experiments obtained through a rigorous modal theory of diffraction [10, 18] that takes into account all of the multiple scattering between rods (in case of an infinit e lattice this is exactly the KKR method), moreover it has been succesfully compared with exp eriments [19]. This method allows to deal with finite structure without periodizing tri cks [20] that may lead to spurious phenomena [21]. As a lattice group GY, we choose the diedral group D6[22], so that the lattice has a hexagonal symmetry. The distance between neig hbouring rods is denoted by d. In order to create a defect mode in that structure, we open a m icrocavity at the centre of the crystal by removing a rod (see [8,9,12,23,24] for stud ies of the properties of defects in photonic crystals). A defect mode appears within the first gap at a complex wavelength λ/d= 2.264 + 0 .06i(using a harmonic time-dependence of e−iωt). Such a structure can be used as a resonator coupled to waveguides [14]. In the first experiment, we choose the same group for the bound ary as that of the lattice (fig.1)(i.e. GΩ=D6). In that case the propagation equation is completely invar iant under Γ (D6). We plot the map of the Poynting vector modulus associated t o the defect mode (fig.1) and the radiation pattern is given in fig.2. Clearly the field s hows a hexagonal symmetry, which is obvious from the invariance of the d’Alembert equat ion. However, when designing light emitting devices, one whishes to control the directio n of light emission. In this example, there are too many directions of emissions: such a device is t o be coupled, for instance to waveguides, and to get a good transmission ratio, one needs t o concentrate the field in a few useful directions. As it has been stated above, the numbe r of authorized directions can be reduced by reducing the global symmetry group D6of the device. This is what we do in the next numerical experiment where we have changed the boun dary so that it has now a 2rectangular symmetry ( GΩ={e, sx, sy, r}, where sdenotes a symmetry with respect to x andyrespectively and ris a rotation of angle π), the device is depicted in fig. 3. In that particular case, the group of the boundary is contained in th e group D6. Then the equation (1) is no longer invariant under Γ ( D6) but solely under Γ ( D6)∩Γ (GΩ) = Γ ( GΩ) which is strictly contained in Γ ( D6). All the other transformations are now forbidden. That way , we expect a strong reduction of the directions of propagation o f the field. Indeed, the map of the Poynting vector of the defect mode (fig. 3) as well as the radiation pattern (fig. 4) shows a strong enhancement of the vertical di rection by forbidding the transverse directions linked to the rotations and the obliq ue symmetries. We have designed a resonator that permits to couple the radiated field in up and down directions with a better efficiency. It should be noted that a group theoretic analysis gives only informations on the possi- ble directions of emission, the actual directions on which t he field concentrates cannot be obtained by this mean: a rigorous computation involving a fin ite structure is then needed. Nevertheless, we have demonstrated that it was possible to s trongly increase the efficiency of resonators by simply taking into account the symmetry of the boundary of the device. This remark can be used rather easily in experimental situations and could lead to a dramatic enhancement of the output of PBG based devices. 3REFERENCES [1]Microcavities and photonic bandgap material: Physics and A pplications , J. Rarity, C. Weisbuch (Eds), Kluwer Academic Publisher, Series E: Appli ed Sciences, Vol. 234. [2] P.R.Villeneuve, S.Fan and J.D.Joannopoulos, Microcavities in photonic crystals, in Mi- crocavities and Photonic Bandgaps: Physics and Applicatio n, NATO, series E, vol.324. [3]Special issue on Photonic Band Structure , J. Mod. Opt 41171 (1994). [4]Development and Applications of Materials Exhibiting Phot onic Band Gaps, Special Issue, J. Opt. Soc. Am. B 10(1993). [5] J. D. Joannopoulos, R. D. Meade, J.N. Winn, Photonic Crystals , Princeton University Press, Princeton, 1995. [6] P. Pottier & al., J. Light. Tech. 112058 (1999). [7] R.D. Meade & al., Phys. Rev. B 4410961 (1991). [8] G. Tayeb, D. Maystre, J. Opt. Soc. Am. A 143323 (1998). [9] E. Centeno, D. Felbacq, J. Opt. Soc. Am. A 162705 (1999), J. Opt. Soc. Am. A. 17 (2000) at press. [10] D. Felbacq, G. Tayeb, D. Maystre, J. Opt. Soc. Am. A 112526 (1994). [11] D. Cassagne, C. Jouanin, D. Bertho, Phys. Rev. B 52R 2217 (1995), Phys. Rev. B 53 7134 (1996). [12] P.R. Villeneuve, S. Fan, J. D. Joannopoulos, Phys. Rev. B547837 (1996). [13] P.R. Villeneuve, M. Pich´ e, Phys. Rev. B 464969 (1992). [14] S. Fan, P.R. Villeneuve, J. D. Joannopoulos, Phys. Rev. B5411245 (1996). [15] S. Fan, P. R. Villeneuve, H. A. Haus, Phys. Rev. B 5915882 (1999). [16] M. Qiu, S. He, Phys. Rev. B 6010610 (1999). [17] V. Heine, Group Theory in Quantum Mechanics , Pergamon Press, New-York, 1964. [18] L. M. Li, Z. Q. Zhang, Phys. Rev. B 589587 (1998). [19] P. Sabouroux, G. Tayeb, D. Maystre, Opt. Com. 16033 (1999). [20] R.C. McPhedran, L. C. Botten, C. M. de Sterke, Phys. Rev. E607614 (1999). [21] D. Felbacq, F. Zolla, in preparation. [22] H. Eyring, J. Walter, G. Kimball, Quantum Chemistry , John Wiley and Sons, New-York, 1944. [23] A. Figotin, A. Klein, J. Opt. Am. A 151435 (1998). [24] S. Y. Lin, V. M. Hietala, S. K. Lyo, Appl. Phys. Lett. 683233 (1996) 4Figure captions Figure 1: Map of the Poynting vector modulus of the defect mod e. Both the lattice and the boundary have the same symmetry group ( GΩ=D6). The red line represents the hexagonal symmetry of the boundary of the crystal. The de fect mode possesses all transformations of the hexagonal point group. The ratio of t he rod radius to the spatial periode is r/d= 0.15 and the optical index is n= 2.9. Figure 2: Radiation pattern of the defect mode for the crysta l defined in figure 1. The radiated power is invariant by the hexagonal point group. Figure 3: Map of the Poynting vector modulus of the defect mod e. The global symmetry of the crystal is given by the subgroup GΩ={e, sx, sy, r}. The red line represents the rectangular symmetry of the boundary of the crystal. The def ect mode is invariant under GΩ. The ratio of the fiber radius to the spatial periode is r/d= 0.15 and the optical index isn= 2.9. Figure 4: Radiation pattern of the defect mode for the crysta l defined in figure 3. The radiated power is invariant by the subgroup GΩ. 5Figure 1Figure 2 0.2 0.4 0.6 0.8 1 30 21060 24090 270120 300150 330180 0Figure 3Figure 4 0.2 0.4 0.6 0.8 1 30 21060 24090 270120 300150 330180 0

arXiv:physics/0003016v1 [physics.bio-ph] 7 Mar 2000On the genealogy of a population of biparental individuals Bernard Derrida∗, Susanna C. Manrubia †, and Dami´ an H. Zanette ‡ ∗Laboratoire de Physique Statistique de l’ ´Ecole Normale Sup´ erieure 24 rue Lhomond, F-75231 Paris 05 Cedex, France †Fritz-Haber-Institut der Max-Planck-Gesellschaft Faradayweg 4-6, 14195 Berlin, Germany ‡Consejo Nacional de Investigaciones Cient´ ıficas y T´ ecnic as Centro At´ omico Bariloche e Instituto Balseiro 8400 S.C. de Bariloche, R´ ıo Negro, Argentina (February 2, 2008) SUMMARY If one goes backward in time, the number of ancestors of an ind ividual doubles at each generation. This exponential growth very quickly exceeds the population size, when this s ize is finite. As a consequence, the ancestors of a given individual cannot be all different and most remote ancestors are repeated many times in any genealogical tree. The statistical properties of these repetitions in genealogic al trees of individuals for a panmictic closed population of constant size Ncan be calculated. We show that the distribution of the repet itions of ancestors reaches a stationary shape after a small number Gc∝logNof generations in the past, that only about 80% of the ancestr al population belongs to the tree (due to coalescence of branches), and tha t two trees for individuals in the same population become identical after Gcgenerations have elapsed. Our analysis is easy to extend to t he case of exponentially growing population. I. INTRODUCTION In the case of sexual reproduction, the ancestry of an indivi dual is formed by 2 parents, 4 grandparents two generations ago, and in general 2Gindividuals Ggenerations back into the past. The explosive growth of the n umber of ancestors belonging to the genealogical tree of, say, a pr esent human should stop at some point due, at least, to the finite size of previous populations. For instance, only G≃33 generations ago (spanning a period of less than thousand years), the number of potential ancestors in the tr ee of any of us is about 8 .5×109, more than the present population of the Earth, and of course much larger than the po pulation living about year 1000. The answer to this apparent paradox is simple: The branches of a typical geneal ogical tree often coalesce, indicating that many of the ancestors were in fact relatives and appear repeatedly in th e tree (Ohno, 1996; Derrida et al., 1999; Gouyon, 1999). It might be difficult to test the statistical properties of such r epetitions for an actual large, randomly mating population . Nevertheless, some exceptions can be found in royal genealo gy. Since nobles usually married within their own castes, the presence of repeated ancestors in royal genealogical tr ees is far from rare. The example of the English king Edward III, where some ancestors appear up to six times, has been ana lysed in our previous work (Derrida et al., 1999).1 Much attention has been paid in the past to a related problem, namely the statistical properties of branching processes (Harris, 1963) and its applications to the charac teristics of the successive descendants of a single ancesto r (Kingman, 1993). Actually, first applications of the branch ig processes technique go back to the twenties. J.B.S. Haldane (Haldane, 1927) calculated the probability that a m utant allele be fixed in a population through a method developed previously by R.A. Fisher (Fisher, 1922). There, the relevant quantity was the survival probability of the descendants of the first individual carrying the mutation. A ll these studies apply to the vertical transmission of names , to the inheritance of characters coming only from one of the p arents, like mithochondrial DNA or the Y chromosome, or to the fate of a mutant gene, for example, and correspond to an effective monoparental population. The heart of our problem is to take into consideration that reproduction is biparental. The distribution of repetitions of ancestor s described below does however satisfy an equation similar to those which appear in branching processes (Harris, 1963). Our problem of repetitions of ancestors in genealogical tre es is much closer to the counting of the descendants of an individual in a sexual population. For example, in the c ase of a population of constant size, the average number of offspring is twoper couple. Therefore after Ggenerations each individual has on average 2Gdescendants. 1We used the tree of Edward III which can be found at http://uts .cc.utexas.edu/ ∼churchh/edw3chrt.html. 1What prevents the number of descendants from growing expone ntially with Gand to exceed the population size is interbreeding: When 2Gbecomes comparable to the population size, interbreeding h appens between the descendants and different lines of descent coalesce. The problem of the st atistical properties of these coalescences is very similar to our present study of genealogical trees. None of them has – to our knowledge– yet been analysed. In the present work, we study theoretically the problem of re petitions in the genealogical trees in the case of a closed, panmictic population. The study of the properties of a singl e tree with coalescent branches and the comparison of the genealogical trees of two contemporary individuals all ows us to show that 1. There is a finite fraction (about 20% for a population of con stant size N) of the initial population whose descendants becomes extinct after a number of generations Gc∝logN. All the rest of the initial population (about 80%) belongs to all genealogical trees, 2. The distribution of the repetitions of ancestors living m ore than Ggenerations ago reaches a stationary shape after about Gcgenerations, 3. The genealogical trees of two individuals in the same popu lation become identical after a small number of generations Gcback into the past, 4. The similarity between two genealogical trees changes fr om 1% (almost all ancestors in the two trees are different) to 99% (the repetitions of the ancestors in the two trees are a lmost identical) within 14 generations around Gc, independently of the population size N. Our work can be generalized (see section IV) to describe coal escent processes, understood as the study of the gene tree originated when looking for the ancestry of a rando m sample of sequences (Kingman, 1982; Hudson, 1991; Donnelly & Tavar´ e, 1995). In the absence of recombination, each sequence has a single ancestor. The topology of thus reconstructed trees is equivalent to that generated throug h branching processes. Next in complexity, one can consider a two-loci sequence and assume that recombination can occur only between the two loci and with a small probability (meaning correlated genealogies2for the two loci). The statistical properties of such proces s can be estimated until the most recent common ancestor (MRCA) is reached (Hudson, 1 991). Instead, if one faces the study of a chromosome (Wiuf & Hein, 1997; Derrida & Jung-Muller, 1999) or of the who le genome, the number of ancestors grows as one proceeds back in time, since each individual has two parents and, apart from coalescence, also recombination (meaning splitting of the branches in the tree) is frequent. If one considers a population or a sample of individuals with in a population, there are relevant differences between the genealogy of a single gene and the genealogy of a chromoso me or of the whole genome (which we study here). While in the first case, in fact, there exists a MRCA for the sam ple (where the gene tree ends), the genealogical tree of a chromosome or of the genome with two parents proceed s backwards in time and never reduces to a single ancestor. The genealogical tree representing the pedigree of a diploid organism contains a large fraction of the ancest ral population. In this case, one may then talk about the most recent common set of ancestors , and study the similarities among different individuals now within the same population. II. STATISTICAL PROPERTIES OF AN INDIVIDUAL TREE Here we consider a simple neutral model of a closed populatio n evolving under sexual reproduction and with non- overlapping generations.3If the population size is N(G) at generation Gin the past, we form couples at random (by randomly choosing N(G)/2 pairs of individuals) and assign each couple a random numbe rkof descendants. The probability pkof the number kof offspring is given and if the population size is Nat present, its size N(G) at generation Gin the past is given by 2In this paper, we use the term genealogy to refer to the ancestry of a single gene or of a whole set of seq uences. In all cases, thegenealogy is the complete set of ancestors contributing to the present object, this object being an individual (as in section II), a group of individuals (as in section III), a sequence (s ection IV), or a single locus (as quoted here). In this case, correlated genealogies simply means that the different sets of ancestors for the two- loci are not independent. 3The Wright-Fisher model for allele frequencies works in the same set of hypothesis (Wright, 1931; Fisher, 1930). More recently, Serva and Peliti (Serva & Peliti, 1991) obtained a number of statistical results for the genetic distance betw een individuals in a sexual population evolving in the absence o f natural selection. 2N(G) =/parenleftbigg2 m/parenrightbiggG N (1) where the factor mis obtained from m=/summationdisplay kk pk. (2) Form= 2, the population size remains constant in time, whereas fo rm∝negationslash= 2 the number of individuals in the next generation is multiplied by a factor m/2. After a number of generations, the tree of each of the indiv iduals in the youngest generation is reconstructed. To quantify the cont ribution of each of the ancestors to the genealogical tree of an individual, we define the weight w(α) γ(G) of an ancestor γin the tree of individual αat generation Gin the past as w(α) γ(G+ 1) =1 2/summationdisplay γ′children of γw(α) γ′(G) (3) We take w(α) γ(0) = δα,γ, as this ensures that at generation G= 0 all the weight is carried by the individual αitself. The factor 1 /2 in (3) keeps the sum of the weights normalized/summationtextN(G) γ=1w(α) γ(G) = 1, for any past generation G. The weight w(α) γ(G) can be thought of as the probability of reaching ancestor γif one climbs up the reconstructed genealogical tree of individual αby choosing at each generation one of the two parents at rando m. The weights essentially measure the repetitions (see figure 1) in the genealogical tree. With out repetitions, w(α) γ(G) would simply be 2−Gfor each ancestor γin the tree. As an illustration of the previous quantities, we represent in Fig. 1 the result of random matings inside a small closed population of constant size N= 14 (thus m= 2) during 7 generations. The lines link progenitors with th eir offspring. The grey scale gives the weight wγ(G) of each of the individuals in the tree. The numbers on the lef t, all of them of the form r/2G, give the weight of the leftmost individual in each generati on. The denominators simply indicate the potential maximum number of ancestors at each g eneration. As counted by the numerator, each of them would appear repeated rtimes in this tree if all the branches were explicitly shown. We further assume that the probability pkof having kchildren per couple follows a Poisson distribution, pk= mke−m/k! (most of what follows could be easily extended to other choi ces of pk). We represent in Fig. 2 the probability for an English couple to have kmarrying sons during the period 1350-1986 (Dewdney, 1986). The solid line corresponds to a Poisson distribution with average 1 .15 (i.e., the average number of offspring per individual in that period, which corresponds to m= 2.3 in our analysis), and implies that the total population is g rowing. These data spanning six centuries and taken over an homogeneous po pulation support the hypothesis that the number of offspring is indeed Poisson distributed.4 If we define S(α)(G), the fraction of the population (at a generation Gin the past) which does not belong to the genealogical tree of individual α, (i.e. such that w(α) γ(G) = 0) one can show (see the appendix) that S(α)(G+ 1) = exp/bracketleftBig −m+m S(α)(G)/bracketrightBig . (4) This recursion, together with the initial condition S(α)(0) = 1 −1/N, determines this quantity for any G(Derrida et al., 1999). For large Gand for any individual α, this fraction S(α)(G) converges to the fixed point S(∞) of (4). This gives form= 2 (i.e. for a population of constant size) a fraction S(∞)≃0.2031878 ..which becomes extinct, so that the remaining fraction 1 −S(∞)≃80% of the population belongs to the genealogical tree of any individual α. A similar calculation shows that this 80% of the population which is no t extinct after a large number of generations appears in the genealogical trees of all individuals: If S(α,β)(G) is the fraction of the population which does not belong to any of the two trees of two distinct individuals αandβ,S(α,β)(G) satisfies the same recursion (4) as S(α)(G), and converges to the same fixed value S(∞). Thus, within this neutral model, an individual either bec omes extinct (with a probability of 20%) or becomes an ancestor of the whole popu lation after a large number of generations (with a probability of 80%). For an exponentially growing populati on with m= 2.3 as in figure 2, the results are the same except for the precise value of S(∞) (for m= 2.3, one finds S(∞)≃14%). 4Nonetheless, deviations from this distribution induced by a social transmission of the reproductive behaviour have be en reported (Austerlitz & Heyer, 1998). 3When Gis large enough, as shown in the appendix, the whole distribu tionP(w) of the weights w(α) γ(G) reaches a stationary shape, the properties of which can be calculated (Derrida et al., 1999). We show in Fig. 3 the distribution P(w/∝angbracketleftw∝angbracketright) for different values of m. As can be seen, it has a power-law dependence, P(w)∝wξfor small values of the ratio w/∝angbracketleftw∝angbracketright, with an exponent given by ξ=−logS(∞) logm−2, (5) and achieves a maximum value for w/∝angbracketleftw∝angbracketright ≃1. III. SIMILARITY BETWEEN TWO TREES We would like to know how similar are the genealogical trees o f two contemporary individuals and how they evolve in time within the same population. We have seen that a large f raction 1 −S(∞)≃80% of the ancestral population constitutes the pedigree of every present individual. As a n ext step, one can compare two individuals and compute the degree of similarity between their trees, that is, the se t of ancestors appearing at each generation in both trees simultaneously. We will see in particular that the two trees become identical after a number Gcof generations. We start with the definition of the overlap between the genealogical trees of two different individuals ,αandβ. Let w(α) γ(G) be the weight of the ancestor γin the tree of αat generation Gin the past, and similarly let w(β) γ(G) be the weight of the same ancestor γat generation Gforβ. These weights evolve according to (3) with w(α) γ(0) = δγ,α andw(β) γ(0) = δγ,βat generation G= 0. In order to quantify the similarity between the two trees , we introduce the quantities X(α)(G) =N(G)/summationdisplay γ=1/bracketleftBig w(α) γ(G)/bracketrightBig2 and Y(α,β)(G) =N(G)/summationdisplay γ=1w(α) γ(G)w(β) γ(G). Y(α,β)(G) measures the correlation between the two trees at generati onGin the past and X(α)(G) acts as a normal- ization factor. We then define the overlap q(α,β)(G) between the two trees at that generation by q(α,β)(G) =Y(α,β)(G) /bracketleftbig X(α)(G)X(β)(G)/bracketrightbig1/2 This overlap is a measure of the (cosine of the) angle between the two N−dimensional vectors w(α) γ(G) and w(β) γ(G).5 When q(α,β)(G)≃0, the two vectors are essentially orthogonal and the ancest ors of αandβare all different. On the other hand, when q(α,β)(G)≃1, the vectors are almost identical (as for brothers). For a large enough population, the fluctuations of X(α)(G) and Y(α,β)(G) are small around the population averaged values ∝angbracketleftX(G)∝angbracketrightand∝angbracketleftY(G)∝angbracketrightfor almost all choices of αandβ. Of course, if αandβare brothers, Y(α,β)(G) =X(α)(G), a value very different from its average ∝angbracketleftY(G)∝angbracketright; it is however very unlikely to get brothers, sisters or even cousins if one picks up two individuals at random from a large populatio n. The averages ∝angbracketleftX(G)∝angbracketrightand∝angbracketleftY(G)∝angbracketrightcan be calculated from the evolution of the weights (3). Init ially,X(0) = 1 and Y(0) = 0 since the individuals αandβin any pair are different. Using the fact that for large Nthe fluctuations of 5Similar quantities have been proposed as an indicator of the amount of evolutionary divergence between populations (Ki mura, 1983). The quantity analogous to our weight w(α) γin the population genetics approach is the frequency of the s ampled alleles, the number of ancestors γcorresponds to the number of genes (that is the dimension of t he space in which the vector w(α) γis embedded), and our individuals αandβcorrespond to the compared populations (Cavalli-Sforza & C onterio, 1960). 4X(α)(G) and Y(α,β)(G) are small, the expected value of the overlap q(G) between two randomly chosen individuals is given by q(G)≃∝angbracketleftY(G)∝angbracketright ∝angbracketleftX(G)∝angbracketright=1 1 +mGc−G(6) where Gc=log((m−1)N) logm−1. (7) This expression is derived in the appendix. Of course Eq (6) i s only valid with probability one with respect to the random choice of αandβand with respect to the dynamics. We see that for large N, the overlap q(G) is essentially zero for a number of generations of order Gc≃logN/logmand then within a number of generations ∆ Gwhich does not depend on N, it becomes equal to unity. Fig. 4 displays the averaged over lapq(G) as a function of the number of generations Gfor different values of N. We have chosen m= 2 so that the population remains constant in size. We see that changing Ndoes not change the Gdependence except for a translation of the curve. In particu lar the range ∆ Gon which the overlap changes from 1% to 99% does not depend on N. It is easy to check from (6) that for m= 2, the overlap should satisfy q(G+ 1) =2q(G) 1 +q(G)(8) (plain line in the insert). The fixed point q(G) = 0 is unstable for this map. All the trajectories finally con verge to the stable fixed point q(G) = 1 for large G. Also the quantity ∆ Gcan be estimated by counting how many generations are required for the overlap to change from 1 % to 99 % and this g ives from (6) ∆G≃log(104)/logm, that is ∆ G≃14 for m= 2 and ∆ G≃11 for m= 2.3 as in figure 2. Typical values of GcareGc≃20 for a population of constant size N= 106. For a population increasing with m= 2.3 as in figure 2, one gets Gc= 21 if the size in the last generation is N=N(0) = 75 millions. The previous analysis can be easily extended to the hypothet ical case of having an arbitrary number pof parents instead of 2. As is shown in the appendix, the statistical pro perties of genealogical trees in a population of constant size but arbitrary pare the same as for a population with only two parents and an ex panding or shrinking size according to Eq. (1). The described statistical properties are thus equivalent in (i) a system with sexual reproduction and a growth rate m=pand (ii) a system with constant population size but a number mof genders. The existence of a generation Gcaround which the genealogical similarity among individual s changes from 0 to 1 and which grows logarithmically with the size of the populat ion is one of our main results. This has to be compared with the number of generations required for the population t o become genetically homogeneous (Donnelly & Tavar´ e, 1991; Harpending et al., 1998), which grows proportionally to N. The difference is that when Gc≪G≪N, all the overlaps are 1, i.e all the genealogical trees in the populat ion have the same ancestors with the same weights, but the genomes are still very different: This is just an extensio n of the situation of brothers who have exactly the same genealogical tree but different genomes. IV. SIMPLE MODEL FOR THE CONTRIBUTION OF THE ANCESTORS TO THE GENOME The evolution of a set of sequences subject to coalescence an d recombination was first described by Hudson (1983). In this case, evolution proceeds until the most recent commo n ancestor for each set of homologous sites has been found. The set of MRCA sites does not necessarily belong to th e genome of a single ancestor, on the contrary, it is in general spread on a finite fraction of the original populat ion (Wiuf & Hein, 1997; 1999). In this section, we focus our attention on the statistical properties of the ancestry of a single extant genome. In particular, we calculate the equilibrium distribution for the fraction of material cont ributed by each ancestor. Consider the whole set of genes that a present diploid organi sm has inherited from its parents. Although both parents contributed 50% each, it is no longer true that grandparents contributed 25% each, since independent assortment of chromosomes plus crossing over mixed in each of the parent al gametes the material inherited from the previous generation. As a rough approximation to the output of geneti c recombination, one might consider that each sequence 5is obtained as the addition of a fraction fof the genetic material of one parent and a fraction 1 −fof the genetic material of the other parent with f∈(0,1). This would be true if the length Lof the sequence was long enough (or infinitely long), so that there would be no restriction on the number of times it could be divided, and if one could forget the linear structure of the sequence. The process of coalesc ence and recombination (for small N) is schematically represented in Fig. 5. We can now repeat the analysis done previously to the present extension. We will discard the correlations between the values of fcoming from a couple. This is equivalent to our assumption th at fixing the pairs for koffspring or choosing the parents of each individual at random only has eff ects of order O(N−1) (see the appendix), and we can therefore work in the simplest realization of the process. H ence, we assume that the fraction ftakes independent values for each parent. The recursive equations (3) for the w eights become w(α) γ(G+ 1) =/summationdisplay γ′children of γfγ′w(α) γ′(G), (9) where the weight w(α) γ(G) means now the fraction of the genetic material of individua lαinherited from ancestor γat generation G. The random fraction fis chosen anew for each offspring from a distribution ρ(f) (with average value ∝angbracketleftf∝angbracketright= 1/2). This implies that now even brothers would have different w eights for their ancestors, and hence brings us slightly closer to the real genetic process. Following the procedure described in the appendix, one can c alculate the fraction Sof ancestors without lines of descent in the present (as we also show in Sec. II) and the expo nentξfor the distribution P(w). In general, given the distribution ρ(f) for the contributions of the parents, we get S(∞) =emS(∞)−m(10) 1 =S(∞)m2+ξ∝angbracketleftf∝angbracketright1+ξ/integraldisplay f−ξ−1ρ(f)df . (11) as one can easily show from (9) that the generating function hG(λ) defined by hG(λ) =∝angbracketleftexp[λw(G)/∝angbracketleftw(G)∝angbracketright]∝angbracketrighthas a limith∞(λ) for large Gwhich satisfies h∞(λ) = exp/bracketleftbigg −m+m/integraldisplay ρ(f)df h/parenleftbiggλf m∝angbracketleftf∝angbracketright/parenrightbigg/bracketrightbigg Fig. 6 summarizes the changes in the distribution P(w) for different distributions ρ(f) of the random variable f. We have considered a simple case of a population of constant s ize (i.e. m= 2) and with ρ(f) = 1/(2δ) uniform in the interval (1 /2−δ,1/2+δ). In this particular case, an implicit relation between δand the exponent ξcan be obtained, δξ=S/bracketleftBigg/parenleftbigg1 2−δ/parenrightbigg−ξ −/parenleftbigg1 2+δ/parenrightbigg−ξ/bracketrightBigg . (12) Asδvaries, P(w) remains a power law at small w(i.e.P(w)∝wξ), and the exponent ξmonotonously decreases withδ. In particular, for δ≃0.35,ξchanges sign: The maximum of P(w) moves discontinuously from w/∝angbracketleftw∝angbracketright ≃1 to w/∝angbracketleftw∝angbracketright ≃0. The exponents obtained through simulations of the proces s are represented in Fig. 7 together with the numerical solution of Eq. (12), showing a good agreement. V. DISCUSSION We have analysed the statistical properties of genealogica l trees generated inside a closed sexual population. We focused our interest on the distribution of the repetitions of ancestors in the trees and on the amount of genetic material contributing to an extant genome. The precise valu es ofξ, S(∞), Gcand ∆ Gdepend only weakly on the details of the model and do not change qualitatively if for in stance a non Poissonian distribution of offspring is used. Moreover, we have shown how our results can be extended to the hypothetical case of having an arbitrary number p of parents: Indeed, this case proves to be equivalent to a bip arental population with a growth rate m/2 =p/2. The problem analysed here presents a number of connections t o other fields. Equations similar to (3) appear also in the distribution of constraints in granular media where t he variables wrepresent the force acting on each grain and 6the recursion (3) expresses the way in which constraints are transmitted from one layer to the next (Coppersmith et al., 1996). In this case, p∝negationslash= 2 and even fluctuating pwould be perfectly realistic. The fact that the overlap chan ges from 0 to 1 within a small number of generations ∆ Gindependent of the size of the population and after Gc≃logN generations is also very reminiscent of the sharp cutoff phen omenon characteristic of some natural mixing processes modelled by Markov chains. One example of such systems is the shuffling of cards, where the stationary state in which the system has lost almost all information about the in itial ordering of the ncards is reached through a sharp cutoff after about log nriffle shuffles (Diaconis, 1996). It is clear that the study of the interplay between the weight s calculated in our generalized model and the structure of the genome would require more sophisticated approaches ( Derrida & Jung-Muller, 1999; Wiuf & Hein, 1997; 1999). We have discarded the correlations between the histo ry of neighboring sites in a sequence and assumed the independence of the factors f. Actually, the closer in the sequence two positions are, the more correlated are their genealogical histories (Kaplan & Hudson, 1985). This fact c onstrains the possible breaking points for our simulated sequences, implying that the random factors fin (9) are a crude approximation to reality. Since we have faced the problem from a statistical perspecti ve, our results represent the average, typical behaviour, and are only valid with probability one when the population s ize is large. We did not study fluctuations due to the finite size of the population. Nonetheless, we hope that our r esults contribute to a better understanding of the role of genealogy in the degree of diversity of finite-size interb reeding populations. APPENDIX In this appendix we have regrouped the technical aspects of t he derivations of the main equations (4,5,6,10,11) presented in the body of the paper. One may consider several variants of the model which all give a Poisson distribution for the number of offspring when the size of the population is large. For instance, the po pulation size could be strictly multiplied by a factor m/2 at each generation or it could fluctuate (if we take the number of offspring from the Poisson distribution). One might decide that each individual has two parents chosen at random in the previous generation or form fixed couples and assign each couple some children. All these variants do not c hange the results when the population size is large, but might affect some finite size corrections that we compute in th is appendix. We will choose the following version of the model, which make s the calculation of the finite size corrections not too difficult. Our population has a given size N(G) at each generation Gin the past, and we will assume that all the N(G) are very large, at least in the range of generations Gthat we will consider. Now, to construct the ancestors of all the N(G) individuals at generation Gin the past, we choose for each of them a pair of parents at rand om among theN(G+1) individuals at the previous generation (to facilitate t he calculation, we do not even exclude that the two parents might be equal). Within this model, the number kof children of an individual at generation G+ 1 is random and can be written as k=2N(G)/summationdisplay i=1zi where zi= 1 with probability 1 /N(G+ 1) and zi= 0 otherwise. It follows that the whole distribution of kcan be calculated. The probability pkthat an individual at generation ( G+1) has exactly kchildren is given by the binomial distribution pk=(2N(G))! k! (2N(G)−k)!/parenleftbigg1 N(G+ 1)/parenrightbiggk/parenleftbigg 1−1 N(G+ 1)/parenrightbigg2N(G)−k . (13) In particular, ∝angbracketleftk∝angbracketright=2N(G) N(G+ 1) ∝angbracketleftk(k−1)∝angbracketright=2N(G)[2N(G)−1] N(G+ 1)2 ∝angbracketleftk(k−1)(k−2)∝angbracketright=2N(G)[2N(G)−1][2N(G)−2] N(G+ 1)3. (14) If the population size is multiplied by a factor m/2 at each generation, i.e. if N(G) =N(G+ 1)m/2 (asGcounts the number of generations in the past), one recovers from (13 ) the Poisson distribution pk=mke−m/k! for large N(G). 7A. Calculation of the density of individuals without long te rm descendants and derivation of (4) To establish (4), one simply needs to notice that for an indiv idual to have no descendants after G+ 1 generations, all his children should have no descendants after Ggenerations. Let M(G) be the number of individuals with no descendants at generation Gin the past. Given M(G), one can write M(G+ 1) as M(G+ 1) =N(G+1)/summationdisplay γ=1yγ where yγ= 1 if all the children of γare among the M(G) and yγ= 0 otherwise. It can be shown that ∝angbracketleftyγ∝angbracketright=/parenleftbigg 1−1 N(G+ 1)/parenrightbigg2N(G)−2M(G) and ∝angbracketleftyγyγ′∝angbracketright=/parenleftbigg 1−2 N(G+ 1)/parenrightbigg2N(G)−2M(G) forγ∝negationslash=γ′. This gives ∝angbracketleftM(G+ 1)∝angbracketright=N(G+ 1)/parenleftbigg 1−1 N(G+ 1)/parenrightbigg2N(G)−2M(G) (15) ∝angbracketleftM2(G+ 1)∝angbracketright=∝angbracketleftM(G+ 1)∝angbracketright+N(G+ 1)[N(G+ 1)−1]/parenleftbigg 1−2 N(G+ 1)/parenrightbigg2N(G)−2M(G) (16) When all the M’s and N’s are large, we see from (15,16) that the fluctuations of M(G+ 1) are small (as ∝angbracketleftM2(G+ 1)∝angbracketright − ∝angbracketleftM(G+ 1)∝angbracketright2≪ ∝angbracketleftM(G+ 1)∝angbracketright2), and one finds from (15) that the ratio M(G)/N(G)≡S(α)(G) satisfies S(α)(G+ 1) = exp/bracketleftbigg2N(G) N(G+ 1)(S(α)(G)−1)/bracketrightbigg which is identical to (4) for N(G) =N(G+ 1)m/2. B. Time evolution of the distribution of the weights From the recursion (3) and from the known distribution (13) o fkone can write recursions for the moments of the weights ∝angbracketleftw(α) γ(G+ 1)∝angbracketright=∝angbracketleftk∝angbracketright 2∝angbracketleftw(α) γ(G)∝angbracketright (17) ∝angbracketleft[w(α) γ(G+ 1)]2∝angbracketright=∝angbracketleftk∝angbracketright 4∝angbracketleft[w(α) γ(G)]2∝angbracketright+∝angbracketleftk(k−1)∝angbracketright 4∝angbracketleftw(α) γ(G)w(α) γ′(G)∝angbracketright, (18) where γ∝negationslash=γ′. The normalization/summationtext γw(α) γ= 1 allows one to rewrite ∝angbracketleftw(α) γ(G)w(α) γ′(G)∝angbracketright=1 N(G)−1/bracketleftBig ∝angbracketleftw(α) γ(G)∝angbracketright − ∝angbracketleft[w(α) γ(G)]2∝angbracketright/bracketrightBig and together with the known moments (14) gives that 8∝angbracketleftw(α) γ(G+ 1)∝angbracketright=N(G) N(G+ 1)∝angbracketleftw(α) γ(G)∝angbracketright=1 N(G+ 1)(19) ∝angbracketleft[w(α) γ(G+ 1)]2∝angbracketright=/bracketleftbiggN(G) 2N(G+ 1)−N(G) [2N(G)−1] 2N(G+ 1)2[N(G)−1]/bracketrightbigg ∝angbracketleft[w(α) γ(G)]2∝angbracketright +2N(G)−1 2N(G+ 1)2[N(G)−1](20) where γ∝negationslash=γ′. For large N(G), if the ratio N(G+ 1)/N(G) = 2/m, as in the case of a population increasing by a factor m/2 at each new generation, expression (20) becomes simpler and on e gets ∝angbracketleft[w(α) γ(G+ 1)]2∝angbracketright=m 4∝angbracketleft[w(α) γ(G)]2∝angbracketright+m2 4/parenleftbigg1 N(G)/parenrightbigg2 (21) In this limit, we have from (14) that ∝angbracketleftk∝angbracketright=mand∝angbracketleftk(k−1)∝angbracketright=m2, and we see that (21) means that in (17) the weights w(α) γandw(α) γ′are, for large N(G), uncorrelated. The calculation of higher moments of the we ights can be done in the same manner and for large N(G) the weights of different ancestors become again uncorrelat ed. If the population size changes in time, the distribution of t he weights cannot be stationary. This is already visible in the expression (17) which shows that even the first moment o f the weights changes with G. One can however check from (17) and (21) that the ratio ∝angbracketleft[w(α) γ(G)]2∝angbracketright/∝angbracketleftw(α) γ(G)∝angbracketright2which satisfies ∝angbracketleft[w(α) γ(G+ 1)]2∝angbracketright ∝angbracketleftw(α) γ(G+ 1)∝angbracketright2=1 m∝angbracketleft[w(α) γ(G)]2∝angbracketright ∝angbracketleftw(α) γ(G)∝angbracketright2+ 1 (22) has a limit m/(m−1) asGincreases. Moreover, as the initial value of this ratio is N(0), the number of generations Gcto converge to this limit is Gc∼logN(0)/logm. Higher moments of the weights behave in a similar way and one can write recursions for ratios which generalize (22) and wh ich show that all the ratios have limits. This indicates that the distribution of the ratio w/∝angbracketleftw∝angbracketrightbecomes stationary. In the limit of large N(G) (considering that the weights of the different children γ′in (3) can be taken as independent and that the distribution o fkbecomes Poissonian), one finds that the generating function hG(λ) defined by hG(λ) =/angbracketleftBigg exp/bracketleftBigg λw(α) γ(G) ∝angbracketleftw(G)∝angbracketright/bracketrightBigg/angbracketrightBigg (23) satisfies hG+1(λ) =/summationdisplay kpk/bracketleftbigg hG/parenleftbiggλ∝angbracketleftw(G)∝angbracketright 2∝angbracketleftw(G+ 1)∝angbracketright/parenrightbigg/bracketrightbiggk = exp [ −m+m h G(λ/m)]. (24) Recursion (24) generalizes to the case m∝negationslash= 2 (i.e. the case of an exponentially increasing population ) the result of our previous work obtained for a population of constant size ( m= 2). Similar recursions have been studied in the theory of branching processes (Harris, 1963). The use of generatin g functions in population genetics is well illustrated in th e book by Gale (1990), where this method is for example applied to the calculation of the probability of fixation of a mutant allele. It is remarkable, that if one considers an imaginary world wh ere each individual would have pparents (instead of 2), the generating function (23), in the case of a population of c onstant size, would satisfy the recursion (24) with m=p. This means that as long as the distribution of weights is conc erned, the problem of a large population of constant size with mparents per individual is identical to the problem of a popul ation of size increasing at each generation by a factor m/2 with two parents per individual. C. Stationary distribution For large G, if we fix the ratio N(G)/N(G+ 1) = m/2, the generating function hG(λ) converges to h∞(λ) solution of 9h∞(λ) = exp [ −m+m h∞(λ/m)] (25) If one expands the solution around λ= 0, one finds that h∞(λ) = 1 + λ+1 2m m−1λ2+1 6m2(m+ 2) (m2−1)(m−1)λ3+. . . and the comparison with (23) gives for large G ∝angbracketleftw2∝angbracketright ∝angbracketleftw∝angbracketright2→m m−1;∝angbracketleftw3∝angbracketright ∝angbracketleftw∝angbracketright3→m2(m+ 2) (m2−1)(m−1); which means that in principle the whole shape of P(w) can be extracted from (25). In particular, one can predict t he power law of P(w) at small w. Trying to solve (25) for large negative λ, if one writes h∞(λ)−S(∞)≃1 |λ|ξ+1(26) one finds, as expected, that S(∞) is the fixed point of (4). Eq. (26) is equivalent to the asumpt ion that P(w)∼wξ at small w, where the exponent ξshould satisfy 1 =S(∞)mξ+2. This completes the derivation of (5) which was already discu ssed in our previous work (Derrida et al., 1999). D. Overlap between two trees Let us now show how (6) can be derived. Starting from recursio n (3), one obtains by averaging over all the links relating generation Gto generation G+ 1 ∝angbracketleftw(α) γ(G+ 1)w(β) γ(G+ 1)∝angbracketright=∝angbracketleftk∝angbracketright 4∝angbracketleftw(α) γ(G)w(β) γ(G)∝angbracketright+∝angbracketleftk(k−1)∝angbracketright 4∝angbracketleftw(α) γ(G)w(β) γ′(G)∝angbracketright, (27) where γ∝negationslash=γ′and the averages over kare carried out with respect to (13). This gives ∝angbracketleftw(α) γ(G+ 1)w(β) γ(G+ 1)∝angbracketright=m 4∝angbracketleftw(α) γ(G)w(β) γ(G)∝angbracketright+1 4/parenleftbigg m2−m N(G+ 1)/parenrightbigg ∝angbracketleftw(α) γ(G)w(β) γ′(G)∝angbracketright. (28) Using the fact that the sum/summationtext γ′w(β) γ′(G) = 1, so that ∝angbracketleftw(α) γ(G)∝angbracketright= 1/N(G) at all generations, one gets that ∝angbracketleftw(α) γ(G+ 1)w(β) γ(G+ 1)∝angbracketright=m 4∝angbracketleftw(α) γ(G)w(β) γ(G)∝angbracketright +1 4/parenleftbigg m2−m N(G+ 1)/parenrightbigg1 N(G)− ∝angbracketleftw(α) γ(G)w(β) γ(G)∝angbracketright N(G)−1. (29) Keeping only the dominant contributions for large N’s we arrive at ∝angbracketleftw(α) γ(G+ 1)w(β) γ(G+ 1)∝angbracketright=m 4∝angbracketleftw(α) γ(G)w(β) γ(G)∝angbracketright+m2 41 N(G)2. Comparing this expression with (27), one sees that for large N, one could have simply neglected the correlations between the weights of different individuals, (i.e. directl y replaced ∝angbracketleftw(α) γ(G)w(β) γ′(G)∝angbracketrightby∝angbracketleftw(α) γ(G)∝angbracketright ∝angbracketleftw(β) γ′(G)∝angbracketright) and used the Poisson distribution instead of (13)). The previou s recursion can be integrated ∝angbracketleftw(α) γ(G)w(β) γ(G)∝angbracketright=/bracketleftBig ∝angbracketleftw(α) γ(0)w(β) γ(0)∝angbracketright+1 N2m m−1(mG−1)/bracketrightBig/parenleftBigm 4/parenrightBigG , (30) 10and using the fact that ∝angbracketleftw(α) γ(G)w(β) γ(G)∝angbracketrightis equal to Y(G)/N(G) when α∝negationslash=βand to X(G)/N(G) when α=β, one finds (with X(0) = 1 and Y(0) = 0) ∝angbracketleftY(G)∝angbracketright ∝angbracketleftX(G)∝angbracketright=(mG−1)m−Gc 1 + (mG−1)m−Gc where Gcis given by (7). For large N, that is for large Gcthis reduces to (6) in the whole range where the expression departs from 0 or 1, that is for Gof order Gc. Finally, one can check that with the value of Gcgiven by (7), N(G) is always large, as long as Nis large, so that our assumption that all the N’s were large was legitimate. ACKNOWLEDGEMENTS. The authors acknowledge discussions wi th Jordi Bascompte, Ugo Bastolla and Julio Rozas. 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USA 95, 1961-1967. [15]Harris, T.E. (1963) The theory of branching processes . Springer-Verlag OHG. [16]Hudson, R.R. (1983) Properties of the neutral allele model with intergen ic recombination. Theor. Popul. Biol. 28, 183-201. [17]Hudson, R.R. (1991) Gene genealogies and the coalescent process. Oxford Surveys in Evolutionary Biology 7, 1-44. D. Futuyma and J. Antonovics, Oxford University Press. [18]Kaplan, N. & Hudson, R.R. (1985) Statistical properties of the number of recombinati on events in the history of a sample of DNA sequences. Genetics 111, 147-164. [19]Kimura, M. (1983) The neutral theory of molecular evolution . Cambridge University Press, Cambridge. [20]Kingman, J.F.C. (1982) The coalescent. Stochast. Proc. Appl. 13, 235-248. [21]Kingman, J.F.C. (1993) Poisson Processes . Oxford Studies in Probability 3, Oxford University Press. [22]Ohno, S. (1996) The Malthusian parameter of ascents: What prevents t he exponential increase of one’s ancestors? Proc. Natl. Acad. Sci. USA 93, 15276-15278. [23]Serva, M. & Peliti, L. (1991) A statistical model of an evolving population with se xual reproduction. J. Phys. A: Math. Gen24, L705-L709. [24]Wiuf, C. & Hein, J. (1997) On the number of ancestors to a DNA sequence. Genetics 147, 1459-1468. [25]Wiuf, C. & Hein, J. (1999) The ancestry of a sample of sequences subject to recom bination. Genetics 151, 1217-1228. [26]Wright, S. (1931) Evolution in Mendelian populations. Genetics 16, 97-159. 11G 26/128 11/64 5/32 3/16 1/8 1/4 1/2 1w t7 6 5 4 3 2 1 0 FIG. 1. Coalescence of branches in a genealogical tree. We di splay the reconstructed ancestry of a present individual in a small population of constant size N= 14. Numbers on the left side stand for the weight wof the leftmost individual at each generation. The grey scale changes from light grey (small w) to dark grey (large w) proportionally to the logarithm of the weight. The exact values are calculated according to Eq. (2) . The weight is a measure proportional to the number of times that an ancestor appears in a tree, or, equivalently, to the n umber of branches which have coalesced up to that point. 0.0 2.0 4.0 6.0 Number of males10−310−210−1100Probability FIG. 2. Probability for an English couple to have kmarrying sons during the period 1350-1986 (open circles). T he solid line corresponds to a Poisson distribution of average 1 .15. (Data from Dewdney (1986)). 1210−410−310−210−1100101 w/<w>10−510−410−310−210−1100P(w/<w>)m=1.5 m=2 m=3 m=4 FIG. 3. Stationary shape of the distribution P(w//angbracketleftw/angbracketright) for different values of m. We compare the constant population case (m= 2) with shrinking ( m= 1.5), and expanding ( m= 3,4) populations. Parameters are N= 4096, G= 20, and averages over 103independent realizations have been performed. 0 10 20 30 40 50 G00.20.40.60.81q(G) 0 0.5 1 q(G)00.51q(G+1) FIG. 4. The averaged overlap q(G) as a function of the number of generations G. The results of simulations for different sizes of the population N= 100 (open circles), 1000 (solid squares), 10000 (open diam onds), 100000 (solid triangles) agree with this prediction, up to small finite size corrections only vis ible for N= 100. The insert shows the results of simulations and the prediction (8). 131234G 0/j34/j34/j34/j34/j34/j34/j34/j34/j34 /j34/j34/j34/j34/j34/j34/j34/j34/j34 /j34/j34/j34/j34/j34/j34/j34/j34/j34 /j65/j65/j65/j65/j65/j65/j65/j65/j65 /j62/j62/j62/j62/j62/j62/j62/j62/j62 /j62/j62/j62/j62/j62/j62/j62/j62/j62 /j65/j65/j65/j65/j65/j65/j65/j65/j65 /j65/j65/j65/j65/j65/j65/j65/j65/j65 /j40/j40/j40/j40/j40/j40/j40/j40/j40 /j40/j40/j40/j40/j40/j40/j40/j40/j40 /j40/j40/j40/j40/j40/j40/j40/j40/j40 /j40/j40/j40/j40/j40/j40/j40/j40/j40 /j40/j40/j40/j40/j40/j40/j40/j40/j40 /j40/j40/j40/j40/j40/j40/j40/j40/j40 /j51/j51/j51/j51/j51/j51/j51/j51/j51 /j65/j65/j65/j65/j65/j65/j65/j65/j65 /j62/j62/j62/j62/j62/j62/j62/j62/j62 /j67/j67/j67/j67/j67/j67/j67/j67/j67 /j43/j43/j43/j43/j43/j43/j43/j43/j43 /j43/j43/j43/j43/j43/j43/j43/j43/j43 /j34/j34/j34/j34/j34/j34/j34/j34/j34 /j34/j34/j34/j34/j34/j34/j34/j34/j34 /j34/j34/j34/j34/j34/j34/j34/j34/j34 /j65/j65/j65/j65/j65/j65/j65/j65/j65 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/j51/j51/j51/j51/j51/j51/j51/j51/j51/j51/j51/j51 f 1−f FIG. 5. Representation of the first 5 generations of the tree i n Fig. 1 with a random distribution of the weight of an individ ual between its two parents. The fraction fof the weight contributed by each ancestor is randomly chose n from a distribution with average value /angbracketleftf/angbracketright= 1/2. 10−410−310−210−1100101102 w/<w>10−510−410−310−210−1100101P(w/<w>) δ=0.5 δ=0.35 δ=0.2 δ=0.05 FIG. 6. Stationary distribution of weights P(w//angbracketleftw/angbracketright) versus w//angbracketleftw/angbracketrightfor several choices of δ. The fixed population size is N= 4096, and we have averaged over 103independent runs. Values of δare as shown in the legend. 140.0 0.1 0.2 0.3 0.4 0.5 δ−0.4−0.20.00.20.4ξ FIG. 7. Comparison between the predicted values of the expon entξ(solid line) given by (12) and the results of the simulations for different values of δ(circles). Parameters as in Fig. 6. For a value of δ≃0.35, the exponent ξchanges sign. This point is important since the typical contribution of a randomly chos en ancestor changes suddenly in a finite amount. 15

arXiv:physics/0003017v1 [physics.chem-ph] 7 Mar 2000Is there a universality of the helix-coil transition in protein models? Josh P. Kemp †, Ulrich H. E. Hansmann ‡, Zheng Yu Chen † a i†Dept. of Physics, University of Waterloo, Waterloo, Ontari o, N2L 3G1, Canada ‡Dept. of Physics, Michigan Technological University, Houg hton, MI 49931-1291, USA Abstract The similarity in the thermodynamic properties of two compl etely different theoretical models for the helix-coil transition is examin ed critically. The first model is an all-atomic representation for a poly-alanine ch ain, while the second model is a minimal helix-forming model that contains no syst em specifics. Key characteristics of the helix-coil transition, in particul ar, the effective critical exponents of these two models agree with each other, within a finite-size scaling analysis. Pacs: 87.15.He, 87.15-v, 64.70Cn, 02.50.Ng The importance of understanding the statistical physics of the protein-folding problem has been stressed recently [1, 2]. For instance, it i s now often assumed that the energy landscape of a protein resembles a partially roug h funnel. Folding occurs by a multi-pathway kinetics and the particulars of the foldi ng funnel determine the transitions between the different thermodynamic states [1, 3]. This “new view” [1] of folding was derived from studies of minimal protein model s which capture only a few, but probably dominant parameters (chain connectivity , excluded volume, etc.) in real proteins. An implicit yet fundamentally crucial assumption is that th e basic mechanism of structural transitions in biological molecules depends so lely on gross features of the energy function, not on their details, and that a law of corre sponding states can be used to explain dynamics and structural properties of real p roteins from studies of related minimal models. This assumption needs to be proven. An even stronger no- tion in statistical physics is the universality hypothesis for critical phenomena. The 1critical exponents are identical for different theoretical models and realistic systems belonging to the same universality class. Many theoretical concepts in protein fold- ing, such as coil-helix or coil-globular transitions invol ve phase transition or phase transition-like behavior. Thus, one wonders if physical me asurements between two model systems for the same transition would have any “univer sal” properties. The purpose of this article is to examine these questions for the helix-coil tran- sition in homopolymers of amino acids [4, 5]. Traditionaly, the coil-helix transition is described by theories such as the Zimm-Bragg model [6] in w hich the homopoly- mers are regarded as one dimensional systems with only local interactions; as such a true thermodynamic phase transition is impossible. Howev er, recently there have been [4, 5] indications that the coil-helix transition near the transition temperature displays phase-transition like behavior. We use here finite -size scaling analysis, a common tool in statistical physics, to examine the question of universality of the helix-coil transition in two completely different, illumin ating models. On one hand, we have a detailed, all-atomic representation of a homo poly -alanine chain [7]. On the other hand, we have a simple coarse-grained model descri bing the general fea- tures of helix-forming polymers [4]. In this article, our in terest lies in finding out how far the similarity of the two models go. If the two models yiel d the same key physi- cal characteristics, then we at least have one concrete exam ple of the validity of the corresponding state principle or universality hypothesis in biopolymer structures. Poly-alanine is well-known to have high helix-propensitie s in proteins, as demon- strated both experimentally and theoretically [5, 7]. It ha s been well tested and gen- erally believed that approximate force fields, such as ECEPP /2[9] as implemented in the KONF90 program [10], give protein-structure predict ions to a surprisingly de- gree of faithfulness. As our first model, we have “synthesize d” poly-alanine with N residues, in which the peptide-bond dihedral angles were fix ed at the value 180◦for simplicity. Since one can avoid the complications of electr ostatic and hydrogen-bond interactions of side chains with the solvent for alanine (a n on-polar amino acid), we 2follow earlier work [7] and neglect explicit solvent molecu les in the current study. Our second model is a minimalistic view of a helix forming pol ymer [4] with- out atomic-level specifics. A wormlike chain is used to model the backbone of the molecule, while a general directionalized interaction, in terms of a simple square well form, is used to capture the essence of hydrogen like bon ding. The interaction energy between the residue labeled iandjis modeled by, Vij(r) =  ∞r<D −v D ≤r<σ 0σ≤r(1) wherev=ǫ[ˆ ui·ˆrij]6+ǫ[ˆ uj·ˆrij]6,ˆ ui= (ˆri+1,i)×(ˆri,i−1),ˆrijis the unit vector between monomeriandj,D= 3/2ais the diameter of a monomer, σ=/radicalBig 45/8ais the bonding diameter, and ais the bond length while bond angle is fixed at 60◦. To obtain the thermodynamic properties, we have conducted m ulticanonical Monte Carlo simulations for both models. In the low-tempera ture region where most of the structural changes occur, a typical thermal ener gy of the order kBTis much less than a typical energy barrier that the polymer has t o overcome. Hence, simple canonical Monte Carlo or molecular dynamics simulat ions cannot sample statistically independent configurations separated by ene rgy barriers within a finite amount of available CPU time, and usually give rise to bias st atistics. One way to overcome this problem is the application of generalized ensemble techniques [11], such as the multicanonical algorithm [12] used here, to the protein folding problem, as has recently been utilized and reported[13]. In a multicanonical algorithm [12] conformations with ener gyEare assigned a weightwmu(E)∝1/n(E),n(E) being the density of states. A simulation with this weight generates a random walk in the energy space; since a la rge range of energies are sampled, one can use the re-weighting techniques [14] to calculate thermody- namic quantities over a wide range of temperatures by ∝angbracketleftA∝angbracketrightT=/integraldisplay dxA(x)w−1 mu(E(x))e−βE(x) /integraldisplay dxw−1 mu(E(x))e−βE(x), (2) 3wherexstands for configurations and βis the inverse temperature. In the case of poly-alanine chains, up to N= 30 alanine residues were consid- ered. The multicanonical weight factors were determined by the iterative procedure described in Refs. [12] and we needed between 4 ×105sweeps (for N= 10) and 5×105sweeps (for N= 30) for estimating the weight factor approximately. All thermodynamic quantities were measured from a subsequent p roduction run of M Monte Carlo sweeps, where M=4×105, 5×105, 1×106, and 3 ×106sweeps for N= 10, 15,20, and 30, respectively. In the minimal model, chai n lengths up to 39 monomers were considered. In this model a single sweep invol ves a rotation of a group of monomers via the pivot algorithm[15]. For the weigh t factors the similar number of iterative procedure was used, and for the producti on run 1 ×108sweeps was used in all cases. We obtain the temperature dependence of the specific heat, C(T), by calculating C(T) =β2∝angbracketleftE2 tot∝angbracketright − ∝angbracketleftEtot∝angbracketright2 N, (3) whereEtotis the total energy of the system. We also analyze the order pa rameterq which measures the helical content of a polymer conformatio n and the susceptibility χ(T) =1 N−2(∝angbracketleftq2∝angbracketright − ∝angbracketleftq∝angbracketright2). (4) associated with q. For poly-alanine qis defined as q= ˜nH (5) where ˜nHis the number of residues (other than the terminal ones) for w hich the dihedral angles ( φ,ψ) fall in the range ( −70±20◦,−37±20◦). For our worm-like chain model the order parameter qis defined as q=N−1/summationdisplay i=2ui·ui+1 (6) In both cases the first and last residues, which can move more f reely, are not counted in the procedure. 4From a finite-size scaling analysis of the heights and width o f specific heat and susceptibility we can extract a set of effective critical exp onents which characterize the helix-coil transition in these two models [16]. For inst ance, with CMAXdefined to be the maximum peak in the specific heat, we have CMAX∝Nα dν. (7) In a similar way, we find for the scaling of the maximum of the su sceptibility χMAX∝Nγ dν. (8) For both quantities we can also define the temperature gap Γ = T2−T1(whereT1< TMAX<T2) chosen such that C(T1) =bCMAX=C(T2), andχ(T1) =bχ(Tc) =χ(T2) wherebis a fraction. The temperature gap obeys Γ =T2−T1∝N−1 dν, (9) as has been suggested in Ref. [16] . The analysis should be ins ensitive to the actual fraction,b, ofCMAX(χMAX) considered for defining T1andT2which was verified from our numerical data fitting of poly-alanine chains. The scaling exponents, α,ν, andγ, have their usual meaning in critical phe- nomena; however, the above scaling relations also hold form ally for the case of a first-order transition, with effective scaling exponents dν=α=γ= 1 [16, 17]. Note thatdis the dimensionality of the system, and it always appears in the combination dν. Without knowing further the effective dimensionality of ou r systems, we use the combination dνas a single parameter in the fit. It then becomes straightforward to use the above equation an d the values given in Table 1 to estimate the critical exponents. We obtain for p oly-alanine from the scaling of the width of the specific heat 1 /dν= 1.02(11) with a goodness of fit (Q= 0.9) (see Ref. [18] for the definition of Q), for chains of length N= 15 to N= 30. Inclusion of N= 10 leads to 1 /dν= 0.84(7), but with a less acceptable 5fit (Q= 0.1). Similarly, we find from the scaling of the width of the susc eptibility 1/dν= 0.98(11) (Q= 0.5) for chains of length N= 15 toN= 30 and 1/dν= 0.81(7) (Q= 0.2) when the shortest chain N= 10 is included in the fit. Hence, we present as our final estimate for the correlation exponent of poly-al aninedν= 1.00(9). This value is in good agreement with the estimate dν= 0.93(5) obtained from the partition function zero analysis in Ref. [8]. The results for the exponent αgiveα= 0.89(12) (Q=0.9) when all chains are considered, and α= 0.86(10) (Q= 0.9) when the shortest chain is excluded from the fit. Analyzing the peak in the susceptibility we find γ= 1.06(14) (Q= 0.5) for chain lengthsN= 15−30 andγ= 1.04(11) (Q= 0.5) for chain lengths N= 10−30. We summarize our final estimates for the critical exponents in T able 2. The scaling plot for the susceptibility is shown in Fig. 1: curves for all leng ths of poly-alanine chains collapse on each other indicating the validity of finite size scaling of our poly-alanine data. The same procedure can be applied to analyze the data from the minimal model. All calculation has been done with the omission of the shorte st chain. Using the widths of the specific heat a b= 80% of the peak height we obtain 1 /dν= 1.03(7), (Q= 0.2). The width of the peak at half maximum is more unreliable in this case as the coil-helix transition is complicated by the additional collapsing transition to a globular state in the vicinity of the coil-helix transition [4]. This exponent agrees with that calculated from the susceptibility widths, 1 /dν= 0.89(9), (Q= 0.3). Hence, our final estimate for this critical exponent in our second mo del isdν= 0.96(8). These values are in good agreement with those of the poly-ala nine model. From theCMAXdata in Table 1 and using the above given value for the exponen t dνwe findα= 0.70(16) (Q= 0.3) which is somewhat smaller than that of the poly-alanine model. The susceptibility exponent as calcul ated from the data in Table 1 yields a value of γ= 1.3(2) (Q= 0.5), which agrees with the previous estimation within the error bar. The scaling plot for the sus ceptibility is shown in 6Fig. 2. While curves corresponding to large polymer sizes co llapse into the same curve, the N= 13 case shows small disagreement, indicating that the finit e size scaling are valid only for longer chain lengths in the minima l model. Comparing the critical exponents of our two models as summar ized in Table 2 we see that the estimates for the correlation exponent dνagrees well for the two models. Within the error bars, the estimates for the suscept ibility exponent γalso agree. The estimates for the specific heat exponent αseem disagree within the error ranges. However, in view of the fact that both analyses are based on small system size the true error ranges could be actually larger th an the ones quoted here. Using these rather crude results, we have already demonstra ted a striking similarity in finite-size scalings of the two model. Therefore, we can co nvincingly make the conjecture that minimal model can be used to represent the st ructural behavior of real helix-forming proteins. Our analysis should tell us also whether the helix-coil tran sition in our models is of first or second order. In the former case we would expect dν=α=γ= 1 which seems barely supported by our data due to the rather large err or bars associated with the estimate of the exponents. We have further explored the nature of the transition from another perspective, by considering the ch ange in energy crossing a small temperature gap (taken to be within 90% of CMAX) from the original data, ∆E= (Etot(T2)−Etot(T1))/N (10) This value should approach either a finite value or zero as N−1goes to zero. A finite value would indicate a first order transition while a ze ro value a second order transition. In the case of a first order transitions the inter cept would indicate the latent heat. Now, the assumption is that this energy change s cales linearly as N−1 goes to zero. Figure 3 shows a plot of the data from both the ato mic-level and minimal models, where nonzero intercepts can be extrapolat ed atN−1=0. Hence, our results seem to indicate and finite latent heat and a first- order like helix-coil transition. However, we can not exclude the possibility tha t the true asymptotic 7limit of |E|is zero, and some of the results of Ref. [5] point for the case o f poly-alanine rather towards a second-order transition. Further simulat ions of larger chains seem to be necessary to determine the order of the helix-coil tran sition without further doubts. In summary, we conclude that in view of the similarity of the t wo models ex- amined here, a corresponding state principle can be establi shed for the coil-helix transition. Examining the finite size scaling analysis allo ws us to calculate esti- mators for critical exponents in the two models which indica te “universality” of helix-coil transitions. Acknowledgments : Financial supports from Natural Science and Engineering R e- search Council of Canada and the National Science Foundatio n (CHE-9981874) are gratefully acknowledged. References [1] K.A. Dill and H.S. Chan, Nature Structural Biology 4, 10 (1997). [2] E.I. Shakhnovich, Curr. Opin. Struct. Biol. 7, 29 (1997); T. Veitshans, D. Klimov and D. Thirumalai, Fold.Des. 2, 1 (1997). [3] J.D. Bryngelson and P.G. Wolynes, Proc. Natl. Acad. Sci. U.S.A. 84, 524 (1987);J.N. Onuchic, Z. Luthey-Schulten, P.G. Wolynes, An nual Reviews in Physical Chemistry 48, 545 (1997). [4] J.P. Kemp and Z.Y. Chen, Phys. Rev. Lett. 81, 3880 (1998). [5] U.H.E. Hansmann and Y. Okamoto, J. Chem. Phys. 110, 1267 (1999); 111 (1999) 1339(E). [6] B.H. Zimm and J.K. Bragg, J. Chem. Phys. 31, 526 (1959). [7] Y. Okamoto and U.H.E. Hansmann, J. Phys. Chem. 99, 11276 (1995). 8[8] N.A.Alves and U.H.E. Hansmann, Phys. Rev. Let. 84(2000) 1836. [9] M.J. Sippl, G. N´ emethy, and H.A. Scheraga, J. Phys. Chem .88, 6231 (1984), and references therein. [10] H. Kawai et al., Chem. Lett. 1991, 213 (1991); Y. Okamoto et al., Protein Engineering 4, 639 (1991). [11] U.H.E. Hansmann and Y. Okamoto, in: Stauffer, D. (ed.) Annual Reviews in Computational Physics VI ,(Singapore: World Scientific), p.129. (1998). [12] B.A. Berg and T. Neuhaus, Phys. Lett. 267, 249 (1991). [13] U.H.E. Hansmann and Y. Okamoto, J. Comp. Chem. 14, 1333 (1993); 18, 920 (1997). [14] A.M. Ferrenberg and R.H. Swendsen, Phys. Rev. Lett. 61, 2635 (1988); Phys. Rev. Lett. 63, 1658(E) (1989), and references given in the erratum. [15] N. Madras and A.D. Sokal, J. Stat. Phys. 50, 109 (1988). [16] M. Fukugita, H. Mino, M. Okawa and A. Ukawa J. Stat. Phys. 59, 1397 (1990), and references given therein. [17] K. Binder, D.W. Heermann, Monte Carlo Simulation in Statistical Physics Springer-Verlag, Berlin, 1988 [18] W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Fl annery, Numerical Recipes , 2nd ed. (Cambridge University Press, New York, 1992) p. 657 . 9Figure Captions 1. Scaling plot for the susceptibility χ(T) as a function of temperature T, for poly-alanine molecules of chain lengths N= 10,15,20 and 30. 2. Scaling plot of χ(T) as a function of temperature T, for the minimum model of chain lengths N= 13,19,26,33 and 39. 3. Scaling of energy gap and transition width at 80% and 90% of CMAX. Here we have used ∆ E80%(△for all-atom model, 3for minimal model), ∆ E90%(2 for all-atom model, ∝circlecop†rtfor minimal model). 10Table 1: Shown are the location of the specific heat maximum TMAX, the maximum of specific heat CMAX, susceptibility χMAX, the width of the half peak in specific heat Γ C, and width of the half peak of susceptibility Γ χfor various chain lengths. NTMAXCMAX ΓCχMAX Γχ All-Atomic Model 10427(7) 8.9(3) 160(7) 0.49(2) 140(7) 15492(5) 12.3(4) 119(5) 0.72(3) 110(5) 20508(5) 16.0(8) 88(5) 1.08(3) 78(5) 30518(7) 22.8(1.2) 58(4) 1.50(8) 56(3) Minimal Model 131.25(1) 1.088(2) 1.22(2) 0.232(2) 2.20(2) 191.17(1) 1.424(5) 1.12(2) 0.353(3) 0.81(2) 261.16(1) 1.789(8) 0.89(2) 0.553(8) 0.57(2) 331.13(1) 2.08(1) 0.73(2) 0.78(1) 0.45(2) 391.12(1) 2.27(2) 0.61(2) 0.96(2) 0.41(2) Table 2: Summary of the critical exponents obtained for the t wo models. All-atomic Minimal dν 1.00(9) 0.96(8) α0.89(12) 0.70(16) γ1.06(14) 1.3(2) 1100.010.020.030.040.05 -10000 -5000 05000 10000N=10 N=15 N=20 N=30 (T-TMAX)N1/dνχ(T)N-γ/dν Fig. 1 1250 0 50 (T-TMAX)N1/dν0.0000.0020.0040.0060.008χ(T)N-γ/dνN = 13 N = 19 N = 26 N = 33 N = 39 Fig. 2 130.00 0.02 0.04 0.06 0.08 1/N0.00.40.81.21.6ΔE Fig. 3 14

arXiv:physics/0003018v1 [physics.optics] 8 Mar 2000Ultra-refraction phenomena in Bragg mirrors D. Felbacq, B. Guizal LASMEA UMR 6602 Complexe des C´ ezeaux 63177 Aubi` ere Cedex France F. Zolla LOE ESA 6079 Facult´ e des Sciences de St-J´ erˆ ome 13397 Marseille Cedex 01 France (February 2, 2008) Abstract We show numerically for the first time that ultra-refractive phenomena do exist in one-dimensional photonic crystals: we exhibit the main features of ultra-refraction, that is the enlargement and the splittin g of an incident beam. We give a very simple explanation of these phenomena in terms of the photonic band structure of these media. It has recently been shown numerically as well as experiment ally that near a band edge, photonic crystals could behave as if they had an effective per mittivity close to zero [1–3]. Such a property induces unexpected behaviors of light usual ly called ultra-refractive optics. The main phenomena are the splitting or the enlargment of an i ncident beam, or a negative Goos-H¨ anschen effect [4]. The common explanation of these f acts lie on the study of the photonic dispersion curves. Though appealing, it seems diffi cult to turn this explanation into a rigorous one as the notion of group velocity in a strongly sc attering media seems doubtful apart in the homogenization sense which is not the situation for ultrarefractive optics. In our opinion, these surprising and beautiful phenomena main ly rely on the rapid change in the behavior of the field inside the structure when crossing a band edge. In this article, we provide a rather simple explanation of some of these phenome na (splitting and enlargment of an incident beam), which implies that they should be obser ved with one dimensional structures (as foreseen by [1]). Indeed, we show by numerica l experiments that it is the case in Bragg mirors (the simplest photonic crystals). From a theoretical point of view, we consider a periodic one d imensional medium charac- terized by its relative permittivity ε(x), which is assumed to be real, illuminated by a plane wave. It is well known that the band structure is determined b y the monodromy matrix Tof one layer [5,6], that is, the matrix linking the field and it s derivative over one period. This matrix is a function of λandθ. The main quantity is then φ(λ, θ) =1 2tr(T(λ, θ)). When |φ(λ, θ)|is inferior to 1 then ( λ, θ) belong to a conduction band, and when |φ(λ, θ)| 1is superior to 1 then ( λ, θ) belong to a gap. In fig. 1 we give a numerical example for a Bragg Mirror with ε1= 1,ε2= 4,h1=h2= 1 (the lengths are given in λunits). Now let us use a Gaussian beam as the incident field. Let us supp ose that the mean angle of the beam is zero (normal incidence) and that its wave length is very near a band edge. Then two things may happen. Reasoning on the oriented w avelengths axis, if the beam is centered on the left side of the gap (the dispersion di agram is given in the plane (λ, θ), if one uses frequencies instead of wavelengths one has to e xchange left and right), the center of the beam belongs to a conduction band and the edges o f the beam belong to the gap. Consequently, after propagation in the medium, the tra nsmitted field has a narrowed spectral profile, and therefore the beam is spatially enlarg ed (figures 1,2). Conservely, if the beam is centered on the right side of the gap, then the center o f the beam belongs to the gap, and the edges of the beam belong to the conduction band. T herefore, the transmitted field has two well separated peaks and the beam is splitted in t wo parts (figures 1,3). The fundamental remark here is that ultra-refractive phenomen a are due to the rapid variation of the conduction band with respect to the angle of incidence ,in complete contradiction with the habitual requested properties of photonic crystals , which are expected to have a dispersion diagram quite independent of the angle of incidence. Let us now check numerically the above explanations. We stil l use the previous Bragg Mirror. The numerical experiments are done with an s-polari zed incident field of the form: ui(x, y) =/integraldisplay A(α)exp ( iαx−iβ(α)y)dα (1) with α=ksinθ, α =ksinθ0, β(α) =√ k2−α2andk= 2π/λ,|A(α)|= W 2√πexp/parenleftBigg −(α−α0)2W2 4/parenrightBigg . In all numerical experiments W= 0.5, the variable θ0is the mean angle of incidence. In the first numerical experiment, we set λ= 2.7 and θ0= 0◦. We have plotted in figure (4a) the transmission coefficient as well as the spectral profi le of the transmitted beam. Obviously, this profile is much narrower than the incident on e. The map of the electric field is given in figure (4b). The incident field is coming from below . As expected, we observe a strong enlargement of the transmitted beam. For the second numerical experiment, we use λ= 3 and θ0= 0◦. This time, the center of the beam belong to the gap. We have plotted in figure (5a) the tr ansmission coefficient as well as the spectral profile of the incident and transmitted fi elds. It appears that there are two isolated peaks, and therefore the transmitted field is sp litted spatially into two parts, as shown in figure (5b). At that point it is easily seen that by swi tching the incident beam it is possible to keep only one transmitted beam. This is done in th e last experiment, where we setθ0= 10◦. As it can been seen on fig 6 (a), only the right part of the beam i s significantly transmitted, and thus there is only one transmitted beam (fig . 6 (b)). If Snell-Descartes law is directly applied to this situation, then it seems that the medium has an optical index that is inferior to 1. As a conclusion, we have shown both theoretically and numeri cally that ultra-refractive phenomena do happen in one-dimensional Bragg mirrors, or mo re generally in one dimen- sional photonic crystals. They may be well explained by mean s of the intersection of the support of the incident beam with the gaps and the conduction bands. It must also be 2noted that, though one dimensional photonic crystals exhib it ultra-refractive properties, bidimensional or three dimensional ones should show a bette r efficiency due their richer band diagrams. Nevertheless, doping 1-D structure or using quasi-crystals may enable a fair control over the width of the gaps and conduction bands, thus leading to the design of practical devices. Finally, it should also be noted that suc h a surprising phenomenon as a negative Goos-H¨ anchen effect does not seem to be possible in 1D structures. 3Figure captions: figure 1: Dispersion diagram of a Bragg mirror, with ε1= 1, ε2= 4, h1= 1, h2= 1. The double arrowed lines indicate the width of the Gaussian beam s. figure 2: Sketch of the behavior of the beam when spatially enl arged. figure 3: Sketch of the behavior of the beam when splitted. figure 4: (a) Transmission through the Bragg mirror vs. angle of incidence (dotted line), spectral amplitude of the incident beam (solid line) and spectral amplitude of the transmitted beam (thick line) ( λ= 2.7, θ0= 0). (b) Map of the intensity of the electric field above and below t he Bragg mirror in the case of figure 2 (above: transmitted field, below: incid ent field). figure 5: (a) same as fig. 4 (a) in the case of figure 3 ( λ= 3, θ0= 0◦). (b) Map of the intensity of the electric field above and below t he Bragg mirror in the case of figure 3 (above: transmitted field, below: incid ent field). figure 6: (a) same as fig. 4 (a) in the case of figure 3.( λ= 3, θ0= 10◦). (b) Map of the intensity of the electric field above and below t he Bragg mirror in the case of figure 3 (above: transmitted field, below: incid ent field). 4REFERENCES [1] J. P. Dowling, C. M. Bowden, J. Mod. Opt. 41, 345 (1994). [2] S. Enoch, G. Tayeb, D. Maystre, Opt. Com. 161, 171.(1999) [3] H. Kosaka & al., Phys. Rev. B 58, 10096 (1998). [4] B. Gralak, G. Tayeb, S. Enoch, in preparation. [5] J. Lekner, J. Opt. Soc. Am. A. 11, 2892 (1994). [6] D. Felbacq, B. Guizal, F. Zolla, Opt. Com. 152, 119 (1998) 5Enlargment of the beam Splitting of the beam Figure 1Angle of incidenceAngle of incidence1 Transmitted energyIncident beam Transmitted beam Figure 2Angle of incidenceAngle of incidence1Transmitted energy Incident beam Transmitted beam Figure 3-80-60-40-20 02040608000.10.20.30.40.50.60.70.80.91 (a) (b) Figure 4(a) (b) Figure 5-80-60-40-20 02040608000.10.20.30.40.50.60.70.80.91 angle of incidence(a) (b) Figure 6-80-60-40-20 02040608000.10.20.30.40.50.60.70.80.91 angle of incidence

arXiv:physics/0003019v1 [physics.space-ph] 8 Mar 2000Auroral field-aligned currents by incoherent scatter plasma line observations in the E region Ingemar H¨ aggstr¨ om1,Mikael Hedin2,Takehiko Aso1, Asta Pellinen-Wannberg2and Assar Westman2 1National Institute of Polar Research, 1-9-10 Kaga, Itabash i-ku, Tokyo 173-8515, Japan. 2Swedish Institute of Space Physics, Box 812, S-981 28, Kirun a, Sweden. Abstract The aim of the Swedish-Japanese EISCAT campaign in February 1999 was to mea- sure the ionospheric parameters inside and outside the auro ral arcs. The ion line radar experiment was optimised to probe the E-region and low er F-region with as high a speed as possible. Two extra channels were used for the plasma line measure- ments covering the same altitudes, giving a total of 3 upshif ted and 3 downshifted frequency bands of 25 kHz each. For most of the time the shifte d channels were tuned to 3 (both), 4 (up), 5.5 (down) and 6.5 (both) MHz. Weak signals are seen whenever the radar is probing the diffus e aurora, corre- sponding to the relatively low plasma frequencies. At times when auroral arcs pass the radar beam, significant increases in return power are obs erved. Many cases with simultaneously up and down shifted plasma lines are rec orded. In spite of the rather active environment, the highly optimised measureme nts enable investigation of the properties of the plasma lines. A modified theoretical incoherent scatter spectrum is used t o explain the mea- surements. The general trend is an upgoing field-aligned cur rent in the diffuse au- rora, confirmed with a full fit of the combined ion and plasma li ne spectra. There are also cases with strong suprathermal currents indicated by large differences in signal strength between up- and downshifted plasma lines. 1 Introduction The incoherent scatter spectrum consists mainly of two line s, the widely used and relatively strong ion line and the very weak easily forgotte n broadband electron line. There is also another line present, the plasma line, du e to scattering from high frequency electron waves, namely Langmuir waves. From the downgoing and upgoing Langmuir waves, two plasma lines can be detected by the radar. The frequency shift from the transmitted signal is the frequ ency of the scattered Langmuir wave plus the Doppler shift caused by electron drif t. Plasma lines can be used to measure the electron drift and hence the line-of-s ight electric current. The problem in ion line analysis with the uncertainty of the r adar constant can be solved by the plasma line frequency determination and when t hat is done the speed of measurement can be significantly increased by including t he plasma line in the ion line analysis. However, since the frequency of the plasm a lines is not known beforehand, and the frequency is varying with height, it is d ifficult to measure them with enough resolution. There have been a number of reports on plasma line measuremen ts and their interpretation. Most of them have discussed the frequency s hift from the transmit- ted pulse and the scattering has mainly been from the F-regio n peak, e.g. Showen (1979), Kofman et al. (1993) and Nilsson et al. (1996a). The l atter two showed also that the simple formula for the Langmuir wave frequency , f2=f2 p(1 + 3 k2λ2 D) +f2 csin2α (1) where fpis the plasma frequency, kthe wave number fcthe electron gyro frequency, λDthe electron Debye length and αthe angle between the scattering wave and2 the magnetic field, is valid to within a few kHz and thus enough to set the radar system constant. To be able to deduce any electron drifts, or current, out of the positions of the lines these authors also show that Eq. (1) is not sufficient, and it is necessary to carry out more accurate calculations. Hagfo rs and Lehtinen (1981) had also to go to further expansions in deriving the ambient e lectron temperature from the plasma lines. The fact that so many reports deal with the F region peak is due to that the altitude profile of the plasma line frequenc y according to Eq. (1) will also show a peak around that height. The measurements ca n thus be made relatively easily using rather coarse height resolution bu t good frequency resolution and only detect the peak frequency. Measurements using the s ame strategy, but at other heights, have been made with a chirped radar by match ing the plasma line frequency gradient and the transmitter frequency grad ient (Birkmayer and Hagfors, 1986; Isham and Hagfors, 1993). This technique all ows determination of the Langmuir frequency with very high frequency resoluti on, but do not use the radar optimally, since the chirped pulse cannot be used f or anything else than plasma line measurements. The enhancement of the plasma lines, which occurs in the pres ence of suprather- mal electron fluxes (Perkins and Salpeter, 1965), either pho toelectrons or secon- daries from auroral electrons, has been investigated by Nil sson et al. (1996b), where they also calculate predictions of plasma line streng th for different incoher- ent scatter radars and altitudes. They also show that the pow er of the plasma line is rather structured with respect to ambient electron densi ty, depending on fine structure in the suprathermal distributions due to excitat ions of different atmo- spheric constituents. Incoherent scatter plasma lines in auroral are more difficult to measure since the variations in the plasma parameters are high with large time and spatial gradients. Reports of auroral plasma lines in the aurora are also more ra re, and most of them are based on too coarse time resolution (Wickwar, 1978; Kofm an and Wickwar, 1980; Oran et al., 1981; Valladares et al., 1988), with resol utions ranging from 30 seconds up to 20 minutes. The enhancements over the thermal l evel were high, but consistent with what could be expected of model calculat ions of suprathermal electron flux. They also tried to calculate currents and elec tron temperature from the frequency shifts of the plasma lines but with very large e rror bars. Kirkwood et al. (1995) used the EISCAT radar and the filter bank techniq ue and recorded much higher intensities of the plasma lines, since they got d own to resolutions of 10 seconds, and showed also that the plasma-turbulence model p roposed by Mishin and Schlegel (1994) was not consistent with the data, but cou ld be explained by reasonable fluxes of suprathermal electrons. In this paper we present data obtained with the high resoluti on alternating code technique (Lehtinen and H¨ aggstr¨ om, 1987), as was also don e for F-region plasma lines by Guio et al. (1996), with even higher intensities due to the time resolutions of 5 seconds. An interesting, but at the time of the experimen t not realisable at EISCAT, technique would have been the type of coded long pu lses used by Sulzer and Fejer (1994) for HF-induced plasma lines. From re lative strengths between up- and downshifted lines we detect a general trend o f upgoing field- aligned currents in the diffuse aurora carried by the suprath ermal electrons. We propose a generalisation of the theoretical incoherent sca tter spectrum, to include multiple electron distributions, and in one example we do a f ull 7-parameter fit of the incoherent scatter spectrum, including the enhanced plasma lines assuming shifted Maxwellian secondary electrons, resulting in the fi rst radar measurement of its flux and an upward current carried by the thermal electr ons. 2 Experiment The measurements we present were collected by the 930 MHz EIS CAT UHF in- coherent scatter radar, with its transmitter located at Ram fjordmoen in Norway3 (69.6◦N, 19.2◦E, L=6.2). The signals scattered from the ionosphere were re ceived at stations in Kiruna, Sweden and Sodankyl¨ a, Finland as wel l as at the transmit- ting site. General descriptions of the radar facility are gi ven by Folkestad et al. (1983) and Baron (1984). Local magnetic midnight at Ramfjor dmoen is at about 2130 UT. In the Swedish-Japanese EISCAT campaign in Februar y 1999, the aim was to measure the ionospheric parameters inside and outsid e the auroral arcs. For this a 3 channel ion line alternating code (Lehtinen and H ¨ aggstr¨ om, 1987) experiment, optimised to probe the E-region and lower F-reg ion with as high a speed as possible, was developed. The 16 bit strong conditio n alternating code with bitlengths of 22 µs was used, giving 3 km range resolution, and with a sample rat e of 11 ms the range separations in consecutive spectra were 1. 65 km. Fig. 1 shows the transmission/reception scheme of the first 20 ms of the ra dar cycle. The whole alternating code sequence takes about 0.3 s to complete. Dur ing this period the incoherent scatter autocorrelation functions (ACF) at the probed heights should not change significantly for the alternating codes to work. I n order to keep this as short as possible, the short pulses, normally used for zer olag estimation, were dropped and instead a pseudo zero lag, obtained from decodin g the power profiles of the different codes in the alternating code sequence, was u sed. Fig. 2 shows the range-lag ambiguity function for this lag, together wit h the one for the lag centred at 22 µs. The range extents are rather similar but the power is, of co urse, considerably lower for the pseudo zero lag. Nevertheless, t his is taken care of in the analysis and this lag is rather important in events with h igh temperatures giving broad ion line spectra or narrow ACFs. The transmitti ng frequencies were chosen to give maximum radiated power for a given high voltag e setting. 0 1 2 3 4 5 6 7 8 9 10111213141516171819203 4 5 Time (ms)Channel numberSP−SW−ALIS Fig. 1. Transmission (black) and reception (dark gray) scheme for p art of SP-SW-ALIS. A 16-bit alternating code, 22 µs bits, is cycled over 3 frequencies. The interscan period is 9 ms and the total cycle takes 300 ms. The plasma line channels were se t to sample the same range extent as the ion lines. The monostatic plasma line part of the experiment used two ch annels covering the same ranges as the ion line but with 3.3 km range separatio ns between the spectra. The frequency setup for the experiment, illustrat ed in Fig. 3, gives the4 Lag (µs)Range (µs)Plasma line pseudo zero lag −20 020−20−1001020 Lag (µs)Plasma line lag 1 0204060−20−1001020 00.511.5Range (µs)Ion line pseudo zero lag −20 020−20−1001020Ion line lag 1 0204060−20−1001020 00.511.5 Fig. 2. Range-lag ambiguity function for the first two lags, a) ion li ne and b) plasma line. The range, given in µs, can be converted to km with multiplication of 0.15. The diff erences between the ion and plasma line ambiguity functions are due to the use of different receiver filters. 9269279289299309319329339349353 MHz 4 MHz 6.5 MHzChannel setup for SP−SW−ALIS 990214 Frequency (MHz)6.5 MHz 5.5 MHz 3 MHzIon line Plasma line Fig. 3. The frequency setup for the experiment. The plasma line chan nels were fixed to 926 and 934.5 MHz, giving ”simultaneously” up and downshifted f requencies at 3 and 6.5 MHz. In addition there is an upshifted line at 4 and a downshifted at 5 .5 MHz.5 possibility for 3 upshifted and 3 downshifted bands. For 3 MH z and 6.5 MHz frequency shifts, both the up- and downshifted plasma lines were measured. In addition there was a 4 MHz upshifted and a 5.5 MHz downshifted band. The width of these bands should have been set to match the bit leng th of the codes used, 50 kHz, but unfortunately this was not the case and 25 kH z wide filters were erroneously used. This gave naturally less signal thro ughput, and in Fig. 2 the corresponding range/lag ambiguity functions for the pl asma line channels are included for the first two lags. The decoding still works, giv ing just slightly in- creased unwanted ambiguities, but above all the pseudo zero lag is moved out to a larger lag value. This fact, with one exception, almost rul ed out the possibilities to measure plasma line spectra because, as will be shown late r in this paper, they are likely to be rather wide, due to large time and height grad ients of the plasma parameters. This lead to use mainly the undecoded zerolag, w hich after integra- tion over the different codes, almost resembles the shape of a 352µs (16 ×22µs) long pulse. The experiment contained a large number of antenna pointing s in order to follow the auroral arcs, but as this day was cloudy over northern Sca ndinavia the trans- mitting antenna was kept fixed along the local geomagnetic fie ld line. The remote sites, receiving only ion lines, were monitoring the same pu lses as the transmitting site, and were used to measure the drifts in the F-region, to d erive the electric field. Thus, these antennas intersected the transmitted beam at th e F-region altitude giving the best signal, for this day mostly at 170 km. 3 Measurements 3.1 Ion line The experiment started at 1900 UT on 14 February 1999 and cont inued until 2300 UT. Fig. 4 shows an overview of the parameters deduced fr om the ion line measurements, which were analysed using the on-line integr ation time, 5 seconds, in order to be compared to the plasma line data. The analysis w as done using the GUISDAP package (Lehtinen and Huuskonen, 1996), but a co rrection of 45% of the radar system constant in package had to be invoked to fit the plasma line measurements according to Eq. (1). This short integration t ime was possible due to the highly optimised mode used, with all the transmitter p ower concentrated to the E-region. In range, some integration was done, so that at lower heights 2 range gates were added together and with increasing height t he number of gates added together increases to 15 in the F-region. There is a rather strong E-region from the start, but no real a rcs, and we interpret this as diffuse aurora. The peak electron density s hows some variation, but as time goes the E-region ionisation decreases until 205 0 UT, where it is almost gone. The density peak during this time was at around 120 km al titude and the lower edge of the E-region at 110 km, but at times the ionisati on reaches down to 100 km. At 2050 UT and onwards until 2240 UT the ionosphere abo ve Tromsø became more active and several auroral arcs passed the beam. Around some of the arcs there were short-lived enhancements of electric field, seen as F-region ion and E-region electron temperature increases. In the last 10 min utes of the experiment the arc activity disappeared and again there were diffuse aur ora. From the field aligned ion drifts it is evident there is a rather strong wave activity in the diffuse aurora until 2050 UT, while it is not so clear in the continuat ion of the experiment. The last panel with the inferred electron density from the ps eudo zero lag shows the same features as the fitted density panel but with highest possible resolution since no height integration is made.6 Fig. 4. Summary of results from the ion line experiment. All panels s hows the parameters in a altitude versus time fashion with the antenna directed a long the geomagnetic field line. Panels from top: Electron density, electron temperature, i on temperature, line-of-sight ion drift velocity and electron density based on only the returned pow er. The data were analysed with 5 s integration due to the active conditions.7 3.2 Plasma line Since the analysis of the plasma lines was forced to handle th e undecoded zerolags of the alternating codes, it was necessary to analyse their p rofile shape. Fig. 5 shows how this analysis was performed. From the fitted parame ters of the ion line, electron density and temperature, a profile of the appr oximate Langmuir frequency can be calculated using Eq. (1). When probing at fix ed frequency, there will be scattered signal only from heights where the pr obing and Langmuir frequencies match each other. The effect of the undecoded zer olag is similar to the one where a normal long pulse is used, but with a lower signal s trength. So, the profile shape will be a square pulse centred on the correspond ing altitude, since no gating is performed. Because the signal strengths are rat her weak compared to the system temperature, there is a great deal of noise in the p rofile shape. To be able to extract the altitude and signal power, a fitting proce dure was performed and the measured plasma line profile was convolved with the pu lse shape to have a triangular shape, in order to get a continuous function wit h a peak. 2 3 4 5100150200250300 fp(1+3k2λD2)1/2 Langmuir and probing frequency (MHz) Altitude (km)a) 0 0.5 1100150200250300 Scattered signal from short pulseb) 0 0.5 1100150200250300 Scattered signal from long pulseAltitude (km)c) 0 0.5 1100150200250300 Scattered signal convolved with puls shaped) Fig. 5. Description of the analysis procedure. a) Langmuir frequen cy profile together with the probing frequency of 3 MHz shift. b) The echo profile using a sh ort pulse. c) The echo profile using a long pulse. d) The echo profile convolved with the puls e shape. The plasma line part of the experiment is overviewed in Fig. 6 . The signal strengths shown should be compared with the UHF system tempe rature of about 90 K. At first glance there is almost nothing in the upshifted p art, but a more careful look shows weak signals between 1940 and 2030 UT and a fter 2250 UT, corresponding to the occurrence of diffuse aurora. Similar e choes can also be seen in the downshifted channel, and are due to plasma lines at 3 MH z offset from the transmitted frequencies. In the downshifted part after 210 0 UT, frequent events of rather strong signals in phase with auroral arcs pass the bea m. Most of these are from the 5.5 MHz shift, but some of them also are due to plasma l ines at 6.5 MHz. Such events are less frequent and for most of them there are al so signals in the upshifted part. Due to the fact that the same channel is used for several frequ encies, there can be several altitudes that fulfill the matching condition between Langmuir and probing frequency. This complicates the analysis somewhat , and there have to be8 Fig. 6. Overview of the plasma line measurements. The panels show th e undecoded upshifted (top panel) and downshifted returns, given as signal streng ths in Kelvin. several fits with different numbers of triangles superposed o n each other. Some examples of this analysis are shown in Fig. 7, from only one pl asma line to several both up- and downshifted lines. Using a lower limit of 2 K sign al power, the total number of 5 second events with enhanced plasma lines for this evening was 256, and a total of 468 plasma line echoes were detected, divided i nto 220 on 3 MHz, 19 on 4 MHz, 157 on 5.5 MHz and 72 on 6.5 MHz. 4 Theory In order to relate the plasma line measurements to physical q uantities it is neces- sary to investigate the spectrum of the incoherent scatter p rocess. The Nyquist theorem approach, derived in a long series of papers by Dough erty and Farley (1960, 1963), Farley et al. (1961), Farley (1966) and finally Swartz and Farley (1979), arrives at σ(ω) =Ner2 esinδ π·|ye|2/summationtext iηiℜ(yi) ω−k·vi+/vextendsingle/vextendsinglejk2λ2 D+/summationtext iµiyi/vextendsingle/vextendsingle2ℜ(ye) ω−k·ve |ye+jk2λ2 D+/summationtext iµiyi|2, (2) where ηi=niq2 i Nee2, (3) µi=ηiTe Ti, (4) and the index estands for electrons and ifor the different ion species. Nandn are the densities, rethe classical electron radius, vthe bulk velocity, qthe charge, ethe electron charge and Tthe temperature. The complex normalised admittance function, y, contains most of the physics with the plasma dispersion fun ction and9 02461001502002501953:10 1 1Altitude (km)a) 02461001502002502019:20 1 0b) 01020301001502002502211:25 0 1Altitude (km) Signal strength (K) and Frequency (MHz)c) 0 10 201001502002502238:15 1 2d) Fig. 7. Examples of profiles and the fitted triangle(s) to the data. In cluded on the plots are the profiles of the Langmuir frequency (black) calculated from t he parameters of ion lines fit. The value on the x-axis represent signal strength (K) and freque ncy (MHz) respectively. The numbers in the title shows the time of measurement and how many echoes was detected for the up- (red) and downshifted (green) frequencies. A lower limit of 2 K rec eived signal power (2% SNR) was used. a) Simultaneous up and downshifted echoes at 3 MHz in di ffuse aurora. b) An upshifted 3 MHz echo in diffuse aurora. c) A strong downshifted echo at 5. 5 MHz in an auroral arc. d) Simultaneous up and downshifted echoes at 6.5 MHz and a downs hifted 5.5 in an auroral arc.10 have as main arguments the collision frequency and magnetic field. The same result was also reached by Rosenbluth and Rostoker (1962), u sing the dressed particle approach. In Fig. 8 there is an example of the spectr um, showing clearly the strong ion line around zero offset frequency and the rathe r weak plasma lines at rather large offsets. The plasma lines become enhanced by a photo electron or auroral electron produced suprathermal electron distri bution, and in order to simulate what this extra distribution does to the spectrum a modification to the formula has to be made. First, a rewriting of Eq. (2) followin g Swartz (1978) has to be performed in order to separate the electron and ion cont ributions to the spectrum: S(f) =1 π·/vextendsingle/vextendsingle/vextendsingleNeye Te/vextendsingle/vextendsingle/vextendsingle2/summationtext iniZ2 iℜ(yi) f+kvi/2π+/vextendsingle/vextendsingle/vextendsinglejCD+/summationtext iniZ2 iyi Ti/vextendsingle/vextendsingle/vextendsingle2Neℜ(ye) f+kve/2π/vextendsingle/vextendsingle/vextendsingleNeye Te+jCD+/summationtext iniZ2 iyi Ti/vextendsingle/vextendsingle/vextendsingle2, (5) where CD=k2ǫK e2, (6) fis the frequency shift, Z=q/e,ǫis the dielectricity and Kis the Boltzmann constant. Here, a normalisation of the spectrum has also bee n performed, so that the zero lag of the corresponding ACF is the raw electron dens ity and the vector velocity is replaced by the line-of-sight velocity. Using a treatment in analogy to the ion contribution, it is now possible to rewrite the spe ctrum to support a number of Maxwellian electron distributions as S(f) =1 π·/vextendsingle/vextendsingle/vextendsingle/summationtext eNeye Te/vextendsingle/vextendsingle/vextendsingle2/summationtext iniZ2 iℜ(yi) f+kvi/2π+/vextendsingle/vextendsingle/vextendsinglejCD+/summationtext iniZ2 iyi Ti/vextendsingle/vextendsingle/vextendsingle2/summationtext eNeℜ(ye) f+kve/2π/vextendsingle/vextendsingle/vextendsingle/summationtext eNeye Te+jCD+/summationtext iniZ2 iyi Ti/vextendsingle/vextendsingle/vextendsingle2.(7) In Fig. 9 the effect on the plasma lines of a suprathermal distr ibution with a reasonable density of 107m−3and width of 10 eV is shown, being lower than the ionisation energy for most ions. The plasma lines grow co nsiderable and the integrated power over the bandwidth used in the experiment b ecomes comparable to the power of the ion line. It may also be noted that there is n o effect at all seen in the ion line. Perkins and Salpeter (1965) have shown similar calculations, but when their method was based on large expansions to allow non-Maxw ellian distributions one can here more directly superpose a few Maxwellian distri butions to explain the measurements and even make fits of the spectra taken to get estimates of the suprathermal distributions, which is of great importance i n auroral measurements. A consequence of Eq. (7) is that it is also possible to derive t he spectrum assuming currents carried by the suprathermal distribution, since i t allows different drift velocities on the various distributions. Indeed, Fig. 10, s hows differential strengths on the two plasma lines, with upgoing electrons enhancing ma inly the downshifted line and downgoing ones the upshifted line. 5 Discussion Plasma line measurements in the active auroral ionosphere a re not an easy task, due to the large variations in the ionospheric parameters. T he Langmuir frequen- cies are largely dependent on the ambient electron density, making the line move considerably as the density changes, which it does on time sc ales of seconds. More- over, the density height gradient makes the lines very broad when measured over a specific height interval, and at times even broader than the receiver band. A chirped radar would solve only a part of the problem at the cos t of transmitter power. These complications make it hard to draw any conclusi ons on the power in11 −10−8−6−4−2 024681010−2100102104106108 Frequency (MHz)m−3sIncoherent scatter spectrum at 140 km Fig. 8. The wide frequency incoherent scatter spectrum. The parame ters used: Ne= 4 · 1011m−3,Te= 700 K, Ti= 600 K, vi=ve= 0 ms−1and collision frequencies νeandνiusing the MSIS90e model (Hedin, 1991) for 140 km altitude. Logarit hmic scale for the strength is used to be able to emphasise the different lines of the spectrum. the lines, as one does not know for sure the scattering volume or the time duration of the scattering. Bearing this in mind and to at least minimi se these effects, one can nevertheless look at the different distributions in heig ht and power for the different lines to get an idea of their nature, using the on-li ne integration time of 5 seconds. The altitude distributions for the different frequencies ar e shown in Fig. 11. One must note that the signal levels shown are not corrected f or range, as the scattering volumes are not known, and signals from a higher a ltitude are in fact stronger than the corresponding signal from a lower height. It is clearly seen that the strongest signal is the 5.5 MHz line, followed by 6.5, 4 an d 3 MHz. The altitude distribution shows more or less the expected depen dence on range, but there are some exceptions: In one point at 188 km in the 3 MHz ba nd and for the 5.5 MHz band the strong values between 130 and 140 km seem t o be stronger than the others, even taking into account the range effect. Ho wever, the number of points are too few to be used as evidence on altitude effects . Most of the echoes are coming from around 120-150 km altitude and Fig. 12 shows a simulation of the expected strength of the plasma line for given backgroun d electron density. It shows a peak at around 5.5 MHz and this is also what the exper iment shows. A more realistic suprathermal distribution will decrease t he returned power for a number of frequencies and one should see the figure as an upper -limit estimate. Indeed, Nilsson et al. (1996b), have made predictions of the expected strength of Langmuir waves for different heights and carefully derived d istributions. These predictions are in rather good agreement with the present me asurements showing a strong peak between 5 and 6.5 MHz. The most interesting thing with incoherent plasma lines is, of course, the possi- bility to derive differential drifts between ions and electr ons, and from these deduce ionospheric currents. For this, one needs to measure the up- and downshifted lines12 10−210−1100101100105m−4s eVElectron distribution −15 −10 −5 0 5 10 1500.511.52x 107 kHzm−3sIon line 5780 5785 5790 5795 5800 5805 58100123456x 107 kHzm−3sPlasma line up down Fig. 9. Close-up of the ion and the plasma lines. Top panel shows the e lectron distribution, middle panel the ion line and bottom panel the plasma lines, w here the frequency scale of the downshifted line have been reversed. The figure shows the lines with two different electron distributions, thick line for the normal thermal distribut ion and thin line for the same distribution together with an suprathermal distribution.13 5780 5785 5790 5795 5800 5805 581000.511.522.533.54x 107 kHzm−3sPlasma line up down Fig. 10. The plasma lines based on an upgoing (thick line) and downgoi ng 10 eV beam of suprathermal electrons at a current of 1 µAm-2 and temperature of 5 eV. 0 50100110120130140150160170180190200 Power (K)Altitude (km)3 MHz 0 50100110120130140150160170180190200 Power (K)5.5 MHz 0 50100110120130140150160170180190200 Power (K)6.5 MHz 0 50100110120130140150160170180190200 Power (K)4 MHz Fig. 11. Scattered plots of echo altitude and strength for the differe nt frequency shifts. The altitudes range from 100 to 210 km, E and lower F region.14 200030004000500060007000800090001000002468x 107 m−3sPlasma line 20003000400050006000700080009000100000246x 1010 Frequeny (kHz)m−3Plasma line power Fig. 12. Simulation of the plasma line strength at 140 km altitude wit h an added distribution of secondaries for different electron densities from 0.5 to 10 ·1011m−3. The envelope in the lowest panel shows the total power of the plasma lines versus the Lan gmuir frequency. simultaneously. Although the time of the measurement for bo th lines were not exactly the same in this experiment, the time shifts between them are so small (3-6 ms) compared to the total cycle time (300 ms) of the codes , that this effect is of minor importance. In Fig. 13, the strengths and altitud es of the two concur- rently recorded up- and downshifted plasma lines are shown. In general, there are stronger up- than downshifted lines for the 3 MHz case, where as no such trend can be seen in the 6.5 MHz band. However, there are exceptions to these overall trends and on occasions there are large differences in the sig nal strengths between the lines. Almost all of the 3 MHz plasma lines were recorded i n diffuse aurora and this evident difference in signal power needs a closer examin ation. When there is a drift of the thermal electrons, the plasma lines shift, and w hen probing at a fixed frequency, the scatter may not come from the same altitude fo r the up- and down- shifted lines respectively. The strength of the plasma line is also rather altitude dependent due to the damping by the collisions of electrons w ith ions and neutrals. But Fig. 13 shows no general height difference between the up- and downshifted 3 MHz lines, so this difference in strength cannot be explaine d by thermal electron bulk drifts. To simulate the effect of current carried by supr athermal electrons, Fig. 14 illustrates the strength of the 3 MHz plasma lines for Maxwellian electron beams of different energies. With no net current the lines are of almost the same strength, and the difference is mainly dependent on where in t he receiver band the lines are. However the experiment shows stronger upshif ted lines, thus it is evident that diffuse aurora this night contained fluxes of dow ngoing suprathermal electrons or, in other words, there was an upgoing current ca rried by suprathermal electrons. For the 6.5 MHz bands there may be a slight differen ce with respect to the altitude, and that is most likely due to thermal curren ts causing frequency shifts of the plasma lines. This effect is not very clear, but a s lower heights have higher density, or Langmuir frequency, and as the downshift ed line is at a slightly15 lower height, this is most probably an effect of a downgoing cu rrent carried by thermal electrons. The correction due to heat-flow in the pla sma dispersion func- tion, discussed by Kofman et al. (1993) and confirmed later al so by Nilsson et al. (1996a) and Guio et al. (1996), but not taken into account her e, would also show the same effect in altitude difference between the lines. The b and widths used here, 25 kHz, are much wider than the effect of heat-flow, which is less than or around 1 kHz in the F-region and much lower in the E-region, so that cannot explain the 6.5 MHz height differences. A recent paper by Guio et al. (1998) with proper calculation of the dispersion equation investigate s the heat-flow and finds that it is not necessary to invoke the effect at all. 051015051015 Downshifted (K)Upshifted (K)Power of 3 MHz plasmalines 050100150200050100150200 Downshifted (km)Upshifted (km)Altitude of 3 MHz plasmalines 051015051015 Downshifted (K)Upshifted (K)Power of 6.5 MHz plasmalines 050100150200050100150200 Downshifted (km)Upshifted (km)Altitude of 6.5 MHz plasmalines Fig. 13. Scatter plots of power and altitude for the two simultaneous ly measured up and downshifted lines at 3 and 6.5 MHz. The triangles and circles for the power of the 3 MHz case, in the upper left, show lines detected before and after magne tic midnight respectively — no difference can be seen. All of the above discussions on currents are only on directio ns of currents, but to get any quantitative numbers one should look at the spectr a themselves. Of the 256 events of enhanced plasma lines, there is only one tha t is good enough to investigate. All the other are too broad either due to the L angmuir frequency gradient smearing or the fact that the Langmuir frequency is not constant during the 5 second time slots. Another reason is that the stationar ity condition for the alternating code technique is not fulfilled, due to changes o f Langmuir frequency, and the spectra change too much within the 300 ms cycle. Anywa y, there is one, and the up- and downshifted bands, together with the ion line band, are shown in Fig. 15. The shifts in this case are around 3 MHz as deduced f rom the ion line analysis. These spectra were then fitted to the theoretical s pectrum in Eq. (7) to get the ionospheric parameters. At the first glance the fit on t he ion line seems rather poor, but the fit was done as usual in the time domain wit h proper weighting on the different lags of the ACF, and in the FFT process to produ ce Fig. 15 these16 0 5 10 15 2000.10.20.30.4 Mean energy (eV)Signal ratio plasma/ion lineUpgoing suprathermals Upshifted Downshifted 0 5 10 15 2000.10.20.30.4 Mean energy (eV)Signal ratio plasma/ion lineDowngoing suprathermals Upshifted Downshifted Fig. 14. Simulation of 3 MHz plasma line strength at 140 km with an elec tron beam with varying mean energy from 0 to 20 eV and temperature of 10 eV, co rresponding to a few mAm−2 field-aligned currents, depending on energy. The top panel s hows the ratio between plasma and ion line strengths for upgoing beams and the bottom for downg oing beams. statistical properties are lost. The fit looks actually bett er in the time domain, but the frequency domain was chosen in the figure to be more inf ormative. The current density j=Nee(vi−ve), and with the fitted parameters, the field aligned current carried by the thermal electrons, amount to 12 µAm−2upward. This is consistent with the general current seen in the plasma line s trength for the 3 MHz band. Again, heat-flow is not taken into account, but that sho uld increase the current somewhat. The deduced suprathermal distribution i s in good agreement with those Kirkwood et al. (1995) derived from the precipita ting flux of primaries; of course in the present case the distribution is Maxwellian and without the fine structure due to the atmospheric constituents, but the numb ers are comparable. 6 Conclusions We have made measurements of plasma lines in the active auror al E-region. During 256 periods of 5 second integration we found a total of 468 pla sma line echoes, divided into 220 on 3 MHz, 19 on 4 MHz, 157 on 5.5 MHz and 72 on 6.5 MHz. It may seem strange to try to measure plasma lines at such a low fr equency as 3 MHz giving low signal levels, but in fact most of the echoes and th e most interesting results came from this frequency offset. The strongest echoes were found at the 5.5 MHz line, and somew hat weaker ones at 6.5 MHz, inside auroral arcs. One must, however, note that the strength measured inside the arcs is mostly a low-limit estimate due t o the active environ- ment. The integration period used, 5 s, is rather long in auro ral arc conditions, and changes typically occur on shorter time scales. Therefo re, effects of gradients in the Langmuir frequency profile, and hence scattering volu mes, have not been taken into account. The Holy Grail in incoherent scatter plasma lines is the poss ibility to measure17 −3050 −3000 −2950024Power (K/kHz) Frequency (kHz)990214 1909:05−1909:10 at 152−154 km altitude −50 0 50024Power (K/kHz) Frequency (kHz) 2950 3000 3050024Power (K/kHz) Frequency (kHz) Fig. 15. Measured spectra of the both plasma lines and the ion line and the best 7-parameter fit of the theoretical spectrum. The parameters fitted were Ne= 9.7·1010m−3,Te= 631 K, Ti= 697 K, vi=−100 ms−1,ve= 657 ms−1and a suprathermal distribution with ne= 0.8·109m−3 andT= 11 eV.18 currents, and in this case, the field-aligned currents in aur ora. The simulations carried out here, the extended full IS spectrum with multi-M axwellian distributions of electrons, show that the strength of the lines is determin ed by the suprathermal part of the electron distribution, and the frequency mainly by the thermal part. For simultaneous up- and downshifted plasma line difference s in intensity we can deduce currents carried by suprathermals, and for differenc es in frequency, currents carried by thermals. The simultaneous up- and downshifted frequencies of the 3 MH z line in the diffuse aurora show, on average, an upward field-aligned supr athermal current during the two main periods when they were detected, 1940-20 30 UT and 2250- 2300 UT. In the arcs in general, there is an indication of down ward thermal current as seen from the altitudes of the 6.5 MHz echoes. Of course, no rule is without exceptions, and there are cases where one line is much strong er than the other or the other line is not at all enhanced, indicating strong curr ents. In the full 7-parameter fit of the incoherent scatter spectru m with the ion line and the both enhanced plasma lines, we obtained a thermal cur rent consistent with the general suprathermal current for diffuse aurora and a rea sonable suprathermal distribution of electrons. Acknowledgements. One of the authors (I.H.) was working under a contract from NI PR and is grateful to the Director-General of NIPR for the support. We are indebted to the Director and staff of EISCAT for operating the facility and supplying t he data. EISCAT is an Interna- tional Association supported by Finland (SA), France (CNRS ), the Federal Republic of Germany (MPG), Japan (NIPR), Norway (NFR), Sweden (NFR) and the Unit ed Kingdom (PPARC). References Baron, M., The EISCAT facility, J. Atmos. Terr. Phys. ,46, 469, 1984. Birkmayer, W. and Hagfors, T., Observational technique and parameter estimation in plasma line spectrum observations of the ionosphere by ch irped incoherent scatter radar, J. Atmos. Terr. 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arXiv:physics/0003020v1 [physics.acc-ph] 8 Mar 2000Asynchronous accelerator with RFQ injection for active longitudinal compression of accelerated bunches A.R. Tumanyan(∗), Yu.L. Martirosyan(∗), V.C. Nikhogosyan(∗), N.Z. Akopov(∗), Z.G. Guiragossian(∗∗), R.M Martirosov(∗), Z.N. Akopov(∗) (∗) Yerevan Physics Institute (YerPhI), Yerevan, Br. Alikani an St.2, 375036, Republic of Armenia (∗∗) Guest Scientist at YerPhI Abstract An asynchronous accelerator is described, in which the prin ciple of its operation permits the active longitudinal bunch comp ression of accelerated proton beams, to overcome the space charge limi tation ef- fects of intense bunches. It is shown that accelerated bunch es from an RFQ linac can be adapted for Asynchronac injection for a mu l- tiple of choices in the acceleration frequencies of the RFQ a nd the Asynchronac. The offered new type of accelerator system is es pecially suitable to accelerate proton beams for up to 100 MeV energy and hundreds of mAaverage current. 1 Introduction According to its principle of operation, the asynchronous a ccelerator (Asyn- chronac) can be viewed to be a machine in between the followin g two cases. The Asynchronac can be viewed as a linear accelerator wrappe d into a spiral, in which the harmonic number of the acceleration voltage and the equilibrium phase of a bunch relative to the acceleration field can be chan ged indepen- dently. Also, it can be viewed as a separate orbit cyclotron ( SOC) [1][2][3] and an asynchronous cyclotron, which has been described ear lier [4][5]. 1It is simpler to present the concept of the Asynchronac as a mo dification of the parameters and the operating mode of a separate orbit c yclotron, without changing its structure (see Figure 1). If R, the radius and f, the acceleration voltage frequency in a SOC are selected to be la rge, such that q, the harmonic number between acceleration cavities is a lar ge integer at injection: q=h Nc=2πRf rf NcV(1) (where, Ncis the number of resonators in a cyclotron stage and Vis the speed of particles), qwould decrease during acceleration as the speed of particles increased, presenting the following possibilit ies. First, it is possible to restrict the increase of radii from i njection to ejec- tion orbits, by not limiting the increase of the average radi us of the machine. Second, it becomes possible to reduce the strength of the ben ding magnetic fields without restrictions. Third, it is possible to increa se the length of drift spaces between bending magnets. And fourth, it become s possible to independently set the RF equilibrium phase during the accel eration process. Thus, the creation of non-isochronous or asynchronous mode of SOC op- eration is provided by having the inter-cavity harmonic num ber,qdiscretely change on integer values as the beam transits through the sec tors and turns. By modifying the magnetic path length in the sectors the hopp ing of integer qvalues is accomplished. These are the modifications of the pa rameters and the operating mode of a separate orbit cyclotron, which conv ert it to become an Asynchronac, with additional new useful qualities. One s uch important quality is now the ability to axially compress and thus to mon ochromatize the accelerated proton or ion beam bunches. Only at low energies rapid changes occur in the speed of parti cles. Be- cause it is now possible to hop over integer values of qin a reasonably sized accelerator radius, the Asynchronac concept can best be app lied at energies below 100 MeV, to produce average beam currents in the hundreds of mA. Such accelerators, in addition to their stand-alone use for specific applica- tions, can be more useful as the initial injector stage for th e production of intense bunches, which are injected into higher energy prot on accelerators, such as synchrotrons, linacs or cyclotrons, to produce high and super-high energy intense beams. 2In the present study a valuable implementation of the Asynch ronac con- cept is based on the multiple use of modern RFQ linacs (genera ting large currents at small energy spread) [6]. 2 Conditions to creat bunch compression in the asynchronac The creation of bunch compression in an Asynchronac is made p ossible by the inherently available independent setting of the RF acceler ation equilibrium phase, from one resonator to another. In this case, the conti nuous differential equation for synchrotron oscillations does not apply. Cons equently, the value of the equilibrium phase in each resonator is determined by t he expression: cosϕe=2∆Es±∆Er 2UnTzsin Ψ(2) In equation (2) ϕeis the acceleration phase for the synchronous particle in a bunch and ∆ Esis the half width of the natural energy spread of particles, induced in a bunch, ∆ Eris the full width of the desired energy spread to be induced in a bunch as it passes through the periodic resonato rs. In an injected bunch, a particle having a higher energy than t he energy of the synchronous particle is found normally at the head of the bunch and those having lower energy than the central one are normally at the t ail. For this normal case, ∆ Eris used with the plus sign in equation (2). However, for the reverse situation where particles having higher energy tha n the synchronous particle appear at the tail of the bunch, ∆ Erin equation (2) is used with the minus sign. Unis the amplitude of the acceleration voltage in the n-th reso nator. Using electrical and mechanical methods to regulate each ca vity channel, the details of which will be described in a separate paper, the ac celeration field in any n-th sector resonator’s particle orbit channel can be independently tuned. In equation (2) 2Ψ is the full phase width of a bunch and Tzis the transit-time factor determined by 3T=sin ∆Φ /2 ∆Φ/2(3) where, ∆Φ =2πfrfLgap βc(4) ffris the frequency of the acceleration field in resonators, Lgapis the full acceleration gap in resonators, βis the relativistic velocity factor, cis the speed of light. Normally Tzcan be maintained at a constant value if the acceleration gap in a resonator is increased in proportion t o the speed of the accelerated beam, which in turn reduces the need to increase the amplitude of the acceleration voltage Un. The energy gain of a central particle in the bunch ∆ Eeafter exiting a resonator, is given by ∆Ee=UnTzsinϕe (5) The energy gain for an edge particle, a, located at the head of the bunch and an edge particle, b, located at the tail of the bunch are determined by ∆Ea=UnTzsin(ϕ−Ψ) (6) ∆Eb=UnTzsin(ϕ+ Ψ) (7) The equilibrium phase for acceleration is set within the lim its of 0< ϕ e<(π 2−Ψ) (8) so that the phase of off-center particles always remains belo wπ/2, and a Gaussian bunch distribution is maintained. The bunch duration τfat the end of any sector (at the space located between two adjacent resonators) is given by τf=τs+ ∆τ (9) ∆τ=Ss c(1 βb−1 βa) (10) 4where τ= ΨT/π,Ssis the orbital path length of particles in the sector andTis the RF period duration. In equation (10) if a positive sign is obtained for ∆ τ, which is when βa> βb, this describes bunch elongation and an increase in τf, and if a negative sign is obtained, it describes axial bunch compression, wit h a corresponding decrease in the value of τf. The last case is possible only for the reverse particle config uration, which is when βa< β b. Here the normal positioning of particles in a bunch is altered. The particles a, at the head of a bunch, now have smaller energy with respect to particles at the center and the tail, a second condition that supports bunch compression to occur. If in equation (9) a negative sign is obtained for τf, it means that over- compression has occurred, after which the normal distribut ion of particles in a bunch will be restored. If in a bunch, conditions are found by the proper selection of the equilib- rium phase to constantly support the reverse particle distr ibution, the bunch duration will constantly decrease and overcome the space ch arge effect, which drives the elongation and transverse expansion of the bunch . From equation (10) it is evident that to obtain large and nega tive values of ∆τit is necessary to have large sector path lengths, Ssand a large negative difference in the reciprocals of β. These are the third and fourth conditions to achieve bunch compression in the Asynchronac. The values of beam injection energy Ei, initial energy spread ∆ Es,i, in- jected beam emittance, bunch duration τs,i, the injection radius Ri, the num- ber of acceleration cavities Nc, the amplitude of acceleration voltage Unand the RF frequency frf, are selected during the conceptual design of the acceler- ator, based on various technical feasibility consideratio ns. These parameters mustϕeselected beforehand, based on the simultaneous solution of equations (9), (10), (5) and (2), to determine the required value of the acceleration equi- librium phase ϕe, so that it produces bunch compression in a given sector and a specific orbital turn. However, the analytical solutions of the combined equation s appear suf- ficiently cumbersome. Consequently, these cannot be reprod uced here, but are found in the computer codes for the calculation and optim ization of these parameters. 5The control algorithm for the steering of bunches from one se ctor to the other in the Asynchronac is as follows. As a function of th e measured energy spread ∆ Es,iof the injected beam, the duration of bunches τi, the design value of the mean path length of the beam in the first sec torSs1and the selected RF acceleration voltage in the first resonator, the RF phase is set, such that an equilibrium phase ϕeis produced on the rising side of the acceleration voltage, which after the passage of a bunch in t he first resonator will cause at once the inversion of particles to occur. Also, this particle inversion must be sufficiently intense, t o reduce the duration of bunches τf1close to zero at the end of the beam orbit in the first sector. Thus, there are two cases, one which conserves t he inverted distribution, and the other which induces over-compressio n and then restores the normal distribution of particles in a bunch. For the first case, to obtain a monoenergetic beam in the follo w-on second sector, it is necessary to adjust the path length of particle s in the first sector Ss1and to set the equilibrium phase in the second resonator, so t hat the acceleration takes place on the falling side of the RF field. F or the second case, the rising side of the RF voltage is used to obtain a beam of zero energy spread. In this case ∆ Esin equation (2) takes on a negative sign and ∆ Eris set to zero. In the desired case of preserving inverse distri bution of particles, it is necessary to work only on the rising side of the RF accele ration field. The choices are determined by the magnitude of ∆ Es. Hence, at the start of the second sector the beam will be monoe nergetic and have a bunch length equal to τf1, and at the end of the sector the bunch duration τf2will increase due to the space charge effect. At the same time this provides the possibility to estimate the effect experim entally. The equilibrium phase at the third resonator must be set such that the acceleration occurs again on the rising side of the RF voltag e, to obtain an inverse particle distribution in bunches, so that at the end of the third sector the bunch duration τf3reduces again to almost zero. Thus, the process repeats itself, with and without alternating the accelerat ion mode on the rising and falling side of the RF voltage. For a more graphic p resentation the numerical modeling in the following section is provided . The path length of particles in sectors is set by the paramete rs of the bending magnetic system, which essentially must change fro m sector to sec- tor, to provide the desired values of qandϕe. To have the possibility of 6precise tuning and to relieve the maintenance of different me chanical tol- erances of the accelerator components and their alignment, we propose to place in the sectors different correction elements. This is i n addition to the magnetic lenses for transverse focusing of the beam and a num ber of beam monitors. In particular, wiggler type magnets will be insta lled in the straight sections to correct the path length of particles. Finally, e valuations show that in having large beam orbit separation steps and in the other f eatures of the Asynchronac, the mechanical tolerances of accelerator ele ments and their alignment are relatively relaxed, in the order of 10−3. The important relaxation of tolerances in the Asynchronac i s one of its main advantages, in comparison to other similar acceler ator structures, namely, as compared with the isochronous separate orbit cyc lotrons [1][2]. In these the necessity of strictly maintaining the isochronis m of particle motion reduces to having tight tolerances, which in practice are di fficult to imple- ment. Other important advantages are due to the features of l ongitudinal bunch compression and strong transverse focusing. As such, it is possible to consider the acceleration of bunches at an equilibrium ph ase close to 900, which in turn, increases the efficiency of acceleration, decr eases the number of turns, decreases the beam losses, and increases the numbe r of accelerated particles in bunches. Basically, the Asynchronac’s deficiency is the uniqueness o r unprece- dented nature of the sector bending magnetic system. This co mplicates the standardization of their manufacture and tuning, and somew hat increases the initial commissioning time and manufacturing cost of the ac celerator. How- ever, some technical innovations already made, essentiall y facilitate the solu- tion of these problems. This concern, the fabrication of mag net yokes from iron sheets with the ability to mechanically change the magn etic lengths and the remote control of the magnetic alignment in each sector a nd turn. The individual feeding of the DC bending magnets and partially t he magnetic fo- cusing lenses is straightforward to implement, using moder n electronics and computers. The issues of beam transverse focusing in this study are not c onsidered, since known standard solutions can be utilized, as normally found in strong focusing synchrotrons. In particular, the separate functi on periodic magnetic structure can be of the FODO type. In the Asynchronac the main difference will be the possibility of having a slowly changing betatron oscillation fre- 7quency, in going from one period to another. This will allow t o compensate the frequency shift of these oscillations, which is due to di fferent effects, including the space charge effect. 3 Configuring RFQ beams for injection into cyclotron The method of forming short duration bunches from modern hig h frequency RFQ’s, which produce large current and small energy spread b eams, can be modified to produce longer duration bunches at longer inter- bunch spacing, that becomes acceptable for injection in the lower frequenc y Asynchronac. This technique is based on time compressing the RFQ-produce d beam, in which the compression is completed downsteam, at the point o f injection into the Asynchronac, as described in Figure 2(a).Figure 2(b) shows the resulting single longer bunch produced from a train of short er RFQ-produced bunches. Our proposed scheme to produce the required beam compressio n is as follows. RFQ-produced bunches are initially steering in a R F deflector with a saw-tooth time varying voltage. The saw-tooth period is eq ual to the period of the Asynchronac’s driving RF frequency. The sequentiall y more and more deflected bunches pass through a 1800shaping magnet with different path lengths, as seen in Figure 2(a). After which all RFQ bunches within the saw- tooth period coincide at the time focal point, producing ful l compression of the bunch train into a single longer bunch, at the injection p oint of the Asynchronac. The RF deflector and the injection point of the A synchronac are located at conjugate points about the 1800shaping magnet. If the duration of the short bunches in a RFQ is designated by τRFQand the period between RFQ bunches is TRFQ, the time-compression of the bunch train produces a single longer bunch τcycfor injection into the Asynchronac, given by τcyc=mτRFG (11) where the period between injected bunches will be Tcyc=mTRFG (12) 8in which m is the number of RFQ bunches in a train length equal t o the period of the driving saw-tooth ramped voltage. The path length of any k-th bunch in the train, starting from t he RF deflector up the time-focused injection point of all the idea lized paths, is obtained by Lk= 2[a cosαk+R(1−cosαk+π 2−αk)−atgαk+b] (13) under the conditions of R≥(atgαmax)and b ≤(Rcosαmax) (14) The significance of the quantities R, a,b,αare exhibited in the geometry ofFigure 2(a). The maximum path length of particles will be Lmax= 2[a + b+ (πR 2)] (15) while the minimum path length is Lmin= 2[a cosαmax+R(π 2−αmax)] (16) The separation of maximum and minimum path lengths, under th e opti- mum condition of R= atgαmaxand b =Rcosαmax (17) will be ∆Lmax= 2R[ctgα max+ cosαmax+αmax−cosecαmax] (18) However, from primary considerations, the separation of ma ximum and minimum path lengths is ∆Lmax=βcTcyc (19) Knowing the value of ∆ Lmaxfrom equation (19) and inverting equation (18) produces the optimal turning radius Rtrof beam tracks in the time com- pression shaping magnet, whereby the overall dimensions an d the magnetic field strength are obtained. Thus, the optimum bending radiu s is given by 9Rtr=0.5∆Lmax ctgα max−cosecαmax+ cosαmax+αmax(20) 4 Results of numerical calculations A numerical example is worked out to present the key performa nce features and to indicate a rough cost estimate of the Asychronac. The f ollowing pa- rameters are used in the calculation of the numerical exampl e accelerator model. The RFQ linac’s RF system operates at 350 MHz, producing a pro- ton beam of 2 .0MeV energy, an energy spread of ∆ Es= 2% and a CW current of up to 100 mA. The frequency of the acceleration voltage in the Asynchronac is chosen to be 50 MHz, i.e. to have a seven-fold difference in the frequencies of the acceleration fields, between the RFQ a nd the Asyn- chronac. This means that the number of RFQ bunches to be compr essed into a single bunch is m= 7. However, in our example calculation, in order to be able to us e two fun- neled RFQ’s for injection, we have assumed 14 bunches to be co mpressed into a single bunch for injection, and for the maximum steeri ng angle of the RF deflector, αmax= 200is selected. Whereby, the following parameter values are obtained τcyc= 4.0ns T cyc= 40.0ns∆Lmax= 78.2cm R≈35.15cm b ≈33.0cma= 96 .6cm The total maximal path length including the magnetic shapin g structure will be 330 cm, and the track length up to the middle of the first resonator will be approximately 4 .4m. The duration of bunches at the end of the total path length will increase due to the beam’s energy spre ad ∆Es, by approximately 2 .2ns, so that the bunch length in the first resonator of the Asynchronac will be τcyc= 4.0 + 2.2 = 6.2ns In the given example of the Asynchronac, operating at a RF acc eleration frequency of 50 MHz, the inter-bunch separation is Tcyc= 20ns. To match with this spacing, the use of two RFQ’s will be required, each injecting at an inter-bunch spacing of 40 ns, which when initially combined in a RFQ funnel [7], will yield the required 20 nsinter-bunch spacing. 10Incidentally, the Asynchronac geometry permits further in creasing the number of injector RFQ linacs in a manner analogous to the con ventional method of multi-turn injection. RFQ’s with own bunch-train compressors can inject beams at each sector of the first or subsequent turn. Ho wever, each must have a different injected beam energy that matches the or bit’s energy at the point of injection. Thus, successive RFQ’s must have c orrespondingly higher beam energies. The bunch-train-compressed beam from a RFQ injector, throu gh the 1800 bending magnet enters the Asynchronac’s first resonator. Figure 1 schemat- ically depicts the Asynchronac structure and the beam orbit s, only for the central particles of the first three turns. The orbit radius a t injection is Ri= 3.0m, the orbit-to-orbit separation is ∆ R= 25cmand the number of resonators is Nc= 4. The number of sectors is also equal to Ns= 4 per beam turn, and since the number of turns is 17, the number of in dependent channels and magnets is 4 ·17 = 68. The key design parameters of the Asynchronac for the numeric al example are summarized in Table 1. Room temperature resonators are used in the design, which increase the machine’s radius. Following the development and operation of modern cyclotron resonators at the Paul Scherr er Institute [8] and the related designs and models [9], the operation of thes e resonators at RF frequencies of 40 −50MHz and peak voltages of up to 1 .1MVcan be made available. These resonators have a length of approxima tely 6 m, height of 3mand width along the beam of 0 .3m, and provide a radial operating clearance for orbits of up to 4 m. Table 1. Key Parameter Values of an Asynchronac 11PARAMETER UNIT VALUE Beam Spacie Proton EiInjected Beam Energy MeV 2.0 EeExtracted Beam Energy MeV 50.0 RiInjected Beam Radius m 3.0 ReExtracted Beam Radius m 7.0 NcNumber of Acceleration Cavities 4 NmNumber of Sector Magnets 66 H Field Strength in Sector Magnets T 0.11−0.85 ∆E Energy Gain per Turn MeV 0.04−3.60 ∆R Orbit Turn-to-Turn Separation cm 25.0 n Number of Turns 16 .5 h Harmonic Number 52 −58 LmLength of Sector Magnets m 0.75−5.13 LfLength of Drift Spaces m 2.2−9.7 τfDuration of Bunches ns 6.2−0.5 N0Number of Protons per Bunch 2 .5·1010 Thus, an injected bunch duration of 6 .2nsis compressed down to2 .5 nsat the end of the first sector, using the parameter values of Un= 130 KeV,Tz= 0.95,ϕ= 55.80and setting the equilibrium phase in the first resonator at ϕe= 18.40. Under these conditions, particles at the head of the bunch will have energy equal to 2 .104KeV, while at the tail of the bunch, particle energy will be 1 .975KeV. Particles at the equilibrium phase will have an energy of 2 .039KeV and the energy spread of the bunch will be ∆Es= 2.104KeVNext, particle inversion will take place. At the end of the first sector, ∆ τwill be 8 .9ns, consequently τf= 6.2−8.9 =−2.7ns, whereby the inversion has been completed and over-compression has o ccurred. The subsequent processes are easier to observe in Figures 3−10, right up to the achievement of the final proton beam energy of approximat ely 50 MeV, which is produced after 16 .5 turns in the Asynchronac. It is observed from the numerical results in these figures, th at the process of effective longitudinal beam compression has terminated a fter the first three turns. The duration of bunches has attained an almost statio nary value of about 0 .5nsand the final energy spread of the beam ∆ Esends up to be zero in the Asynchronac. We now roughly estimate the maximum particle population in a proton 12bunch, as a function of bunch shortening. In our simplified ap proach we ignore the intra-beam scattering and wake field effects on bun ch lengthening and the axial focusing from the RF acceleration cavities. In the proton bunch’s rest frame, the energy spread due to the bunch electr ic self-field is given by ∆P2 2m=−e/integraldisplay Ezdz (21) where zis the longitudinal coordinate and ∆ Pis the momentum spread. The longitudinal space-charge electric field of the bunch is obtained as [10] Ez=−e 4πε01 γ2(1 + 2 lnb a)∂λ(z) ∂z(22) where is the absolute dielectric constant, a and b are the rad ii of the proton beam and the vacuum chamber, respectively, and λ(z) is the particle linear density in the bunch. Taking a Gaussian distribution for the bunch linear density λ(z) =N0√ 2πσez2 2σ2 (23) withσ= 0.7zas the standard value of the bunch length, and inserting expression (22) into equation (21), and after integration w ithin the bounds of the bunch’s initial z iand final zfhalf-lengths, the following formula is obtained for the maximum particle population in a bunch N0=∆P2 2m4πε0 e2γ2 e−z2 f 2σ2−e−z2 i 2σ2√ 2πσ 1 + 2 lnb a(24) Using the parameters in the above, and taking for the beam rad iusa= 0.5 cmand the vacuum chamber radius b= 5.0cm, the estimated maximum number of protons per bunch is N0≈5·1011. From these figures it is seen that the magnetic field in each sec tor and turn has sufficiently different and not necessarily optimized parameters. To simplify the numerical calculations we assumed that the ben ding magnets in each sector and turn consist of single whole units, instea d of being a number of shorter modular magnets that would serve the requi red purpose. In making the conceptual design of a specific Asynchronac, th e sector and 13turn magnets will be modularized with optimized parameters , such that the differences among modular magnets will be few, to permit stan dardizing the manufacturing process. Rather low values of 0 .1−0.85Tesla are required for the magnetic field strength in each sector and turn. The fabrication of small-s ized modular bending magnets at these field strengths is a standard matter . Should the beam’s vacuum chamber need an aperture full width of as much a s 10cm, this can be easily accommodated, since the turn-to-turn orb it separations will all be equal at ∆ R= 25cm. The lengths of the remaining free drift spaces in sectors and turns, after the allocation of resonators and bending magnets, will be more than 2 .0m. This will allow not only to freely install focusing magnetic elements and beam monitoring apparatus, but also t o provide easily 100% extraction of the beam. A rough estimation of the cost to build and operate the Asynch ronac shows that the cost per megawatt of proton beam produced from the Asyn- chronac is much less expensive by an order of magnitude, as co mpared to a megawatt of proton beam produced from the high and super-hi gh energy accelerators. 5 Conclusion The primary objective of this paper is to show that the innova ted acceler- ator type, which we refer to as the Asynchronac, has sufficient feasibility features for its implementation, in which an accelerated pr oton beam of up to 100 MeV energy and hundreds of mAcurrent can be effectively bunch- compressed. Consequently, it is important to expedite the e xtension of fur- ther multifaceted studies on this concept, to improve the qu ality of future machines for scientific and applied applications, potentia lly using this alter- native. References [1] J. A. Martin et. al. The 4-MeV Separated-Orbit Cyclotron , IEEE Trans- actions on Nuclear Science, v. NS-16, N3, part 1, p.479, 1969 14[2] U. Trinks, Exotic Cyclotrons - Future Cyclotrons , CERN Accelerator School, May 1994, CERN Report 96-02, 1996 [3] O. Brovko et.al. Conceptual Design of a Superferric Separated Orbit Cy- clotron of 240 MeV Energy , Proceedings of the 1999 Particle Accelerator Conference, vol. 4, p. 2262, Brookhaven, NY [4] A. R. Tumanian, Kh. A. Simonian and V. Ts. Nikoghosian, Powerful Asynchronous Multi-Purpose Cyclotron , Physics Proceedings of the Ar- menian National Academy of Sciences, No. 4, vol. 32, p. 201, 1 997, Yerevan, Armenia [5] A. R. Tumanyan, G. A. Karamysheva and S. B. Vorozhtsov, Asyn- chronous Cyclotrons , Communication of the Joint Institute for Nuclear Research, Report E9-97-381, 1997, Dubna, Russia [6] A. Schempp, H. Vorrman. Design of a High Current H- RFQ Injec- tor, Proceedings of the 1997 Particle Accelerator Conference, vol. 1, p. 1084, Vancouver, B.C., Canada; A. Lombardi et al. Comparison Study of RFQ Structures for the Lead Ion Linac at CERN , Proceedings of EPAC, Berlin, 1992 [7] K.F.Johnson et. al. A Beam Funnel Demonstration; Experiment and Simulation , Particle Accelerator Conference, Vols.37-38, p.261, 199 2 [8]Proceedings of the LANL Workshop on Critical Beam Intensity Issues in Cyclotrons , Santa Fe, NM, December 4-6, 1995, p.358 [9] N. Fietier and P. Mandrillon, A Three-Stage Cyclotron for Driving the Energy Amplifier , Report CERN/AT/95-03(ET), Geneva, 1995 [10] H.Wiedeman, Particle Accelerator Physics, v.2, p.344 , 1995 15CA1 CA2 CA3 CA4 IB - Injection Beam BM - Bending Magnet CA - Cavity Axis BM1 BM5 BM5 BM2 BM6 BM10 BM3 BM7 BM11 BM4 BM8 BM12 IB 3m Figure 1 Trajectory of central particle in Asynchronac on first three turn R F Q RF DEFLECTOR MAGNET YOKE b to Asynchronac a/cos α max a*tg α max a α max R Figure 2b RFQ and Asynchronac bunch time sequence t [ns] 0 10 20 30 40 50 I t [ns] 0 10 20 30 40 50 I Figure 2a Layout of RFQ bunch compression scheme RFQ AA 0.286 ns 2,86 ns 4.0 ns N sector 0 1 2 3 4 5 6 7 8 9 10 11 12 13 τ (ns) 0 1 2 3 4 5 6 7 Figure 3 Bunch duration 2 nd turn 3 rd turn 1 st turn N sector 0 1 2 3 4 5 6 7 8 9 10 11 12 13 ϕ 10 20 30 40 50 60 70 80 90 Figure 4 Acceleration equilibrium phase 1 st turn 2 nd turn 3 rd turn N sector 0 1 2 3 4 5 6 7 8 9 10 11 12 13 U [kV] 100 150 200 250 300 350 400 450 500 1 st turn 2 nd turn 3 rd turn Figure 5 Acceleration voltage amplitude N sector 0 1 2 3 4 5 6 7 8 9 10 11 12 13 E k [MeV] 1 2 3 4 5 6 N sector 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 E k [MeV] 0 10 20 30 40 50 60 a) b) 1 st turn 2 nd turn 3 rd turn Figure 6 (a,b) Kinetic energy + 3.2% _ N sector 0 1 2 3 4 5 6 7 8 9 10 11 12 13 q 9 10 11 12 13 14 N sector 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 q 6 7 8 9 10 11 12 13 14 1 st turn 2 nd turn 3 rd turn a) b) Figure 7 (a,b) Intercavity harmonic number N sector 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 S n [m] 4 6 8 10 12 14 Figure 8 (a,b) Beam trajectory length N sector 0 1 2 3 4 5 6 7 8 9 10 11 12 13 S n [m] 4 5 6 7 8 a) b) 1 st turn 2 nd turn 3 rd turn N sector 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 L m [m] 0 1 2 3 4 5 6 Figure 9 (a,b) Length of sector magnets a) 1 st turn 2 nd turn 3 rd turn N sector 0 1 2 3 4 5 6 7 8 9 10 11 12 13 L m [m] 1 2 3 4 a) Figure 10 (a,b) H-field in sector magnets N sector 0 1 2 3 4 5 6 7 8 9 10 11 12 13 H m [T] 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 N sector 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 H m [T] 0.0 0.2 0.4 0.6 0.8 1.0 1 st turn 2 nd turn 3 rd turn a) b)

arXiv:physics/0003021v1 [physics.acc-ph] 8 Mar 2000February 2, 2008 Diffraction Radiation Diagnostics for Moderate to High Ener gy Charged Particle Beams Abstract Diffraction radiation (DR) is produced when a charged partic le passes through an aperture or near a discontinuity in the media in which it is tr aveling. DR is closely related to transition radiation (TR), which is produced whe n a charged particle tra- verses the boundary between media with different dielectric constants. In contrast to TR, which is now extensively used for beam diagnostic purpos es, the potential of DR as a non-interceptive, multi-parameter beam diagnostic re mains largely undeveloped. For diagnostic measurements it is useful to observe backwar d reflected DR from an cir- cular aperture or slit inclined with respect to the beam velo city. However, up to now, well founded equations for the spectral-angular intensiti es of backward DR from such apertures have not been available. We present a new derivati on of the spectral angular intensity of backward DR produced from an inclined slit for t wo orientations of the slit axis, i.e. perpendicular and parallel to the plane of in cidence. Our mathematical approach is generally applicable to any geometry and simple r than the Wiener Hopf method previously used to calculate DR from single edges. Ou r results for the slit are applied to the measurement of orthogonal beam size and diver gence components. We discuss the problem of separating the simultaneous effects o f these beam parameters on the angular distribution of DR and provide solutions to this difficulty. These include use of the horizontal and vertical polarization components of the radiation from a sin- gle slit and interferences from two inclined slits. Example s of DR diagnostics for a 500 MeV beam are presented and the current limitations of the tec hnique are discussed. R. B. Fiorito Catholic University of America, Washington, DC 20064 email : rfiorito@rocketmail.com D. W. Rule Carderock Division, Naval Surface Warfare Center, W. Bethe sda, MD 208171 Introduction The new generation very high power radiation devices such as short wavelength free electron lasers and single pass spontaneous emission (SASE) devices for the generation of intense uv and x-ray beams requires the development of linear accelera tors which produce optical quality charged particle beams. The high power, small size and low em ittance of these beams present an enormous challenge for both the diagnostic measurement o f beam parameters and the accurate positioning and control of these beams. Conventio nal screens or other interceptive probes are incompatible with the operation of such accelera tors in many instances. Non interceptive devices such as wall monitor arrays, give limi ted beam information. Other types of monitors, such as synchrotron monitors, while non- invasive, cannot be used in linear beam line geometries. Hence the development of low co st, compact, nondestructive monitors capable of measuring multiple beam parameters wou ld be very useful for many high power beam applications. We have investigated how the properties of diffraction radia tion (DR) can be used to measure beam divergence, energy, position, transverse bea m size and emittance. DR devices such as slits or circular apertures through which the beam pa sses offer minimal perturbation to the beam, respond rapidly to changes in beam parameters an d are inherently compact, so that they can be implemented at many places in the beam line . Diffraction radiation (DR) is produced when a charged partic le passes through an aper- ture or near an edge or interface between media with different dielectric properties. DR is closely related to transition radiation (TR), which is pr oduced when a charged particle traverses the boundary between media with different dielect ric constants. DR has been studied theoretically since the early 1960’s, [1–5]. Howev er, in contrast to TR, which has been well investigated experimentally and is now used exten sively for beam diagnosis, we are aware of only two experimental studies of DR in the literatur e [6],[7]. In only one of these investigations [6] was DR actually used as a beam parameter d iagnostic, i.e. to measure the beam bunch length. Furthermore, in this study forward an d 1800backward directed DR and TR were observed simultaneously. This situation comp licated the comparison of measured to theoretically predicted DR properties. Theref ore, to avoid such difficulties and for experimental convenience, it is advantageous to observ e the radiation from an inclined aperture, e.g. a slit inclined at 450, which produces DR both in the forward direction and in the direction of specular reflection of the virtual photon s associated with the particle, i.e. at 90 degrees with respect to the beam velocity. The detailed nature of the far field angular distribution (AD ) offorward directed DR has been derived and discussed by Ter-Mikaelian [4] for an el ectron normally incident on a circular aperture and a slit in an infinite, thin plane. The fie lds and AD of DR from an electron incident on a single edge (half plane) inclined at a n arbitrary angle with respect to the velocity vector of the particle have also been derived an d discussed in Refs. [2], [4] and [8]. Although some authors have surmised that reflected DR from an inclined circular aperture or slit has properties similar to reflected TR [9-10], a deriv ation, from first principles, of the spectral-angular distribution of reflected DR from such ape rtures has not been previously 1available in the literature. We present here solutions for t he fields and the horizontally and vertically polarized intensities of DR from an inclined sli t for two orientations of the slit edge, i.e.parallel and perpendicular with respect to the pl ane of incidence. Previous theoretical investigations of forward directed D R generated from electron beams passing through circular apertures and slits [9-12] have ex amined the potential use of the far field angular distribution (AD) to diagnose beam properties such as position with respect to the center of the aperture, beam size, trajectory angle, div ergence and energy. The potential advantage and difficulty in using the AD of DR for beam diagnost ics is that it depends on the beam position (size) as well as the beam divergence and en ergy. In contrast, the AD of TR depends only on the beam divergence and energy. The chal lenge for the application of the AD of DR as a diagnostic then is to devise a means to separ ate out the competing effects of divergence and beam size on this observable. We have developed a number of ways to address and solve this pr oblem. We demon- strate here how the measurements of the horizontal and verti cal polarization components of backward reflected DR from a single slit can be used to separ ate out the effects of beam size perpendicular to the slit edge and the component of dive rgence parallel to the slit edge. Furthermore, we show how the measured AD’s for two different o rthogonal orientations of the slit axis, can be used to determine the components of beam size and divergence in two orthogonal directions, i.e. the directions perpendicular and parallel to the plane of incidence. In addition we suggest how the interferences from two slits c an be used to make a sensitive measurement of divergence alone. 2 Diffraction Radiation Properties When a charged particle or ensemble of particles passes thro ugh an aperture or near an edge at a distance d, appreciable DR is generated when the condition: d/lessorsimilarγ¯λis fulfilled, where ¯λ=λ/2π, and λis the observed wavelength. Table I. presents typical wavel engths of DR observed from a slit whose width a= 2d= 2mmfor a variety of lepton and hadron beams. Table I. shows, for example, that DR is produced in the optica l part of the spectrum (i.e. IR to visible) for electrons with energies in the range of 0.5 - 5 GeV. Table I. Typical wavelengths of diffraction radiation generated fro m a 2-mm wide slit for various accelerators E(GeV) γλ(µm)Accelerator 0.5 1036.30 APS 5.0 1040.630 CEBA 50.0 1050.063 SLAC 250.0 250 25.00 RHIC 1031036.30 FNAL 2When the observed DR wavelength is much smaller than the tran sverse or longitudinal size of a charged particle bunch, incoherent DR is produced i ndependently from each particle and the total intensity I∼N, the number of particles in the bunch. For wavelengths comparable to the bunch dimensions coherent DR is observed w ith an intensity I∼N2. The DR diagnostic methods we will present can utilize either inc oherent or coherent DR. However, for simplicity and to illustrate and emphasize the basic met hodology we will concern ourselves here only with the analysis of incoherent DR. Furthermore, s ince we have discussed DR diagnostics using circular apertures elsewhere [10], we wi ll focus our attention in this paper to the generation of DR from slits. An extended analysis of co herent DR from a slit will be presented in a future work. 3 Theory of Backward Diffraction Radiation from an Inclined Slit 3.1 Parallel orientation of slit with respect to the plane of inc idence A relativistic charged particle passing through a slit in a c onducting screen emits diffraction radiation in the forward and backward directions, the latte r in the direction of specular reflection of the virtual photons associated with the relati vistic particle. The peak intensity of diffraction radiation, like transition radiation, occur s at an angle θ∼γ−1, where θis measured from the direction of the particle’s velocity vect orvfor forward DR, and from the direction of specular reflection for backward DR. Figure 1A. shows a side view, and Figure 1B. a top view, of the c one of backward diffraction radiation emitted by a charge passing through a s lit which is inclined at the angle Ψ with respect to the velocity vector and oriented such that i ts edge is horizontal or parallel to the plane of incidence ( n, vplane), where nis the unit vector normal to the plane of the slit, i.e. the ( x′, y′) plane. Additionally, the particle passes through the slit with an offset δin the vertical ( y′) direction measured from the center of the slit which is defin ed by the x′axis. For simplicity, we have not drawn the forward directed cone o f DR which is also generated in the direction v. Figures 1A. and 1B. show the coordinate system ( x, y, z ) used to describe the radiation fields associated with the wave vector kobserved in the backward direction (i.e. directed along the zaxis). The slit is assumed to be an infinitely thin conducting screen with infinite extent in the x′andy′directions beyond the slit aperture. Our calculation of the backward reflected DR fields employs th ree different concepts (a) the Huygens-Fresnel diffraction approach, following the tr eatment given by Ter-Mikaelian [4] for the case of forward DR, (b) the concept of backward reflect ed pseudo photons, used by Wartski [13] to calculate reflected transition radiation an d (c) a form of Babinet’s principle. Our approach is quite different and mathematically much less complex than the exact Wiener Hopf method used by previous researchers to calculate backw ard DR from a single edge [2,4]. In addition, our method is generally applicable to other geo metries where the Wiener Hopf technique is very difficult to implement. 3Consider a thin reflecting metal screen S∞with infinite extent. A charged particle crossing this surface produces the well known forward and backward tr ansition radiation. Compare this to the case of Fig. 1., where the slit has an area S1and the conducting screen has an areaS2. Babinet’s principle states that the transition radiation (TR) field from the infinite conducting screen, S∞, can be obtained from a Huygen’s Fresnel integral over the in finite areaS∞=S1+S2, so that the TR field can be expressed as− →E∞=− →E1+− →E2where− →E1 and− →E2are those parts of the field which are obtained from a Huygen’s Fresnel integral over the areas S1andS2, respectively [8]. If material of area S1is removed, then the field− →E2=− →E∞−− →E1, where− →E2is the desired backward DR field. Note that if the area S1→0,− →E2→− →E∞, and the DR field approaches that of TR. The incident field in the Huygens-Fresnel integral is the fiel d associated with the rela- tivistic particle,− →Ei. Using Ter-Mikaelian’s approach to calculate the diffracti on radiation field and Wartski’s method of calculating backward reflected radiation, we can express the xandycomponents of the radiation field as Ex,y(kx, ky) =r/bardbl,⊥(ω,Ψ)1 4π2/integraldisplay/integraldisplay Eix,iy(x, y)e−i− →k·− →ρdxdy (1) where Eix,iyare the components of the field of the incoming charged partic le and kxandkyare the components of the wave vector kin the plane normal to the direction zshown in Figure 1A. and 1B. As is the case for TR, there are generally three com ponents to the radiation: direct, reflected and transmitted. However, for an opaque, h ighly conducting screen, the dominant term for backward TR or DR production is the reflecte d component. The fields Ex andEyfor backward TR and DR are thus proportional to the Fresnel re flection coefficients r/bardblandr⊥, respectively. In Eq. (1), the integral over the area S2is done in terms of the coordinates ( x, y) of Figures 1 and 2. The far field reflected spectral intensity of DR is found by integrating the flux of pseudo photons reflected off the surf aceS2, which pass through the plane that is perpendicular to the direction z, i.e. the direction of specular reflection of incident pseudo photons having a wave vector k,which is parallel or closely parallel to the velocity vector v. The phase term in Eq. (1), which is a function of− →ρ= (x′, y′), a vector lying in the plane of the screen, can be related to the coordin ates (x, y) by the transformations x=x′sin(Ψ), y=y′, and the differential elements of area by dxdy =dx′dy′sin(Ψ). The detailed solution of the integral in Eq. (1) is presented in t he Appendix. The results are: Ex(kx, ky) =ier/bardbl(ω,Ψ) 4π2vkx f/bracketleftbigg1 (f−iky)e−a1(f−iky)+1 (f+iky)e−a2(f+iky)/bracketrightbigg , (2) which is identical to Eq. (31.18) of Ref. [4], except for the r eflection coefficient r/bardbl(ω,Ψ), and the scalar quantity kx≡kx−[k/(2γ2)] cotΨ, and Ey(kx, ky) =er⊥(ω,Ψ) 4π2v/bracketleftbigg1 (f−iky)e−a1(f−iky)−1 (f+iky)e−a2(f+iky)/bracketrightbigg , (3) 4which is identical to Eq. (31.19) of Ref. [4], except for the f actor r⊥(ω,Ψ). Here a1,2= a/2±δ,f= [k2 x+ω2/(v2γ2)]1/2,v=|v|,kx=ksinθx,ky=ksinθyandθx,yare the projected angles of the vector kinto the x, zandy, zplanes, respectively. The horizontally polarized intensity (i.e. parallel to the slit edge) and vertically polarized intensity (i.e. perpendicular to the slit edge), which are o bserved in the plane perpendicular to the direction of specular reflection, are defined in terms o f thexandycomponents of the fields: d2Nhoriz. dωdΩ=2π2k2c /planckover2pi1ω|Ex|2(4) =/vextendsingle/vextendsingler/bardbl/vextendsingle/vextendsingle2αk2k2 xe−af 4π2ωf2(f2+k2y)[cosh(2 fδ) + sin( aky+ Φ(kx, ky))] and d2Nvert. dωdΩ=2π2k2c /planckover2pi1ω|Ey|2(5) =|r⊥|2αk2 4π2ωe−af (f2+k2 y)[cosh(2 fδ)−sin(aky+ Φ(kx, ky))] respectively, where Φ(kx, ky) = sin−1[(f2−k2 y)/(f2+k2 y)] = cos−1[−2fky/(f2+k2 y)], (6) andα≃1 137is the fine structure constant. The equations for vertically polarized forward diffraction radiation intensity, first ob- tained by Ter-Mikaelian [4] and later rewritten and present ed in Ref. [9], are both equiva- lent to Eq. (5 ) without the factor |r⊥|2. Note, however, that the equations for phase term, Φ(kx, ky) presented in Ref. [9] are incorrect. The correct relations hips are given in Eq. (6). 3.1.1 Limiting Forms for Backward DR at 450 Under the usual small angle approximation for observation o f TR and DR from relativistic particles, i.e. v≈c, θ∼γ−1≪1, the quantities kx≈kx≈kθx,ky≈kθy, and kz≈k. We then can consider several limiting forms for backward DR. (1)Single Edge Limit :When either a1ora2→ ∞ , the sinusoidal term in Eqs. (4) and (5), which represents the interference between the inte nsities from each edge of the slit, disappears and expressions for the horizontal and vertical intensities of DR from a single edge are obtained. The addition of the horizontal and vertic al intensities taken in this limit produces an expression for the total DR intensity which is id entical to that obtained using the exact Wiener Hopf approach (see Ref. [8], Eq. (14)). The c orrespondence of the results of our calculations to those calculated using exact theory i s a strong confirmation of the validity of our method for calculating backward reflected DR . 5(2)Small Displacement Limit : When δ,the particle displacement from the center of the slit, is small, i.e. δ≪γ¯λ∼a, the hyperbolic cosine in Eqs. (4) and (5) can be expanded in a Taylor series. Retaining terms only up to second order, a nd using the small angle approximations given above, Eqs. (4) and (5) become: d2Nhoriz. dωdΩ=/vextendsingle/vextendsingler/bardbl/vextendsingle/vextendsingle2α 4π2ωγ2X2 (1 +X2)e−R(1+X2)1/2 (1 +X2+Y2)(7) ·[1 + 2(δ γ¯λ)2(1 +X2) + sin( RY+ Φ′(X, Y))] and d2Nvert. dωdΩ=|r⊥|2α 4π2ωγ2e−R(1+X2)1/2 (1 +X2+Y2)(8) ·[1 + 2(δ γ¯λ)2(1 +X2)−sin(RY+ Φ′(X, Y))] where we have introduced the new variables X=γθxandY=γθy, thexandyprojected angles scaled in units of γ−1,R≡a/γ¯λand the reduced phase term Φ′(X, Y)) = sin−1[(1+X2−Y2)/(1+X2+Y2)] = cos−1[−2(1+X2)1/2Y/(1+X2+Y2)].(9) (3)TR Limit : Additionally, when R << 1 and δ << γ ¯λ, Eqs. (7) and (8) each reduce to one half the intensity of TR, which is the expected correct li mit. In this regime the particle radiates as if the slit in the screen were absent. 3.2 Perpendicular orientation of the slit with respect to the pl ane of incidence Figures 2A. and 2B. illustrate the case in which the slit is or iented such that the edge is perpendicular to the plane of incidence, i.e. the plane cont aining vandnthe normal to the plane of the screen. In this case, the offset εof the particle velocity vector from the center of the slit is in the direction Y′and the radiation fields ExandEycan be shown to be the same as those given above in Eqs. (2) and (3) but with a1→a1sin Ψ, a2→a2sin Ψ and δ→εsin Ψ. Also, Ex∝r⊥(ω,Ψ) and Ey∝r/bardbl(ω,Ψ) since, for this orientation of the slit,− →Exis perpendicular and− →Eyis parallel, respectively, to the plane of incidence. Then Exand Eytake the same forms as Eqs. (2) and (3), with the coefficients r⊥andr/bardblinterchanged. Similarly the vertical and horizontal intensity component s take on the same forms as Eqs. (4) and (5), with the coefficients |r⊥|2and/vextendsingle/vextendsingler/bardbl/vextendsingle/vextendsingle2interchanged. In Ref. [8] it has been shown that an exact calculation of refle cted DR produced by a particle traveling with a velocity with a parallel componen t along the direction of the edge of a semi-infinite screen [2] reduces, in the limit γ >> 1, to the same result as that of a particle whose velocity vector is purely perpendicular to t he edge (see Eqs.(7) and (14) of 6Ref.[8]). The former situation corresponds to the geometry of Figure 1. as discussed above in Section 3.1; the latter situation corresponds to the geomet ry of Figure 2., which is discussed in this Section. As in the parallel case, the single edge resu lts for the geometry of Fig. 2. are completely reproduced using our approach in the limit a1ora2→ ∞. Furthermore, in the perpendicular incidence case the small angle, relativi stic limit produces the same forms for the horizontal and vertical intensities as the parallel incidence case, i.e. Eqs. (7) and (8), but with the coefficients |r⊥|2and/vextendsingle/vextendsingler/bardbl/vextendsingle/vextendsingle2interchanged. In addition, when R << 1, the horizontal and vertical intensities each go to the same TR li mit as in the parallel incidence case. These results provide further evidence of the soundne ss of our theoretical approach. For the sake of simplicity and to illustrate the main results of our analysis of DR from slits, we will present numerical examples only for the case o f the slit oriented with its edge parallel to the plane of incidence . However, numerical results can ea sily be obtained for the perpendicular orientation by simple substitution of varia bles as listed above. 4 Discussion and Computational Results 4.1 General properties of DR Eqs. (5) and (6) indicate that the highest yield of backward D R observed in the IR to visible part of the spectrum will be generated when the surface of the slit is a highly conducting mirrored surface. In the case |r⊥|2≃/vextendsingle/vextendsingler/bardbl/vextendsingle/vextendsingle2≃1. and for γ >> 1, Eqs. (7) and (8) show that the polarization components are the same as forward DR. These equations also reveal some interesting characteristics of DR from a slit, which ha ve not been fully discussed in previous studies, and which can be easily visualized with th e help of three dimensional plots of intensity observed in the X, Yplane. Figs. 3A. and 3B. show the horizontal and vertical component s, respectively, of the intensity of DR for a value of the ratio R=a/(γ¯λ) = 0.5. Similarly, Figs. 4A. and 4B. show the component intensities for R= 2.0 . The intensity axis ( Z) in these Figures is scaled in proportion to the corresponding intensity component of tra nsition radiation. For example, Fig. 3A indicates that the peak intensities of horizontal an d vertical DR, respectively, are about 25% and 80% that of TR , when R= 0.5.Figures 3. and 4. also show that diffraction fringes are present in both polarization components, but ar e only observed in the Ydirection (perpendicular to the slit edge) for both the orientations s hown in Figs. 1 and 2. This can be deduced directly by an examination of the term sin( RY+Φ′(X, Y)) in Eqs. (7) and (8), i.e. the frequency of oscillation is determined solely by the val ue of kycomponent of the wave vector represented through the variable R.Secondly, the inteferences in the horizontal and vertical intensities are 1800out of phase. This is evident by the presence of the difference in the sign of the sinusoidal terms in each component. Thus, i n general, the maxima of the vertical component along the Yaxis occur at the minima of the horizontal component. One further observes that the equation for horizontal inten sity, Eq.(7) contains the factor X2/(1 +X2) , which is not present in Eq.(8) for the vertical intensity. This term, which 7is similar to the form of TR, forces the horizontal component to have a null at X= 0 for every value of Y. A comparison of the ratio of vertical to horizontal intensi ties reveals that the vertical component is larger than the horizontal compon ent, and that ratio becomes smaller as Rdecreases approaching unity as R→0 , i.e. the transition radiation limit. Also, as Rbecomes large the DR intensity drops off exponentially. Thus the value of R should be kept to a minimum to maximize the observed intensit y of DR. We have chosen the values R= 0.5 and 2 .0 to show that measurable horizontal and vertical intensity levels can be achieved for reasonable experimental conditions and to demonstrate that the ratio of vertical to horizontal intensity depends on the value of R. For example, a value of R= 2 applies to the case when the beam energy Eb= 500 MeV, the slit width a= 2mmand the wavelength λ≃3µm. By change of the parameters aandλ, or by use of different values ofR, other workable combinations are possible for a given beam e nergy. Note that as the beam energy increases, a smaller value of Rcan be achieved for the same wavelength and slit size. Since the intensity of DR, like TR, is proportiona l toγ2(see Eqs. (7) and (8)), DR becomes progressively more intense as hence easier to detec t as the beam energy increases. In general, the sum of horizontal and vertical polarization intensity components will be observed by a detector in the X, Y plane. Thus, if the angular distribution of DR is imaged, e.g. by using a camera with a lens focussed at infinity , the sum of horizontal and vertical distributions will be superimposed. However, eit her distribution can be imaged separately by placing a rotatable polarizer in front of the l ens. We will show that each of these intensity components provides information about t he beam size and divergence in each of two orthogonal directions and, furthermore, that al l of these beam parameters can be measured by proper analysis of the horizontally and verti cally polarized intensities. 4.2 Effect of Beam Parameters on the Angular Distribution of DR 4.2.1 Beam Size The above discussion shows that both the horizontal and vert ical DR spectral-angular dis- tributions are functions of the beam position δrelative to the center of the slit. This effect has been previously considered [9,11] as a possible diagnos tic of the beam position and size. In Ref. [9] it was shown that the effects of beam size and offset o n the angular distribution of DR are the same. Also, in Ref. [11] it was shown that the DR in tensity is a minimum for a beam centered in the slit, i.e. zero offset. Since the beam ca n be centered by monitoring and steering the beam to minimize the total intensity, the off set can be nullified or otherwise determined, for example, by use of a standard wall current mo nitor. The remaining effects on the angular distribution of the DR will then be the beam size i n the direction perpendicular to the slit edge and the two orthogonal components of the beam divergence. To determine the effect of beam size on the intensity componen ts, we assume that the beam centroid has been centered in the slit by means of one of t he previous methods described above and that any particle within the finite beam spatial dis tribution has the same effect as the offset of a single particle. We further assume that the b eam has a separable spatial distribution given by S=S1(δ)·S2(ε), where δandεare in the directions y′andx′, 8respectively. For the parallel orientation of the slit, δandεare perpendicular and parallel, respectively, to the slit edge. Integration of the distribu tionSoverδandεproduces the average beam sizes ∝angbracketleftδ∝angbracketrightand∝angbracketleftε∝angbracketright. Because of the simple dependence of the above expressions for horizontal and vertical intensity on the variable δ, an integration of Sover the intensity components is mathematically equivalent to a simple replac ement the variable δwith its average value ∝angbracketleftδ∝angbracketright. To simplify the notation in the remainder of the text we will identify and refer to δandεas the othogonal rms beam size components, which are perpend icular and parallel, respectively, to the plane of incidence. 4.2.2 Beam Divergence Ref. [9] has examined the effect of beam size perpendicular to the edge of a slit on the vertical intensity component of forward DR in the absence of divergen ce. A discussion of the effect of beam size on the horizontal component was dismissed becau se of its much lower intensity in comparison to the vertical intensity - an effect which was d ue to the rather large value of R= 2πused in the analysis. However, for any realistic beam, the di vergence is not negligible and affects both the horizontal and vertical intensities of D R. Furthermore, in general, the horizontal component of DR is not negligible in comparison t o the vertical component and, as we will show, this component provides important informat ion on the component of beam divergence parallel to the slit edge. It is therefore import ant to analyze both the effect of beam size and divergence on the vertical and horizontal inte nsity components to fully assess the diagnostic potential of DR. The effect of divergence must be taken into account by perform ing a two dimensional convolution of a distribution of particle trajectory angle s projected in the X, ZandY, Z planes (e.g. separable Gaussian distributions in θxandθy). However, we have shown by numerical calculation that the effect of the rms divergence δ′on the horizontal component of the DR intensity, and the effect of the rms divergence ε′on the vertical component is insignificant for δ′, ε′/lessorsimilar0.2, a value which is large by the standards of most high quali ty accelerators. Then, to a good approximation, a one dimensio nal convolution of a line scan of the horizontal or vertical intensity calculated either i n the plane defined by Y=const. orX=const. can in principal be used to predict the effects of the beam dive rgences ε′ andδ′, respectively. Inversely, these divergences can be determ ined by fitting the convolved intensities to measured data. Horizontal Intensity Component The effect of the divergence ε′, the component par- allel to the incidence plane, on the horizontal intensity co mponent can be estimated by neglecting the beam size term. Consider the term X2in the numerator of the horizontal component Eq.(7). Because of the presence of this term, a con volution of the horizontal intensity component with a distribution of particle angles G(X, ε′) will have the maximum effect on this component. Furthermore, the effect of ε′will be maximized near the cen- ter of the pattern where the horizontal intensity goes to zer o in the absence of divergence. One can estimate this effect by setting the variable Yin Eq. (7) equal to a constant value 9Y=δ′=const. , and noting the effect on a line scan of the one dimensional con volution of the horizontal intensity with e.g. the Gaussian distributi on:G(X, ε′) =1√ 2πε′2exp(−X2 2ε′2). Figure 5. shows two such normalized scans of a 1D convolution of horizontal intensity (Eq. (7)) with Gfor two different values of the variable, Y=δ′= 0.2 and Y= 0. In this calculation, R= 2.0 and ε′= 0.1. As Fig. 5. shows, there is no discernible difference between the two patterns. Then only a small error will be made by setting the variable Y equal to a constant and performing a one dimensional convolu tion over Xto infer the effect of the divergence ε′.Inversely, it is possible to measure the divergence, ε′from line scans of horizontally polarized DR for a number of Yvalues by comparing the measured intensity scans to theoretically convolved line scans in the variable X, which are parameterized by the variable ε′. Now consider the effect of the beam size δmeasured perpendicular the plane of incidence, which is also perpendicular to the slit edge for the orientat ion shown in Fig. 1., on the horizontal intensity. Note that the effect of beam size εor any displacement parallel to the slit edge has no effect on this component. To compare the effect of the beam size δwith effect of the divergence ε′, consider a line scan of the horizontal intensity component observed over a finite bandwidth (∆ ω) taken in the plane, Y= 0. Eq. (7) then reduces to: dNHoriz. dΩ=/vextendsingle/vextendsingler/bardbl/vextendsingle/vextendsingle2αγ2 2π2∆ω ωX2 (1 +X2)2e−R(1+X2)1/2·[1 + (δ γ¯λ)2(1 +X2)]. (10) A convolution of this expression with the Gaussian distribu tion,G(X, ε′), evaluated at X= 0, to first order in ε′2, gives dNHoriz. dΩ⊗G(X, ε′)∝ε′2[1 + (δ γ¯λ)2]. (11) Thus, while it is still present, the effect of the beam size δon the horizontal component will be much smaller than that of the divergence ε′. Figure 6. shows how a change in the divergence ε′affects the horizontal intensity for a fixed value of the beam s izeδ= 300 µ. As predicted from Eq.(11), the effect of ε′is maximum near the origin. Conversely, the shape of pattern shape in the vicinity of the origin can be used to se parate out and measure the divergence ε′. Figure 7. shows the effect of the beam size, δon a line scan of the horizontal intensity (Eq. (12)) taken in the X, Zplane, i.e. Y= 0, when ε′= 0 and R= 2. Figure 7. shows that a change in δhas no effect on the horizontal component in the region near X= 0, but does affect the peak value of the intensity observed at |X| ≈1 and the fall off of intensity for|X|>1, for δ >50µ. Numerical calculations indicate similar variation of the intensities with beam size and divergence for R= 0.5. Therefore, for simplicity, we will only present below numerical results for a single value of the parameter R= 2. Vertical Intensity Component A procedure similar to the one described above can be used to show that the effect of the divergence ε′has a minimal effect on the vertical intensity 10distribution. Figure 8. shows two scans of the vertical inte nsity observed in Figure 4B in the planes X=ε′= 0.2 and X= 0. As similarly shown in Fig. 5., the difference in the two scans is very small. Thus, it should be possible to use a one di mensional convolution to infer the effect of the divergence δ′, on line scans of vertically polarized DR (measured in the Y direction) for a number of Xvalues. By comparing the measured intensity scans to a set of one dime nsional convolutions in Yparameterized by the divergence component δ′, one would hope to be able to measure this quantity. However, for the vertical intensity compone nt the effect of the beam size δis comparable to that of the divergence δ′. To see this consider an observation of the vertical intensity component (Eq. (8)) in the Y, Zplane i.e. X= 0. For a finite bandwidth (∆ ω) measurement, Eq. (8) produces the vertical angular distrib ution dNV ert. dΩ=|r⊥|2α 4π2∆ω ωγ2e−R (1 +Y2)(12) ·[1 + 2(δ γ¯λ)2+2Ysin(RY)−(1−Y2) cos(RY) (1 +Y2)]. To compute the effect of the divergence δ′one must take the convolution of Eq. (12) with a distribution over the variable Y, e.g. the Gaussian distribution. G(Y, δ′) =1√ 2πδ′2exp(−Y2 2δ′2). We have performed this computation numerically to see how th e effects of beam size and divergence compare. Figure 9. shows a vertical intensity sc an taken at X= 0, for a fixed value of the beam size δ=200µandR= 2.0 for several values of divergence δ′. Figure 10. shows the effect of a change in the beam size, δon similar scans for the divergence value δ′= 0. A comparison of Figs. 9. and 10. clearly indicates that th e effect of divergences δ′<0.2 will compete significantly with the effect of beam sizes δ/lessorsimilar300µon the angular distribution of the vertical intensity component. Thus, in general, neither effect can be neglected and the two effects are not separable for this compo nent. 4.3 Strategies for separating beam size and divergence effec ts 4.3.1 Use of slit oriented perpendicular to the plane of inci dence In order to help separate out the competing effects of beam siz e and divergence, we have considered two alternative strategies. One method is to rot ate the slit (or insert another into the beam line) so that the slit edge is perpendicular to t he plane of incidence (see Fig. 2A.). In this configuration the horizontal intensity compon ent of the DR, convolved with the Gaussian function, G(X, δ′) =1√ 2πδ′2exp(−X2 2δ′2),can be used to measure the divergence δ′, which is now the component of divergence measured parallel to the slit edge. For this orientation the horizontal component is highly sensitive t oδ′and only weakly depend on the beam size ε, which is now the component of the beam size perpendicular to the slit edge. Evaluating the convolved horizontal intensity at X= 0, we obtain dNHoriz. dΩ⊗G(X, δ′)∝δ′2[1 + (ε γ¯λ)2]. (13) 11Eq.(13) is the direct analog of Eq.(11) above which, as shown in the previous section, can be used to determine ε′. For the perpendicular orientation of the slit edge, howeve r, the center of the horizontal AD pattern is sensitive to, and can be used t o measure, the divergence component δ′. The vertical DR intensity component for this orientation of the slit is: dNV ert. dΩ=/vextendsingle/vextendsingler/bardbl/vextendsingle/vextendsingle2α 4π2∆ω ωγ2e−R (1 +Y2)(14) ·[1 + 2(ε γ¯λ)2+2Ysin(RY)−(1−Y2) cos(RY) (1 +Y2)], which is the direct analog of Eq. (12). A one dimensional conv olution of Eq. (14) with the Gaussian distribution G(Y, ε′) =1√ 2πε′2exp(−Y2 2ε′2), produces a pattern that is sensitive to the divergence ε′as well as the beam size ε. Thus for the perpendicular slit orientation the divergence ε′and beam size εhave comparable effects and it is not possible to distinguish between them using the vertical intensity component alone. 4.3.2 DR Interferometry The second strategy we have devised to separate out and measu re divergence is to use the interference of DR produced from two slits inclined at 450with respect to the beam velocity in a configuration which is the direct DR analogy to a Wartski O TR interferometer[13]. In such a system forward DR from the first slit reflects from the second slit surface and interferes with backward DR generated from the second slit. The interferences produced by the two intensities will be superimposed on the single slit D R intensity. Figure 11. shows one possible configuration for a DR interferometer composed of two slits oriented with their edges parallel to the plane of incidence. The equation for in terference DR, in analogy to TR, is obtained by multiplying Eqs. (7) and (8) by an addition al interference term which is due to the difference in phase between forward and backward DR . The expressions for the horizontal and vertical intensity components are of the for m: dN(I) Horiz,V ert dΩ= 4dN(S) Horiz,V ert dΩsin2(L 2LV) (15) where the superscript Irefers to the two slit (interferometer), and Sto the single slit angular intensity distributions, respectively, LV≡λ/π(γ−2+θ2) is the coherence length in vacuum for TR and DR, which respresents the distance over which the p article’s field and the TR or DR photon differ in phase by one radian and Lis the path length between the slits. For a fixed wavelength and energy the single slit term varies s lowly with the angle θ. Then a convolution of Eq.(15) with a distribution of beam angles w ill chiefly effect the interference term sin2(L 2LV). The interference fringe visibility is then a function of t he angular divergence and is independent of beam size effects. Thus it should be poss ible to use DR interferences to obtain a measurement of the beam divergence alone. Also as is the case with interference 12OTR, measurements of the two polarized components of interf erence DR can be used to measure orthogonal components of the beam divergence[14]. The interference fringes, which are superimposed on the single slit angular distribution, p rovide increased sensitivity to angular differences in the particle trajectory angles and th erefore smaller values of divergence (δ′,ε′/lessorsimilar0.05) can be measured with an interferometer than with a single slit alone. Figures 12. and 13. show the vertical and horizontal compone nts of intensity, respectively, from a DR interferometer produced by an electron beam with en ergyE= 500 MeV, R= 2, λ=3.2µm,δ=200µandδ′=0.05. The separation distance L= 6LV= 3 meters for this beam energy. In any DR interferometer the angular distribution o f the forward DR from the first foil will be partially cut off by the presence of the second sli t in such a configuration. If the path length between the slits is large, the total angular fiel d will only be cut off in the center of the pattern by a small amount. Thus only a small fraction of the interference patterns (1/6γfor the parameter range specified above) will be cut out. This effect is not shown in Figs. 12. and 13. In contrast to the very weak effect of a small value of divergen ce, i.e. δ′= 0.05,on single slit DR (see Figs. 7. and 9.), the effect of this divergence on t he DR interference fringe visibility is clearly visible. Since the fringes are well mo dulated out to Y∼4, the cutoff of the field of view due to the second aperture will have a negligi ble effect on a measurement of the divergence, which uses the interference pattern. Wit h the help of a second camera or imager, a simultaneous measurement of backward DR from the fi rst slit can be made and thereby provide addition single slit data for extracting th e beam size. At the present time the use of the angular distribution of DR f or diagnostics is practically limited to beams with moderate to high energies by the vacuum coherence length LV∼γ2λ. The DR produced by upstream sources such as beam line discont inuities or other apertures will destructively interfere with backward DR generated fr om a diagnostic aperture when the distance between the upstream source and the aperture is much less than LV. Also, the successful application of interference DR to measure beam d ivergences δ′, ε′/lessorsimilar0.05 requires that the inter-aperture distance L/greaterorsimilar5LV. Therefore, the diagnostic techniques described in this paper are limited to observation wavelengths and beam e nergies which do not give rise to an impracticably long coherence length for the accelerat or facility being used. The means to overcome this limit must be developed before the methods d escribed here can be applied to beams with very high energy. 5 Summary We have developed a new method to calculate backward reflecte d diffraction radiation from any type of aperture. Using this method we have derived the eq uations for backward hori- zontal and vertically polarized spectral-angular intensi ties of DR from a slit inclined at an arbitrary angle with respect to the particle velocity. We ha ve obtained results for two orthog- onal orientations of the slit edges, i.e. parallel and perpe ndicular to the plane of incidence. The results of our calculations reduce in the single edge lim it precisely to those previously 13derived using the exact Wiener Hopf method. However, in cont rast to the Wiener Hopf approach, which has been successfully used to calculate DR f rom one type of aperture only, i.e. the single edge, our approach is readily applicable to a ny type of aperture and is much simpler to employ. From an analysis of backward DR for the two orientations of th e slit mentioned above, we have developed a number of strategies to determine four un known beam parameters of interest: the two orthogonal beam divergences, δ′,ε′, which are perpendicular and parallel to the incidence plane, respectively, and the two correspon ding orthogonal components of the beam size: δ, ε. These strategies require the measurement of the horizonta lly and ver- tically polarized intensities from a single slit or a double slit DR interferometer. We have demonstrated that one dimensional convolved scans of the ho rizontal intensities in either theX= const. or Y= const. plane can be used to separately measure δ′andε′, respec- tively. With the divergences known, the corresponding beam size components δandεcan be inferred. Since each of these scans can be produced at mult iple angles in the plane of observation, a large amount of data is available which can be used to reduce the error in the measurement of all the beam parameters of interest. The stat egies we have developed are useful for moderate to high energy lepton or hadron beams. 6 Appendix To proceed to evaluate the integral presented in Eq. (1) we ex pressEix(x, y) by its Fourier transform, so that we have Ex(kx, ky) =ie 8π41 vr/bardbl(Ψ)/integraldisplay/integraldisplayk′ x (k′2 x+k′2 y+ ¯α2)ei(− →k′−− →k)·− →ρdk′ xdk′ ydxdy. (16) Here, ¯ α≡1/(βγ¯λ), and the phase term appearing in Eq. (16) above is (− →k′−− →k)·− →ρ= (k′ x−¯kx)x′sin Ψ + ( k′ y−ky)y′(17) where ¯kx≡kx+ (kz−− →k′·− →v v) cotΨ. (18) Note that− →k′·− →v=ω, where ωis the frequency of the Fourier component of the field. This phase term takes into account the variation in phase wit h position− →ρof the− →k′Fourier component of the radiation field. For relativistic electron s, typically, γ >> 1 and− →kis nearly along the direction Zin Fig. 1., which is the direction into which a ray along− →v would be reflected if the screen were replaced by a mirror. The vector− →k=− →kx+− →ky+− →kz≈ k(θxˆx+θyˆy+ ˆz) (see the x, y, z coordinate system in Fig. 1.), where θ2=θ2 x+θ2 yand θx,θy<<1 . For Ψ = 450, we have ¯kx≈k(θx−γ−2/2)≈kθx. Therefore, in general, the radiation pattern is shifted by an angle γ−2/2<<1 . 14Thexandyintegrations are straight forward. The variable y′ranges from −∞toa1and a2to +∞, where the electron trajectory along− →vis taken to be a distance a1from the lower edge and a distance a2from the upper edge of the slit so that a=a1+a2in Fig. 1. The x integration yields 2 πδ(k′ x−¯kx) so that the k′ xintegral can be done trivially. Using ¯kx≈kx, when γ >> 1 , we obtain Ex(kx, ky) =ie 2π21 vr/bardbl(Ψ)/braceleftbigg1 2π[I1+I2] +kx (k2x+k2y+ ¯α2)/bracerightbigg (19) where I1= (−i)/integraldisplay+∞ −∞dk′ ykx (k2 x+k2 y+ ¯α2)1 k′ y−kye−i(k′ y−ky)a1, (20) and I2= (+i)/integraldisplay+∞ −∞dk′ ykx (k2 x+k2 y+ ¯α2)1 k′ y−kyei(k′ y−ky)a2. (21) Now we can write k2 x+k2 y+ ¯α2= (k′ y−if)(k′ y+if), (22) where f2≡k2 x+ ¯α2. (23) We see that I1andI2have poles at k′ y=±if,and at k′ y=ky, where I1=I∗ 2(a2→a1). These integrals can be done using the contours shown in Fig. 1 4. The results are I1=π/braceleftbiggkx f1 (f−iky)e−a1(f−iky)−kx (k2 x+k2 y+ ¯α2)/bracerightbigg , (24) and I2=π/braceleftbiggkx f1 (f+iky)e−a2(f+iky)−kx (k2x+k2y+ ¯α2)/bracerightbigg . (25) Substituting these expressions for I1andI2into Eq. (19) above, one obtains Eqs.(2). Eq.(3), the expression for Ey(kx, ky) can be derived in a similar manner. Note that when a1,2→0, Eq. (19) gives the result for the transition radiation field a s expected. 15References [1] F. G. Bass, V. M. Yakovenko, Sov. Phys. Uspecki 8 (3) (1965 ), 420. [2] A. P. Kazantsev, G. I. Surdutovich, Sov.Phys. Dokl. 7 (19 63) 990. [3] E. Keil, Nuc. Instrum. and Meth. 100 (1972), 419. [4] M. L. Ter-Mikaelian, High Energy Electromagnetic Proce sses in Condensed Media, Wiley-Interscience, New York, 1972. [5] N. J. Maresca and R. L. Liboff, Can. J. Phys. 53 (1975), 62. [6] Y. Shibata, et al., Phys. Rev. E 52 (1995), 6787. [7] I. Vnukov, et.al. JETP Letter, 67,(1998), 802. [8] A. P. Potylitsin, Nuc. Instr. and Meth. B, 145 (1998), 169 . [9] M. Castellano, Nuc. Instr. Meth. A 394 (1997), 275. [10] D.W. Rule, R.B. Fiorito and W.D. Kimura, Nondestructiv e Beam Diagnostics Based on Diffraction Radiation, AIP Conf. Proc. 390, (1997), pp. 510-517 ; R. B. Fiorito, D. W. Rule and W. D. Kimura, Noninvasive Beam Positi on, Size, Divergence and Energy Diagnostics Using Diffraction Radiation, in CP47 2,Advanced Accelerator Concepts: Eighth Workshop, ed. W. Lawson, C. Bellamy and D. Brosius, AIP (1999), 725. [11] M.J. Moran and B. Chang, Nucl Instr. Meth. Phys. Res. B40 /41, (1989), 970. [12] A. P. Potylitsin and N.A. Potylitsyna, Feasibility of B ackward Diffraction Ra- diation for Nondestructive Beam Diagnostics of Relativist ic Charge Particle Beams, Tomsk Polytechnic University, Tomsk, Russia, submitted to Phys. Lett.; arXiv, physics/0002034. [13] L. Wartski, et al., J. Appl. Phys.,46 (1975), 3644; L. Wa rtski, Ph.D.Thesis, Uni- versite de Paris-Sud, Centre d’Orsay, France (1976), unpub lished. [14] R. B. Fiorito and D. W. Rule, OTR Beam Emittance Diagnost ics, in Conf. Proc. No. 319 Beam Instrumentation Workshop, R. E. Shafer, ed., AI P (1994), 21. 16FigureCaptions Figure 1 . A side view(1A), and a top view (1B), of backward diffraction radiation emitted by a charge passing through a slit which is oriented with its e dge parallel to the plane of incidence ( n, vplane). Figure 2 . A side view(2A), and a top view (2B), of backward diffraction radiation emitted by a charge passing through a slit which is oriented with its e dge perpendicular to the plane of incidence ( n, vplane). Figure 3. Horizontal (3A) and vertical components (3B) of the intensi ty of DR for R= a/γ¯λ= 0.5; the intensity (Z) axis is scaled in proportion to the corre sponding intensity component of TR. Figure 4. Horizontal (4A) and vertical components (4B) of the intensi ty of DR for R= 2.0; the intensity axis is scaled in proportion to the correspond ing intensity component of TR. Figure 5 . Effect of the divergence δ′on the horizontal component of the intensity. Shown are line scans in the variable Xfor two values of the variable Y=δ′= 0.2 and Y=δ= 0,for a fixed value of the divergence ε′= 0.1 and R= 2.0. Figure 6 . Effect of divergence ε′on the horizontal intensity. Shown are line scans in the variable Xfor a fixed value of the beam size δ= 300 µandR= 2. Figure 7 . Effect of beam size δon the horizontal intensity. Shown are line scans taken in theX, Zplane, i.e. Y= 0, for a fixed value of divergence ε′= 0 and R= 2.0. Figure 8 . Effect of the beam divergence ε′on the vertical intensity. Shown are line scans in the variable Yfor two values of the variable X=ε′= 0.2 and X=ε′= 0, for a fixed value of the divergence δ′= 0.1, and R= 2. Figure 9 . Effect of divergence δ′on the vertical intensity. Shown are line scans taken at X= 0 for a fixed value of the beam size, δ=200µandR= 2.0. Figure 10 . Effect of beam size δon the vertical intensity. Shown are line scans taken at X= 0 for a fixed value of the divergence, δ′= 0 and R= 2.0. Figure 11 . Configuration of a DR interferometer composed of two slits o riented with the edges parallel to the plane of incidence. Figure12 . Vertical component of intensity from a DR interferometer f or an electron beam with energy E= 500 MeV; R= 2,λ= 3.2µm,δ= 200 µ,δ′= 0.05 and separation distance L= 6Lv= 3 meters. Figure13. Horizontal component of intensity from a DR interferometer for an electron beam with same parameters as in Fig. 12. 17Figure 14 . Contours used to evaluate the integrals I1andI2defined by Eqs. (20) and (21), respectively.mo 18Figure 1Y ’ ( out ) X ( /G35 ) Knδ Ψ V a Y (/GB5 , out )X ’Y(/GB5)Y ’ X ’ ZΨ Z ( 1 A ) ( 1 B )V X ( /G35 ) 19Figure 2ε sinΨ a/2 sinΨ X ’(out) εΨ V Y (/G35)X ’ a/2 sinΨY ’ X (/GB5, out)a ZKX(/GB5) Y(/G35)Y ’ Ψ Z ( 2A ) ( 2B )V 202122X = γθx-4-3-2-101234Arbitrary Units 0.00.20.40.60.81.0 Y = δ' = 0 Y = δ' = 0.2 Figure 5 23X = γ θx-4-3-2-101234No. Photons per 10% BW per sterad per electron 01020304050607080 ε' = 0.05 ε' = 0.1 ε' = 0.2 Figure 6 24X = γ θx-4-3-2-101234No. Photons per 10% BW per sterad per electron 01020304050607080 δ = 50 µ δ = 100 µ δ = 200 µ δ = 300 µ Figure 7 25Y = γθy-4-3-2-101234Arbitrary Units 0.00.20.40.60.81.0 X = ε' = 0 X = ε' = 0.2 Figure 8 26Y = γθY-4-3-2-101234No. Photons per 10% BW per steradian per electron 050100150200250300350 δ' = 0 δ' = 0.1 δ' = 0.2 Figure 9 27Y = γ θy-4-3-2-101234No. Photons per 10% BW per sterad per electron 050100150200250300350 δ = 300 µ δ = 200 µ δ = 100 µ δ = 50 µ Figure 10 28Figure 11a YY ’ δ Ψ X ’ KV aLY ’ X ’ X 29Y = γθy-5-4-3-2-1012345Arbitrary Units 01234 Single Slit DR Two Slit IDR Figure 12 30X = γθx-5-4-3-2-1012345Arbitrary Units 0.00.20.40.60.81.0 Single Slit DR Two Slit IDR Figure 13 31Figure 14Re k'y+if -ifIm k'y ky I1I2I2 I1 32

arXiv:physics/0003022v1 [physics.gen-ph] 9 Mar 2000COMMENTS ON THE PAPER ”ON THE UNIFICATION OF THE FUNDAMENTAL FORCES...” B.G. Sidharth Centre for Applicable Mathematics & Computer Sciences B.M. Birla Science Centre, Hyderabad 500 063 Abstract In this brief paper we justify observations made in El Naschi e’s pa- per ”On the Unification of the Fundamental Forces...”, on the Planck scale, fractal space time and the unification of interaction s, from dif- ferent standpoints. In a recent paper El Naschie[1] has emphasized the intimate c onnection be- tween the Planck length, the Compton wavelength and a unifica tion of the fundamental interactions within the framework of complex t ime and a frac- tal Cantorian space. We will now make some observations whic h justify the contentions made in the above paper, from different standpoi nts. 1. Complex Time: It has been pointed out[2, 3, 4] that Fermion s can be thought of as Kerr-Newman Black Holes, in the context of quan tized space time: There are minimum space time intervals, and when we ave rage over these, Physics arises. Within the minimum intervals, we enc ounter un- physical Zitterbewegung effects, which also show up as a comp lexification of coordinates[5] - indeed they are the double Weiner proces s discussed by Abbott and Wise, Nottale and others. It may be mentioned that the transi- tion from the Kerr metric in General Relativity to the Kerr-N ewman metric is obtained by precisely such a complex shift of coordinates , a circumstance which has no clear meaning in Classical Physics[6]. On the ot her hand it is this ”Classical” Kerr-Newman metric which describes the fie ld of an electron 1including the Quantum Mechanical anomalous gyro magnetic r atio,g= 2. This has been discussed in detail in references[2, 3]. 2. The Unification of Electromagnetism and Gravitation and t he Planck Scale: It is in the context of point 1 that we arrive at a unified picture of electromagnetism and gravitation (Cf.ref.[4]). The poi nt is that at the Compton wavelength scale we have purely Quantum Mechanical effects like Zitterbewegung, spin half and electromagnetism, while at t he Planck scale, we have a purely classical Schwarschild Black Hole. However the Planck scale is the extreme limit of the Compton scale, where electr omagnetism and gravitation meet (Cf.ref.[4], and [7]). This is because for a Planck mass mP∼10−5gmswe have Gm2 P e2∼1 (1) whereas for an elementary particle like an electron we have t he well known equation Gm2 e2∼1√ N≡10−40(2) where N∼1080is the number of elementary particles in the universe. Another way of expressing this result is that the Schwarzsch ild radius for the Planck mass equals its Compton wavelength. This is where Quantum Mechanics and Classical Physics meet. This point can be analysed further[7]. From equations (1) an d (2) it can be seen that we obtain the Planck mass, when the number of partic les in the universe is 1. Indeed as has been pointed out by Rosen[8], the Planck mass can be considered to be a mini universe, in the context of the S chrodinger equation with the gravitational interaction. 3. The above brings us to another interesting aspect discuss ed by El Naschie in[1]. This is the fact that the Planck mass is intimately rel ated to the Hawk- ing radiation, and infact from the latter consideration we c an deduce that a Planck mass ”evaporates” within about 10−42secs, which also happens to be its Compton time! On the other hand, as pointed out in[5, 7], an elementary part icle like the pion is intimately related to Hagedorn radiation which lead s to a life time of the order of the age of the universe. The above two conclusions have been obtained on the basis of a background Zero Point Field, the Langevin equation and space time cut off s leading to a 2fluctuational creation of particles at the Planck scale and t he Compton scale respectively. 4. Resolution and the Unification of Interactions: El Naschi e[1] has referred to the fact that there is no apriori fixed length scale (the Bie denharn conjec- ture). Indeed it has been argued by the author in the above con text[9, 10] that depending on our scale of resolution, we encounter elec tromagnetism well outside the Compton wavelength, strong interactions a t the Compton wavelength or slightly below it and only gravitation at the P lanck scale. The differences between the various interactions are a manifest ation of the reso- lution. 4. The Universe as a Black Hole: As pointed out in[1] by El Nasc hie and the author[11, 12], the universe can indeed be considered to be a black hole. Prima Facie this is clear from the fact that the radius of the u niverse is of the order of the Schwarzchild radius of a black hole with the s ame mass as the universe. Also as pointed out in[11]) the age of the unive rse coincides with the time taken by a ray of light to travel from the horizon of a black hole to its centre or vice versa. 5. The ”Core” of the Electron: El Naschie refers to the core of the electron ∼10−20cms, as indeed has been experimentally noticed by Dehmelt and Co - workers[13]. It is interesting that this can be deduced in th e context of the electron as a Quantum Mechanical Kerr-Newman Black Hole. It was shown that[2] for distances of the order of the Compton wavelength the potential is given in its QCD form V≈ −βM r+ 8βM(Mc2 ¯h)2.r (3) For small values of rthe potential (3) can be written as V≈A re−µ2r2, µ=Mc2 ¯h(4) It follows from (4) that r∼1 µ∼10−21cm. (5) Curiously enough in (4), rappears as a time, which is to be expected because at the horizon of a black hole randtinterchange roles. One could reach the same conclusion, as given in equation (5) from a different 3angle. In the Schrodinger equation which is used in QCD, with the potential given by (3), one could verify that the wave function is of the typef(r).e−µr 2, where the same µappears in (4). Thus, once again one has a wave packet which is negligible outside the distance given by (5). It may be noted that Brodsky and Drell[14] had suggested from a very dif- ferent viewpoint viz., the anomalous magnetic moment of the electron, that its size would be limited by 10−20cm. The result (5) was experimentally con- firmed by Dehmelt and co-workers [13]. Finally, it may be remarked that it is the fractal double Wein er process referred to earlier that leads from the real space coordinat e,xsay, to the complex coordinate x+ıct(Cf.ref.[5]), which is the space and time divide: As pointed out by Hawking[15] and others, an imaginary time w ould lead to a ”static” Euclidean four geometry, rather than the Minko wski world, a concept that has been criticised by Prigogine[16]. It is in t he above fractal formulation (Cf.ref.[5]), on the contrary, that we see the e mergence of the space and time divide, that is, time itself. Thus in conclusion, it may be said that the recognition of a fr actal quantized underpinning of space time ties together several apparentl y disparate facts. References [1] M.S. El Naschie, ”On the unification of the fundamental fo rces and complex time.......”, to appear in Chaos Solitons and Fract als. [2] B.G. Sidharth, Ind.J. Pure & Appld.Phys., Vol.35, 1997, pp.456- 471.IJPAP [3] B.G. Sidharth, Int.J. of Mod.Phys.A, 13(15), pp2599ff, 1 998. [4] B.G. Sidharth, Gravitation & Cosmology, 4 (2) (14), 158ff , 1998. [5] B.G. Sidharth, ”Space Time as a Random Heap”, to appear in Chaos, Solitons and Fractals. [6] E.T. Newman, J.Math.Phys.14 (1), 1973, p102. [7] B.G. Sidharth, ”Emergence of Planck Scale”, to appear in Chaos Soli- tons and Fractals. 4[8] N. Rosen,International Journal of Theoretical Physics , 32 (8), 1993, 1435-1440. [9] B.G. Sidharth, ”Universe of Chaos and Quanta”, Chaos, So litons and Fractals, 11 (8), 2000, 1269-1278. [10] B.G. Sidharth, ”Instantaneous Action at a Distance in a Holistic Uni- verse”, to appear in IAAD Pro & Contra, Eds. A Chubykalo et al, Nova Science Publishers, New York, 1999. [11] B.G. Sidharth, ”Fluctuational Cosmology” in Quantum M echanics and General Relativity” in Proceeding of the Eighth Marcell Gro ssmann Meeting on General Relativity, Ed., T. Piran, World Scienti fic, Singa- pore, 1999, pp.476ff. [12] B.G. Sidharth, ”The Scaled Universe”, to appear in Chao s Solitons and Fractals. [13] H. Dehmelt, Physica Scripta T22, 1988, pp102ff. [14] S.J. Brodsky, and S.D. Drell, Phys. Rev. D 22(9), 1980, p p2236ff. [15] S.W.H. Hawking, ”A Brief History of Time”, Bantam Books , New York, 1988. [16] I. Prigogine, ”The End of Certainity”, Free Press, New Y ork, 1997. 5

1 The periodic table of elementary particles Ding-Yu Chung P.O. Box 180661, Utica, Michigan 48318, USA All leptons, quarks, and gauge bosons can be placed in the periodic table of elementary particles. The periodic table is derived from dualities of string theoryand a Kaluza-Klein substructure for the six extra spatial dimensions. As amolecule is the composite of atoms with chemical bonds, a hadron is thecomposite of elementary particles with hadronic bonds. The masses ofelementary particles and hadrons can be calculated using the periodic table withonly four known constants: the number of the extra spatial dimensions in the superstring, the mass of electron, the mass of Z°, and α e. The calculated masses are in good agreement with the observed values. For examples, the calculated masses for the top quark, neutron, and pion are 176.5 GeV, 939.54MeV, and135.01MeV in excellent agreement with the observed masses, 176 ± 13 GeV,939.57 MeV, and 134.98 MeV, respectively. The masses of 110 hadrons arecalculated. The overall average difference between the calculated masses and theobserved masses for all hadrons is 0.29 MeV. The periodic table of elementaryparticles provides the most comprehensive explanation and calculation for themasses of elementary particles and hadrons.PACS number(s): 11.25.Sq, 11.30.Pb, 12.10.Kt, 12.40.YxKeywords: string field theory, supersymmetry, mass relations, hadron massmodels and calculations2 1. Introduction The mass hierarchy of elementary particles is one of the most difficult problems in particle physics. There are various approaches to solve the problem.In this paper, the approach is to use a periodic table. The properties of atoms canbe expressed by the periodic relationships in the periodic table of the elements.The periodic table of the elements is based on the atomic orbital structure. Doelementary particles possess orbital structure? In this paper, the orbital structurefor elementary particles is constructed from the superstring. The extra dimensionsin the superstring become the orbits with the Kaluza-Klein substructure. In theKaluza’s model of five-dimensional spacetime, the fifth dimension is considered tobe a one-dimensional circle associated with every point in ordinary flat four-dimensional spacetime. Similarly, in the Kaluza-Klein substructure for the extraspatial dimensions, the fifth dimension is a one-dimensional circle associated withevery point in ordinary flat four-dimensional spacetime, the sixth dimension circlesthe circle of the fifth dimension, and in the same way, every higher dimensioncircles the circle of the lower dimension. The energy is assumed to be distributeddifferently in different dimensions in such way that the energy increases withincreased number of spacetime dimensions. This Kaluza-Klein substructure isderived from the duality relations for the superstring. These duality relationsprovide the explanation for the co-existence of leptons and quarks, the structure ofthe dimensional orbits, and the compositions of elementary particles.3 All elementary particles are placed in the periodic table of elementary particles [1]. As a molecule is the composite of atoms with chemical bonds, ahadron is the composite of elementary particles with hadronic bonds. The massesof elementary particles and hadrons can be calculated using only four knownconstants: the number of the extra spatial dimensions in the superstring, the mass of electron, the mass of Z°, and α e. The calculated masses of elementary particles and hadrons are in good agreement with the observed values. For examples, the calculated masses for the top quark, neutron, and pion are 176.5 GeV,939.54MeV, and 135.01MeV in excellent agreement with the observed masses,176 ± 13 GeV [2], 939.57 MeV, and 134.98 MeV, respectively. 2. Dualities A membrane can be described as two dimensional object that moves in an eleven dimensional space-time [3]. This membrane can be converted into the tendimensional superstring with the extra dimension curled into a circle to become aclosed superstring. If the membrane is split into two membranes which are thenconverted into two superstrings connected by the extra dimensions. Themembrane becomes a dual superstring. It is proposed that the dual superstring isthe lepton-quark dual superstring consisting lepton superstring and quarksuperstring as shown in Fig. 1.4 lepton string 11D S-duality 1D 10D 1D 11D 10D split membrane quark string Fig. 1. S-duality There is S-duality between the split membrane and the dual superstring. Themembrane interacts strongly at the high energy, when the dual lepton-quarksuperstring interacts weakly. The four dimensional version of S-duality isMontonen-Olive duality [4] where the magnetic dipole is split into magneticmonopoles, and electrons and quarks arise as solitons. M. Duff and J. Lu [5] suggested the U-duality between the four dimensional space-time solitonic string in six dimensional space-time and the fundamental tendimensional superstring. In the same way, the dual lepton-quark superstring canalso have two four dimensional space-time solitonic strings in two sevendimensional space-time. There is U-duality between the dual solitonic string andthe dual lepton-quark superstring as shown in Fig. 2. 6D space-time U-duality 1D 1D 6D space-time Fig. 2. U-duality The Type II dual string can be compacified into dual D-branes in the dual 7D tori for the dual seven extra dimensions to form orbitfolds [6]. It is proposed5 that the energy associated with each extra dimension increases incrementally from the fifth to the eleventh dimension as shown in Fig. 3. 5D 5D T-duality 11D 11D Fig. 3. T-duality This process generates the Kaluza-Klein substructure for the dual tori. In the Kaluza-Klein substructure for the seven extra spatial dimensions, the fifthdimensional orbit is a one-dimensional circle associated with every point inordinary flat four-dimensional space-time, the sixth dimensional orbit circles thefifth dimensional orbit, and in the same way, every higher extra dimensional orbitwith higher energy circles the lower dimensional orbit with lower energy. Each extra space-time dimension can be described by a fermion and a boson. The masses of fermion and its boson partner are not the same. Thissupersymmetry breaking is in the form of a energy hierarchy with increasingenergies from the dimension five to the dimension eleven as F 5 B5 F6 B6 F7 B7 F8 B8 F9 B9 F10 B10 F11 B11 where B and F are boson and fermion in each spacetime dimension. Theprobability to transforming a fermion into its boson partner in the adjacent dimension is same as the fine structure constant, α, the probability of a fermion emitting or absorbing a boson. The probability to transforming a boson into its6 fermion partner in the same dimension is also the fine structure constant, α. This hierarchy can be expressed in term of the dimension number, D, E D-1, B = E D,F αD,F , (1) E D, F = E D, B αD, B , (2) where E D,B and E D,F are the energies for a boson and a fermion, respectively, and αD, B or αD,F is the fine structure constant, which is the ratio between the energies of a boson and its fermionic partner. All fermions and bosons are related by theorder 1/α. (In some mechanism for the dynamical supersymmetry breaking, the effects of order exp(-4 π 2/ g2 ) where g is some small coupling, give rise to large mass hierarchies [7].) Assuming αD,B = αD,F , the relation between the bosons in the adjacent dimensions, then, can be expressed in term of the dimension number, D, E D-1, B = E D, B α2 D,( 3 ) or EE DBDB D,,=−1 2α,( 4 ) where D= 6 to 11, and E 5,B and E 11,B are the energies for the dimension five and the dimension eleven, respectively. The lowest energy is the Coulombic field, E 5,B E 5, B = α M6,F = α Me,( 5 )7 where M e is the rest energy of electron, and α = αe , the fine structure constant for the magnetic field. The bosons generated are called "dimensional bosons" or"B D". Using only αe, the mass of electron, the mass of Z 0, and the number of extra dimensions (seven), the masses of B D as the gauge boson can be calculated as shown in Table 1. Table 1. The Energies of The Dimensional Bosons BD = dimensional boson, α = αe BD ED GeV Gauge Boson Interaction B5 Me α 3.7x10-6A electromagnetic B6 Me/α 7x10-2π1/2 strong B7 E6/αw2 cos θw91.177 Z L0weak (left) B8 E7/α21.7x106XR CP (right) nonconservation B9 E8/α23.2x1010XL CP (left) nonconservation B10 E9/α26.0x1014ZR0weak (right) B11 E10/α21.1x1019 In Table 1, αw is not same as α of the rest, because there is symmetry group mixing between B 5 and B 7 as the symmetry mixing in the standard theory of the electroweak interaction, and sin θw is not equal to 1. As shown latter, B 5, B6, B7, B8, B9, and B 10 are A (massless photon), π1/2, ZL0, XR, XL, and Z R0, respectively, responsible for the electromagnetic field, the strong interaction, the weak (left8 handed) interaction, the CP (right handed) nonconservation, the CP (left handed) nonconservation, and the P (right handed) nonconservation, respectively. The calculated value for θw is 29.690 in good agreement with 28.70 for the observed value of θw [8]. The total energy of the eleven dimensional superstring is the sum of all energies of both fermions and bosons. The calculated total energy is 1.1x1019 GeV in good agreement with the Planck mass, 1.2x1019 GeV, as the total energy of the superstring. As shown later, the calculated masses of all gaugebosons are also in good agreement with the observed values. Most importantly,the calculation shows that exactly seven extra dimensions are needed for allfundamental interactions. In the Kaluza-Klein substructure, the bosons and the fermions in the dimensions > 5 gain masses in the four dimensional spacetime by combining withscalar Higgs bosons in the four dimensional spacetime. Without combining withscalar Higgs bosons, neutrino and photon remain massless. Graviton, themassless boson from the vibration of the superstring, remains massless. 3. The Internal Symmetries And The Interactions All six non-gravitational dimensional bosons are represented by the internal symmetry groups, consisting of two sets of symmetry groups, U(1), U(1), andSU(2) with the left-right symmetry. Each B D is represented by an internal symmetry group as follows.B 5: U(1), B 6: U(1), B 7: SU(2) L, B 8: U(1) R, B9: U(1) L, B 10: SU(2) R9 The additional symmetry group, U(1) X SU(2) L, is formed by the "mixing" of U(1) in B5 and SU(2) L in B 7. This mixing is same as in the standard theory of the electroweak interaction. As in the standard theory to the electroweak interaction, the boson mixing of U(1) and SU(2) L is to create electric charge and to generate the bosons for leptons and quarks by combining isospin and hypercharge. The hypercharges for both e+ and ν are 1, while for both u and d quarks, they are 1/3 [9]. The electric charges for e+ and ν are 1 and 0, respectively, while for u and d quarks, they are 2/3 and - 1/3, respectively. There is no strong interaction from the internal symmetry for the dimensional bosons, originally. The strong interaction is generated by"leptonization" [10], which means quarks have to behave like leptons. Leptonshave integer electric charges and hypercharges, while quarks have fractionalelectric charges and hypercharges. The leptonization is to make quarks behavelike leptons in terms of "apparent" integer electric charges and hypercharges.Obviously, the result of such a leptonization is to create the strong interaction,which binds quarks together in order to make quarks to have the same "apparent"integer electric charges and hypercharges as leptons. There are two parts for thisstrong interaction: the first part is for the charge, and the second part is for thehypercharge. The first part of the strong interaction allows the combination of aquark and an antiquark in a particle, so there is no fractional electric charge. It involves the conversion of π 1/2 boson in B 6 into the electrically chargeless meson10 field by combining two π1/2, analogous to the combination of e+ and e- fields, so the meson field becomes chargeless. (The mass of π, 135 MeV, is twice of the mass of a half-pion boson, 70 MeV, minus the binding energy. π1/2 becomes pseudoscalar up or down quark in pion.) In the meson field, no fractional charge of quark can appear. The second part of the strong interaction is to combine threequarks in a particle, so there is no fractional hypercharge. It involves the conversion of B 5 (π1/2) into the gluon field with three colors. The number of colors (three) in the gluon field is equal to the ratio between the lepton hypercharge andquark hypercharge. There are three π 1/2 in the gluon field, and at any time, only one of the three colors appears in a quark. Quarks appear only when there is a combination of all three colors or color-anticolor. In the gluon field, no fractionalhypercharge of quarks can appear. By combining both of the meson field and thegluon field, the strong interaction is the three-color gluon field based on the chargeless vector meson field from the combination of two π 1/2's. The total number of π1/2 is 6, so the fine structure constant, αs, for the strong force is ααS e= =6 01191 .(6) which is in a good agreement with the observed value, 0.124 [11]. The dimensional boson, B 8, is a CP violating boson, because B 8 is assumed to have the CP-violating U(1) R symmetry. The ratio of the force constants between the CP-invariant W L in B 8 and the CP-violating X R in B 8 is11 8 7722 W W 82 -10G G = E E = 5.3 X 10 ,α αcosΘ (7) which is in the same order as the ratio of the force constants between the CP- invariant weak interaction and the CP-violating interaction with ∆S = 2. The dimensional boson, B 9 (XL), has the CP-violating U(1) L symmetry. The lepton (l 9) and the quark (q 9) are outside of the three families for leptons and quarks, so baryons in dimension nine do not have to have the baryon numberconservation. The baryon which does not conserve baryon number has thebaryon number of zero. The combination of the CP-nonconservation and the baryon number of zero leads to the baryon, p 9− , with the baryon number of zero and the baryon, p 9 + , with the baryon number of 1. The decay of p9− is as follows. p9 − → l9− l9° The combination this p9 − and p 9+ as well as leptons, l 9° l9° results in p9 + p9 − + l 9° l9°→ p 9 + + l9− l9° + l9° l9° → p9 + + l9− + radiation + l9° → n9 ° + radiation Consequently, excess baryons, n 9 °, are generated in the dimension nine. These excess baryons, n 9 ° , become the predecessors of excess neutrons in the low energy level.12 The ratio of force constants between X R with CP-independent baryon number and X L with CP-dependent baryon number is 9 882 92 -9G G = E E = 2.8 X 10 ,α α (8) which is in the same order as the observed ratio of the numbers between the left handed baryons (proton or neutron) and photons in the universe. The direct transition of the superstring to q 9 without going through the last two extra dimensions (B 11, F11, B10, and F 10) may account for the inflation in the inflationary universe model [12]. This inflation may be the first of the two inflations.The second inflation may involve the direct transition of the superstring from q 7 to the dimension four without going through the first two extra dimensions (B 6 = π1/2, F6 = e, B 5 = A, and F 5 = ν). It involves the symmetry mixing between dimensions five and seven, so the first two dimensions ( π1/2, e, ν, and A) are merged with the dimension seven, and become parts of force fields and decay modes for the fermions (heavy leptons and baryons) of F 7 with seven auxiliary dimensional orbits. Furthermore, during this second inflation, integer charge leptons, fractional chargequarks, the strong force field, and electroweak force field are generated.13 3. The periodic table of elementary particles Leptons and quarks are complementary to each other, expressing different aspects of superstring. The model for leptons and quarks is shown in Fig. 4. Theperiodic table for elementary particles is shown in Table 2. Lepton ν e e νµ ντ l 9 l10 µ7 τ7 µ8 D = 5 6 7 8 9 10 11 a = 0 1 2 3 4 5 0 1 2 d 7 s7 c7 b 7 t 7 b 8 t8 u 7 u d 3 µ µ′ q 9 q 10 Quark Fig. 4. Leptons and quarks in the dimensional orbits D = dimensional number, a = auxiliary dimensional number Table 2. The Periodic Table of Elementary Particles D = dimensional number, a = auxiliary dimensional number D a = 0 12a = 0 12345 Lepton Quark Boson 5 l5 = νe q5 = u = 3νe B5 = A 6l 6 = e q 6 = d = 3e B6 = π1/2 7 l7 = νµµ7τ7 q7 = 3µ u7/d7s7c7b7t7 B7 = Z L0 8 l8 = ντµ8 q8 = µ' b8t8 B8 = X R 9l 9 q9 B9 = X L 10 B 10 = Z R0 11 B 11 D is the dimensional orbital number for the seven extra space dimensions. The auxiliary dimensional orbital number, a, is for the seven extra auxiliary space14 dimensions. All gauge bosons, leptons, and quarks are located on the seven dimensional orbits and seven auxiliary orbits. Most leptons are dimensionalfermions, while all quarks are the sums of subquarks from the loops around the tori from the compacified extra dimensions. ν e, e, νµ , and ντ are dimensional fermions for dimension 5, 6, 7, and 8, respectively. All neutrinos have zero mass because of chiral symmetry. The dimensional fermions for D = 5 and 6 are neutrino (l 5) and electron (l 6). Other fermions are generated, so there are more than one fermion in the same dimension. These extra fermions include quarks and heavy leptons ( µ and τ). To generate a quark whose mass is higher than the lepton in the same dimension is to add the lepton to the boson from the combined lepton-antilepton, so the mass ofthe quark is three times of the mass of the corresponding lepton in the samedimension. The equation for the quark mass is MM qlDD=3 (9) The mixing between the fifth and the seventh dimensional orbits which is same as the symmetry mixing in the standard model allows the seventhdimensional orbit to be the starting dimensional orbit for the additional orbits,auxiliary orbits with quantum number “a.” (The total number of the auxiliary orbitsis also seven as the seven dimensional orbits.) A heavy lepton ( µ or τ) is the combination of the dimensional leptons and the auxiliary dimensional leptons. In the same way, a heavy quark is the combination of the dimensional quarks from Eq.(9) and the auxiliary quarks. The15 mass of the auxiliary dimensional fermion (AF for both heavy lepton and heavy quark) is generated from the corresponding dimensional boson as follows. MM aAFB a aa DaD ,,, =− =∑10 4 0α(10) where αa = auxiliary dimensional fine structure constant, and a = auxiliary dimension number = 0 or integer. The first term, MB aD−10, α, of the mass formula (Eq.(10)) for the auxiliary dimensional fermions is derived from the mass equation, Eq. (1), for the dimensional fermions and bosons. The second term, a aa 4 0=∑ , of the mass formula is for Bohr-Sommerfeld quantization for a charge - dipole interaction in a circular orbit as described by A. Barut [13]. 1/ αa is 3/2. The coefficient, 3/2, is to convert the dimensional boson mass to the mass of the auxiliary dimensional fermion in the higher dimension by adding the boson mass to its fermion masswhich is one-half of the boson mass. The formula for the mass of auxiliarydimensional fermions (AF) becomes D,aD-1,0 D DAFB a a=0a 4 B aa FD aaM = M a Ma Maα α∑ ∑ ∑= =− = =3 2 3 210 04 0 4 0, ,(11) When the mass of this auxiliary dimensional fermion is added to the sum of masses from the corresponding dimensional fermions (zero auxiliary dimension16 number) with the same electric charge and the same dimension, the fermion mass formula for heavy leptons and quarks is derived as follows. MM M MM a MM aFF A F FB aa FF D aaDa D Da DD DD,, , ,, ,,=+ =+ =+∑ ∑∑ ∑∑− = =0 01 0 003 2 3 24 0 4 0α(12) Each fermion can be defined by dimensional numbers (D's) and auxiliary dimensional numbers (a's). Heavy leptons and quarks consist of one or more D's and a's.The compositions and calculated masses of leptons and quarks are listed in Table 3.17 Table 3. The Compositions and the Constituent Masses of Leptons and Quarks D = dimensional number and a = auxiliary dimensional number Da CompositionCalc. Mass Leptons Da for leptons νe 50 νe 0 e6 0 e 0.51 MeV (given) νµ 70 νµ 0 ντ 80 ντ 0 µ 60 + 7 0 + 7 1 e + νµ + µ7 105.6 MeV τ 60 + 7 0 + 7 2 e + νµ + τ7 1786 MeV µ' 60 + 7 0 + 7 2 + 8 0 + 8 1e + νµ + µ7 + ντ + µ8 136.9 GeV Quarks Da for quarks u5 0 + 7 0 + 7 1 u5 + q 7 + u 7 330.8 MeV d6 0 + 70 + 7 1 d6 + q 7 + d 7 332.3 MeV s6 0 + 70 + 7 2 d6 + q 7 + s 7 558 MeV c5 0 + 7 0 + 7 3 u5 + q 7 + c 7 1701 MeV b6 0 + 70 + 7 4 d6 + q 7 + b 7 5318 MeV t5 0 + 7 0 + 7 5 + 8 0 + 8 2u5 + q 7 + t7 + q 8 + t8 176.5 GeV The lepton for dimension five is νe, and the quark for the same dimension is u5, whose mass is equal to 3 M νe from Eq. (9). The lepton for the dimension six is e, and the quark for this dimension is d 6. u5 and d 6 represent the “light quarks” or “current quarks” which have low masses. The dimensional lepton for the dimensions seven is νµ. All ν's become massless by the chiral symmetry to preserve chirality. The auxiliary dimensional leptons (Al) in the dimension sevenare µ 7 and τ7 whose masses can be calculated by Eq. (13) derived from Eq. (11).18 7, a 6, 0 2/1Al B a=0a 4 a=0a 4M = 3 2 M a = 3 2 M a ,∑ ∑π(13) where a = 1, 2 for µ7 and τ7, respectively. The mass of µ is the sum of e and µ7, and the mass of τ is the sum of e + τ7, as in Eq.(12). For heavy quarks, q 7 (the dimensional fermion, F 7, for quarks in the dimension seven) is 3 µ instead of massless 3 ν as in Eq. (9). According to the mass formula, Eq. (11), of the auxiliary fermion, the mass formula for the auxiliary quarks, Aq 7,a , is as follows. where α7 = αw , and a = 1, 2, 3, 4, and 5 for u 7/d7, s7, c7, b7, and t 7, respectively. The dimensional lepton for the dimension eight is ντ , whose mass is zero to preserve chirality. The heavy lepton for the dimensional eight is µ‘ as the sum of e, µ, and µ8 (auxiliary dimensional lepton). Because there are only three families for leptons, µ' is the extra lepton, which is "hidden". µ' can appear only as µ + photon. The pairing of µ + µ from the hidden µ' and regular µ may account for the occurrence of same sign dilepton in the high energy level [14]. For the dimension7, a 7Aq q 7 a=0a 4 w a=0a 4M = 3 2 M a = 3 2 (3M) a ,α αµ∑ ∑(14)19 eight, q 8 (the F 8 for quarks) is µ' instead of 3 µ', because the hiding of µ' allows q 8 to be µ'. The hiding of µ' for leptons is balanced by the hiding of b 8 for quarks. The calculated masses are in good agreement with the observed constituent masses of leptons and quarks [2,15]. The mass of the top quark foundby Collider Detector Facility is 176 ± 13 GeV [2] in a good agreement with thecalculated value, 176.5 GeV. 4. Hadrons As molecules are the composites of atoms, hadrons are the composites of elementary particles. Hadron can be represented by elementary particles in manydifferent ways. One way to represent hadron is through the nonrelativisticconstituent quark model where the mass of a hadron is the sum of the masses of quarks plus a relatively small binding energy. π 1/2 (u or d pseudoscalar quark), u, d, s, c, b, and t quarks mentioned above are nonrelativistic constituent quarks. On the other hand, except proton and neutron, all hadrons are unstable, and decayeventually into low-mass quarks and leptons. Is a hadron represented by allnonrelativistic quarks or by low-mass quarks and leptons? This paper proposes a dual formula: the full quark formula for all nonrelativistic constituent quarks ( π 1/2, u, d, s, c, and b) and the basic fermion formula for the lowest- mass quark ( π1/2 and u) and lepton (e). The calculation of the masses of hadrons requires both formulas. The full quark formula sets the initial mass for a hadron. This initialmass is matched by the mass resulted from the combination of various particles20 in the basic fermion formula. The mass of a hadron is the mass plus the binding energy in the basic fermion formula. M full quark formula ≈ M basic fermion formula M hadron = M basic fermion formula + binding energy (15) The full quark formula consists of the vector quarks (u, d, s, c, and b) and pseudoscalar quark, π1/2 (70.03 MeV), which is B 6 (dimension boson in the dimension six). The strong interaction converts B 6 (π1/2) into pseudoscalar u and d quark in pseudoscalar π meson. The combination of the pseudoscalar quark (π1/2) and vector quarks (q) results in hybrid quarks (q' ) whose mass is the average mass of pseudoscalar quark and vector quark. Mq' = 1/2 ( Mq + M π1/2)( 1 6 ) Hybrid quarks include u', d', and s' whose masses are 200.398, 201.164, and 314.148 MeV, respectively. For baryons other than n and p, the full quarkformula is the combination of vector quarks, hybrid quarks (u' and d'), and pseudoscalar quark ( π 1/2). For example, Λ° (uds) is u'd's3, where 3 denotes 3 π1/2. For pseudoscalar mesons (J = 0), the full quark formula is the combination of π1/2 and q' (u', d' and s') or π1/2 alone. For vector mesons (J > 0), the full quark formula is the combination of vector quarks (u, d, s, c, and b) and π1/2 . For examples, π° is 2, η (1295,J =0) is u'u'd'd'8, and K 1 (1400, J=1) is ds8. The compositions of hadrons from the particles of the full quark formula are listed in Table 4.21 Table 4. Particles for the full quark formula π1/2 u’, d’ s’ u, d s c, b mass (MeV) 70.025 200.40, 201.16314.15 330.8, 332.3558 1701, 5318 n and p √ baryons other than n and p√√ √√ mesons (J = 0) except cc and bb√√√ √ mesons (J > 0) and cc and bb√ √√√ Different energy states in hadron spectroscopy are the results of differences in the numbers of π1/2. The higher the energy state, the higher the number of π1/2. The number of π1/2 attached to q or q' in the full quark formula is restricted. The number of π1/2 can be 0, ±1, ±3, ±4, ±7, and ±3n (n>2). This is the “π1/2 series.” The number, 3, is indicative of a baryon-like (3 quarks in a baryon) number. The number, 4, is the combination of 1 and 3, while thenumber, 7, is the combination of 3 and 4. Since π 1/2 is essentially pseudoscalar u and d quarks, the π1/2 series is closely related to u and d quarks. Since s is close to u and d, the π1/2 series is also related to s quark. . Each presence of u, d, and s associates with one single π1/2 series. The single π1/2 series associates with baryons and the mesons with u u dd ss, ss, dc, uc, db, ub, sc, and sb. The combination of two single π1/2 series is the double π1/2 series (0, ±2, ±6,22 ±8, ±14, and ±6n) associating with u d and ds mesons. For the mesons (c c and bb) without u, d, or s, the numbers of π1/2 attached to quarks can be from the single π1/2 series or the double π1/2 series. The basic fermion formula is similar to M. H. MacGregor's light quark model [17], whose calculated masses and the predicted properties of hadronsare in very good agreement with observations. In the light quark model, themass building blocks are the "spinor" (S with mass 330.4 MeV) and the massquantum (mass = 70MeV). S has 1/2 spin and neutral charge, while the basicquantum has zero spin and neutral charge. For the basic fermion formula, S is u, the quark with the lowest mass (330.77 MeV), and the basic quantum is π 1/2 (mass = 70.025 MeV). In the basic fermion formula, u and π1/2 have the same spins and charge as S and the basic quantum, respectively. For examples, in the basic fermion formula, neutron is SSS, and K 1 (1400, J=1) is S 2 11, where S 2 denotes two S, and 11 denotes 11 π1/2 's. In additional to S and π1/2, the basic fermion formula includes P (positive charge) and N (neutral charge) with the masses of proton and neutron. As in the light quark model, the mass associated with positive or negative charge is theelectromagnetic mass, 4.599 MeV, which is nine times the mass of electron.This mass (nine times the mass of electron) is derived from the baryon-likeelectron that represents three quarks in a baryon and three electron in d 6 quark as in Table 2. This electromagnetic mass is observed in the mass difference23 between π° (2) and π+ (2+) where + denotes positive charge. The calculated mass different is one electromagnetic mass, 4.599 MeV, in good agreement withthe observed mass difference, 4.594 MeV, between π° and π +. (The values for observed masses are taken from "Particle Physics Summary "[18].) The particles in the basic fermion formula are listed in Table 5. Table 5. Particles in the basic fermion formula S π1/2 N P electromagnetic mass mass(MeV)330.77 70.0254 939.565 938.272 4.599 Hadrons are the composites of quarks as molecules composing of atoms. As atoms are bounded together by chemical bonds, hadrons are bounded by“hadronic bonds,” connecting the particles in the basic fermion formula. Thesehadronic bonds are similar to the hadronic bonds in the light quark model. The hadronic bonds are the overlappings of the auxiliary dimensional orbits. As in Eq (11), the energy derived from the auxiliary orbit for S (u quark) is E a = (3/2) (3 M µ) αw = 14.122 MeV (17) The auxiliary orbit is a charge - dipole interaction in a circular orbit as described byA. Barut [13], so a fermion for the circular orbit and an electron for the charge areembedded in this hadronic bond. The fermion for the circular orbit is thesupersymmetrical fermion for the auxiliary dimensional orbit according to Eq (2).24 M f = E a αw (18) The binding energy (negative energy) for the bond (S !S) between two S's is twice of 14.122 MeV minus the masses of the supersymmetrical fermion and electron. E S -S = - 2 (E a - M f - M e) = -26.384 MeV (19) It is similar to the binding energy (-26 MeV) in the light quark model. An example of S!S bond is in neutron (S − S − S) which has two S − S bonds. The mass of neutron can be calculated as follows. Mn = 3M S + 2E S-S = 939.54 MeV, (20) which is in excellent agreement with the observed mass, 939.57 MeV. The massof proton is the mass of neutron minus the mass difference (three times ofelectron mass = M 3e) between u and d quark as shown in Table 2. Proton is represented as S − S − (S -3e). The calculation of the mass of proton is as follows. Ea for (S-3e) = (3/2) (3 (M µ - M 3e)) αw M f = E a αw Mp = 2 M S + M (S-3e) + E S!S + E S!(S-3e) = 938.21 MeV (21) The calculated mass is in a good agreement with the observed mass, 938.27MeV.25 The binding energy for π1/2 − π1/2 bond can be derived in the same way as Eqs (17), (18), and (19). Ea = (3/2) M π1/2 αw M f = E a αw E π1/2 - π1/2 = - 2 (E a - M f - M e) = -5.0387 MeV (22) It is similar to the binding energy (-5 MeV) in the light quark model. An example for the binding energy of π1/2 − π1/2 bond is in π°. The mass of π° can be calculated as follows. Mπ° = 2 Mπ1/2 + E π1/2 - π1/2 = 135.01 MeV. (23) The calculated mass of π° is in excellent agreement with the observed value, 134.98 MeV. There is one π1/2 − π1/2 bond per pair of π1/2 ‘s, so there are two π1/2 − π1/2 bonds for 4 π1/2 ‘s, and three π1/2 − π1/2 bonds for 6 π1/2 ‘s. Another bond is N − π1/2 or P − π1/2, the bond between neutron or proton and π1/2. Since N is SSS, N − π1/2 bond is derived from S − π1/2. The binding energy of S − π1/2 is the average between S −S and π1/2 − π1/2. E S - π1/2 = 1/2 ( E S-S + E π1/2 - π1/2)( 2 4 ) An additional dipole (e +e -) is needed to connected S − π1/2 to neutron. E N - π1/2 = E S - π1/2 + 2 M e = - 14.689 MeV. (25)26 It is similar to -15 MeV in the light quark model. An example for P − π1/2 is Σ+ which is represented by P4 whose structure is 2 − P − 2. The 4 π1/2 's are connected to P with two P − π1/2 bonds. The mass of Σ+ is as follows. MΣ+ = M P + 4 Mπ1/2 + 2 E N - π1/2 = 1189.0 MeV. (26) The calculated mass is in a good agreement with the observed mass, 1189.4 ± 0.07 MeV. There is N – N hadronic bond between two N’s. N has the structure of S – S – S. N – N has a hexagonal structure shown in Fig 5. S SSSS S Fig 5. The structure of N - N There are two additional S – S for each N. The total number of S – S bonds between two N’s is 4. An example is Ω° c (ssc) which has the basic fermion formula of N 2 S9 with the structure of 3 – N – N – 6 S. The mass of Ω°c can be calculated as follows. M = 2M N + M S + 9Mπ1/2 + 4 E S – S + 2 E N - π1/2 = 2705.3 MeV (27)27 The calculated mass for Ω°c is in a good agreement with the observed mass (2704 ± 4 MeV). For baryons other than p and n, there are two or three N - π1/2 or P - π1/2 bonds per baryon. It is used to distinguish J = ½ and J ≥ 3/2. For J = ½ which have asymmetrical spins (two up and one down), there are two bondsfor three S in a baryon to represent the asymmetrical spins. For J ≥ 3/2, there are three bonds for three S in a baryon except if there is only one π 1/2, only two bonds exists for a baryon. Among the particles in the basic fermion formula, there are hadronic bonds, but not all particles have hadronic bonds. In the basic fermion formula,hadronic bonds appear only among the particles that relate to the particles in thecorresponding full quark formula. The related particles are the “core” particlesthat have hadronic bonds, and the unrelated particles are “filler” particles thathave no hadronic bonds. In the basic fermion formula, for baryons other than p and n, the core particles are P, N, and π 1/2. For the mesons consisting of u and d quarks, the core particles are π1/2, S, and N. For the mesons containing one u, d, or s along with s, c, or b, the core particles are S and N, and no hadronicbond exist among π 1/2's , which are the filler particles. For the mesons (c c and bb), the only hadronic bond is N − N. The occurrences of hadronic bonds are listed in Table 6.28 Table 6. Hadronic bonds in hadrons S –S π1/2 - π1/2N(P) - π1/2 N –N binding energy (MeV) -26.384 -5.0387 -14.6894 2 S−S per N baryons other than n and p √√ √ mesons with u and d only √√ √ mesons containing one u, d, or s along with s, c, or b√√ cc or bb mesons √ An example is the difference between π and f 0. The decay modes of f 0 include the mesons of s quarks from K meson. Consequently, there is no π1/2 - π1/2 for f 0. The basic fermion formula for f 0 is 14. The mass of f 0 is as follows. M = 14M π1/2 = 980.4 MeV (28) The observed mass is 980 ± 10 MeV. In additional to the binding energies for hadronic bonds, hadrons have Coulomb energy (-1.2 MeV) between positive and negative charges and magnetic binding energy ( ±2MeV per interaction) for S − S from the light quark model [17]. In the light quark model, the dipole moment of a hadron can be calculated from the magnetic binding energy. Since in the basic fermion formula,magnetic binding energy becomes a part of hadronic binding energy as shown inEq (19), magnetic binding energy for other baryons is the difference in magneticbinding energy between a baryon and n or p. If a baryon has a similar dipole29 moment as p or n, there is no magnetic binding energy for the baryon. An example for Coulomb energy and magnetic binding energy is Λ (uds, J=1/2) whose formula is P3- with the structure of 2 − P − 1- where "-" denotes negative charge. The dipole moment of Λ is –6.13 µN, while the dipole moment of proton (P) is 2.79 µN [18]. According to the light quark model, this difference in dipole moment represents -6 MeV magnetic binding energy. The Coulomb energy between the positive charge P and the negative charge 1- is –1.2 MeV. The electromagnetic mass for 1- is 4.599 MeV. The mass of Λ is calculated as follows. MΛ = M P + 3Mπ1/2 + M e.m. + 2 E N - π1/2 + E mag + E coul = 1116.4 MeV (29) The observed mass is 1115.7 ± 0.0006 MeV. An example of the dual representation of the full quark formula and the basic fermion formula as expressed by Eq. (15) is Λ (uds). The full quark formula for Λ is u’d’s’3 with mass of 1169.9 MeV. This mass is matched by the mass of the basic fermion formula, P3- with the mass of 1152.9 MeV. The finalmass of Λ is 1152.9 MeV plus the various binding energies. Table 7 is the results of calculation for the masses of baryons selected from Ref. (18). The hadrons selected are the hadrons with precise observedmasses and known quantum states such as isospin and spin.30 Table 7. The masses of baryons Baryons I(JP ) Full quark formulaBasic fermionformulaCalculated massObserved massDifference n and p n1 / 2 ( ½+) udd SSS 939.54 939.57 -0.03 p½ ( ½+) uud SSS-3e 938.21 938.27 -0.06 uds, uus, dds Λ° 0(½+) u'd's3 P3- 1116.4 1115.7 0.7 ∑+1(½+) u'u's4 P4 1189.0 1189.4 -0.4 ∑° 1(½+) u'd's4 P4- 1192.4 1192.6 -0.2 ∑-1(½+) d'd's4 N4- 1194.9 1197.4 -2.5 ∑+1(3/2+) u'u's7 P7 1384.4 1382.8 1.6 ∑° 1(3/2+) u'd's7 P7- 1381.8 1383.7 -1.9 ∑-1(3/2+) d'd's7 N7- 1390.3 1387.2 3.1 Λ° (s01)0(½-) u'd's7 P7- 1402.5 1407.0 -4.5 Λ° (D03)0(3/2-) u'd's9 N9 1525.7 1519.5 6.2 uss, dss Ξ° ½(½+) u'ss NS1 1311.0 1314.9 -3.9 Ξ-½(½+) d'ss NS1- 1315.6 1321.3 -5.7 Ξ° ½(3/2+) u'ss4 N9+- 1533.7 1531.8 1.9 Ξ-½(3/2+) d'ss4 N10- 1530.3 1535.0 -4.7 Ξ° ½(3/2-) u'ss9 N 2 1+- 1822.2 1823.0 -0.8 udc, ddc, uuc Λ+ c 0(½+) u'd'c3 PS15 2290.0 2284.9 5.1 ∑c01(½+) d'd'c7 N 210 2452.5 2452.1 0.4 ∑c++1(½+) u'u'c7 N 210++ 2452.5 2452.9 -0.4 ∑c+1(½-) u'd'c7 N 210+ 2449.1 2453.5 -4.4 Λ+ c 0(½-) u'd'c9 N 212+ 2589.1 2593.6 -4.5 usc, dsc Ξ+ c ½(½+) u'sc1 NS 38+ 2467.3 2465.6 1.7 Ξ°c ½(½+) d'sc1 N S 38+- 2470.7 2470.3 0.4 sss Ω-0(3/2+) sss NS6- 1665.7 1672.5 -6.8 ssc Ω°c 0(½+) ssc N 2S9 2705.2 2704.0 1.2 udb Λ°b 0(½+) u'd'b1 N 2S 913+- 5639.5 5641.0 -1.531 The full quark formula for baryons other than p and n involves hybrid quarks (u’ and d’), vector quarks (s, c, and b), and pseudoscalar quark ( π1/2 ). The whole spectrum of baryons (other than p and n) follows the single π1/2 series with 0, 1, 3, 4, 7, and 3n. π1/2 is closely related to u and d, so the lowest energy state baryons ( Λand Σ) that contain low mass quark and high number of u or d have high number of π1/2. . On the contrary, the lowest energy state baryons ( Ω, Ωc, and Λb) that contain high mass quarks and no or low number of u and d have low number of π1/2 or no π1/2. π1/2 's are added to the lowest energy state baryons to form the high-energy state baryons. The basic fermion formula includes P, N, S, and π1/2. If the dominant decay mode includes p, the basic fermion formula includes P. If the dominant decay mode includes n, the basic fermion formula includes N. If the decaymode does not include either n or p directly, the baryon with the dipole momentsimilar to p or n has the basic fermion formula with P or N, respectively. If thedipole moment is not known, the designation of P or N for the higher energystate baryon follows the designation of P or N for the corresponding lower energystate baryon. Only N is used in the multiple nucleons such as N 2 (two N’s). The result of calculated masses for light unflavored mesons is listed in Table 8. Light unflavored mesons have zero strange, charm, and bottomnumbers.32 Table 8. Light unflavored mesons Meson I (J pc ) Full quark formulaBasic fermionformulaCalculated mass.Observed massDifference J =0, only u, d, or lepton in decay mode π° 1 (0-+) 2 2 135.01 134.98 0.04 π±1 (0-)2 2± 139.61 139.57 0.04 η 0 (0-+) 8 8+- 548.0 547.5 0.5 η’ 0 (0-+) 14 14+- 958.1 957.8 0.3 η 0 (0-+) u'u'u'd'd'd'2 S 3 5+- 1292.6 1295.0 -2.4 f0 0 (0++) u'u'u'd'd'd'6 S 4 4 1514.0 1503.0 11.0 J >0, only u, d, or lepton in decay modes ρ 1 (1--) ud2 S 2 2 770.2 768.5 1.7 ω 0 (1--) ud2 S 2 2+- 785.4 781.9 3.5 h1 0 (1+-) ud8 S 2 8 1176.2 1170.0 6.2 ω 0 (1--) uudd2 S 3 7+- 1415.6 1419.0 -3.4 ω 0 (1--) uudd6 S 4 6 1650.0 1649.0 1.0 ω3 0 (3--) uudd6 S 4 6 1662.0 1667.0 -5.0 J = 0, u and d in major decay modes, s in minor decay modes f0 0 (0++) u'd's'4 14 980.4 980.0 0.4 aO 1 (0++) u'd's'4 14 985.4 983.5 1.9 η 0 (0-+) u'u'd'd's's' S 211+- 1418.5 1415.0 3.5 J >o, u and d in major decay modes, s in minor decay mode a 1 1 (1++) uds S13 1232.7 1230.0 2.7 b 1 1 (1+-) uds S13 1232.7 1231.0 1.7 f2 0 (2++) uds1 S 2 9 1278.4 1275.0 3.4 f1 0 (1++) uds1 S 2 9+- 1278.4 1282.2 -3.8 a2 1 (2++) uds1 S14 1316.2 1318.1 -1.9 f1 0 (1++) uds4 S 3 7 1429.7 1426.8 2.9 ρ 1 (1--) uds4 S 2 12 1475.5 1465.0 10.5 π2 1 (2-+) uds7 S 2 15 1685.5 1670.0 15.5 ρ3 1 (3--) uds7 S 2 15 1693.5 1691.0 3.5 ρ 1 (1--) uds7 S 2 15+- 1698.6 1700.0 -1.4 f4 0 (4++) uds12 S 2 20+- 2043.7 2044.0 -0.3 s in major decay modes Φ 0 (1--) ss-1 S 3 +- 1017.6 1019.4 -1.8 f1 0 (1++) ss7 S 4 4 1524.0 1512.0 12.0 f2’0 (2++) ss7 S 4 4+- 1532.0 1525.0 7.0 Φ 0 (1--) ss9 S 4 6+- 1672.1 1680.0 -7.9 Φ 3 0 (3--) ss12 S 6 1852.7 1854.0 -1.3 f2 0 (2++) ss15 S 6 2 2000.7 2011.0 -10.3 f2 0 (2++) ss18 N S 3 6 2299.3 2297.0 2.3 f2 0 (2++) ss18 N S 4 2+- 2331.5 2339.0 -7.533 The full quark formulas for light unflavored mesons are different for different decay modes. There are three different types of the full quark formulafor light unflavored mesons. Firstly, when the decay mode is all leptons or all mesons with u and d quarks, the full quark formula consists of π 1/2, u’, d’, u, and d quarks, Secondly, when u, d, and leptons are in the major decay modes, and sis in the minor decay modes, the full quark formula consists of π 1/2, u’, d’, s’, u, d, and s quarks. The most common full quark formula for such mesons is uds,which is essentially ½ (u u dd ss). Finally, when s quarks are in the major or all decay mode, the full quark formula consists of π 1/2 ‘s and s quarks. When J = 0, the full quark formula includes π1/2 ‘s and hybrid quarks (u’, d’, and s’). When J > 0, the full quark formula includes π1/2 ‘s and vector quarks (u, d, and s). Since there are double presence of u and d quarks for the full quark formula with u andd quark, it follows the double π 1/2 series: 0, 2, 6, 8, 14, and 6n. The full quark formula with odd presence of u, d, and s follows the single π1/2 series: 0, ±1, ±3, ±4, ±7, and ±3n. The basic fermion formula includes π1/2 ‘s, S, and N. π1/2 − π1/2 hadronic bond exists in the full quark formula with u and d quarks, and does not exist inthe full quark formula with s quark. S − S bond exists in all formula. Light unflavored mesons decay into symmetrical low mass mesons, such as 2 γ, 3π o, and π02γ from η, or asymmetrical high and low mass mesons, such as asymmetrical π+πη, ρ0γ, and π0π0η from η’ . To distinguish the asymmetrical34 decay modes from the symmetrical decay modes, one “counter π1/2 − π1/2 hadronic bond” is introduced in η’ [17]. The binding energy for the counter π1/2 − π1/2 hadronic bond is 5.04 MeV, directly opposite to –5.04 MeV for the π1/2 − π1/2 hadronic bond. All light unflavored mesons with asymmetrical decay modes include this counter π1/2 − π1/2 hadronic bonds. For an example, the mass of η’ (14+-) is calculated as follows. Mη’ = 14 M π1/2 + 2 M e.m. + E e.m. + 7Eπ1/2 − π1/2 - Eπ1/2 − π1/2 = 958.2 MeV (30) The observed mass is 957.8 ± 0.14 MeV. The result of the mass calculation for the mesons consisting of u, d, or s with s, c, or b is listed in Table 9.35 Table 9. Mesons with s, c, and b Meson J pcFull quark formulaBasic fermionformulaCalculated mass.Observed massDifference Light strange mesons K±0-7 7± 494.8 493.7 1.1 K° 0-7 7+- 498.2 497.7 0.5 K* 0+d's'14 S 3 7 1429.7 1429.0 0.7 K* 1-us S8± 895.6 891.6 4.0 K* 1-ds S8 899.0 896.1 2.9 K 1 1+ds6 S 2 9 1273.4 1273.0 0.4 K 1 1+ds8 S 2 11 1405.4 1402.0 3.4 K* 1-ds8 S 2 11+- 1413.4 1412.0 1.4 K*±2+us8 N7± 1434.3 1425.4 8.9 K* 2+ds8 N7+- 1437.7 1432.4 5.3 K* 1-ddss S 3 11+- 1717.8 1714.0 3.8 K 2 2-ddss N12 1779.9 1773.0 6.9 K 3*3-ddss N12 1779.9 1780.0 -0.1 K 2 2-ds14 S 4 8+- 1812.1 1816.0 -3.9 K 4*4+ds18 S 5 7+- 2046.5 2045.0 1.5 Charmed mesons D° 0-u'c1 S 6 +- 1860.7 1864.5 -3.8 D±0-d'c1 S6 ± 1857.3 1869.3 -12.0 D * o1-uc1 S 6 2+- 2000.7 2006.7 -6.0 D*±1-dc1 S6 2± 1997.4 2010.0 -12.6 D 1 1+uc7 S 6 8+- 2420.9 2422.2 -1.3 D 2*2+uc7 N S 4 4 2463.6 2458.9 4.7 D 2*±2+dc7 N S 4 4± 2468.2 2459.0 9.2 Charmed strange mesons Ds 0-s'c1 S 5 6 1968.5 1968.5 0.0 Ds1 1+sc4 NS18 2535.4 2535.4 0.0 Bottom mesons B±0-u'b N4 S20± 5283.1 5278.9 4.2 B° 0-d'b N 4 S 20 5278.5 5279.2 -0.7 B* 1-db N 6 5320.8 5324.8 -4.0 Bs 0-s'b N 5 S3 5373.5 5369.3 4.2 The full quark formula for these mesons contains π1/2’s, u’, d’, s’, u, d, s, c, and b. The full quark formula for the mesons with J = 0 contains π1/2, hybrid36 quarks (u’, d’, and s’), c, and b quarks. The full quark formula with J > 0 contains π1/2’s and vector quarks (u, d, s, c, and b). Strange mesons have double presence of d and s, so it follows the double π1/2 series. All other mesons have single presence of u, d, or s, so they follow the single π1/2 series. The basic fermion formula consists of π1/2’s, S, and N. There is no π1/2 − π1/2 bond. There are S − S and N − N bonds. For example, the mass of B s (N5 S3) is calculated as follows. M= 5 M N + 3M S + (10+ 2) M S−S = 5373.5 MeV (31) There are 10 S − S bonds for N 5 (two S − S bonds for each N) and 2 S − S bonds for S 3. The observed mass is 5369.3 ± 2 MeV. The result of mass calculation for c c and bb mesons is listed in Table 10.37 Table 10. cc and bb mesons Meson J pcFull quark formulaBasic fermionformulaCalculated mass.Observed massDifference cc mesons ηc (1s) 0-+cc-6 N S3 15 2982.3 2979.8 2.5 J/ψ 1--cc-3 N 2 S4 3096.7 3096.9 -0.2 χc (1p) 0++cc2 N 2 S2 14 3415.5 3415.1 0.4 χc’ (1p) 1++cc3 N 2 S4 6 3516.8 3510.5 6.5 χc2 (1p) 2++cc4 N 2 S216 3555.5 3556.2 -0.7 U2s 1--cc6 N 3 S10 3691.4 3686.0 5.4 ψ 1--cc7 N 3 S11 3761.4 3769.9 -8.5 ψ 1--cc12 N 4 7 4037.4 4040.0 -2.6 ψ 1--cc14 N 4 S4 4158.1 4159.0 -0.9 ψ 1--cc18 N 4 S2 3 4418.8 4415.0 3.8 bb mesons ϒ (1s) 1--bb-12 N 7 S 6 18 9452.7 9460.4 -7.7 χb (1p) 0++bb-4 N 10 S 3 9860.3 9859.8 0.5 χb1 (1p) 1++bb-3 N 10 S10 9899.0 9891.9 7.1 χb2 (1p) 2++bb-3 N 10 15 9918.4 9913.2 5.2 ϒ (2s) 1--bb-1 N 10 S2 7 10019.7 10023.3 -3.6 χb0 (1p) 0++bb2 N 10 S2 10 10229.8 10232.1 -2.3 χb1 (2p) 1++bb2 N 10 S4 1 10261.1 10255.2 5.9 χb2 (2p) 2++bb2 N 10 S4 1 10261.1 10268.5 -7.4 ϒ (3s) 1--bb3 N 10 S3 7 10350.5 10355.3 -4.8 ϒ (4s) 1--bb7 N 11 S7 10575.7 10580.0 -4.3 ϒ 1--bb12 N 12 3 10851.6 10865.0 -13.4 ϒ 1--bb14 N 11 S3 4 11027.2 11019.0 8.2 The full quark formula contains c or b. Since it contains no u, d, or s related to π1/2, it follows a mixed π1/2 series from both the single π1/2 series and the double π1/2 series. Since c and b are high mass quarks unlike the low mass u and d quarks that relate to π1/2, the π1/2 series actually starts from a negative π1/2. The basic fermion formula contains π1/2’s, S, and N. The only hadronic bond is N − N.38 An example is J/ ψ (N2 S4) whose mass (3096.9 ± 0.04 MeV) is calculated as follows. M= 2 M N + 4 M S + 4 M S−S = 3096.7 MeV (32) 5. Conclusion The periodic table of elementary particles is derived from S-duality between the eleven dimensional membrane and dual lepton-quark superstring (Fig. 1), U-duality between the superstring and solitonic string (Fig. 2), and T-duality betweenType II string and compacified string. (Fig. 3). The periodic table of elementaryparticles (Fig. 4 and Table 2) is constructed from the Kaluza-Klein substructure.All leptons, quarks, and gauge bosons can be placed in the periodic table. Themasses of elementary particles can be calculated using only four known constants:the number of the extra spatial dimensions in the eleven dimensional membrane, the mass of electron, the mass of Z°, and α e. The calculated masses (Tables 1 and 3) of elementary particles derived from the periodic table are in good agreement with the observed values. For an example, the calculated mass (176.5GeV) of the top quark has an excellent agreement with the observed mass (176 ±13 GeV). A hadron can be represented by the full quark formula (Table 4) and the basic quark formula (Table 5). The full quark formula consists of all quarks,pseudoscalar quark in pion, and the hybrid quarks. The basic quark formula39 consists of the lowest mass quarks. The relation between these two formulas is expressed by Eq. (15). As a molecule is the composite of atoms with chemicalbonds, a hadron is the composite of elementary particles with "hadronic bonds"(Table 6) which are the overlappings of the auxiliary dimensional orbits. Thecalculated masses (Tables 7, 8, 9, and 10) of hadrons are in good agreement withthe observed values. For examples, the calculated masses for neutron and pionare 939.54 and 135.01MeV in excellent agreement with the observed masses,939.57 and 134.98 MeV, respectively. The overall average difference betweenthe calculated masses and the observed masses for all hadrons (110 hadrons) is0.29 MeV, and the standard deviation for such differences is 5.06 MeV. The average observed error for the masses of the hadrons is ± 6.41 MeV. The periodic table of elementary particles provides the most comprehensive explanation and calculation for the masses of elementary particles and hadrons.40 References [1] D. Chung, Speculations in Science and Technology 20 (1997) 259; Speculations in Science and Technology 21(1999) 277 [2] CDF Collaboration, 1995 Phys. Rev. Lett 74 (1995) 2626[3] C.M.Hull and P.K. Townsend, Nucl. Phys. B 438 (1995) 109; E. Witten, Nucl. Phys. B 443 (1995) 85 [4] C. Montonen and D. Olive, Phys. Lett. B 72 (1977) 117[5] M.J. Duff and J.X. Lu, Nucl. Phys. B 354 (1991) 141[6] M. Dine, P. Huet, and N. Seiberg, Nucl. Phys. B 322 (1989) 301[7] E. Witten 1982 Nucl. Phys. B202 (1981) 253; T. Bank, D. Kaplan, and A. Nelson A 1994 Phys. Rev. D49 (1994) 779 [8] P. Langacher, M. Luo , and A. Mann, Rev. Mod. Phys. 64 (1992) 87[9] E. Elbaz, J. Meyer, and R. Nahabetian, Phys. Rev. D23. (1981)1170[10] A. O. Barut and D. Chung, Lett. Nuovo Cimento 38 (1983) 225[11] SLD Collaboration, Phys. Rev. D50 (1994) 5580-5590[12] A. Guth A. 1981. Phys. Rev. D23 (1981) 347; K. Olive, Phys. Rep. 190 (1990) 307 [13] A.O. Barut, Phys. Rev. Lett. 42.(1979) 1251[14] L. Hall, R. Jaffe, J. Rosen1985, Phys. Rep. 125 (1985) 105[15] C.P. Singh, Phys. Rev. D24 (1981) 2481; D. B. Lichtenberg Phys. Rev. D40 (1989) 367541 [16] G.E. Brown, M. Rho, V. Vento, Phys Lett 97B (1980); G.E. Brown, Nucl. Phys. A374 (1982) 630 [17] M.H. MacGregor, Phys. Rev. D9 (1974) 1259; Phys. Rev. D10 (1974) 850; The Nature of the Elementary Particles (Springer-Verlag, New York, 1978) [18] Particle Data Group, Rev. of Mod. Phys. 68 (1996) 611

arXiv:physics/0003024v1 [physics.gen-ph] 10 Mar 2000From the Neutrino to the Edge of the Universe B.G. Sidharth Centre for Applicable Mathematics & Computer Sciences B.M. Birla Science Centre, Hyderabad 500 063 Abstract Two recent findings necessitate a closer look at the existing stan- dard models of Particle Physics and Cosmology. These are the dis- covery of Neutrino oscillation, and hence a non zero mass on t he one hand and, on the other, observations of distant supernovae w hich in- dicate that contrary to popular belief, the universe would c ontinue to expand for ever, possibly accelerating in the process. In th is paper it is pointed out that relatively recent studies which indicat e a stochas- tic, quantum vacuum underpinning and a fractal structure fo r space time, reconcile both of the recent observations, harmoniou sly. 1 Introduction In the recent years, there have been two significant findings w hich necessitate a closer look at the existing standard models of Particle Phy sics and Cosmol- ogy. The first is the Superkamiokande experiment[1] which de monstrates a neutrino oscillation and therefore a non zero mass, whereas , strictly going by the standard model, the neutrino should have zero mass. The o ther finding based on distant supernovae observations[2, 3, 4] is that th e universe will continue to expand without deceleration and infact possibl y accelerating in the process. We will now demonstrate how a recent model of fractal, quanti zed space time 0Partly based on an invited talk at the National Workshop on Ne utrino Physics, Uni- versity of Hyderabad, 1998. 1arising from the underpinning of a quantum vaccuum or Zero Po int Field, reconciles both the above facts, in addition to being in agre ement with other experimental and observational data. 2 Neutrino Mass According to a recent model, elementary particles, typical ly leptons, can be treated as, what may be called Quantum Mechanical Black Hole s (QMBH)[5, 6, 7, 8, 9], which share certain features of Black Holes and al so certain Quan- tum Mechanical characteristics. Essentially they are boun ded by the Comp- ton wavelength within which non local or negative energy phe nomena occur, these manifesting themselves as the Zitterbewegung of the e lectron. These Quantum Mechanical Black Holes are created out of the backgr ound Zero Point Field and this leads to a consistent cosmology, wherei n usingN, the number of particles in the universe as the only large scale pa rameter, one could deduce from the theory, Hubble’s law, the Hubble’s con stant, the ra- dius, mass, and age of the universe and features like the hith erto inexplicable relation between the pion mass and the Hubble constant[5]. T he model also predicts an ever expanding universe, as recent observation s do confirm. Within this framework, it was pointed out that the neutrino w ould be a massless and charge less version of the electron and it was de duced that it would be lefthanded, because one would everywhere encounte r the psuedo spinorial (”negative energy”) components of the Dirac spin or, by virtue of the fact that its Compton wavelength is infinite (in practise ver y large). Based on these considerations we will now argue that the neutrino w ould exhibit an anomalous Bosonic behaviour which could provide a clue to the neutrino mass. As detailed in [6] the Fermionic behaviour is due to the non lo cal or Zitterbe- wegung effects within the Compton wavelength effectively sho wing up as the well known negative energy components of the Dirac spinor wh ich dominate within while positive energy components predominate outsi de leading to a doubly connected space or equivalently the spinorial or Fer mionic behaviour. In the absence of the Compton wavelength boundary, that is wh en we en- counter only positive energy or only negative energy soluti ons, the particle would not exhibit the double valued spinorial or Fermionic b ehaviour: It would have an anomalous anyonic behaviour. 2Indeed, the three dimensionality of space arises from the sp inorial behaviour outside the Compton wavelength[10]. At the Compton wavelen gth, this dis- appears and we should encounter lower dimensions. As is well known[11] the low dimensional Dirac equation has like the neutrino, only t wo components corresponding to only one sign of the energy, displays hande dness and has no invariant mass. The neutrino shows up as a fractal entity. Ofcourse the above model strictly speaking is for the case of an isolated non interacting particle. As neutrinos interact through the we ak or gravitational forces, both of which are weak, the conclusion would still be approximately valid particularly for neutrinos which are not in bound stat es. We will now justify the above conclusion from other standpoi nts: Let us first examine why Fermi-Dirac statistics is required in the Quant um Field Theo- retic treatment of a Fermion satisfying the Dirac equation. The Dirac spinor has four components and there are four independent solution s corresponding to positive and negative energies and spin up and down. It is w ell known that [12] in general the wave function expansion of the Fermion sh ould include solutions of both signs of energy: ψ(/vector x,t) =N/integraldisplay d3p/summationdisplay ±s[b(p,s)u(p,s)exp(−ıpµxµ/¯h) +d∗(p,s)v(p,s)exp(+ıpµxµ/¯h) (1) whereNis a normalization constant for ensuring unit probability. In Quantum Field Theory, the coefficients become creation and annihilation operators while bb+anddd+become the particle number operators with eigen values 1 or 0 only. The Hamiltonian is now given by[13]: H=/summationdisplay ±s/integraldisplay d3pEp[b+(p,s)b(p,s)−d(p,s)d+(p,s)] (2) As can be seen from (2), the Hamiltonian is not positive defini te and it is this circumstance which necessitates the Fermi-Dirac statisti cs. In the absence of Fermi-Dirac statistics, the negative energy states are n ot saturated in the Hole Theory sense so that the ground state would have arbi trarily large negative energy, which is unacceptable. However Fermi-Dir ac statistics and the anti commutators implied by it prevent this from happeni ng. From the above, it follows that as only one sign of energy is en countered for thev, we need not take recourse to Fermi-Dirac statistics. 3We will now show from an alternative view point also that for t he neutrino, the positive and negative solutions are delinked so that we d o not need the negative solutions in (1) or (2) and there is no need to invoke Fermi-Dirac statistics. The neutrino is described by the two component Weyl equation [14]: ı¯h∂ψ ∂t=ı¯hc/vector σ·/vector∆ψ(x) (3) It is well known that this is equivalent to a massless Dirac pa rticle satisfying the following condition: Γ5ψ=−ψ We now observe that in the case of a massive Dirac particle, if we work only with positive solutions for example, the current or expecta tion value of the velocity operator c/vector αis given by (ref.[12]), J+=<cα> =<c2/vector p E>+ =<vgp>+ (4) in an obvious notation. (4) leads to a contradiction: On the one hand the eigen values ofc/vector αare±c. On the other hand we require, <vgp><1. To put it simply, working only with positive solutions, the D irac particle should have the velocity cand so zero mass. This contradiction is solved by including the negative solutions also in the description of the particle. This infact is the starting point for (1) above. In the case of mass less neutrinos however, there is no contra diction because they do indeed move with the velocity of light. So we need not c onsider the negative energy solutions and need work only with the positi ve solutions. There is another way to see this. Firstly, as in the case of mas sive Dirac par- ticles, let us consider the packet (1) with both positive and negative solutions for the neutrino. Taking the zaxis along the /vector pdirection for simplicity, the acceptable positive and negative Dirac spinors subject to t he above stated condition are u= 1 0 −1 0 v= 0 −1 0 1  4The expression for the current is now given by, Jz=/integraldisplay d3p/braceleftigg/summationdisplay ±s[|b(p,s)|2+|d(p,s)|2]pzc2 E +ı/summationdisplay ±s±s′b∗(−p,s′)d∗(p,s)−¯u(−p,s′)σ30v(p,s) −ı/summationdisplay ±s±s′b(−p,s′)d(p,s)−¯v(p,s′)σ30u(−p,s)  (5) Using the expressions for uandvit can easily be seen that in (5) the cross (or Zitterbewegung) term disappears. Thus the positive and negative solutions stand delinked in c ontrast to the case of massive particles, and we need work only with positiv e solutions (or only with negative solutions) in (1). Finally this can also be seen in yet another way. As is known (r ef.[14]), we can apply a Foldy-Wothuysen transformation to the mass less Dirac equation to eliminate the ”odd” operators which mix the components of the spinors representing the positive and negative solutions. The result is the Hamiltonian, H′= Γ◦pc (6) Infact in (6) the positive and negative solutions stand deli nked. In the case of massive particles however, we would have obtained instea d, H′= Γ◦/radicalig (p2c2+m0c4) (7) and as is well known, it is the square root operator on the righ t which gives rise to the ”odd” operators, the negative solutions and the D irac spinors. Infact this is the problem of linearizing the relativistic H amiltonian and is the starting point for the Dirac equation. Thus in the case of mass less Dirac particles, we need work onl y with solutions of one sign in (1) and (2). The equation (2) now becomes, H=/summationdisplay ±s/integraldisplay d3pEp[b+(p,s)b(p,s)] (8) As can be seen from (8) there is no need to invoke Fermi-Dirac s tatistics now. The occupation number bb+can now be arbitrary because the question of 5a ground state with arbitrarily large energy of opposite sig n does not arise. That is, the neutrinos obey anomalous statistics. In a rough way, this could have been anticipated. This is beca use the Hamil- tonian for a mass less particle, be it a Boson or a Fermion, is g iven by H=pc Substitution of the usual operators for Handpyields an equation in which the wave function ψis a scalar corresponding to a Bosonic particle. According to the spin-statistics connection, microscopic causality is incom- patible with quantization of Bosonic fields using anti-comm utators andr Fermi fields using commutators[13]. But it can be shown that this do es not apply when the mass of the Fermion vanishes. In the case of Fermionic fields, the contradiction with micro scopic causality arises because the symmetric propogator, the Lorentz invar iant function, ∆1(x−x′)≡/integraldisplayd3k (2π)33ωk[e−ık.(x−x′)+eık.(x−x′)] does not vanish for space like intervals ( x−x′)2<0,where the vacuum expectation value of the commutator is given by the spectral representation, S1(x−x′)≡ı<0|[ψα(x),ψβ(x′)]|0>=−/integraldisplay dM2[ıρ1(M2)∆x+ρ2(M2)]αβ∆1(x−x′) Outside the light cone, r >|t|,wherer≡ |/vector x−/vector x′|andt≡ |x0−x′ 0|,∆1is given by, ∆1(x′−x) =−1 2π2r∂ ∂rK0(m√ r2−t2), where the modified Bessel function of the second kind, K0is given by, K0(mx) =/integraldisplay∞ 0cos(xy)√m2+y2dy=1 2/integraldisplay∞ −∞cos(xy)√m2+y2dy (cf.[15]). In our case, x≡√ r2−t2, and we have, ∆1(x−x′) =const1 x/integraldisplay∞ −∞ysinxy√m2+y2dy As we are considering massless neutrinos, going to the limit asm→0, we get,|Ltm→0∆1(x−x′)|=|(const. ).Ltm→01 x/integraltext∞ −∞sinxydy |<0(1) x. That is, as 6the Compton wavelength for the neutrino is infinite (or very l arge), so is |x| and we have |∆1|<<1. So the invariant ∆ 1function nearly vanishes every- where except on the light cone x= 0, which is exactly what is required. So, the spin-statistics theorem or microscopic causality is no t violated for the mass less neutrinos when commutators are used. The fact that the ideally, massless, spin half neutrino obey s anomalous statis- tics could have interesting implications. For, given an equ ilibrium collection of neutrinos, we should have if we use the Bose-Einstein stat istics[16]. PV=1 3U, (9) instead of the usual PV=2 3U, (10) whereP,VandUdenote the pressure, volume and energy of the collection. We also have, PVαNkT,N andTdenoting the number of particles and tem- perature respectively. On the other hand for a fixed temperature and number of neutrin os, com- parison of (9) and (10) shows that the effective energy U′of the neutrinos would be twice the expected energy U. That is in effect the neutrino acquires a rest mass m. It can easily be shown from the above that, mc2 k≤≈√ 3T (11) That is for cold background neutrinos mis about a thousandth of an evat the present background temperature of about 2◦K: 10−9me≤m≤10−8me (12) This can be confirmed, alternatively, as follows. As pointed out by Hayakawa, the balance of the gravitational force and the Fermi energy o f these cold background neutrinos, gives[17], GNm2 R=N2/3¯h2 mR2, (13) whereNis the number of neutrinos. Further as in the Kerr-Newman Black Hole formulation equati ng (13) with the energy of the neutrino, mc2we immediately deduce m≈10−8me 7which agrees with (11) and (12). It also follows that N∼1090, which is correct. Moreover equating this energy of the quantum mecha nical black hole tokT, we get (cf.also (11)) T∼1◦K, which is the correct cosmic background temperature. Alternatively, using (11) and (12) we get from (13), a backgr ound radiation of a few millimeters wavelength, as required. So we obtain not only the correct mass and the number of the neu trinos, but also the correct cosmic background temperature, at one stro ke. Indeed the above mass of the neutrino was predicted earlier[ 18]. 3 Cosmology The above model of quantized space time ties up with the model of fluctua- tional cosmology discussed in several papers[8]. We observe that the ZPF leads to divergences in QFT[19] if no l arge fre- quency cut off is arbitrarily prescribed, e.g. the Compton wa velength. We argue that it is these fluctuations within the Compton wavele ngth and in time intervals ∼¯h/mc2, which create the particles. Thus choosing the pion as a typical particle, we get[19, 5], (Energy density of ZPF) Xl3=mc2(14) Using the fact there are N∼1080such particles in the Universe, we get, Nm=M (15) whereMis the mass of the universe. We equate the gravitational potential energy of the pion in a three dimen- sional isotropic sphere of pions of radius R, the radius of the universe, with the rest energy of the pion, to get, R=GM c2(16) whereMcan be obtained from (15). We now use the fact that the fluctuation in the particle number is of the order 8√ N[17, 16, 5], while a typical time interval for the fluctuation s is∼¯h/mc2 as seen above. This leads to the relation[5] T=¯h mc2√ N (17) whereTis the age of the universe, and dR dt≈HR (18) Strictly speaking the above equations are order of magnitud e relations. So from (18), a further differenciation leads to the conclusion that a cosmological constant cannot be ruled out such that Λ≈≤0(H2) (19) (19) explains the smallness of the cosmological constant or the so called cosmological problem[20]. To proceed it can be shown that the above equations lead to[21 ] G=β T≡G0(1−t t0) (20) wheret0is the age of the universe and Tis the time that has elapsed in the present epoch. It can be shown that (20) can explain the prece ssion of the perihelion of Mercury[21]. We could also explain the correct gravitational bending of l ight. Infact in Newtonian theory also we obtain the bending of light, though the amount is half that predicted by General Relativity[22]. In the Newto nian theory we can obtain the bending from the well known orbital equations , 1 r=GM L2(1 +ecosΘ) (21) whereMis the mass of the central object, Lis the angular momentum per unit mass, which in our case is bc,bbeing the impact parameter or minimum approach distance of light to the object, and ethe eccentricity of the trajectory is given by e2= 1 +c2L2 G2M2(22) 9For the deflection of light α, if we substitute r=±∞, and then use (22) we get α=2GM bc2(23) This is half the General Relativistic value. We also note that the effect of time variation on ris given by (cf.ref.[21]) r=r0(1−t t0) (24) Using (24) the well known equation for the trajectory is give n by (Cf.[23],[24],[25]) u” +u=GM L2+ut t0+ 0/parenleftbiggt t0/parenrightbigg2 (25) whereu=1 rand primes denote differenciation with respect to Θ. The first term on the right hand side represents the Newtonian contribution while the remaining terms are the contributions due to (24). The solution of (25) is given by u=GM L2/bracketleftbigg 1 +ecos/braceleftbigg/parenleftbigg 1−t 2t0/parenrightbigg Θ +ω/bracerightbigg/bracketrightbigg (26) whereωis a constant of integration. Corresponding to −∞<r< ∞in the Newtonian case we have in the present case, −t0<t< t 0, wheret0is large and infinite for practical purposes. Accordingly the analog ue of the reception of light for the observer, viz., r= +∞in the Newtonian case is obtained by takingt=t0in (26) which gives u=GM L2+ecos/parenleftbiggΘ 2+ω/parenrightbigg (27) Comparison of (27) with the Newtonian solution obtained by n eglecting terms ∼t/t0in equations (24),(25) and (26) shows that the Newtonian Θ is replaced byΘ 2, whence the deflection obtained by equating the left side of ( 27) to zero, is cosΘ/parenleftbigg 1−t 2t0/parenrightbigg =−1 e(28) 10whereeis given by (22). The value of the deflection from (28) is twice the Newtonian deflection given by (23). That is the deflection αis now given not by (23) but by the correct formula, α=4GM bc2, We now come to the problem of galactic rotational curves (cf. ref.[22]). We would expect, on the basis of straightforward dynamics that the rotational velocities at the edges of galaxies would fall off according t o v2≈GM r(29) However it is found that the velocities tend to a constant val ue, v∼300km/sec (30) This has lead to the postulation of dark matter. We observe th at from (24) it can be easily deduced that a≡(¨ro−¨r)≈1 to(t¨ro+ 2˙ro)≈ −2ro t2 o(31) as we are considering infinitesimal intervals tand nearly circular orbits. Equation (31) shows (Cf.ref[21] also) that there is an anoma lous inward ac- celeration, as if there is an extra attractive force, or an ad ditional central mass. So, GMm r2+2mr t2 o≈mv2 r(32) From (32) it follows that v≈/parenleftigg2r2 t2 o+GM r/parenrightigg1/2 (33) From (33) it is easily seen that at distances within the edge o f a typical galaxy, that is r<1023cmsthe equation (29) holds but as we reach the edge and beyond, that is for r≥1024cmswe havev∼107cmsper second, in agreement with (30). 11Thus the time variation of G given in equation (20) explains o bservation without invoking dark matter. Interestingly a background Zero Point Field of the type disc ussed above, is associated with a cosmological constant in General Relativ ity[26]. We can reconcile this latter view with the above considerations. F or this we observe that the variation in G, is small so that over a small period of time the General Relativistic equations hold approximately. Thus w e have ¨R(t) =−4πρ(t)GR(t)/3 + ΛR(t)/3 (34) In (34) we use equation (20), to get on using the above conside rations Λ∼Gρ√ N(35) On the other hand the Zero Point Field leads to a cosmological constant (Cf.ref.[26]) Λ∼G<ρ vac> (36) In the above fluctuational cosmological picture, as√ Nparticles are created we get ρ∼√ Nρvac (37) (35) and (36) can be seen to be identical upon using (37). This ofcourse should not be surprising, because in both case s we have effec- tively a cosmological constant which is a manifestation of v accuum energy. 4 Comments It must be mentioned that the value of the neutrino mass as ded uced in equation (12) rules out the neutrino as a candidate for dark m ass, so that there is no contradiction with the observed ever continuing expansion of the universe. It must also be mentioned that the value of the cosm ological con- stant from vacuum energy as deduced by Zeldovich (Cf.ref.[2 6]) was adhoc and unclear. The effective cosmological constant which we ha ve deduced, however, is consistent. Interestingly, by reversing the steps in Section 3 we can con clude that a small cosmological constant would imply a variable G. 12It may be mentioned that what was called the ether and later th e quan- tum vacuum has been the concept that has survived the whole of the twen- tieth Century, through the works of Physicists like Dirac[2 7], Vigier[28], Nelson[29], Prigogine[30], and more recently through the w orks of Rueda and co-workers[31], the author[5] and even string theories ts like Wilzeck. We also remark that the considerations of Section 2 (Cf. equa tions (1) and (2)), show that a Fermion while spread out is localized to wit hin the Compton wavelength. On the other hand the neutrino can be considered to be a truly point particle–the double connectivity of the space, the di vide between the region within the Compton wavelength of ”negative energy” s olutions, and the region without disappears. The neutrino is the divide be tween Fermions and Bosons. Finally, it may be mentioned that such a space time cut off is at the heart of a fractal picture of space time, studied by Nottale, Ord, E l Naschie, the author and others (Cf.ref.[32] and references therein). References [1] Website http://www.phys.hawaii.edu:80/jglnuosc-st ory.html. [2] S. Perlmutter, et. al., Nature, 391 (6662), 1998. [3] R.P. Kirshner, Proc. Natl. Acad. Sci. Vol.96, 1999, pp.4 224-4227. [4] I. Zehavi, A. Dekel, Nature, 401 16 September 1999, p.252-254. [5] B.G. Sidharth, ”Universe of Fluctuations”, Int.J.of Mo d.Phys. A 13(5), 1998, pp599ff. [6] B.G. Sidharth, ”Quantum Mechanical Black Holes:Toward s a Unifica- tion of Quantum Mechanics and General Relativity”, IJPAP, 3 5, 1997. [7] B.G. Sidharth, Gravitation & Cosmology, Vol.4, No.2, 19 98. [8] B.G. Sidharth, International Journal of Theoretical Ph ysics, 37 (4), 1307-1312, 1998. [9] B.G. Sidharth, in ”Frontiers of Fundamental Physics”, E ds., Lim, S.C., et al. Springer Verlag, Singapore, 1998. 13[10] C.W. Misner, K.S. Thorne and J.A. Wheeler, ”Gravitatio n”, W.H. Free- man, San Francisco, 1973. [11] A. Zee, ”Unity of Forces in the Universe”, Vol.II, World Scientific, Sin- gapore, 1982, and several papers reproduced and cited there in. [12] J.D. Bjorken and S.D. Drell, ”Relativistic Quantum Mec hanics”, McGraw-Hill Inc., New York, 1964. [13] J.D. Bjorken and S.D. Drell, ”Relativistic Quantum Fie lds”, McGraw- Hill Inc., New York, 1965. [14] S.S. Schweber, ”An Introduction to Relativistic Quant um Field The- ory”, Harper and Fow, New York, 1961. [15] B.G. Sidharth, Journal of Statistical Physics, 95(3/4 ), May 1999. [16] K. Huang, ”Statistical Mechanics”, Wiley Eastern, New Delhi, 1975. [17] S. Hayakawa, Suppl of PTP, 1965, 532-541. [18] B.G. Sidharth, ”Quantum Mechanical Black Holes: Issue s and Ramifi- cations” xxx.lanl.gov quant-ph/9803048. [19] R.P. Feynman, and A.R. Hibbs, ”Quantum Mechanics and Pa th Inte- grals”, McGraw Hill, New York, 1965, p245ff. [20] S. Weinberg, Reviews of Modern Physics, Vol.61, No.1, J anuary 1989, p.1-23. [21] B.G. Sidharth, ”Effects of Varying G” to appear in Nuovo C imento B. [22] J.V. Narlikar, Foundations of Physics, Vol.13. No.3, 1 983. [23] P.G. Bergmann, ”Introduction to the Theory of Relativi ty”, Prentice- Hall (New Delhi), 1969, p248ff. [24] H. Goldstein, ”Classical Mechanics”, Addison-Wesley , Reading, Mass., 1966. [25] H. Lass, ”Vector and Tensor Analysis”, McGraw-Hill Boo k Co., Tokyo, 1950, p295 ff. 14[26] Ya. B. Zeldovich, JETP Lett. 6, 316, 1967. [27] P.A.M. Dirac, ”Principles of Quantum Mechanics”, Clar endon Press, Oxford, 1958. [28] N.C. Petroni and J.P. Vigier, Fun. Phys 13 (2), 1983, 253ff. [29] E. Nelson, Phys. Rev., 150, 1966, pg.1079ff. [30] I. Prigogine, ”End of Certainity”, Free Press, New York , 1997. [31] Haisch, B., Rueda, A., and Puthoff, H.E., Phys. Rev. A49 (2), 1994, pp 678-694. [32] B.G. Sidharth ”Space Time as a Random Heap”, to appear in Chaos, Solitons and Fractals. 15

arXiv:physics/0003025v1 [physics.atom-ph] 13 Mar 2000Generalized oscillator strength for Na 3s−3ptransition Zhifan Chen and Alfred Z. Msezane Center for Theoretical Studies of Physical Systems, and Dep artment of Physics Clark Atlanta University, Atlanta, Georgia 30314, U. S. A. ABSTRACT Generalized oscillator strengths (GOS’s) for the Na 3 s−3ptransition have been investigated using the spin-polarized technique of th e random phase ap- proximation with exchange (RPAE) and the first Born approxim ation (FBA), focussing our attention on the position of the minimum. Inte rshell correla- tions are found to influence the position of the minimum signi ficantly, but hardly that of the maximum. The RPAE calculation predicts fo r the first time the positions of the minimum and maximum at momentum tra nsfer, K values of 1.258 a.u. and 1.61 a.u., respectively. The former value is within the range of values extracted from experimental measurements, K= 1.0−1.67 a.u.. We recommend careful experimental search for the mini mum around the predicted value for confirmation. PACS number(s):34.10.+x, 34.50Fa, 31.50.+w1 Introduction The generalized oscillator strength(GOS) is an important p roperty of the atom, since Bethe introduced it [1]. To study this property, the sodium atom has been chosen as the subject in many experimental and t heoretical investigations because its electronic configuration has an inert core and a single valence electron which is similar to that of the hydro gen atom. The differential cross sections (DCS’s) and GOS’s for the Na 3 s−3ptransition were measured by Shuttleworth at al [2] using a high-resolution electron spectrometer over the angular range of 1◦-20◦at the incident electron energies of 54.4, 100, 150 and 250 eV. The measurement observed a GOS mi nimum at the momentum transfer value of K= 0.67 a.u. (or K2= 0.45 a.u.). Buckman and Teubner [3] measured data for the same transitio n using a modulated crossed-beam technique at the incident energies of 54.4 , 100 , 150 and 217.7 eV, covering the extended angular range of 2◦to 145◦. Their data are in good agreement with those of Shuttleworth et alat small angles but did not show a GOS minimum around K2= 0.45 a.u.. The Shuttleworth’s et al minimum was also not confirmed by the ex- periment of Srivastava and Vuskovic [4], which utilized a cr ossed-electron -beam-metal-atom-beam scattering technique and performe d measurements at incident energies of 10, 20, 40, and 54.4 eV. The data of Sri vastava and Vuskovic are in poor agreement with those of Buckman and Teub ner at large 1angles. Srivastava and Vuskovic implied that the conflict wa s caused by the inproper geometrical correction factor in Buckman and T eubner’s exper- iment. To resolve the discrepancy Teubner et al[5] remeasured the Na 3 s−3p transition at 22.1 and 54.4 eV and found a probable source of s ystematic er- ror in the measurement of Srivastava and Vuskovic. All the me asurements did not observe the GOS minimum around K= 0.67 a.u. as predicted by Shuttleworth et al. Some recent measurements on the Na 3 s−3ptransition by Bielschowsky et al[6] at impact energy of 1 keV and Marinkovic et al[7] at 10, 20 and 54.4 eV did not report GOS minima in the momentum t ransfer regions they considered. Theoretically, Shimamura [8] predicted GOS minima to appea r between K2= 0.72 and 0.93 a.u., depending on the choice of the exponent in th e Slater orbitals. Miller [9] calculated the GOS minimum in th e Na 3 s−3p transition within the FBA, employing hydrogenlike orbital s with effective nuclear charge and predicted a minimum at K2= 0.71 a.u.. The FBA calculation from Bielschowsky et alshowed a minimum around K2= 2.0 a.u.. In summary the GOS minimum for the Na 3 s−3ptransition observed in Shuttleworth’s et alexperiment was not confirmed by other measurements. Also, the wide range of positions of the minimum predicted by the theoretical calculations at K2= 0.71, 0.72, 0.93 and 2.0 a.u. are not observed by the experments. Obviously, the GOS and the position of its minim um for the Na 23s−3ptransition is still an unresolved and interesting problem. In this paper we have used the spin-polarized technique of th e random phase approximation with exchange(RPAE) to investigate th e GOS for the Na 3s−3ptransition. The major objective of our calculation has been the unambiguous identification and location of the positions of the minimum and maximum. As a result, values of Kwere carefully selected. We found for the first time the positions of the minimum and the maximum to be at around K= 1.258 a.u. (or K2= 1.582 a.u.) and K= 1.61 a.u. (or K2= 2.59 a.u.), respectively. After careful analysis of the exper imental data, we found the GOS minimum at K= 1.258 a.u. is supported by some previous measurements. The intershell correlations are found to hav e significant effect to the position of the minimum. 2 THEORY In the FBA the generalized oscillator strength, f, for dipol e allowed transi- tions in the length form can be calculated [10] as f=2(2l+ 1)Nlw (2li+ 1)K2|dα|2(1) where Nlis the number of electrons in the excited state, liis the initial orbital angular momentum of the excited electron, lis the total angular momentum of the electron-hole pair, which satisfies triangle rule |lf+li|> l > |li−lf|, 3wis the excitation energy (a.u.). The dipole matrix element, dαcan be calculated from < φf|dα|φi>=/radicalBig (2li+ 1)(2 lf+ 1) lfl li 0 0 0 /integraldisplay∞ 0Pi(r)Pf(r)jl(Kr)dr(2) where Pi(r), Pf(r) are the radial wave functions of the initial and final states , respectively, jl(Kr) is the spherical Bessel function. For the dipole allowed transition, the calculations performed in this paper are wi thl= 1 and Pi(r), Pf(r) represented by Hartree-Fock wave functions. Each channel , such ass−ptransition, includes three discrete excited states, 3 p,4p,5p, and sev- enteen continuum states. The radial part of the wave functio n for each state was represented by 700 points. According to the semiempirical Hund rule, the total spin of a shell in the ground state reaches the largest value permitted by the P auli principle. Therefore, in the semiclosed shell all the electron spin vec tors are collinear, and their projections on to an arbitrary fixed direction are e qual. Therefore, every shell can be devided into two spin subshells, each havi ng a certain spin direction, ↑or↓. Because of this the wave functions double. Electrons in the subshell interact with other electrons in two different ways , with or without exchange. Since the Coulomb interaction will not change the spin direction, only the electrons having the same spin direction can intera ct with exchange. The equation for the dipole matrix element in the spin-polar ized technique 4of the RPAE [11] is (D↑ α, D↓ α) = (d↑ α, d↓ α) +/summationdisplay α′(D↑ α′χ↑ α′, D↓ α′χ↓ α′) u↑↑ α′αu↑↓ α′α u↓↑ α′αu↓↓ α′α  (3) where uα′αis the Coulomb inter-electron potential, d↑ αrepresents the am- plitude for the direct excitation of 3 s↑−3p↑andχα′is the electron-vacancy propagator. If the states of ↑and↓electron are equivalent, the dipole matrix element D↑ αin the RPAE for the 3 s↑−3p↑transition can be obtained from < ǫf|D↑ α|ǫi>=< ǫf|d↑ α|ǫi>+(/summationdisplay ǫ3≤F,ǫ4>F−/summationdisplay ǫ3>F,ǫ4≤F) < ǫ4|D↑ α|ǫ3>< ǫ 3ǫf|u↑↑,↑↓,|ǫ4ǫi> w−ǫ4+ǫ3+iη(1−2n4)(4) where ǫ3andǫ4represent the virtual excitation states, iηgives the direction of tracing the pole while integrating over the energy, η→+0,Fis the Fermi energy of the atom, and n4is the Fermi step: n4= 1,ǫ4≤F;n4= 0, ǫ4> F. The symbol/summationtextdenotes summation over discrete and integration over continuous states. A similar equation for the D↓ αof the 3 s↓−3p↓transition can be obtained. It is important to remember that only the sta tes with the same spin direction can have exchange interaction in the sum of Eq. (4). Finally, the GOS for Na 3 s−3ptransition can be written as f=2(2l+ 1)Nlw (2li+ 1)K2(D↑ α2+D↓ α2) (5) Eqs. (1),(2),(4) and (5) are the basic equations used in this paper to calculate the GOS’s in FBA and RPAE. 53 RESULTS AND DISCUSSION The results of our calculation are given in Fig. 1, Fig. 2 and T able 1. Table 1 lists the positions of the minimum and maximum for the Na 3 s−3ptran- sition obtained by different authors. The results of our FBA c alculation are in excellent agreement with that of Bielschowsky et al. The data of Shuttle- worth et alshowed a GOS minimum around K= 0.67 a.u.. However, this minimum was not observed in all other measurements. After an alyzing the experimental data obtained by Shuttleworth et al[2], Buckman and Teub- ner [3], Srivastava and Vuskovic [4], Teubner et al[5], and Marinkovic et al [7], we found that the GOS minima from the experimental data a re around K= 1.0−1.67 a.u. (or K2= 1.0−2.80 a.u.) at impact energies of 20 and 54.4 eV. These minima were not originally noticed by the expe rimentlists. 6Fig. 1 shows the GOS’s versus K2for the Na 3 s−3ptransition at 20 eV. The circles are from Srivastava and Vuskovic, black dots are from Marinkovic 7et al, and squares are from Teubner et al. The solid and dotted lines represent our RPAE and FBA calculations, respectively. All three meas urements show the GOS minimum around K= 1.0−1.18 a.u. (or K2= 1.0−1.39 a.u.) which are close to our results. 8Fig. 2 is the same as Fig.1, except that the data are at 54.4 eV a nd the crosses are from Buckman and Teubner and the triangles ar e from Shut- 9tleworth et al. The black dots and circles show the GOS minimum around K= 1.67 a.u. (or K2= 2.80 a.u.), while the crosses give a minimum at K= 1.51 a.u. (or K2= 2.29 a.u.). Since the position of the GOS min- imum for the Na 3 s−3ptransition occurs at large momentum transfer, K= 1.0−1.67 a.u., which corresponds to large angles in the measuremen ts; for example θ= 37◦at the impact energy of 54.4 eV( K= 1.258 a.u.). There- fore it is easy to miss the GOS minimum in the measurement by ta king large angular steps at large angles as is the usuall practice. From our calculation it is suggested to reperform the experiment for Na 3 s−3ptransition and pay particular attention to the position of the minimum we indic ated in Table 1. We believe more experimental data will be obtained to confirm our results. It is interesting to compare the intershell correlations fo r the transitions between the Na 3 s−3pand the Ar 3 p−4s. The position of the GOS mini- mum for the later is influenced insignificantly by correlatio ns. The difference between positions of the minimum from the RPAE and FBA is less than 0.7%. However, in the Na 3 s−3ptransition the difference is more than 11%. This is because the Na 3 plevel is only 2.1 eV above the 3 sground state; therefore Na has an enormous dipole polarizability (23 .6×10−24cm3). In RPAE the many-electron correlation effects are essentially due to polariza- tion of the electron shell by the external field. Results from RPAE will show a significant difference from that of FBA if the atoms have larg e polarizabil- 10ity. Therefore, we can expect that Li, K and Rb atoms will yiel d results that are similar to those of Na, while Ne, Kr and Xe atoms will exhib it results similar to those of Ar if we compare the positions of GOS minim um from RPAE and FBA. Correlations between intershell electrons are strongly aff ected by the separation of the subshells. The 3 s↑ −3p↑excitation energy will be 0.07267(a.u.), 0.07255(a.u.), 0.07190(a.u.), and 0.0717 5 (a.u.) if channels (s↑−p↑), (s↑−p↑+p↑−s↑+p↓−s↓), (s↑−p↑+p↑−d↑+p↓−d↓) and (s↑−p↑+p↑−d↑+p↓−d↓+p↑−s↑+p↓−s↓) are included in the RPAE calculations. This indicates that the influence of the 3 d↑↓electrons upon the 3p↑electron is larger than that from the 4 s↑↓electrons. In conclusion, the positions of the minimum and the maximum f or the Na 3s−3ptransition have been calculated and found for the first time f rom RPAE at the momentum transfer values of K= 1.258 a.u. and 1.61 a.u., respectively. Furthermore, the many-electron correlatio ns are found to play an important role in the determination of the position of the minimum. We recommend that experiments search carefully for predicted minimum, using the value obtain here as a guide. 114 Acknowledgments Research was supported in part by the US DoE, Division of Chem ical Sci- ences, Office of Basic Energy Sciences, Office of Energy Researc h and NSF. 12References [1] H. A. Bethe, Ann. Phys. 5, 325 (1930) [2] T. Shuttleworth, W. R. Newell and A. C. H. Smith, J. Phys. B 101641, (1977) [3] S. J. Buckman and P. J. O. Teubner, J. Phys. B 12, 1741 (1979) [4] S. K. Srivastava and L. Vuskovic, J. Phys. B 13, 2633 (1980) [5] P. J. O. Teubner, J. L. Riley, M. J. Brunger and S. J. Buckma n, J. Phys. B19, 3313 (1986) [6] C. E. Bielschowsky, C. A. Lucas, G. G. B. de Souza, and J. C. Nogueira, Phys. Rev. A 435975 (1991) [7] B. Marinkovic, V. Pejcev, D. Fillpovic, I. Cadez, and L. V uskovic, J. Phys. B 255179 (1992) [8] Isao Shimamura, J. Phys. Soc. Jap 30824 (1971) [9] K. J. Miller, J. Chem. Phys. 595639 (1973) [10] M Ya Amusia and L V Chernysheva, Computation of Atomic Pr ocess, Institute of Physics Publishing, Bristol and Philadelphia , 1998. 113 13[11] M. Ya. Amusia, Atomic Photoeffect, Plenum Press, New Yor k, 1990. 136-140 14Figure Captions Fig. 1. GOS’s versus K2for the Na 3 s−3ptransition at 20 eV. Circles are from Srivastava and Vuskovic, black circles are from Mar inkovic et al, and squares are from Teubner et al. Solid and dotted lines represent our RPAE and FBA calculations, respectively. Fig. 2. GOS’s versus K2for the Na 3 s−3ptransition at 54.4 eV. All the symbols have the same meaning as in Fig. 1, except that crosse s are from Buckman and Teubner and triangles are from Shuttleworth et al. 15Table 1: Positions of the GOS minimum and the maximum for the e xcitation of Na 3 s−3p K(a.u.) for minimum K(a.u.) for maximum Atom Authors Expt. Theory Theory Shuttleworth et al[2] 0.67 Marinkovic et al[7] 1.18-1.67 Buckman and Teubner [3] 1.51 Srivastava and Vuskovic [4] 1.0-1.7 Teubner et al[5] 1.15 Shimamura 0.847-0.965 Na Miller 0.84 Bielschowsky et al 1.41 Present RPAE 1.258 1.61 Present H-F 1.401 1.69 16

arXiv:physics/0003026v1 [physics.plasm-ph] 12 Mar 2000Electron-drift driven ion-acoustic mode in a dusty plasma w ith collisional effects R. Singh and M. P. Bora∗ Institute for Plasma Research, Bhatt, Gandhinagar-382 428 , India. Instabilities of ion-acoustic waves in a dusty plasma with e lectron-drift, collisional, and dust charge fluctuations effects, have been investigated. The reg imes are clearly marked out where the theory is applicable. The critical electron-drift velocit y required to drive the instability is predicted. It is also shown that electron thermal conductivity and char ged grains concentration enhance the growth of the ion-acoustic mode whereas ion-viscosity, ion -thermal conductivity, and dust charge fluctuations have a stabilizing effect. I. INTRODUCTION Dusty plasmas have acquired considerable importance1 because of their applications to astrophysics and plan- etary physics2, problems of plasma processing3, and physics of strongly coupled systems4. Experimentally, one of the important areas of investigation is the study of low frequency ( ω≤Ωi) fluctuations in a plasma with dust grains. In a recent experiment by Barkan et al.5, it is shown that the phase velocity of ion-acoustic wave in- creases with the density of negatively charged dust grains and thus ion Landau damping becomes less severe in a plasma with charged dust grains5or with negative ions6. The ion-acoustic wave is a typical compressional mode in which ions provide the inertia and electrons bring in the pressure effects through their shielding cloud. In a magnetized plasma, it propagates with acoustic phase speed along the magnetic field lines. Earlier, the ki- netic theory of the current-driven ion-acoustic instabil- ity, in a fully ionized, collisional7,8and collisionless9two- component plasma, has been investigated by various au- thors. In the limit of strong collisions (i.e. mean free path smaller than the wavelength), Coppi et al.10and Rognlien et al.11have used the two-fluid equations for an investigation of the current-driven ion-acoustic insta - bility. Kulsrud and Shen12have investigated the effect of weak collisions on ion-acoustic wave using a Fokker- Plank model, while St´ efant13used kinetic theory and a collision integral of Bhatnagar-Gross-Krook type to show that ion-viscosity leads to a damping of the ion-acoustic instability, whereas, the electron-ion collisions tend to in- crease the growth rate. Recently, Rosenberg7has carried out the investigation of current-driven ion wave instabil- ity in a fully ionized collisionless dusty plasma using ki- netic theory to predict the critical electron-drift veloci ty relative to the ions, required to drive the instability. Interpretations of ion-acoustic wave excitation in a dusty plasma experiment have typically relied on the theories based on collisionless inverse electron Landau damping10,11. A realistic analysis of the experimentalsituation shows that in a typical experimental setup5,6, the plasma density maybe quite high ∼109-1010cm−3 when the plasma temperature is quite low. As an ex- ample, when plasma density ∼1010cm−3, with elec- trons and ions having approximately equal temperatures Te∼Ti∼0.1 eV, the electron-ion collision mean free path ( λe) may be comparable or even smaller than the plasma scale length ( L) along the magnetic field, i.e. λe<∼L. Under these conditions, the electrons no longer follow the Boltzmann relation and a collisionless theory of the electron-drift driven ion-acoustic instability is n ot justified. It is therefore important to re-examine this ion wave instability in a collisional dusty plasma. In these experiments cited above, however, ion-acoustic waves are launched by applying a sinusoidal voltage to a grid im- mersed in plasma rather than by drawing an electric current. Nevertheless, the effect of Coulomb collisions in presence of current driven ion-acoustic waves in such a plasma, as is considered in this work, remains to be seen in a laboratory experiment. In another experiment by Barkan et al.14, current driven ion-acoustic waves in a dusty plasma are studied. But, this experiment was conducted at a relatively low plasma densities ∼106- 107cm−3. In this paper, we investigate an electron-drift (or elec- tron current) driven ion-acoustic mode in the presence of negatively charged dust grains. The dust charged grains are considered to be massive particles in a multicompo- nent plasma. This is valid when a≪d≪λD, where ais the average dust radius, dis the average distance between the dust particles, and λDis the plasma Debye length. The paper is organized as follows. In Sec.II, the regimes of interest, where the theory is applicable are discussed. Sec.III deals with the linear instability theor y. Sec.IV contains the conclusions. ∗Permanently at Physics Department, Gauhati University, Guwahati-781 014 , India. Electronic mail : mbora@gw1.dot.net.in 1II. REGIMES OF INTEREST At first, we refer to regimes of interest where the present calculations are relevant. We consider a typi- cal experimental situation such as considered in exper- iments by Barkan et al.5,14and Song et al.6. In such conditions, the typical parameters in these experiments are : potassium plasma (K+) with mi/mp∼40,miand mpbeing the ion and proton masses respectively, dust mass md∼10−12gm, gas pressure <∼10−5Torr, the corresponding neutral ( n) density nn≃2×1011cm−3, equal electron ( e) and ion ( i) temperatures Te∼Ti∼ 0.1 eV, and dust grain temperature Td∼0.03 eV (as- sumed to be at room temperature). The plasma is confined radially by a magnetic field B∼0.4 T and the plasma length along the magnetic field L∼40 cm with a radius rp∼2 cm for the plasma column. The characteristic frequency of the waves f∼20-80 kHz which corresponds to ω= 2πf∼1-5×105rad/s. The other plasma parameters are : electron-thermal veloc- ityce= (Te/me)1/2∼1.3×107cm/s, ion-thermal ve- locity ci= (Ti/mi)1/2∼5×104cm/s, the ion-acoustic speed cs= (Te/mi)1/2∼5×104cm/s, the dust-thermal velocity cd= (Td/md)1/2∼2 cm/s, the ion gyrofre- quency Ω i=eB/m ic∼106s−1, the ion Larmor radius ai=ci/Ωi∼7×10−2cm, the collision frequency of electrons and ions with the neutrals νen=nnσence∼ 1.3×104s−1,νin=nnσinci∼50 s−1(where σin∼σen∼ 5×10−15cm2), with the corresponding mean free paths λen∼λin∼103cm. The e-icollision frequency ( νei) andi-icollision frequency ( νii) in the form presented by Braginskii9are calculated for the present parameters and are found to be as νei= 1/τe∼8.4×106s−1and νii= 1/τi∼2.8×104s−1and the mean free paths are λe=ce/νei∼1 cm and λi=ci/νii∼1 cm, where τe= 3.5×104T3/2 e0/nλ,τi= 3.0×106(mi/2mp)1/2T3/2 i0/nλ, andλ= 23.4−1.15 log n+ 3.45 log T∼9.5 (for T < 50 eV) is the Coulomb logarithm. From these calculations the following assumptions can now be made (1)the collision mean free paths of electrons and ions are comparable to or even smaller than the plasma scale length along the magnetic field i.e. λe,i< L, so that we can make use of Braginskii’s fluid equa- tions, (2)sinceω < ν ei, we can neglect the electron inertia, (3)sinceai< rpand assuming a homogeneous plasma, the ion motion transverse to the magnetic field can be neglected (the polarization drift), (4)since νei≫νen, νedandνii≫νin, νid, so the mo- mentum and energy losses of electrons and ions to the neutrals and dust grains can also be neglected in the fluid equations, (5)since the phase velocity of the observed ion-acoustic wave is much larger than the dust-thermal speed i.e.vp≫cd, we can neglect the dust dynamics in the analysis. However, we shall incorporate the self-consistent fluctuation of charge on dust grains,which is shown to cause a damping effect on the ion-acoustic wave15,16. The charge fluctuation on the surface of the dust grains, which depends cru- cially on the dust size and the plasma density, is important when charging rate is comparable to the growth of the mode. (6)we consider negatively charged grains as an addi- tional charged plasma species of uniform massive particles similar to a multicomponent plasma. This remains valid as long as a≪d≪λD. III. INSTABILITY ANALYSIS We write down the basic equations in order to de- rive the linear dispersion relation of electron-drift driv en ion-acoustic wave instability including the effects of dust charge fluctuations, electron and ion temperature perturbations, ion-viscosity etc. We consider a one- dimensional, plane, homogeneous plasma with low β (β= 4πnT e/B2≪1 is the ratio of kinetic energy to mag- netic energy). The resultant governing equations for elec- trons and ions are the Braginskii’s fluid equations viz., equations of continuity, momentum, and energy. We write down the equilibrium equations that define this specific regime of interest and confine ourselves to the parallel motions only. They are the ion and electron continuity equations ∂ni ∂t+∇/bardbl·(niui/bardbl) = 0, (1) ∂ne ∂t+∇/bardbl·(neue/bardbl) = 0, (2) parallel electron and ion momentum equations minidui/bardbl dt=−∇/bardblpi+eniE+µ/bardbl∇2 /bardblui/bardbl, (3) menedue/bardbl dt=−∇/bardblpe−eneE+R, (4) where Rrepresents the momentum gained by the elec- trons through collision with the ions9, R=Ru+RT, (5) Ru≃ −meneνei(0.51u/bardbl), (6) RT≃ −0.71ne∇/bardblTe, (7) u/bardbl=ue/bardbl−ui/bardbl. (8) In the above equations, Eis the electric field driving the instability and µ/bardbl= 0.96ni0Ti0/νiiis the parallel ion-viscosity coefficient9. In the ion momentum equa- tion, however, we have neglected the collision term10in the limit ui/bardbl>(me/mi)(νei/ω)u/bardbl. We have further ne- glected the electron viscosity term in Eq.(4). The re- maining equations are the energy equations9, 23 2nidTi dt+pi(∇/bardbl·ui/bardbl) =∇/bardbl·(χi /bardbl∇/bardblTi), (9) 3 2nedTe dt+pe(∇/bardbl·ue/bardbl) =∇/bardbl·(χe /bardbl∇/bardblTe) −0.71∇/bardbl·(neTeu/bardbl),(10) where χi/bardbl= 3.9ni0Ti0/miνiiandχe/bardbl= 3.2ne0Te0/meνei are the parallel ion and electron-thermal conductivities, respectively. In the above equations, the terms ∼ω−1 ce,i have neglecetd owing to the fact that ω < ω ce,i. We consider a small electrostatic perturbation and lin- earize the above equations. The linearized equations are, ∂ni1 ∂t+∇/bardbl(ni0ui/bardbl1) = 0, (11) ∂ne1 ∂t+∇/bardbl(ne0ue/bardbl1+ne1ue/bardbl0) = 0, (12) where equilibrium and perturbed quantities are defined by the subscripts 0 and 1, respectively and ue/bardbl0is the zeroth order electron-drift velocity with respect to theions and dust grains. The electrons drift with respect to the ions and dust particles, so that ui0=ud0= 0. The ion motion along the magnetic field Bis given by the linearized parallel momentum equation mini0∂ui/bardbl1 ∂t=−∇/bardbl(eni0ϕ1+ni0Ti1+ni1Ti0) +µ/bardbl∇2 /bardblui/bardbl1, (13) where ϕ1is the perturbed electrostatic potential. For ω < ν ei, the parallel electron momentum equation be- comes 0 =∇/bardbl(ene0ϕ1−ne1Te0−ne0Te1)−0.71ne0∇/bardblTe1 −0.51meνei0(ne0ue/bardbl1−ne0ui/bardbl1) −0.51mene0ue/bardbl0νe1, (14) where νe1=νei0(ne1/ne0−3Te1/2Te0) and the equilib- rium drift velocity ue/bardbl0=−eE/bardbl0/meνei. The perturbed electron and ion temperature equations are given by 3 2ni0∂Ti1 ∂t+ni0Ti0∇/bardblui/bardbl1=χi/bardbl∇2 /bardblTi1, (15) 3 2ne0/parenleftbigg∂ ∂t+ue/bardbl0∇/bardbl/parenrightbigg Te1+ne0Te0∇/bardblue/bardbl1=χe/bardbl∇2 /bardblTe1−0.71ne0Te0∇/bardbl(ue/bardbl1−ui/bardbl1) −0.71ue/bardbl0∇/bardbl(ne0Te1+ne1Te0), (16) Finally, we write the equations for dust charge fluctua- tions as given by Jana et al.17,18and the quasineutrality condition as /parenleftbigg∂ ∂t+η/parenrightbigg Qd1=−|Ie|/parenleftbiggni1 ni0−ne1 ne0/parenrightbigg ,(17) e(ne1−ni1) +nd0Qd1= 0, (18) where Qd0=eZd0,Zd0is the equilibrium charge num- ber on the surface of the dust grains, Qd1=eZd1,Zd1 is the charge fluctuation, η=e|Ie|(T−1 e0−W−1 0)/Cwith W0=Ti0−eφf0,φf0is the potential at the dust surface, |Ie| ∼Ii∼eni0πa2csis the equilibrium electron (or ion) current at the dust surface, and C∼abeing the grain capacitance. In writing Eq.(18), we have neglected thedust density fluctuations as the ion transit time ( ∼L/cs) is much shorter than the dust transit time19[∼L/cd∗, cd∗∼(Z2 d0nd0Ti0/ni0md)1/2] along the magnetic field. It should, however, be noted that in writing the above equations, we have assumed a stationary back- ground equilibrium. On the other hand, a non-stationary background equilibrium may lead to stabilization of the ion-acoustic instability in a low temperature collisional plasma20. We now take perturbations of the form f(z, t)∼f1e−i(ωt−k/bardblz)(19) and write Eqs.(11-18) as ˜ni=k/bardblui/bardbl1 ω, (20) ˜ne=k/bardblue/bardbl1 (ω−ω/bardbl0), (21) k/bardblui/bardbl1=k2 /bardblc2 s (ω+iˆµ/bardbl)/parenleftBigg ˜ϕ+˜ni τ+˜Ti τ/parenrightBigg , (22) ˜ne+/parenleftbigg 1.71 +i3 2ω/bardbl0 ˆχe/parenrightbigg ˜Te−˜ϕ=ik/bardbl ˆχe(ue/bardbl1+ue/bardbl0˜ne−ui/bardbl1), (23) /parenleftbigg3 2ω+iˆχi/bardbl/parenrightbigg ˜Ti=k/bardblui/bardbl1, (24) 3/parenleftbigg3 2ω−2.21ω/bardbl0+iˆχe/bardbl/parenrightbigg ˜Te=k/bardbl(1.71ue/bardbl1−0.71ui/bardbl1) + 0.71ω/bardbl0˜ne, (25) where τ=Te0/Ti0, ˜ne=ne1/ne0, ˜ni=ni1/ni0,˜Ti=Ti1/Ti0,˜Te=Te1/Te0, ˜ϕ=eϕ1/Te0are the normalized variables with ˆχe≃k2 /bardblc2 e/0.51νei, ˆµ/bardbl= 0.96k2 /bardblc2 i/νii,ω/bardbl0=k/bardblue/bardbl0, ˆχe/bardbl= 3.2k2 /bardblc2 e/νei, and ˆ χi/bardbl= 3.9k2 /bardblc2 i/νii. We can now combine Eqs.(17) and (18) as /parenleftBigg 1 +iZd0nd0 ne0ˆI0 (ω+iη)/parenrightBigg ˜ne=/parenleftBigg ni0 ne0+iZd0nd0 ne0ˆI0 (ω+iη)/parenrightBigg ˜ni, (26) where ˆI0=|Ie|/eZd0andλD= (Te0/4πne0e2)1/2is the plasma Debye length. In the limit ω∼ω/bardbl0<ˆχe/bardbli.e. 3.2(k/bardblλe)2νei/ω >1, Eq.(25) can be re-written, using Eq.(21), as ˜Te=−i(1.71ω−ω/bardbl0) ˆχe/bardbl˜ne+i0.71ω ˆχe/bardbl˜ni. (27) From Eqs.(20) and (24), we get ˜Ti=ω/parenleftbig3 2ω+iˆχi/bardbl/parenrightbig˜ni. (28) Substituting Eqs.(22) and (28) into Eq.(20), we obtain /bracketleftBigg 1−k2 /bardblc2 i ω(ω+iˆµ/bardbl)−k2 /bardblc2 i (ω+iˆµ/bardbl)/parenleftbig3 2ω+iˆχi/bardbl/parenrightbig/bracketrightBigg ˜ni=k2 /bardblc2 s ω(ω+iˆµ/bardbl)˜ϕ. (29) Using Eqs.(20), (21), and (27), Eq.(23) can be written as /bracketleftbigg 1−iω ˆχe−i1.71(1.71ω−ω/bardbl0) ˆχe/bardbl/bracketrightbigg ˜ne−˜ϕ=−i1.75ω ˆχe˜ni. (30) In the limit of ω > η , Eqs.(26), (29), and (30) can be combined to obtain the follo wing dispersion relation, /bracketleftBigg 1−k2 /bardblc2 i ω(ω+iˆµ/bardbl)−k2 /bardblc2 i (ω+iˆµ/bardbl)/parenleftbig3 2ω+iˆχi/bardbl/parenrightbig/bracketrightBigg/bracketleftBigg 1 +iZd0nd0 ne0ˆI0 ω/parenleftbigg 1−ne0 ni0/parenrightbigg/bracketrightBigg =ni0 ne0k2 /bardblc2 s ω(ω+iˆµ/bardbl)/bracketleftbigg 1−iω ˆχe−i1.71(1.71ω−ω/bardbl0) ˆχe/bardbl +i1.75ω ˆχene0 ni0/braceleftBigg 1 +iZd0nd0 ne0ˆI0 ω/parenleftbigg 1−ne0 ni0/parenrightbigg/bracerightBigg/bracketrightBigg . (31) ForTi0∼Te0andk2 /bardblc2 e/ωνei>1, it is obvious that ω >ˆµ/bardbl,ˆχi/bardbl(i.e.k2 /bardblc2 i/ωνii<1). We now write Eq.(31) as ǫ(k/bardbl, ω) = 0 and ω=ωr+iγwithωr> γ. Taking ε= ˆµ/bardbl/ω(0) ras an expansion parameter, the real part of ω, to the first order, can be written as, ω2 r≃(ω(0) r)2+ 1.71k2 /bardblc2 s/parenleftbiggˆµ/bardbl ω(0) rni0 ne0ω/bardbl0 ˆχe/bardbl/parenrightbigg , (32) where ω(0) ris given by ω(0) r=k/bardblcs/parenleftbiggni0 ne0+5 3Ti0 Te0/parenrightbigg1/2 . (33) The growth rate can now be written as γ≃ −ˆµ/bardbl 2−2 9k2 /bardblc2 i ω2rˆχi/bardbl−Zd0nd0 2ne0ni0 ne0k2 /bardblc2 s ω2rˆI0/parenleftbigg 1−ne0 ni0/parenrightbigg + 1.71ni0 ne0k2 /bardblc2 s 2ˆχe/bardbl/parenleftbiggω/bardbl0 ωr−1−Zd0nd0 ni0ˆχe/bardbl ˆχe/parenrightbigg . (34) 4(c)(b)(a) ˆkˆγ 6 5 4 3 2 10.3 0.25 0.2 0.15 0.1 0.05 0 FIG. 1. Normalized growth rate ˆ γof the ion-acoustic wave vs. normalized wave number ˆkfor different values of the pa- rameter δ=ne0/ni0. The curves labeled as ( a), (b), and ( c) are for δ= 1 (no dust), 0.4, and 0.3, respectively. The other parameters are ˆ u= 6.0,ˆI= 0, and ˆλe,i= 0.01.(d)(c)(b)(a) ˆkˆγ 6 5 4 3 2 10.2 0.15 0.1 0.05 0 FIG. 2. Behavior of ˆ γvs.ˆkfor (a)ˆI= 0, (b) 0.01, ( c) 0.05, and (d) 0.1. Dust density parameter δ= 0.3 and ˆλe,i= 0.01. We note that, at the zeroth order, the phase veloc- ity of the ion-acoustic wave is modified with a factor of/parenleftbiggni0 ne0+5 3Ti0 Te0/parenrightbigg1/2 . The presence of dust (decreasing ra- tio of ne0/ni0) leads to an increase in the phase velocity of the ion-acoustic wave. In the expression for growth rate of the ion-acoustic wave Eq.(34), the collisional damping is represented by the first two factors, the parallel ion-viscosity and parall el ion-thermal conductivity. The third term causes damp- ing due to charge fluctuation on the surface of the dust grains, which is proportional to the electron/ion current (ˆI0) on the dust surface. We note that in absence of dust charge fluctuation (i.e. ˆI0= 0), increasing nega- tively charged dust density with respect to electron and ion density, results in an increase of the growth rateof the ion-acoustic instability. This can be seen from Fig.1, where we have plotted the normalized growth rate ˆγ=γL/c sagainst normalized wave number ˆk=k/bardblL, as obtained from the solution of the full dispersion relation Eq.(31), at different negatively charged dust concentra- tionδ=ne0/ni0. Note that δ= 0.3 means 70% of the electronic charges is now carried by the massive dust par- ticles and δ= 1 means no negatively charged dust grains. The other parameters in Fig.1 are ˆ u=ue/bardbl0/cs= 6, ˆλ=λi,e/L= 0.01, and ˆI=ˆI0L/cs= 0. As in ion- acoustic instability without dust particles, the positive growth rate, here also, appears only when the drifting electron velocity ue/bardbl0, exceeds several times the phase velocity. Therefore, the necessary condition for required electron-drift relative to ions and negatively charged dus t grains to excite the ion wave instability is ue/bardbl0> cs/parenleftbiggni0 ne0+5 3Ti0 Te0/parenrightbigg1/2/braceleftbigg/parenleftbigg 1 +Zd0nd0 ni0/parenrightbigg + (k/bardblλe)2ne0 ni0Ti0 Te0νei νii/bracketleftBigg 1.8 +/parenleftbigg5 3+ni0Te0 ne0Ti0/parenrightbigg−1/bracketrightBigg + 1.87/parenleftbiggZd0nd0 ni0/parenrightbigg2ni0 ne0/parenleftbiggni0 ne0+5 3Ti0 Te0/parenrightbigg−1/parenleftbiggπa2λeni0 Zd0/parenrightbigg/parenleftbiggmi me/parenrightbigg1/2/bracerightBigg , (35) where we have used, only the zeroth order expression forωr. Note that the ion-viscosity and ion-thermal con- ductivity raise the critical electron current for the ion- acoustic wave to be unstable [the term in the square bracket in Eq.(35)], whereas the electron-thermal con- ductivity tends to increase the growth rate, as seen from Eq.(34). The effect of increasing negatively charged dust den- sity on growth rate of the ion-acoustic instability may be compensated by the presence of charge fluctuation ondust surface, which has a damping effect, as shown in Fig.2. In Fig.2 we have taken the dust density parame- terδ= 0.3. The other parameters are same as in Fig.1. We show in Fig.3, the behavior of ˆ γversus δfor different values of dust current ˆI= 0, 0.05, 0.1, and 1.0. Other parameters chosen are ˆ u= 6,ˆλ= 0.01, and ˆk= 5.0. The overall behavior is consistent with Fig.1 and Fig.2. 5(d)(c) (b)(a) ˆkˆγ 1 0.9 0.8 0.7 0.6 0.50.1 0.08 0.06 0.04 0.02 0 FIG. 3. Dependence of ˆ γon the dust density parameter δ for different fluctuation level of the dust charge. ( a)ˆI= 0 i.e. no dust charge fluctuation, ( b)ˆI= 0.05, (c)ˆI= 0.1, and (d)ˆI= 1.0. Note the change in behavior of ˆ γas the dust cur- rent increases. The normalized wave number ˆk= 5.0, ˆu= 6.0, andˆλe,i= 0.01. However, it is important to note that, although a de- creasing δcauses peaking up of the growth rate, at any given wave number, a rapid dust charge fluctuation rate (larger ˆI0) causes the wave to damp at higher dust con- centration, which is opposite in behavior to the case of absence of dust charge fluctuation (Fig.1 and 2), as can be seen from all the curves in Fig.3. For typical experimental parameters ne0/ni0∼0.3, Zd0nd0/ni0∼0.7,Te0∼Ti0, and νei/νii∼(mi/me)1/2, the necessary threshold condition for instability reduces to ue/bardbl0>2.24cs/bracketleftBigg 1.7 +/parenleftbiggmi me/parenrightbigg1/2/parenleftBig 0.7k2 /bardblλ2 e + 0.6πa2λeni0 Zd0/parenrightbigg/bracketrightbigg . (36) It is to be noted that the effects of charge fluctuation at the dust surface will tend to play an important role when the mean free paths are comparable to the plasma length along the magnetic field and the term πa2λeni0 Zd0>∼3/parenleftbiggme mi/parenrightbigg1/2 . (37) Also notice that in high density and low temperature plasma limits, the effects ion-viscosity and thermal con- ductivity will be important, typically at wavelengths such thatk/bardblλe∼1.5/parenleftbiggme mi/parenrightbigg1/4 . (38) The condition (36) can be realized in typical labora- tory situations5. For example, with an average size of dust grains to be few microns ( a∼10−4cm) and a cor- responding Zd0∼104, the condition (36) yields ue/bardbl0>∼4cs, (39) for the excitation of the ion-acoustic instability, where we have taken the collisional parameters as λe,i/L≃0.01 andˆk=k/bardblL∼2. The typical parallel electric field re- quired for the corresponding threshold electron current is∼0.1 V/m. IV. CONCLUSIONS We have studied the ion-acoustic instability in a col- lisional dusty plasma with fluid equations, where dust grains are treated as massive and negatively charged com- ponent in a multicomponent plasma. The regimes are clearly marked out where the theory is applicable, espe- cially in a relatively high density (1010cm−3) and low temperature plasma where the effect of collisions cannot be neglected. While treating the negatively charged dust particles, we take into account the effect of charge fluctu- ation on the dust surface. We have shown that in such a plasma, which are routinely produced in laboratory, there is a significant impact of electron-ion collisions, even at large mean free paths (i.e. λf>∼L). We have derived the threshold electron-drift velocity required to drive the io n- acoustic instability. It is shown that the electron-therma l conductivity and the dust charge concentrations reduce the threshold value of electron current for driving the ion- acoustic mode. In particular, the ion-viscosity and ion- thermal conductivity raise the threshold current. And a similar effect due to dust charge fluctuations is also found when the mean free paths are of the order of the plasma length. ACKNOWLEDGMENT It is pleasure to thank A. Sen for fruitful discussions. One of the authors, MPB, would also like to thank A. Sen for kind hospitality at IPR. 1Physics of Dusty Plasmas, Proceedings of the VI workshop in D usty Plasma , edited by P. K. Shukla, D. A. Mendis, and 6V. Chew (World Scientific, Singapore, 1996). 2C. K. Goertz, Rev. Geophys. 27, 271 (1989). 3D. B. Graves, IEEE Trans. Plasma Sci. 22, 31 (1994). 4H. Ikezi, Phys. Fluids 29, 1764 (1986). 5A. Barkan, N. D’Angelo, and R. L. Merlino, Planet. Space Sci.44, 239 (1996). 6B. Song, N. D’Angelo, and R. L. Merlino, Phys. Fluids B3, 284 (1991). 7M. Rosenberg, Planet. Space Sci. 44, 229 (1993). 8N. D’Angelo, S. von Goeler, and T. Ohe, Phys. Fluids 9, 271 (1966). 9S. I. Braginskii, in Reviews of Plasma Physics , edited by M. A. Leontovich (Consultants Bureau, New York, 1965) Vol. 1, pp. 205. 10B. Coppi and E. Mazzucato, Phys. Fluids 14, 134 (1971).11T. D. Rognlein and S. A. Self, Phys. Rev. Lett. 27, 792 (1971). 12P. M. Kulsrud and C. S. Shen, Phys. Fluids 9, 177 (1966). 13R. J. St´ efant, Phys. Fluids 14, 2245 (1971). 14A. Barkan et al., Phys. Lett. A 222, 329 (1996). 15F. Li, O. Havnes, and F. Melandsø, Planet. Space Sci. 42, 401 (1994). 16N. D’Angelo, Planet. Space Sci. 42, 507 (1994). 17M. R. Jana, A. Sen, and P. K. Kaw, Phys. Rev. E 48, 3930 (1993). 18R. K. Varma, P. K. Shukla, and V. Krishan, Phys. Rev. E 47, 3612 (1993). 19P. K. Kaw and R. Singh, Phys. Rev. Lett. 79, 423 (1997). 20P. K. Kaw and A. K. Sundaram, Phys. Lett. A38, 355 (1972). 7

arXiv:physics/0003027v1 [physics.atom-ph] 14 Mar 2000Magnetic Field Dependence of Ultracold Inelastic Collisio ns near a Feshbach Resonance J. L. Roberts, N. R. Claussen, S. L. Cornish, and C. E. Wieman JILA, National Institute of Standards and Technology and th e University of Colorado, and the Department of Physics, University of Colorado, Boulder, Colorado 80309-0440 (February 21, 2014) Inelastic collision rates for ultracold85Rb atoms in the F=2, m f=–2 state have been measured as a function of magnetic field. Dramatic change in the vicinity of a Feshbach resonance at 155 G was observed. Similar to the elastic rate, the inelastic rates s how a high peak and a deep trough. Both two- and three-body processes are important, and individua l contributions have been determined and compared with theory. This work has made it possible to creat e an85Rb Bose-Einstein condensate with a highly adjustable scattering length. Feshbach resonances have recently been observed in a variet y of cold atom interactions, including elastic scattering [1], radiative collisions and enhanced inelastic loss [2], photoassociation [3], and the loss of atoms from a Bose-Eins tein condensate (BEC) [4]. By changing the magnetic field through the resonance, elastic collision rates can be changed by orders of magnitude and even the sign of the atom-atom inte raction can be reversed [5]. The work in Ref. [4] has received particular attention because the BEC loss rate s were extraordinarily high and several proposals for exotic coherent loss processes have been put forward [6–8]. However, even ordinary dipole relaxation and three-body recombination are expected to show dramatic enhancements b y the Feshbach resonance [9–14], and the calculations of these ordinary enhancements have never been tested. This has left many outstanding questions as to the nature of inelastic losses near a Feshbach resonance. How large are the dipole relaxation and three-body recombination near the Feshbach resonance and how accurate are the calcula tions of these quantities? How much of the observed condensate losses in Ref. [4] are due to these more tradition al mechanisms and how much arise from processes unique to condensates? How severe are the“severe limitations” [4] that inelastic loss puts on the use of Feshbach resonances to change the s-wave scattering length in a BEC? The nature of the inelastic losses near Feshbach resonances also has important implications for efforts to create BEC in85Rb, because these losses play a critical role in determining the success or failure of evaporative cooling. Thus it is impera tive to better understand the nature of inelastic collision s between ground state atoms near a Feshbach resonance. In this paper we present the study of the losses of very cold85Rb atoms from a magnetic trap as a function of density and magnetic field. There are two types of inelastic c ollisions that induce loss from a magnetic trap. The first is dipolar relaxation where two atoms collide and change spi n states. The second process is three-body recombination, where three atoms collide and two of those atoms form a molecu le. Measuring the losses as a function of density has allowed us to determine the two-body and three-body inelast ic collision rates, while measuring the variation of these losses as a function of magnetic field has allowed us to find out how the Feshbach resonance affects them. In contrast to the work of Ref. [4] we observe a pronounced dip in the inela stic losses near the Feshbach resonance. This has allowed us to create85Rb BECs with a positive scattering length. To study these losses we needed a cold, dense85Rb sample. This was obtained through evaporative cooling wi th a double magneto-optic trap (MOT) system as described in Ref. [15]. The first MOT repeatedly collected atoms from a background vapor and those atoms were transferred to anoth er MOT in a low-pressure chamber. Once the desired load size was achieved, the MOTs were turned off and a baseball -type Ioffe-Pritchard magnetic trap was turned on, resulting in a trapped atom sample of about 3x108F=2, m f=–285Rb atoms at 45 µK. Forced radio-frequency (rf) evaporation was used to increase the density of the atoms whi le decreasing the temperature. Because of the high ratio of inelastic/elastic collision rates in85Rb and the dependence of the ratio on magnetic field ( B) and temperature ( T), trap conditions must be carefully chosen to achieve efficient cooling. We evaporated to the desired temperature and density (typically 3x1011cm−3and 500-700 nK) at a final field of B=162 Gauss (G). We then adiabatically changed the DC magnetic field to variou s values and measured the density of the trapped atom cloud as a function of time. We observed the clouds with b oth nondestructive polarization-rotation imaging [16] and destructive absorption imaging. In both cases, the clou d was imaged onto a CCD array to determine the spatial size and number. The nondestructive method allowed us to obs erve the time evolution of the number and spatial size of a single sample. The destructive method required us to pre pare many samples and observe them after different delay times, but had the advantages of better signal to noise for a single image and larger dynamic range. A set of nondestructive imaging data is shown in Fig. 1. In addition t o these techniques, we made a redundant check of the 1number by recapturing the atoms in the MOT and measuring thei r fluorescence [17]. We measured the time-dependence of the number of atoms and th e spatial size, and from these measurements determined the density. We assigned a 10% systematic error t o our density determination, based primarily on the error in our measurement of number. The value of Bwas measured in the same way as in Ref. [1]: the rf frequency at which the atoms in the center of the trap were resonantly sp in-flipped was measured and the Breit-Rabi equation was then used to determine the magnitude of B. The magnetic field width of the clouds scaled as T1/2and was 0.39 G FWHM at 500 nK. In each data set we observed the evolution of the sample while a significant fraction (20–35%) of the atoms was lost. The temperature of the sample also increased as a function of time. This heating rate scaled with the inelastic rates and therefore varied with B. Since the volume and temperature in a magnetic trap are dire ctly related, the change in the volume (typically 50%) with time was fit to a polynomial —usually a straight line was sufficient within our precision. The number (N) as a function of time was then fit to t he sum of three-body and two-body loss contributions given by ˙N=−K2/angbracketleftn/angbracketrightN−K3/angbracketleftn2/angbracketrightN−N τ. (1) Here/angbracketleftn/angbracketright=1 N/integraltextn2(x)d3xis the density-weighted density and /angbracketleftn2/angbracketright=1 N/integraltextn3(x)d3x. The time evolution of the volume is contained in the computation of the density as a function o f time. The background loss rate is given by τ. It was independently determined by looking at low-density clouds for very long times. Typically, it was 450 seconds, with some modest dependence on B, particularly near the Feshbach resonance. The change in the combined inelastic rates as a function of Baround the Feshbach resonance is shown in Fig. 2(a). All of the data shown in Fig. 2(a) were taken with initial temp eratures near 600 nK. The points less than 157 G were taken with initial densities within 10% of 1x1011cm−3, while the points higher than 157 G were taken with initial densities that were 2.7x1011cm−3. The decrease in initial density below 157 G was due to rampin g through the high inelastic loss region after forming the sample at 162 G. The shape of the inelastic rate vs. Broughly follows that of the elastic rate vs. B[1]. In particular, the peak of the inelastic rate occurs at 155.4 ±0.5 G, identical to the position of the elastic peak at 155.2 ±0.4 G within the error. Also, just as is the case for the elastic rates, the inelastic loss rates around the Feshbach resonance vary by orders of magnitude. The peak in the inelastic rate is much less symm etric, however. Another interesting feature is that the loss rates not only increase near the elastic rate peak bu t decrease near its minimum. The field where the loss is minimum, 173.8 ±2.5 G, is higher than the minimum of the elastic rate at 166.8 ±0.3 G. Even though the two- and three-body inelastic rates have diff erent density dependencies, it is difficult to separate them. Figure 1 shows how a purely two-body or purely three-bo dy loss curve fits equally well to a typical data set. An excellent signal-to-noise ratio is, or, equivalently, dat a from a large range of density are required to determine whet her the loss is three-body, two-body, or a mixture of the two. To b etter determine density dependence, we decreased the initial density by up to a factor of 10 for several Bvalues. For fields with relatively high loss rates, the signa l-to-noise ratio was adequate to distinguish between two- and three-bo dy loss. However, where the rates were lower this was not the case. Figure 2(b) shows the two-body inelastic rate determined fr om the density-varied data and Fig. 2(c) shows the same for the three-body rate. The character of the inelastic loss clearly changes as one goes from higher to lower field (right to left in Fig. 2). On the far right of the graph, at B= 250 G, the inelastic rate is dominated by a three-body process. From B= 250 G down to B= 174 G the inelastic rate decreases to a minimum and then begi ns to increase again. Near the minimum, we could not determine the loss char acter, but at 162 G the losses are dominated by a two-body process. At 158 G the two-body process is still domi nant and rising rapidly as one goes toward lower field. However, by ∼157 G it has been overtaken by the three-body recombination t hat is the dominant process at 157-145 G. At lower fields, both two-body and three-body rates contri bute significantly to the total loss at these densities. Along with varying the density, the initial temperature was varied at B= 145,156,160, and 250 G. There was no significant rate change in the loss rate for temperatures bet ween 400 and 1000 nKfor the 160 and 250 G points. The combined loss rate increased by a factor of 8 at 156 G from 1 µKto 400 nK, and by a factor of 2 at 145 G. This temperature dependence near the peak is expected in ana logy to the temperature dependence of the elastic rates [18]. The fact that the loss rates near the peak exhibit both a two- and three-body character plus a temperature dependence and considerable heating make interpreting the data challenging. This introduces additional uncertainty that is included in the error bars in Fig. 2(a-c). A calculated dipolar loss rate [12] is compared with the data in Fig. 2(b). Keeping in mind the significance of error bars on a log plot, the agreement is reasonably good. In the region between B= 160 and 167 G where the determination of K 2is not complicated by three-body loss and temperature depen dence, the agreement is particularly good. 2Likewise, we show a prediction of the three-body recombinat ion rate in Fig. 2(c) [10]. It is predicted that the recombination rate scales as the s-wave scattering length t o the fourth power (a4 s) for both positive and negative a s, although with a smaller coefficient for the positive case [9,1 0,14]. Since a svaries across the Feshbach resonance, the recombination rate is expected to change [19]. Temperature dependence has not been included in the prediction. Qualitatively, the main features of the predicted three-bo dy recombination match the data. The three-body rate decreases as a sdecreases, and it increases rapidly where a sdiverges at B= 155 G. From 155 G to 167 G, a sis positive and the three-body loss is much smaller than it is at Bfields with a comparable negative a s, as expected from theory. This overall level of agreement is reasonable g iven the difficulties and approximations in calculating three-body rates. The marked dependence on Bof both the elastic and inelastic rates has important implic ations for the optimization of evaporative cooling to achieve BEC in85Rb. As the density increases through evaporative cooling, t he absolute loss rate, which is already much greater than in87Rb and Na, becomes larger. Not only are atoms then being lost from the sample, but additional heat is being produced, whic h counteracts the cooling. It is true that the absolute elastic collision rate needed for rethermalization and the refore efficient evaporation also increases with density, bu t only up to the density where the mean free path of an atom becom es smaller than the cloud size. Any further increase in density will only increase the ratio of loss rate to rether malization rate. These dependencies add additional complexity to the usual c omplicated optimization of evaporation. However, by using the data presented here as a guide, along with the field d ependence of the elastic cross section of Ref. [1] we have successfully found a suitable evaporation path to reac h BEC. For relatively low densities we evaporate at a field (250 G) well above the Feshbach resonance where elastic rate s are high enough compared to background losses and inelastic rates are fairly low. As the density increases, we move to lower field to obtain reduced inelastic losses. At a final field of 162 G the balance of thermalization and inelasti c loss is such as to allow condensates of several thousand atoms to be created. In contrast with field values away from th e Feshbach resonance, here the scattering length is also positive and so such large condensates are stable [20], and because of the Feshbach resonance their scattering length is easily varied. The85Rb Feshbach resonance has a profound effect on the two- and thr ee-body inelastic rates, changing them by orders of magnitude. The dependence of the inelastic rates o n magnetic field is similar in structure to the dependence of the elastic rate: the maxima in both rates occur at the same field, while the minima are close but do not coincide. The total loss is a complicated mixture of both two- and three -body loss processes. They have different dependencies on field so both have field regions in which they dominate. We are pleased to acknowledge useful discussions and with Er ic Cornell, Jim Burke, Jr., Chris H. Greene, and Carl Williams. We also thank the latter three and their colleague s for providing us with loss predictions. This research has been supported by the NSF and ONR. One of us (S. L. Cornish) ack nowledges the support of a Lindemann fellowship. [1] J. L. Roberts et al., Phys. Rev. Lett. 81, 5109 (1998). [2] V. Vuleti´ c et al., Phys. Rev. Lett. 82, 1406 (1999); V. Vuleti´ c et al., Phys. Rev. Lett. 83, 943 (1999). [3] Ph. Courteille et al., Phys. Rev. Lett. 81, 69 (1998). [4] S. Inouye et al., Nature 392, 151 (1998); J. Stenger et al., Phys. Rev. Lett. 82, 2422 (1999). [5] E. Tiesinga et al., Phys. Rev. A 46, R1167 (1992); E. Tiesinga, B. J. Verhaar, and H. T. C. Stoof, Phys. Rev. A 47, 4114 (1993). [6] E. Timmermans et al., Phys. Rev. Lett. 83, 2691 (1999). [7] F. A. van Abeelen and B. J. Verhaar, Phys. Rev. Lett. 83, 1550 (1999). [8] V. A. Yurovsky et al., Phys. Rev. A 60, R765 (1999). [9] E. Nielsen and J. H. Macek, Phys. Rev. Lett. 83, 1566 (1999). [10] B. D. Esry, C. H. Greene, and J. P. Burke, Jr., Phys. Rev. L ett.83, 1751 (1999). [11] J. P. Burke, Jr. et al., Phys. Rev. Lett. 80, 2097 (1998). [12] C. J. Williams, NIST Gaithersburg, private communicat ion. [13] A. J. Moerdijk, H. M. J. M. Boesten, and B. J. Verhaar, Phy s. Rev. A 53, 916 (1996). [14] P. O. Fedichev, W. M. Reynolds, and G. V. Shlyapnikov, Ph ys. Rev. Lett. 77, 2921 (1996). [15] C. J. Myatt et al., Opt. Lett. 21, 290 (1996). [16] C. C. Bradley, C. A. Sackett, and R. G. Hulet, Phys. Rev. L ett.78, 985 (1997). [17] C. G. Townsend et al., Phys. Rev. A 521423 (1995). [18] For instance, see J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic C ollisions (R. E. Kreiger 3Company, Malabar, Florida, 1987), pp. 89,352-353. [19] The predicted change in a swith field is obtained using the potential determined in Ref. [1] by J. P. Burke, Jr. (NIST, Gaithersburg, private communication). That potential pre dicts a s= -422 bohr away from the Feshbach resonance. Private communications with C. J. Williams indicate that a s= -600 bohr away from the resonance may be more consistent wit h all the available Rb data. [20] P. A. Ruprecht et al., Phys. Rev. A 51, 4704 (1995); See also the review by F. Dalfovo et al., Rev. Mod. Phys. 71, 463 (1999). FIG. 1. Number of atoms versus time ( •) taken by polarization-rotation imaging at 159 G. Fits to bo th purely two-body (dashed line) and purely three-body (solid line) inelastic rates are shown, illustrating the difficulty in separating tw o- and three-body loss processes. FIG. 2. (a) Elastic rate (upper plot and ◦symbol) from Ref. [1] and total loss rates vs. magnetic field. The total loss rate is expressed as a sum of the two- and three-body loss as K 2+βK3. Due to initial density differences caused by ramping across the peak, β= 1.6 x 1011cm−3for the points below B=157 G (filled circles) and 4.0 x 1011cm−3for the points above (filled triangles). The vertical lines represent the positions of t he elastic rate maximum and minimum at B=155 G and B=167 G respectively. (b) The determination of the two-body inelas tic rate for several Bfields. The theory prediction from Ref. [12] is shown as a solid line. The open circles ( ◦) are the two-body rates determined from the total loss from F ig. 2(a) by assuming (not explicitly measuring) that the loss between 162 G and 16 6 G is predominantly two-body. These points are added to aid in comparison with theory. (c) The determination of the thre e-body inelastic rate. The points with down arrows (↓) are to be interpreted as upper limits on the three-body rate. The soli d line here is a prediction of the loss rate from Ref. [10]. The open circles in Fig. 2(c) are similar to those in 2(b), only in 2(c) we assume (but again do not measure) that the loss above 175 G is predominantly three-body. The error bars on the 250 G poin t are relatively small because a large amount of data was take n there. In addition to the statistical errors shown in Fig. 2( a-c), there is another 10% systematic uncertainty in K 2and 20% in K3due to the estimated error in our density determination. 4012345.05.56.06.57.0Number (106 atoms) Time (sec)10012515017520022525010-1410-1310-1210-11K2 + β K3 (cm3/s) Elastic Rate (s-1) Magnetic Field (Gauss)10-310-210-1100101 (a)10012515017520022525010-1510-1410-1310-1210-11 (b)K2 (cm3/s) Magnetic Field (Gauss)10012515017520022525010-2710-2610-2510-2410-2310-22(c)K3 (cm6/s) Magnetic Field (Gauss)

arXiv:physics/0003028v1 [physics.ao-ph] 14 Mar 2000A New Hypothesis for Layers of High Reflectivity Seen in MST Ra dar Observations G. C. Asnani⋆and M. K. Rama Varma Raja⋆⋆ Indian Institute of Tropical Meteorology, Dr. Homi Bhabha R oad , Pune 411 008(India). E-mail: asnani@giaspn01.vsnl.n et.in or Prof.asnani@vsnl.com ⋆⋆Department of Physics, University of Pune, Pune 411 007(Ind ia). E-mail: rama@physics.unipune.ernet.in We have worked with MST Radar located at Gadanki, Tirupati, I ndia(13.47 degree N, 79.18 degree E). Altitude of the location is approximately 100 meters. This is the first and only MST Radar operating in India. It is in Tropical Monsoon region. Monsoon moves northward across Ti rupati around 1st June and withdraws southward across Tirupati in the first week of December. Our interest ha s been to examine the characteristics of vertical velocity and other wind characteristics associated with monsoon. We examined the characteristics of range-compensated Signal-to-Noise ratio ( r2SNR). We call this as Reflectivity. We examined the MST Radar d ata set for 14 months (September 1995 to November 1996 except May 1996). The heigh t range considered for the analysis was between 3.6 km and 21 km above ground. Our findings have been on the followi ng lines: 1) Except on Thunderstorm occasions, high reflectivity-reg ions are in the form of layers 1-2 km thick. One such high-reflectivity layer is always present near tropopause l evel 17 km. Below the tropopause, there is a bunch of such high-reflectivity layers, generally 3-4 in number, between 4 and 11 km. The atmospheric layer between 11 and 15 km is generally free from high reflectivity layer. 2) On a thunderstorm occasion, there is a deep high-reflectiv ity layer extending from 4-11 km. After the passage of the thunderstorm, this deep layer of high reflectivity bre aks up into layers. 3) Layers of high reflectivity occur throughout the year even outside monsoon season, when ITCZ is far away from the region and they cannot be attributed as originating from deep convective clouds. When visual observation and MST observations are taken simultaneously at Gadanki, visu al clouds are estimated to be at the same levels as the high reflectivity layers seen through MST observations. Ind ian satellite pictures over Gadanki also suggest similar heights of clouds as given by MST Radar for high reflectivity l ayers. 4) Satellite pictures in infra-red range show much more exte nsive areas of cloudiness than the pictures in the satellite visible range. In other words, there are extensive sub-visu al clouds in the atmosphere. 5) Every month early morning and late evening, in twilight ho urs we see beautiful cirrus clouds in the form of cloud streets, streaks or sheets on several occasions which may or may not be clearly visible at other hours. 6) On looking at the sky frequently, one gets the impression t hat the sky is not clear even though we do not see clouds or haze layers. In these haze layers, appear clouds in the form of cloud streets, streaks or cloud sheets at some times. The appearance of these clouds gives the clear impres sion that there are waves in the atmosphere which give visible clouds in the region of upward motion associated wit h these waves. Again in each bigger wave cloud, there are smaller and smaller wave clouds and clearances; there ar e waves within waves. When the clouds disappear, they leave a sort of haze. Hence, the clouds form out of haze and lea ve some haze after dissipating. Even when there are no visible clouds during the day or night, the structure o f the atmosphere is patchy in appearance. One gets a clear impression that there are waves and waves, bigger and smaller waves in the atmosphere which are creating patches of haze and sometimes visible clouds in the atmosphe re. These are due to gravity-type waves with a very wide spectrum of horizontal and vertical wavelengths. As we know from theoretical and observational evidence, horizontal wavelengths are 1-2 orders of magnitude larger than vertica l wavelengths. If and when they occur in the atmosphere, they have a tendency to take horizontally spread layered str ucture with vertical depth 1-2 orders of magnitude smaller than horizontal extent. 7) In the atmosphere, we visualize three classes of waves: a. Inertial waves or Rossby-type waves: Their horizontal extent is of the order of a few thousand kilometers and vertic al wavelengths of the order of 10 km. In the mechanism of their formation, we have to consider the rotation of the ea rth and the resulting coriolis force. Their period is of the order of a few days. b. Gravity waves: These arise mainl y from local horizontal pressure gradients arising out of gravitational weight of the overlying air column, air accelerating from higher pressure towards lower pressure. Horizontal accelerations and displacements are accompani ed by appropriate vertical accelerations and displacement s to conform to the requirements of law of conservation of mass i.e., equation of continuity. Vertical displacements and accelerations of air parcels ca n also arise from buoyancy forces. Heavier parcel tends to sink down while lighter air parcels tend to rise up in an envir onment of horizontal density gradients. These are called Brunt-Vaisala oscillations. These horizontal gradients o f density arise out of differences in temperature, humidity a nd hydro-meteor loading. This hydrometeor-loading needs a little elaboration: 1(i) Every parcel of cloud air contains at least one hydromete or in liquid water or solid form. Inside each hydrometeor is an aerosol which acts as a nucleus which has induced conden sation and/or freezing. (ii) Invariably, hydrometeor has higher density than the su rrounding air. As such, it tends to fall down due to gravitational force. (iii) As it descends down, it exchanges sensible heat and moi sture with its environment. Hence, its volume, mass, and density undergo a change during its descent. (iv) As the hydrometeor descends down with gravitational ac celeration through the air parcel, there is frictional resistance/viscous resistance to its vertical motion. soo n, the hydrometeor loses its acceleration and descends down with what is called ”Terminal velocity”. Where has its weigh t gone? Its weight is taken up by the air parcel which is offering resistance to its vertical movement and accelera tion. In other words, the air parcel becomes heavier to the extent it has taken over the gravitational acceleration of t he falling hydrometeor. If the hydrometeor falls with its terminal velocity, it means that its entire weight is taken o ver by the air parcel. As such the air parcel has become heavier by the total weight of the hydrometeor; the density o f the air parcel has effectively increased. If only half of the gravitational acceleration of the descen ding hydrometeor has been taken over by the air parcel, then the weight of the air parcel has increased only by half th e weight of the hydrometeor. The hydrometeor gradually loses its gravitational acceler ation and gives its weight to the surrounding air parcel gradually and not instantaneously. As such, the air parcel t akes the load of the hydrometeor gradually during the downward trajectory of the descending hydrometeor. In many calculations, for the sake of simplicity, it is assum ed that as soon as condensation or freezing takes place in the atmosphere, the load of the hydrometeors is immediate ly taken over by the surrounding air. However, we have to recognize that the hydrometeor-loading occurs graduall y through a finite interval of time. We should also remember that during the fall, the hydrometeo r is simultaneously undergoing a change in its volume, mass and density. Hence, a realistic, quantitative estimat e of hydrometeor loading effect on the air parcel needs carefu l calculation. However, nature takes care of the process and c reates varying density effects on the cloud air parcel as the hydrometeor descends. (v) In addition to buoyancy fall of the hydrometeor, there ar e upward and downward motions of air parcels inside the cloud. These upward and downward motions of air parcels i nside the cloud create further complications in the calculation of hydrometeor-loading effect on air parcel. (vi) In addition to pure buoyancy forces operating in a class of waves called gravity waves, there also occur what are known as Kelvin-Helmholtz waves due to presence of verti cal shear of horizontal winds which is almost always present. (vii) This class of gravity waves of Kelvin-Helmholtz type h ave very small wavelengths of the order of centimeters and meters and correspondingly small periods of the order of a few minutes. Earth’s rotation or Coriolis force does not perceptibly come into the calculations for these waves. c. Inertio-Gravity waves: Between the two extremes of large inertial waves and small gravity waves, there is an intermediate class of waves which may be termed as inertio-g ravity waves in which Coriolis force plays some role along with gravity force. There is literature on the subject of ine rtio-gravity waves (1,2,3, etc.), but more work needs to be done on this class of waves. Orographic influence also come in to play. 8) When we fly in an aircraft, large-scale weather and clouds a re influenced by Rossby-type waves. When we look at the sky through cockpit or through the window near the wind ow-seat, during day time, we immediately get the view of air clouds at different levels and also gravity waves o f various dimensions near the flight level. We see fairly large waves with estimated wavelengths of the order of tens o f kilometers, along with embedded smaller and smaller waves, thick clouds, thin clouds, thinner clouds, space fill ed with haze and space clear of visible clouds. In our view, MST Radar reflectivity pattern gives us spot view of these numerous beautiful waves. 9) We have analyzed the field of MST Radar reflectivity as seen a t Gadanki along with MST Radar measured wind fields. We have interpreted the reflectivity fields within a co nceptual model given below: (i) Inertio-gravity waves in the atmosphere generate layer s of upward/downward motion, high/low humidity and high/low temperature lapse rates. The layers of upward and d ownward motion are regularly seen in the vertical wind field given by MST Radar. The vertical wavelength of these ine rtio-gravity waves has wide spectrum depending on orography and diabatic heating; vertical wavelength of abo ut 5 km is a more frequently observed wavelength. The corresponding horizontal wavelength is of the order of 200 k m, the more dominant wavelength visible in satellite picture is 500 km in the direction of wind and 1000 km across th e wind. The layers of high relative humidity created by inertio-gra vity waves are favorable for the formation of layered clouds, which we call ”Mother Cloud Layers”; these clouds ma y be visible or sub-visible. (ii) Hydrometeors inside a ”Mother Cloud Layer” tend to fall down attaining their respective terminal velocities. During their stay inside the clouds, the hydrometeors excha nge heat, moisture, mass and momentum with the en- vironmental air on small micro-scales. These exchanges bet ween the hydrometeors and the ”Mother Clouds’s air” 2create strong gradients of temperature, humidity, density and momentum. Density variations are also created through hydrometeor-loading. Electrical charges are also generated during the processes of condensation, evaporation, freezing, melting and sublimation. These micro-physical gradients in density along with preva iling wind field in the vertical generate internal gravity waves in the form of Brunt-Vaisala oscillations, Kelvin-He lmholtz waves and other waves of different horizontal and vertical wavelengths. These wavelengths range from a few mi llimeters to tenths of meters in the vertical and from a few meters to about thousand meters in the horizontal. Known law s of physics and dynamics suggest that the wavelengths may be still smaller, equivalent to the distances between ad jacent parcels of air exchanging heat moisture, mass and momentum, with the hydrometeor embedded in the parcel. (iii) The strong gradients of temperature, humidity and den sity created by these micro-physical and micro-dynamical processes in the air surrounding the hydrometeor or an ensem ble of hydrometeors cause strong variations in the refractive index of air parcel, in respect of electro-magne tic lidar and radar beams impinging on the air parcels. In turn, this causes high reflectivity/scatter of the impingin g lidar/radar beam. In respect of VHF MST radar Bragg-type reflection/scatter is a dominant type of reflection/ scatter . We examined the size of air parcels giving the highest values of reflectivity, at Gadanki. We came to the conclusion that their horizontal extent is of the order of 1 km while thei r vertical extent is of the order of 100 m. Indian MST radar beam oriented in vertical has a half-wavele ngth of about 3m. As such, Indian MST radar is capable of detecting reflectivity patterns of vertical wave lengths of the order of about 6m. These small-scale variatio ns in reflectivity appeared in the form of very delicate embroid ery inside the large-scale reflectivity of the ”Mother Cloud Layer”. (iv) These variations in the reflectivity pattern may look li ke turbulent fluctuation in the atmosphere. If we do not associate these fluctuations with clouds and hydrometeo rs inside the clouds, the clouds might appear as clear-air turbulence as has been prevalent in the current explanation appearing in literature connected with MST radars. This prevalent explanation has faced paradoxes, the main parado x being of thin horizontal sheets of turbulence. If we free our thinking from the concept of clear-air turbule nce and turn our thinking in the direction of visible or sub-visible clouds containing hydrometeors, the apparent paradoxes of clear-air turbulence causing high- MST radar reflectivity get immediately resolved. The existence of sub-visible clouds occupying much larger a rea than the visible clouds has now been established beyond question, through latest observations by satellite s in the infra-red range, by aircraft flying through cirrus cl oud air and by lidars operating in high altitude aircraft sendin g their beams through visible and sub-visible cloud layers. (v) Using over 2,50,000 observational data points for MST ra dar reflectivity and vertical wind shear (deduced from corresponding MST radar wind observations) spread over 14 m onths (from 1995 September to 1996 November), we plotted scatter diagrams of MST radar reflectivity versus ve rtical wind shear. We are pleasantly surprised to find that reflectivity decreases almost exponentially as vertic al wind shear increases. If mechanical turbulence was the main cause of high reflectivity, we should see reflectivity in creasing with vertical wind shear, and not decreasing almost exponentially. Scatter diagrams for each of the 14-m onths are presented in4. Also, Scatter diagrams for four representative months (January, April, July and October), for Indian monsoon region, are presented in5,6. This shows that mechanical turbulence is not the principal cause of high-MST radar reflectivity.7had hypothesized that turbulence may not be the primary cause of high-reflectivity seen in MST radars. (vi) Knowing the importance of cirrus clouds, the world scie ntific community launched the programme known as FIRE (First ISCCP Regional Experiment). This FIRE programm e concentrated on the study of layer clouds (Cirrus Clouds in the upper troposphere and low level stratus clouds in the lower troposphere). FIRE I programme was executed during the period 1985-1990 while FIRE II programm e was executed during the period 1990-1995. FIRE III programme is proposed to be executed in the beginning of t his new millennium, with particular emphasis on the tropics. Results of FIRE I have been summarized in a special i ssue of Monthly Weather Review (November, 1990); results of FIRE II have been summarized in a special issue of J ournal of Atmospheric Sciences (December, 1995). This topic is an important component of CLIVAR programme in t he tropics. The results of FIRE I and FIRE II have broadly confirmed that th ere is fine micro structure inside the layered clouds which can be interpreted easily, atleast qualitativ ely, in terms of micro-physical and micro-dynamical proces ses presented above. Also, the latest Numerical Modeling Work on Cirrus Clouds (f or example8,9,10) show that there are fine-scale structures of various in-cloud parameters including tempe rature, humidity and ice-concentrations. In fact in our view these fine scale structures created inside the visible or sub-visible ”Mother Cloud Layers” can cause steep refractive index gradients sufficient to cause hi gh-MST radar reflectivity. 10) We expect that the conceptual model of MST radar reflectiv ity presented above will give a new orientation to the thinking and interpretation of MST radar reflectivity. I t will provide a satisfactory, physically and dynamically acceptable, interpretation of the reflectivity patterns se en in the MST radar observation. 311) A few corollaries follow from this conceptual model: (i) Since, high-MST radar reflectivity layer, 1-2 km thick, i s always observed near the tropopause, it may or may not be directly connected to inertio-gravity waves. The mechan ism for the formation and sustenance of high-reflectivity layer is realized as follows: (a) The temperature near the tropopause level are very cold ( 200 K); temperature lapse rate is stable, 2-4 degree C/km; as such vertical mixing of air is inhibited and is weak. Relative humidity below the tropopause is high11. Water substance and aerosols injected into the upper tropos phere by deep convection near ITCZ remains below the tropopause while the relative humidity is very low above the tropopause. Water substance and aerosols form visible or sub-visible cirrus cloud layer or haze layer below the tro popause. Through the micro-physical and micro-dynamical processes mentioned earlier, the air layer develops strong vertical gradients or discontinuities in the refractive in dex with respect to MST radar beam impinging on the air there. Thi s gives high-reflectivity echoes near the tropopause level on all days of the year. When the cloud gets dissolved, it leaves large number of aero sols suspended there. By themselves, aerosols are not likely to be detected by MST radar. But the layer of high conte nt of aerosols can be detected and has been detecting near the tropopause (12,13,14etc.). In the tropics, ITCZ injects aerosols and moisture which rem ain trapped in the layer immediately below the tropopause. In extra-tropics, this injection is done by ext ra-tropical cyclone waves and polar front. On the same reasoning, as given for tropics, a layer with high content of aerosols and thin cirrus ice crystals will also get formed below the tropopause in extra-tropics. Thus a layer having h igh content of aerosols and thin cirrus cloud is expected to envelope the whole earth’s atmosphere near tropopause. T his has been substantiated by observations of (Nee et al., 1995, 1998; FIRE I and FIRE II observations and their r esults published in the Special issues of Monthly Weather in November 1990 and Journal of Atmospheric Science s in December 1995 respectively). Asnani et al.15had hypothesized the existence of an aerosol layer near tropica l tropopause, based on the same mechanism mentioned above. (ii) As we have stated above, observations confirm existence of alternate layers of upward and downward motion with vertical depth of 2-3 km. We should expect accumulation and depletion of both water substance and aerosol substance near the levels of vertical convergence. Further vertical upward motion tends to create higher relative humidity and higher temperature lapse rate; vertical downw ard motion tends to do the opposite, creating stable lapse rate and drier air. Hence we should expect to find layers of hig h and low relative humidity in the vertical. Sensitive instrumentation is required to detect this type of structur e in the atmosphere; this structure is likely to be missed by the ordinary radiosonde instruments. Such layers have been detected by special effort16. (iii) As stated earlier, these alternate layers of upward an d downward motion are associated with inertio-gravity waves, which are always present in the atmosphere. These ver tical wave motion will also tend to split large convective clouds, particularly in their decaying stage, into layered clouds. While convective instability in the atmosphere ten ds to generate deep convective clouds in the tropical atmosphe re, these inertio-gravity waves inhibit the formation and sustenance of these deep convective clouds in the tropical a tmosphere. (iv) Cirrus clouds have a capacity to retain their existence for a long time, even away from the source of their formation. Hence, visible or sub-visible cirrus clouds are likely to be seen at many places, with or without deep convective clouds. The upward vertical motion associated w ith inertio-gravity waves tends to generate cirrus clouds, visible or sub-visible at many places in the atmosphere. If n othing else, these clouds influence the radiative heat budget of the earth’s atmosphere. Acknowledgment: The authors thank Prof. D. Narayana Rao and his group at Department of Physics, Sri Venkateswara University, Tirupati (India) for considerab le help in Indian MST Radar data collection and analy- sis. Mr. M. K. Rama Varma Raja thanks Dr. (Mrs.) P. S. Salvekar and the Director, Indian Institute of Tropical Meteorology for providing necessary support and facilitie s during the course of research work. 1Muraoka, Y., T. Sugiyama and K. Kawahira, 1988: Cause of a mon ochromatic inertia-gravity wave breaking observed by the MU radar, Geophys. Res. Lett., 15, 1349-1352. 2Wurtele, M. G., Datta, A. and Sharman, R. D., 1996: The propag ation of gravity- inertia waves and Lee waves under a critical level. J. Atmos. Sci., 53, 1505-1523. 3Joseph, B., 1997: Chaotic mixing by internal inertia-gravi ty waves. Phys. Fluids, 9, 945-962. 4Rama Varma Raja, M. K., 1999: ” Parameterization of Cloudine ss Over India and Neighbourhood ”, Ph.D. Thesis submitted to University of Pune, Pune-411007, India. 45Rama Varma Raja, M. K., G. C. Asnani and P. S. Salvekar, 1999a: MST Radar Data can Reveal Micro-Structure of Clouds”. Preprints of 29th Conference on Radar Meteorology (America n Meteorological Society), Montreal, Canada (July 1999), p ages 343-346. 6Rama Varma Raja, M. K., G. C. Asnani, P. S. Salvekar, A. R. Jain , D. Narayana Rao, P. Kishore, S. Venkoba Rao and M. Hareesh, 1999b: Layered Clouds In The Indian Monsoon Regi on. Proc. Ind. Acad. Sci. (Earth and Planetary Sciences), Paper accepted for Publication in the December 1999 issue of the Journal. 7Asnani, G. C., M. K. Rama Varma Raja, D. Narayana Rao, P. S. Sal vekar, P. Kishore, T. Narayana Rao and P. B. Rao, 1998a: Patchy Layered Structure of Tropical Troposphere As Seen By Indian MST Radar. STEP HANDBOOK (As a Proceedings of the 8th International Workshop on Technical and Scientific Aspects of MST Radar held at Bangalore during 15-20 December, 1997), Editor Belva Edwards, SCOSTEP SECRE TARIAT, BOULDER, CO, USA, pages 192-195. 8Sassen, K. and Dodd, G. C., 1989: Haze particle Nucleation Si mulations in Cirrus clouds and Applications for Numerical Lidar Studies. J. Atmos. Sci., 46, 3005-3014. 9Khvorostyanov, V. I. and Sassen, K., 1998a: Cirrus Cloud Sim ulation Using Explicit Microphysics and Radiation. Part I: Model description. J. Atmos. Sci., 55, 1808-1821. 10Khvorostyanov, V. I. and Sassen, K., 1998b: Cirrus Cloud Sim ulation Using Explicit Microphysics and Radiation. Part II : Microphysics, Vapor and Ice mass budgets, and optical and ra diative properties. J. Atmos. Sci., 55, 1822-1845. 11Newell, R. E., Zhu, Y., Reed, W.J., and Waters, J. W., 1997: Re lationship between tropical upper tropospheric moisture and eastern tropical pacific sea surface temperature at seas onal and interannual time scales. Geophys. Res. Lett., 24, 2 5-28. 12Nee, J.B., G.B. Wang, P.C. Lee and S. B. Lin, 1995: Lidar studi es of particles and temperatures of the atmosphere: First results from National Central University lidar. Radio Scie nce, 30, 1167-1175. 13Nee J. B., Len C. N. and Chen, W. N., 1998: Lidar observations o f the cirrus cloud in the tropopause at Chung-Li (25 degree N, 121 degree E). J. Atmos. Sci., 55,2249-2257. 14Schroder, F. and Strom, J., 1997: Aircraft measurements of s ub-micrometer aerosol particles (¡ 7nm) in the midlatitude free troposphere and tropopause region. Atmos. Res.,44, 333-35 6. 15Asnani, G. C., M. K. Rama Varma Raja and P. S. Salvekar, 1998b: Aerosol Layer Near Tropical Tropopause. J. Aerosol Sci. (USA), 29,suppl.1, pp. s649-s650. 16Iselin, J. P. and W. J. Gutowski Jr., 1997: Water Vapour Layer s in Strom-Fest Rawinsonde Observations, Mon. Wea. Rev., 125, 1954-1963. 5

arXiv:physics/0003029v1 [physics.flu-dyn] 14 Mar 2000Wavelet Cross-Correlation Analysis of Turbulent Mixing from Large-Eddy-Simulations S. Sello and J. Bellazzini Enel Research A. Pisano 120, Pisa 56122, ITALY Contact e-mail: sello@pte.enel.it 1 Introduction The complex interactions existing between turbulence and m ixing in a bluff- body stabilised flame configuration is investigated by means of a wavelet cross- correlation analysis on Large Eddy Simulations. The combin ed approach allows to better point out typical features of unsteady turbulent fl ows with mixing through the characterisation of the processes involved bot h in time and scales. The wavelet cross-correlation analysis of the time signals of velocity and mixture fraction fluctuations can be an an effective tool to study the p rocesses involved in turbulent mixing flows which are of great interest in combu stion problems. 2 Generalities on wavelet cross-correlation The continuous wavelet transform of a function f(t) is defined as the convolution betweenfand a dilated function ψcalled wavelet mother: Wf(a,τ) =1√a/integraldisplay+∞ −∞f(t)ψ∗(t−τ a)dt, (1) whereais the dilation parameter, which plays the same role as the fr equency in Fourier analysis, and τindicates the translation parameter corresponding to the position of the wavelet in the physical space. In the pres ent study we use the complex Morlet wavelet ( ψ(t) =eiω0te−t2/2) as wavelet mother. LetWf(a,τ) andWg(a,τ) be the continuous wavelet transforms of f(t) andg(t). We define the wavelet cross-scalogram as Wfg(a,τ) =W∗ f(a,τ)Wg(a,τ), (2)2 where the symbol ∗indicates the complex conjugate. When the wavelet mother is complex, the wavelet cross-scalogram Wfg(a,τ) is also complex and can be written in terms of its real and imaginary parts: Wfg(a,τ) =CoWfg(a,τ)−iQuadWfg(a,τ). (3) It can be shown that the following equation holds if f(t),g(t)∈ L2(ℜ) /integraldisplay+∞ −∞f(t)g(t)dt= 1/cψ/integraldisplay+∞ 0/integraldisplay+∞ −∞CoWfg(a,τ)dτda, (4) where 1/cψis a constant depending on the choice of the wavelet mother. 3 Cross wavelet coherence functions The highly redundant information from a multiscale wavelet analysis of time series must be reduced by means of suitable selective proced ures and quantities, in order to extract the main features correlated to an essent ially intermittent dynamics. In this study, we analysed and compared the proper ties of two com- plementary wavelet local correlation coefficents which are a ble to well evidence peculiar and anomalous local events associated to the vorte x dynamics. More precisely, given two signals f(t) andg(t), we refer to the so-called Wavelet Local Correlation Coefficent (Buresti et. al [1]), defined as: WLCC (a,τ) =CoWfg(a,τ) |Wf(a,τ)||Wg(a,τ)|. (5) This quantity is essentially a measure of the phase coherenc e of the signals. Here we introduce the Cross Wavelet Coherence Function (CWCF) defined as: CWCF (a,τ) =2|Wfg(a,τ)|2 |Wf(a,τ)|4+|Wg(a,τ)|4, (6) which is essentially a measure of the intensity coherence of the signals. Using the polar coordinates we can write the wavelet transforms of Wf(a,τ),Wg(a,τ) andWfg(a,τ) as: Wf(a,τ) =ρfeıθfWg(a,τ) =ρgeıθg(7) Wfg(a,τ) =ρfρgeı(θg−θf), (8) and the Cross Wavelet Coherence Function can be written also as: CWCF (a,τ) =2ρ2 fρ2 g ρ4 f+ρ4g. (9) It is easy to observe the two basic properties of the function (6): CWCF (a,τ) = 0 = ⇒ρf= 0 orρg= 0 (10) 0≤CWCF ≤1∀a,τ. (11)3 4 Numerical simulation We considered a laboratory-scale axisymmetric flame of meth ane-air in a non confined bluff-body configuration. More precisely, the burne r consists of a 5.4 mm diameter methane jet located in the center of a 50 mm diamet er cylinder. Air is supplied through a 100 mm outer diameter coaxial jet ar ound the 50 mm diameter bluff-body. The Reynolds number of the central jet i s 7000 (methane velocity =21 m/s) whereas the Reynolds number of the coaxial jet is 80000 (air velocity =25 m/s). This is a challenging test case for all the turbulence models, as well documented in the ERCOFTAC report (Chatou, 1994) [2] . Moreover, due to the highly intermittent, unsteady dynamics involved and the high turbulence level, especially for the reactive case, the Large Eddy Simu lation (LES) appears as the most adequate numerical approach (Sello et. al [3]). 5 Results and discussion In this analysis we are mainly interested to relations exist ing between evolution of turbulence and mixing, for the reactive case. Previous DN S simulations on coaxial jets at different Reynolds numbers, show the ability of the wavelet cross- correlation analysis to better investigate the relations b etween mixing process and the dynamics of vorticity (Salvetti et. al [4]). Thus, the signals analysed here are velocity fluctuations (for Reynolds stress contributio ns) and mixture fraction fluctuations (for mixing evolution) from LES. As an example, Figure 1 shows the wavelet co-spectrum maps for a significant time interval in t he pseudo-stationary regime of motion. The main contributions to the Reynolds str ess are evidenced by high intensity correlations (red) and anti-correlation s (blue) regions, which evolve intermittently. The dominant frequencies involved are located around 130 Hz. For the mechanisms responsable of the evolution of mi xing, we note that the same regions of high Reynolds stress correspond to h igh correlation, or cooperation, between velocity and mixture fraction fluct uations, suggesting that, at the selected location, the same events of stretchin g and tilting of the vorticity layer, drive both Reynolds stress and mixing evol utions. Note that the large high value region located at low frequencies in the rig ht map is statistically not significant if we assume a proper red noise background spe ctrum. To better investigate the role of the high correlation regions, we per formed a cross section in the wavelet map at the frequency 160 Hz. Figure 2 (left) sho ws the time behaviour of the coherence functions WLCC, eq.(5), and CWCF , eq.(6). Here the phase and intensity coherence of signals are almost equi valent, but we can clearly point out an important anomalous event occurred at a round t=0.19 s, corresponding to a loss of both intensity and phase coherenc e, followed by a change of the correlation sign. The link between this event a nd the dynamics of vorticity is evidenced by Figure 2 (right), which display s the wavelet map of the related vorticity signal. The higher frequency signific ant regions ( ≈730 Hz) result strongly intermittent, with a bifurcation to lower a nd higher values than4 average, followed by a drop of activity, in phase with the ano malous event. Figure 1: Cross-Wavelet co-spectrum maps for axial and radi al velocity fluctu- ations (left) and for axial velocity and mixture fraction flu ctuations (right) at a given spatial point near the edge of the central jet. Figure 2: Coherence functions for axial velocity and mixtur e fraction fluctuations (left) and wavelet map of vorticity time series (right). These few examples support the usefulness of the cross-wave let analysis ap- proach to better investigate turbulent mixing processes in real systems. References [1] G. Buresti and G. Lombardi. Application of continuous wa velet transforms to the analysis of experimental turbulent velocity signals .Proc. of the 1st Int. Symp. on Turb. Shear Flow Phen. , S. Barbara USA, Sept. 1999. [2] EDF Direction des Etudes et Researches. 1stA.S.C.F. Workshop Final Re- sults, Chatou, France, October 1994. [3] S. Sello and G. Mariotti. Large eddy simulation of a bluff b ody stabilised flame. Proc. of the 4thETMM Int. Symp. , Ajaccio France, May 1999. [4] M.V. Salvetti, G. Lombardi and F. Beux. Application of a w avelet cross- correlation technique to the analysis of mixing. AIAA Jour. , 37:1007–1009, 1999.This figure "fig1a_sel.gif" is available in "gif" format from: http://arXiv.org/ps/physics/0003029v1This figure "fig1b_sel.gif" is available in "gif" format from: http://arXiv.org/ps/physics/0003029v1This figure "fig2a_sel.gif" is available in "gif" format from: http://arXiv.org/ps/physics/0003029v1This figure "fig2b_sel.gif" is available in "gif" format from: http://arXiv.org/ps/physics/0003029v1

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/CQ /CT/CT/D2/CU/D3/D9/D2/CS /CQ /DD /CP /D5/D9/CX /CZ/B9/DB/CX/D8/D8/CT/CS /D4/D9/D4/CX/D0/BA /C4/CT/D8 /D9/D7 /D9/D8 /CP /D7/D5/D9/CP/D6/CT /CU/D6/D3/D1 /CP /D7/CW/CT/CT/D8 /D3/CU /D4/CP/D4 /CT/D6 /CP/D2/CS/D1/CP/CZ /CT /CP /D4/CX/D2/CW/D3/D0/CT /CX/D2 /D8/CW/CT /CT/D2 /D8/CT/D6 /D3/CU /D8/CW/CT /D7/D5/D9/CP/D6/CT/BA /C8/D9/D8 /CP /D7/D4/CX/CZ /CT /D3/CU /CP /D4 /CT/D2 /CX/D0 /CX/D2 /D8/CW/CT /CW/D3/D0/CT/CP/D2/CS /D8/D9/D6/D2 /D7/D0/D3 /DB/D0/DD /D8/CW/CT /D7/D5/D9/CP/D6/CT /CZ /CT/CT/D4/CX/D2/CV /CX/D8 /CX/D2/D7/CX/CS/CT /D8/CW/CT /CP/D2/CV/D0/CT /CS/D6/CP /DB/D2 /D3/D2 /CP/D2/D3/D8/CW/CT/D6 /D7/CW/CT/CT/D8 /D3/CU/D4/CP/D4 /CT/D6/BA /CC/CW/CX/D7 /D6/CT/D7/D9/D0/D8/D7 /CX/D2 /CP /D2/CX /CT /D0/D3/D7/CT/CS /D9/D6/DA /CT /DB/CW/CX /CW /CP/D2 /CQ /CT /D9/D7/CT/CS /D8/D3 /CS/CT/D8/CT/D6/D1/CX/D2/CT /D8/CW/CT/D1/CX/D2/CX/D1 /D9/D1 /D4 /D3/D7/CX/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CT/D2 /D8/CT/D6 /D3/CU /D1/CP/D7/D7/BA/BD/CC/CW/CT /D7/D3/D0/D9/D8/CX/D3/D2/BA /C4/CT/D8 /D9/D7 /D2/D3 /DB /CU/D3/D6/D1 /D9/D0/CP/D8/CT /D8/CW/CT /D4/D6/D3/CQ/D0/CT/D1 /D1/CP/D8/CW/CT/D1/CP/D8/CX /CP/D0/D0/DD /B8 /D2/D3/D8 /D6/CT/D7/D8/D6/CX /D8/B9/CX/D2/CV /D3/D9/D6/D7/CT/D0/CU /D8/D3 /CP /D4/CP/D6/D8/CX /D9/D0/CP/D6 /DA /CP/D0/D9/CT /D3/CU /D8/CW/CT /CP/D2/CV/D0/CT Θ /BA /CF/CW/CT/D2 /D8/CW/CT /CP/D2/CV/D0/CT /CX/D7 /D7/CW/CP/D6/D4/B8 /D3/D2/CT /CP/D2 /D7/CT/CT /D6/CT/CP/CS/CX/D0/DD /D8 /DB /D3 /D4 /D3/D7/D7/CX/CQ/D0/CT /D4 /D3/D7/CX/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /D7/D5/D9/CP/D6/CT/B8 /DB/CW/CT/D6/CT /CT/CX/D8/CW/CT/D6 /D8/CW/CT /CS/CX/CP/CV/D3/D2/CP/D0/B4/BY/CX/CV/D9/D6/CT /BE/CP/B5 /D3/D6 /D8/CW/CT /CT/CS/CV/CT /B4/BY/CX/CV/D9/D6/CT /BE/CQ/B5 /D3/CU /D8/CW/CT /D7/D5/D9/CP/D6/CT /D8/D3/D9 /CW/CT/D7 /D8/CW/CT /D7/CX/CS/CT/D7 /D3/CU /D8/CW/CT /CP/D2/CV/D0/CT Θ /BA /B4/CC/CW/CT /D8/CW/CX/D6/CS /D8/D6/CX/DA/CX/CP/D0 /D4 /D3/D7/D7/CX/CQ/CX/D0/CX/D8 /DD /CP/D6/CX/D7/CT/D7 /DB/CW/CT/D2 /CP/D2/CV/D0/CT Θ /CX/D7 /D3/CQ/D8/D9/D7/CT/BA/B5/CC /D3 /CS/CT/D8/CT/D6/D1/CX/D2/CT /D8/CW/CT /D3/D6/CX/CT/D2 /D8/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D3/CQ /CY/CT /D8 /CX/D2/D7/CX/CS/CT /D8/CW/CT /CP/D2/CV/D0/CT Θ /B8 /D0/CT/D8 /D9/D7 /CS/CT/AS/D2/CT/D8/CW/CT /CP/D2/CV/D0/CT α /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /CS/CX/CP/CV/D3/D2/CP/D0 /CP/D2/CS /CP /DA /CT/D6/D8/CX /CP/D0 /D0/CX/D2/CT /CV/D3/CX/D2/CV /CU/D6/D3/D1 /D8/CW/CT /CT/D2 /D8/CT/D6 /D3/CU /D8/CW/CT/D7/D5/D9/CP/D6/CT /B4/BY/CX/CV/D9/D6/CT/D7 /BE/CP/B8 /BE/CQ/B5/BA /CC/CW/CT /DA /CT/D6/D8/CX /CP/D0 /CS/D3 /D2/D3/D8 /CV/CT/D2/CT/D6/CP/D0/D0/DD /D3/CX/D2 /CX/CS/CT /DB/CX/D8/CW /D8/CW/CT /CQ/CX/D7/CT /D8/D3/D6/D3/CU /D8/CW/CT /CP/D2/CV/D0/CT Θ /BA /C8 /D3/D8/CT/D2 /D8/CX/CP/D0 /CT/D2/CT/D6/CV/DD /D3/CU /D8/CW/CT /CW/D3/D1/D3/CV/CT/D2/CT/D3/D9/D7 /D1/CP/D7/D7/CX/DA /CT /D7/D5/D9/CP/D6/CT /CP/D2 /CQ /CT/DB/D6/CX/D8/D8/CT/D2 /CP/D7/BMV=mgh /B8 /DB/CW/CT/D6/CT h /CX/D7 /D8/CW/CT /CW/CT/CX/CV/CW /D8 /D3/CU /D8/CW/CT /CT/D2 /D8/CT/D6 /D3/CU /D8/CW/CT /D7/D5/D9/CP/D6/CT/BA /CC/CW/CT/D4/D6/D3/CQ/D0/CT/D1 /D8/CW /D9/D7 /D6/CT/CS/D9 /CT/D7 /D8/D3 /AS/D2/CS/CX/D2/CV /D8/CW/CT /D5/D9/CP/D2 /D8/CX/D8 /DD h /CP/D7 /CP /CU/D9/D2 /D8/CX/D3/D2 /D3/CU /D8 /DB /D3 /CP/D2/CV/D0/CT/D7 /B4Θ/CP/D2/CSα /B5 /CP/D2/CS /D8/CW/CT /D0/CT/D2/CV/D8/CW /D3/CU /D8/CW/CT /D7/D5/D9/CP/D6/CT /CT/CS/CV/CTa /BA /CC/CW/D6/CT/CT /CP/D7/CT/D7 /D1/CT/D2 /D8/CX/D3/D2/CT/CS /CP/CQ /D3 /DA /CT /D7/CW/D3/D9/D0/CS/CQ /CT /D8/D6/CT/CP/D8/CT/CS /D7/CT/D4/CP/D6/CP/D8/CT/D0/DD /BA/CP/BA /CC/CW/CT /CS/CX/CP/CV/D3/D2/CP/D0 /D8/D3/D9 /CW/CT/D7 /D8/CW/CT /CP/D2/CV/D0/CT /D7/CX/CS/CT/D7 /B4/BY/CX/CV/D9/D6/CT /BE/CP/B5/B8 /CX/BA/CT/BAπ 4+Θ 2≤α≤π 4−Θ 2 /CP/D2/CS 0<Θ≤π 2. /BT/D2/CP/D0/DD/DE/CX/D2/CV △ABE /DB /CT /CW/CP /DA /CT/BM Ψ =π 2−α−Θ 2 /BA /CB/CX/D1/CX/D0/CP/D6/D0/DD △ACD/DD/CX/CT/D0/CS/D7/BM Ψ =π−Φ−Θ /BA /CC/CW /D9/D7 /D3/D2/CT /AS/D2/CS/D7/BM Φ =π 2+α−Θ 2 /BA /BT/D4/D4/D0/DD/CX/D2/CV /D8/CW/CT /D7/CX/D2/CT/D8/CW/CT/D3/D6/CT/D1 /D8/D3 /D8/D6/CX/CP/D2/CV/D0/CT ACD /DB /CT /CV/CT/D8/BMd sin Θ=f sinΨ /CP/D2/CSd sin Θ=ℓ sin Φ /BA /CC/CW/CT/D6/CT/CU/D3/D6/CT/DB /CT /CP/D2 /CT/DC/D4/D6/CT/D7/D7 f /CP/D2/CSℓ /D8/CW/D6/D3/D9/CV/CW /D8/CW/CT /CP/D2/CV/D0/CT/D7 α,Θ /CP/D2/CS /D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6 d=a√ 2 /BA/BY/CX/D2/CP/D0/D0/DD /B8 /D8/CW/CT /CW/CT/CX/CV/CW /D8 h /CP/D2 /CQ /CT /CU/D3/D9/D2/CS /CP/D7 /CP/D2 /CP /DA /CT/D6/CP/CV/CT /D3/CU /CP/D0/D8/CX/D8/D9/CS/CT/D7 /D3/CU /D8/CW/CT /D8/D6/CP/D4 /CT/DE/D3/CX/CS AFGD /BM ha=1 2/parenleftbigg fsinπ−Θ 2+ℓsinπ−Θ 2/parenrightbigg =1√ 2cosαcotΘ 2./CQ/BA /CC/CW/CT /CT/CS/CV/CT /D8/D3/D9 /CW/CT/D7 /D8/CW/CT /CP/D2/CV/D0/CT /D7/CX/CS/CT/D7 /B4/BY/CX/CV/D9/D6/CT /BE/CQ/B5/BM0<Θ≤π 2 /CP/D2/CSπ 4−Θ 2≤α≤π 4 /BA/C1/D2 /D8/CW/CX/D7 /CP/D7/CT /D8/CW/CT /D8/D3/D8/CP/D0 /CW/CT/CX/CV/CW /D8 hb /CP/D2 /CQ /CT /CP/D0 /D9/D0/CP/D8/CT/CS /CP/D7 /CP /D7/D9/D1/BM hb=h′+h′′/BA/BT/D2/CP/D0/DD/DE/CX/D2/CV △ABC /DB /CT /CW/CP /DA /CTΘ + Φ + Ψ = π /BA /CC/CW/CT /D8/D6/CX/CP/D2/CV/D0/CT △ADE /DD/CX/CT/D0/CS/D7/BM π 4+ Ψ +Θ 2+α=π /B8 /D7/D3 /D8/CW/CP/D8Φ =π 4+α−Θ 2 /BA /BT/D4/D4/D0/DD/CX/D2/CV /D8/CW/CT /D7/CX/D2/CT /D8/CW/CT/D3/D6/CT/D1/D8/D3 /D8/CW/CT /D8/D6/CX/CP/D2/CV/D0/CT ABC /DD/CX/CT/D0/CS/D7/BMa sin Θ=ℓ sinΦ /BA /C6/D3 /DB /DB /CT /CP/D2 /CT/DC/D4/D6/CT/D7/D7 ℓ /B8 /CV/CX/DA/CX/D2/CV/BM h′=ℓcosΘ 2. /C7/D2 /D8/CW/CT /D3/D8/CW/CT/D6 /CW/CP/D2/CS/B8 /D3/D2/CT /AS/D2/CS/D7 /D6/CT/CP/CS/CX/D0/DD/BM h′′=√ 2 2acosα /BA /BT/D7 /CP/D6/CT/D7/D9/D0/D8/B8 /D8/CW/CT /D8/D3/D8/CP/D0 /CW/CT/CX/CV/CW /D8 /D8/CP/CZ /CT/D7 /D8/CW/CT /CU/D3/D6/D1/BM hb=acosΘ 2 sin Θsin(π 4+α−Θ 2) +√ 2 2acosα./CC/CW/CT /D7/CP/D1/CT /D4/D6/D3 /CT/CS/D9/D6/CT /CP/D2 /CQ /CT /CP/D6/D6/CX/CT/CS /D3/D9/D8 /CU/D3/D6−π 4≤α≤ −π 4+Θ 2 /BA /BA /CC/CW/CT /CP/D2/CV/D0/CT Θ /CX/D7 /D3/CQ/D8/D9/D7/CT/BMπ 2≤Θ≤π /CP/D2/CS−π 4+Θ 2≤α≤π 4−Θ 2 /BA /C1/D2 /D8/CW/CX/D7 /CP/D7/CT/D8/CW/CT /CW/CT/CX/CV/CW /D8 hc /CS/CT/D4 /CT/D2/CS/D7 /D3/D2/D0/DD /D3/D2 /D8/CW/CT /CP/D2/CV/D0/CT α /BM hc=a√ 2 2cosα./BE/CA/CT/D7/D9/D0/D8/D7 /CP/D6/CT /D4/D0/D3/D8/D8/CT/CS /CU/D3/D6 /D8 /DB /D3 /CS/CX/AR/CT/D6/CT/D2 /D8 /D7/CW/CP/D6/D4 /CP/D2/CV/D0/CT/D7/BM Θ = π/6 /B4/BY/CX/CV/D9/D6/CT/D7 /BF/CP/B5/CP/D2/CSΘ = π/3 /B4/BY/CX/CV/D9/D6/CT/D7 /BF/CQ/B5/BA /CC/CW/CT /D1/D3/D7/D8 /CU/CP/D7 /CX/D2/CP/D8/CX/D2/CV /CU/CT/CP/D8/D9/D6/CT /CX/D7 /D8/CW/CP/D8 /D8/CW/CT /CT/D2/CT/D6/CV/DD/D1/CX/D2/CX/D1 /D9/D1 /D3/CU /D8/CW/CT /D7/DD/D7/D8/CT/D1 /CP/D4/D4 /CT/CP/D6/D7 /CX/D2 /D8/CW/CT /D4 /D3/D7/CX/D8/CX/D3/D2/D7 /B4/BE/B5 /CP/D2/CS /B4/BG/B5 /DB/CW/CX /CW /CQ/D6/CT/CP/CZ /D8/CW/CT/CX/D2/CX/D8/CX/CP/D0 /D7/DD/D1/D1/CT/D8/D6/DD /D3/CU /D8/CW/CT /D7/DD/D7/D8/CT/D1/B8 /D6/CP/D8/CW/CT/D6 /D8/CW/CP/D2 /CX/D2 /D8/CW/CT /D4 /D3/D7/CX/D8/CX/D3/D2/D7 /D4 /D3/D7/CX/D8/CX/D3/D2/D7 /B4/BD/B5/B8 /B4/BH/B5 /D3/D6/B4/BF/B5 /D8/CW/CP/D8 /D4/D6/CT/D7/CT/D6/DA /CT /D8/CW/CX/D7 /D7/DD/D1/D1/CT/D8/D6/DD /BA /C1/D2 /D8/CW/CT /CP/D7/CT/D7 /B4/BE/B5 /CP/D2/CS /B4/BG/B5/B8 /D7/DD/D1/D1/CT/D8/D6/DD /CQ /CT /D3/D1/CT/D7/CQ/D6/D3/CZ /CT/D2 /D7/D4 /D3/D2 /D8/CP/D2/CT/D3/D9/D7/D0/DD /B8 /CQ /CT /CP/D9/D7/CT /DB /CT /CP/D2/D2/D3/D8 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arXiv:physics/0003031v1 [physics.chem-ph] 14 Mar 2000Energy distribution analysis of the wavepacket simulations of CH 4and CD 4scattering R. Milot and A. P. J. Jansen Schuit Institute of Catalysis, ST/SKA, Eindhoven Universi ty of Technology P.O. Box 513, NL-5600 MB Eindhoven, The Netherlands. E-mail: tgakrm@chem.tue.nl, Tel.:+31-40-2472189, Fax:+ 31-40-2455054 Abstract The isotope effect in the scattering of methane is studied by w avepacket simu- lations of oriented CH 4and CD 4molecules from a flat surface including all nine internal vibrations. At a translational energy up to 96 kJ/m ol we find that the scattering is still predominantly elastic, but less so for C D4. Energy distribution analysis of the kinetic energy per mode and the potential ene rgy surface terms, when the molecule hits the surface, are used in combination w ith vibrational exci- tations and the corresponding deformation. They indicate t hat the orientation with three bonds pointing towards the surface is mostly responsi ble for the isotope effect in the methane dissociation. keywords: Computer simulations, Models of surface chemica l reaction, Alka- nes, Low index single crystal surfaces 1 Introduction The dissociation of methane on transition metals is an impor tant reaction in catalysis. It is the rate limiting step in steam reforming to produce syngas.[1] It is also prototypical for C–H activation in other processe s. A large number of molecular beam experiments in which the dissociation energ y was measured as a function of translational energy have already been done on this system.[2– 22] These experiments have contributed much to our understa nding of the mechanism of the dissociation. Some of them observed that vi brationally hot CH4dissociates more readily than cold CH 4, with the energy in the inter- nal vibrations being about as effective as the translational energy in inducing dissociation.[2–8] A more detailed assessment of the impor tance of the inter- nal vibrations could not be made, because of the large number of internal Preprint submitted to Elsevier Preprint 24 July 2013vibrations. A recent molecular beam experiment with laser e xcitation of the ν3mode succeeded in measuring a dramatical enhancement of the dissociation on a Ni(100) surface, but it is still much too low to account fo r the vibrational activation observed in previous studies and indicates that other vibrationally excited modes contribute significantly to the reactivity of thermal samples.[22] Wavepacket simulations are being used more and more to study the dynamics of this kind of molecule surface reactions. The published wa vepacket simula- tions on the methane dissociation reaction on transition me tals have treated the methane molecule always as a diatomic up to now.[23–27] B esides the C–H bond and molecule surface distance, a combination of other c oordinates were included, like (multiple) rotations and some lattice motio n. None of them have looked at the role of the internal vibrations. Various t heoretical stud- ies have obtained reaction pathways and barriers for dissoc iation by DFT calculations,[28–37] but they cannot explain the role of th e vibrational modes in the reaction dynamics either. A nice way to study reaction dynamics is the use of isotopes. T he most re- cent wavepacket simulation on the dissociation probabilit y of CH 4and CD 4 showed a semiquantitative agreement with the molecular bea m experiments of Ref.[5], except for the isotope effect and the extracted vibr ational efficacy.[27] The molecular beam study with laser excitation of the ν3asymmetrical stretch mode shows that the incorrect vibrational efficacy is caused b y the assump- tions in the fit procedure that both stretch modes behaves ide ntical.[22] One of the possible explanation of the incorrect isotope effect can be the role played by the non-included intramolecular vibrations. In a previous paper we reported on wavepacket simulations th at we have done to determine which and to what extent internal vibrations ar e important for the dissociation of CH 4.[38] We were not able yet to simulate the dissocia- tion including all internal vibrations. Instead we simulat ed the scattering of methane, for which all internal vibrations can be included, and used the results to deduce consequences for the dissociation. We used model p otential energy surfaces (PESs) that have been developed with Ni(111) in min d, but our re- sults should hold for other surfaces as well. At a translatio nal energy up to 96 kJ/mol we found that the scattering is almost completely ela stic. Vibrational excitations when the molecule hits the surface and the corre sponding deforma- tion depend on generic features of the potential energy surf ace. In particular, our simulations indicate that for methane to dissociate the interaction of the molecule with the surface should lead to an elongated equili brium C–H bond length close to the surface. We have been using the multiconfigurational time-dependent Hartree (MCTDH) method for our wavepacket simulation, because it can deal wi th a large num- ber of degrees of freedom and with large grids.[39,40] This m ethod has been 2applied successfully to gas phase reactions and reactions a t surfaces.[41–60] In this paper we report wavepacket simulations of CD 4scattering including all internal vibrations for fixed orientations, performed o n the same model PESs as in our previous paper.[38] Translational motion par allel to the sur- face and all rotational motion was neglected. No degrees of f reedom of the surface were included. Experiments show that coupling with these degrees of freedom is dependent on the metal surface. For example, the o bserved surface temperature effect are small on Ni[5], but quite large on Pt[7 ]. As we are only interested in the role of internal vibrations, we have not in cluded degrees of freedom of the surface to keep the simulations as simple as po ssible. We will discuss the vibrational excitation and the deformation of t he CD 4molecule when it hits the surface and compare it with CH 4. Later on we will look at the energy distribution of the kinetic energy per mode and the po tential energy in some terms of the PES with the elongated equilibrium bond l ength close to the surface for both isotopes. The transfer of translatio nal kinetic energy towards vibrational kinetic energy gives an indication abo ut the dissociation probability, since vibrational kinetic energy helps in ove rcoming the dissoci- ation barrier. It gives a better idea too about which modes ar e essential to include in a more accurate wavepacket simulation of methane dissociation. After that we will discuss the implications of this for the di ssociation and give a summary with some general conclusions. 2 Computational details 2.1 The Potential Energy Surfaces We used for the scattering of CD 4the same model PESs as we did for CH 4. Since we expressed the PES in mass-weighted coordinated the parameters in the PESs for CD 4differs from CH 4. We will now give an overview of our model PESs and the corresponding parameters for CD 4. The parameters of CH4for these PESs were already given in Ref. [38], where also mor e detailed information about our assumptions and contour plots of some cross-section of the model PESs can be found. The PESs we used can all be written as Vtotal=Vintra+Vsurf, (1) whereVintrais the intramolecular PES and Vsurfis the repulsive interaction with the surface. For the Vintrawe looked at four different types of PESs. The 3Vintrainclude also for two types changes in the intramolecular pot ential due to interactions with the surface. 2.1.1 A harmonic potential The first one is completely harmonic. We have used normal mode coordinates for the internal vibrations, because these are coupled only very weakly. In the harmonic approximation this coupling is even absent so that we can write Vintraas Vintra=Vharm=1 210/summationdisplay i=2kiX2 i, (2) the summation is over the internal vibrations, Xi’s are mass-weighted displace- ment coordinates and kiare mass-weighted force constants. (see Table 1 for definitions and values); ( X1is the mass-weighted overall translation along the surface normal).[61] The force constants have been obtaine d by fitting them on the experimental vibrational frequencies of CH 4and CD 4.[62,63] We have assumed that the repulsive interaction with the surf ace is only through the deuterium atoms that point towards the surface. We take t hez-axis as the surface normal. In this case the surface PES is given by Vsurf=A NDND/summationdisplay i=1e−αzi, (3) whereNDis the number of deuteriums that points towards the surface, α=1.0726 atomic units and A=6.4127 Hartree. These parameters are chosen to give the same repulsion as the PES that has been used in an MCTDH wavepa cket simulation of CH 4dissociation.[26] If we write Vsurfin terms of normal mode coordinates, then we obtain for one deuterium pointing towards the surface Vsurf=Ae−α1X1e−α2X2e−α3X3e−α4X4, (4) whereAas above, and α’s as given in Table 2. X2,X3andX4correspond all toa1modes of the C 3vsymmetry (see Fig. 1). There is no coupling between the modes X5toX10in theVsurfpart of the PES, which are all emodes of the C 3vsymmetry. For two deuteriums we obtain 4Vsurf=A e−α1X1e−α2X2e−α3X3e−α4X4e−α5X5(5) ×1 2/bracketleftBig eβ3X7e−β3X8eβ5X9e−β5X10 +e−β3X7eβ3X8e−β5X9eβ5X10/bracketrightBig , withAagain as above, α’s andβ’s as given in Table 2. X2,X3,X4andX5 correspond all to a1modes of C 2v.X7,X8,X9andX10correspond to b1and b2modes of C 2v.X6corresponds to the a2mode of C 2vand has no coupling with the other modes in Vsurf. For three deuteriums we obtain Vsurf=A e−α1X1e−α2X2e−α3X3e−α4X4(6) ×1 3/bracketleftBig eβ1X5eβ2X6e−β3X7eβ4X8eβ5X9e−β6X10 +eβ1X5e−β2X6e−β3X7e−β4X8eβ5X9eβ6X10 +e−2β1X5e2β3X7e−2β5X9/bracketrightBig , withAagain as above, α’s andβ’s as given in Table 2. X2,X3andX4 corresponds to a1modes in the C 3vsymmetry (see Fig. 1). Because these last six coordinates correspond to degenerate emodes of the C 3vsymmetry, the β parameters are not unique. 2.1.2 An anharmonic intramolecular potential Even though we do not try to describe the dissociation of meth ane in this and our previous paper, we do want to determine which interna l vibration might be important for this dissociation. The PES should at l east allow the molecule to partially distort as when dissociating. The har monic PES does not do this. A number of changes have therefor been made. The fi rst is that we have describe the C–D bond by a Morse PES. VMorse=De4/summationdisplay i=1/bracketleftBig 1−e−γ∆ri/bracketrightBig2, (7) whereDe= 0.1828 Hartree (the dissociation energy of methane in the gas- phase) and ∆ rithe change in bond length from the equilibrium distance. γwas calculated by equating the second derivatives along one bond of the harmonic and the Morse PES. If we transform Eq. (7) back into n ormal mode coordinates, we obtain VMorse=De4/summationdisplay i=1/bracketleftBig 1−eγi2X2eγi3X3eγi4X4eγi7X7eγi8X8eγi9X9eγi,10X10/bracketrightBig2,(8) 5withDeas above.γ’s are given in Tables 3 and 4. Note that, although we have only changed the PES of the bond lengths, the ν4umbrella modes are also affected. This is because these modes are not only bending, bu t also contain some changes of bond length. The new intramolecular PES now becomes Vintra=Vharm+VMorse −Vcorr, (9) whereVharmis as given in Eq. (2) and Vcorris the quadratic part of VMorse, which is already in Vharm. 2.1.3 Intramolecular potential with weakening C–D bonds When the methane molecule approach the surface the overlap o f substrate orbitals and anti-bonding orbitals of the molecule weakens the C–D bonds. We want to include this effect for the C–D bonds of the deuteriu ms pointing towards the surface. We have redefined the VMorsegiven in Eq. (8) and also replace it in Eq. (9). A sigmoidal function is used to switch f rom the gas phase C–D bond to a bond close to the surface. We have used the f ollowing, somewhat arbitrary, approximations. (i) The point of inflec tion should be at a reasonable distance from the surface. It is set to the turna round point for a rigid methane molecule with translation energy 93.2 kJ/mo l plus twice the fall-off distance of the interaction with the surface. (ii) T he depth of the PES of the C–D bond is 480 kJ/mol in the gas phase, but only 93.2 kJ/ mol near the surface. The value 93.2 kJ/mol corresponds to the height of the activation barrier used in our dissociation.[26] (iii) The exponentia l factor is the same as for the interaction with the surface. If we transform to normal-mode coordinates for the particul ar orientations, we then obtain Vweak=De4/summationdisplay i=1Wi/bracketleftBig 1−eγi2X2eγi3X3eγi4X4eγi7X7eγi8X8eγi9X9eγi,10X10/bracketrightBig2,(10) whereWi= 1 for non-interacting bonds and Wi=1 + Ωe−α1X1+ω 1 +e−α1X1+ω(11) for the interacting bonds pointing towards the surface. α1is as given in Table 2,γ’s are given in Tables 3 and 4, Ω = 1 .942·10−1andω= 7.197. 62.1.4 Intramolecular potential with elongation of the C–D b onds A weakened bond generally has not only a reduced bond strengt h, but also an increased bond length. We include the effect of the elongatio n of the C–D bond length of the deuteriums that point towards the surface due t o interactions with the surface. We have redefined the VMorsegiven in Eq. (8) and also replace it in Eq. (9) for this type of PES. We have used the following ap proximations: (i) The transition state, as determined by Ref. [30] and [64] , has a C–H bond that is 0.54 ˚A longer than normal. This elongation should occur at the tur n around point for a rigid methane molecule with a translation energy of 93.2 kJ/mol. (ii) The exponential factor is again the same as for t he interaction with the surface. If we transform to normal-mode coordinates for the particul ar orientations, then we obtain Vshift=De4/summationdisplay i=1/bracketleftBig 1−eγi2X2eγi3X3eγi4X4eγi7X7eγi8X8eγi9X9eγi,10X10exp[Sie−α1X1]/bracketrightBig2,(12) whereα1is as given in Table 2, γ’s are given in Tables 3 and 4. For orientation with one deuterium towards the surface we obtain; S1= 2.942·102andS2= S3=S4= 0, with two deuteriums; S1=S2= 0 andS3=S4= 1.698·102, and with three deuteriums; S1= 0 andS2=S3=S4= 2.942·102. 2.2 Initial States The exact wave-function of a D-dimensional system, is expressed in the MCTDH approximation by the form ΨMCTDH (q1,...,q D;t) =/summationdisplay n1...nDcn1...nD(t)ψ(1) n1(q1;t)...ψ(D) nD(qD;t).(13) All initial states in the simulations start with the vibrati onal ground state. The initial translational part ψ(tr)is represented by a Gaussian wave-packet, ψ(tr)(X1) = (2πσ2)−1/4exp/bracketleftBigg −(X1−X0)2 4σ2+iP1X1/bracketrightBigg , (14) whereσis the width of the wave-packet (we used σ= 320.248 atomic units), X0is the initial position (we used X0= 11σ, which is far enough from the surface to observe no repulsion) and P1is the initial momentum. Since we used mass-weighted coordinates the Gaussian wavepacket ar e identical for CD 4 and CH 4. We performed simulations in the energy range of 32 to 96 kJ/m ol. 7We here present only the results of 96 kJ/mol (equivalent to P1=−0.2704 atomic units), because they showed the most obvious excitat ion probabilities forVMorse. We used seven natural single-particle states, 512 grid poi nts and a grid-length of 15 σfor the translational coordinate. With this grid-width we can perform simulation with a translational energy up to 144 kJ/mol. Gauss-Hermite discrete-variable representations (DVR)[ 65] were used to rep- resent the wavepackets of the vibrational modes. We used for all simulations of CD4the same number of DVR points as for CH 4, which was 5 DVR points for theν2bending modes and 8 DVR points for the ν4umbrella,ν3asymmetrical stretch, and ν1symmetrical stretch mode for an numerical exact integratio n, except for the simulations with Vshift, where we used 16 DVR points for the ν1 symmetrical stretch mode, because of the change in the equil ibrium position. Also the same configurational basis was used for both isotope s. We did the simulation with one bond pointing towards the surface in eig ht dimensions, because the ν2bending modes X5andX6do not couple with the other modes. We needed four natural single-particle states for modes X2,X3andX4, and just one for the others. So the number of configurations was 71·43·14= 448. The simulation with two bonds pointing towards the surface w as performed in nine dimensions. One of the ν2bending mode ( X6) does not couple with the other modes, but for the other mode X5we needed four natural single-particle states. The number of configurations was 71·44·14= 1792, because we needed the same number of natural single-particle states as mentio ned above for the other modes. We needed ten dimensions to perform the simulat ion with three bonds pointing towards the surface. We used here one natural single-particle state for the modes X5toX10and four natural single-particle states for X2 toX4, which gave us 71·43·16= 448 configurations. 3 Results and Discussion 3.1 Excitation probabilities and structure deformation of CD4 The scattering probabilities for CD 4are predominantly elastic, as we also found in our previous simulations of CH 4scattering.[38] The elastic scattering probability is larger than 0.99 for all orientation of the PE Ss withVMorseand Vweakat a translational energy of 96 kJ/mol. For the PES with Vshiftwe observe an elastic scattering probability of 0.981 for the orientat ion with one, 0.955 with two and 0.892 with three deuteriums pointing towards th e surface. This is lower than we have found for CH 4, which is 0.956 for the orientation with three hydrogens pointing towards the surface and larger than 0.99 for the others. The higher inelastic scattering probabilities of CD 4was expected, because the 8force constants kiof CD 4are decreased up to 50% with respect to those of CH4and the translational surface repulsion fall-off differs onl y little. When we look at the excitation probabilities at the surface f or the PES with VMorseandVweak, then we observe generally an increase for CD 4compared with CH4, except for the ν4umbrella mode in the orientation with two bond point- ing towards the surface. Relevant differences in the structu re deformations are observed only in the bond angles, which are increased for CD 4in the orienta- tions with one and three bonds pointing towards the surface. The bond angle deformation of the angle between the bonds pointing towards the surface in the orientation with two bonds pointing towards the surface is decreased for CD4. We observe again that the PES with Vweakgives larger structure defor- mations than the PES with VMorse, but the differences are smaller for CD 4 than CH 4. For the PES with Vshiftwe do not observe this effect on the bond angle defor- mation. The bond angle deformation for the orientation with two and three deuteriums pointing towards the surface is the same as for CH 4and it is just 0.1◦less for the bond angle at the surface side in the orientation with one deu- terium pointing towards the surface. The excitation probab ilities (see Table 5) for theν2bending and ν4umbrella modes become higher for all orientations for CD 4, which is necessary for getting the same bond angle deformat ions as CH4. The changes in the bond distances for the orientations with o ne and two bonds pointing towards the surface is for CD 4almost the same as for CH 4. For the orientation with three bonds pointing towards the su rface, we found that the maximum bond lengthening of the bonds on the surface side was 0.032˚A less for CD 4than CH 4. We also found that the bond shortening of the bond pointing away from the surface is 0 .010˚A more for CD 4. These are only minimal differences, which also only suggest that the bond de formation for CD4has been influenced slightly more by the ν3asymmetrical stretch mode than theν1symmetrical stretch mode. The observed excitation probabi lities for these modes do not contradict this, but are not reliable e nough for hard conclusions because of their high magnitude. It is also not c lear, beside of this problem, what they really represent. Is the excitation caus ed by a different equilibrium position of the PES at the surface in a mode or is i t caused by extra energy in this mode? To answer these questions we decid ed to do an energy distribution analysis during the scattering for bot h isotopes. 93.2 Energy distribution in CH 4and CD 4 The energy distribution analysis is performed by calculati ng the expectation values of the important term of the Hamiltonian Hfor the wave-function Ψ( t) at a certain time tduring the scattering of CD 4and CH 4for all presented orientations in this and our previous paper.[38] We will pre sent here only the results of the PES with Vshift, because it is the only model PES for which the energy distribution analysis is relevant for the discussio n of the dissociation hypotheses later on. We can obtain good information about the energy distributio n per mode by looking at the kinetic energy expectation values /angbracketleftΨ(t)|Tj|Ψ(t)/angbracketrightper modej (see Table 6), because the kinetic energy operators Tjhave no cross terms like the PESs have. When we discuss the kinetic energy of a mode we n ormally refer to the a1mode of the C 3vor C 2vsymmetry, because in these modes we have observed the highest excitation probabilities and the change in kinetic energy in the other modes is generally small. By looking at the expectation values of some terms of the PES /angbracketleftΨ(t)|Vterm|Ψ(t)/angbracketright (see Table 7), we obtain information about how the kinetics o f the scattering is driven by the PES. The VsurfPES [see Eqs. (4), (5) and (6)] is the surface hydrogen/deuterium repulsion for a given orientation. Vharm(ν2) andVharm(ν4) [see Eq. (2)] are the pure harmonic terms of the intramolecul ar PES of the a1modes in the C 3vand C 2vsymmetry corresponding to a ν2bending and ν4 umbrella modes, respectively. The pure harmonic correctio n terms ofVcorr[see Eq. (9)] are included in them. Vbond(Rup) andVbond(Rdown) are the potential energy in a single C–H or C–D bond pointing respectively towa rds and away from the surface, and they give the expectation value of one b ond term of Vshift [see Eq. (12)]. All given expectation values are the maximum deviation of the initial values, which effectively means the values at the mom ent the molecule hits the surface. The largest changes in expectation values are, of course, in the kinetic energy of the translational mode. The translational kinetic energ y does not become zero as we should expect in classical dynamics. The loss of tr anslational kinetic energy is primary absorbed by the Vsurfterms of the PESs. The expectation values of the Vsurfterms show the ability of the hydrogens or deuteriums to come close to the metal surface, since in real space their exp onential fall-offs are the same for both isotopes. For a rigid molecule the sum of the translational kinetic energy and Vsurfshould be constant, so all deviations of this sum have to be found back in the intramolecular kinetic energy and oth er PES terms. We observe that both the minimum in the translational kineti c energy and the maximum in the Vsurfterms were higher for CH 4than CD 4, so we have to 10find more increase in energy in the intramolecular modes and P ES terms for CD4than CH 4. We indeed do so and that can be one of the reasons we found higher inelastic scatter probabilities for CD 4for the PES with Vshift. For the orientations with one and two bonds pointing towards the surface we observe a large increase of the kinetic energy in the ν3asymmetrical stretch mode. If we compare this with the excitation probabilities, we find that the kinetic energy analysis gives indeed a different view on the d ynamics. For the orientation with two bond pointing towards the surface we ha ve found for both isotopes very high excitation probabilities in the ν1andν3stretch modes. We know now from the kinetic energy distribution that for the ν1symmetrical stretch mode the high excitation probability is caused by th e change of the equilibrium position of the ν1mode in the PES and that for the ν3stretch mode probably the PES also has become narrower. For the orientation with three bonds pointing towards the su rface we also obtain an large increase of the kinetic energy of the ν3asymmetrical stretch mode, but we also find an even larger increase in the kinetic en ergy of the ν1symmetrical stretch mode. The total kinetic energy was extr emely large, because the kinetic energy of the translational mode become s also much larger than for the other orientations. Because of this the Vsurfterms had to be around twice as low as for the other orientations. AllVbond(Rup) terms become lower compared to the initial value, especial ly in the orientation with two bond pointing towards the surface. In the orientation with one bond pointing towards the surface, the Vbond(Rdown) term became higher. This is caused by the repulsion of Vsurfin the direction of the bond. The increase of this PES term value is higher for CD 4than CH 4. In the orientation with three bond pointing towards the surf ace we also observe a higherVbond(Rdown) value, with also the highest increase for CD 4. In relation with the somewhat shorter bond distance for the Rdownof CD 4compared with CH4, which was also a bit lower compared with the other orientati ons, we know now that the hydrogens and especially the deuterium hav e problems in following the minimum energy path of the PES with Vshiftduring the scattering dynamics. This leads to higher kinetic energy in the vibrati onal modes, which results in more inelastic scattering. TheVharm(ν2) term increases in respect to the initial value, but not as mu ch as the increase of the Vharm(ν4) is for the orientation with two bonds pointing towards the surface. The values of Vharm(ν4) for CD 4are even higher than for CH4. We observe also a larger increase of the kinetic energy in th eν4umbrella mode for CD 4than for CH 4. So although there is somewhat more energy transfer to the vibrational modes for CD 4than CH 4, this extra vibrational energy is absorbed especially in the ν4umbrella mode of CD 4. 113.3 Dissociation hypotheses We like now to discuss some possible implications of the scat tering simulation for the isotope effect on the dissociation of methane. In our p revious paper we have already drawn some conclusions about the possible reac tion mechanism and which potential type would be necessary for dissociatio n.[38] We found the direct breaking of a single C–H bond in the initial collision more reasonable than the splats model with single bond breaking after an inte rmediary Ni–C bond formation as suggested by Ref. [4], because the bond ang le deformations seems to small to allow a Ni–C to form. From the simulations wi th CD 4 we can draw the same conclusions. The PES with Vshiftgives the same angle deformations for both isotopes, which is not sufficient for th e splats model. The other potentials give higher bond angle deformations for th e orientation with three deuteriums pointing towards the surface. If the Ni–C b ond formation would go along this reaction path, then the dissociation of C D4should be even more preferable than CH 4, which is not the case. So we only have to discuss the implication of the scattering simulation for the dissoc iation probabilities of CH 4and CD 4for a direct breaking of a single bond reaction mechanism. This reaction mechanism can be influenced by what we will call a direct or an indirect effect. A direct effects is the expected changes in the dissociation p robability between CH4and CD 4for a given orientation. Since we expect that we need for diss o- ciation a PES with an elongation of the bonds pointing toward s the surface, we only have to look at the isotope effect in the simulation for the PES with Vshiftfor different orientations to discuss some direct effect. It i s clear from our simulations that the bond lengthening of CD 4is smaller than CH 4for the ori- entation with three bonds pointing towards the surface. If t his orientation has a high contribution to the dissociation of methane, then thi s can be the rea- son of the higher dissociation probability of CH 4. In this case our simulations also explain why Ref.[27] did not observe a high enough isoto pe effect in the dissociation probability of their simulation with CH 4and CD 4modelled by a diatomic, because we do not observe a change in bond lengthen ing between the isotopes for the orientation with one bond pointing towa rds the surface. The orientation with three bonds pointing towards the surfa ce is also the orientation with the highest increase of the total vibratio nal kinetic energy for the PES with Vshift, because the energy distribution analysis shows besides an high increase of the kinetic energy in the ν3asymmetrical stretch mode also an high increase in the ν1symmetrical stretch mode. Since vibrational kinetic energy can be used effectively to overcome the dissoc iation barrier, the orientation with three bonds indicates to be a more preferab le orientation for dissociation. Moreover the relative difference in kinetic e nergy between both isotopes is for the ν1stretch mode larger than for the ν3stretch mode. If the 12kinetic energy in the ν1stretch mode contributes significantly to overcoming the dissociation barrier, then it is another explanation fo r the low isotope effect in Ref.[27]. An indirect effect is the expected changes in the dissociatio n probability be- tween CH 4and CD 4through changes in the orientations distribution caused by the isotope effect in the vibrational modes. This can be the case if the favourable orientation for dissociation is not near the ori entation with three bonds pointing towards the surface, but more in a region wher e one or two bonds pointing towards the surface. These orientations do n ot show a large difference in deformation for the PES with Vshift. We can not draw immediate conclusion about the indirect effect from our simulations, s ince we did not include rotational motion, but our simulation show that an i ndirect isotope effect can exist. For the PES with VMorsein the orientation with three bonds pointing towards the surface, we observe that CD 4is able to come closer to the surface than CH 4. So this rotational orientation should be more preferable f or CD4than for CH 4. On the other hand, if the PES is in this orientation more likeVshiftthe dissociation probability in other orientation can be de creased for CD4through higher probability in inelastic scattering channe ls. So for both effects the behaviour of the orientation with thre e bonds point- ing towards the surface seems to be essential for a reasonabl e description of the dissociation mechanism of methane. A wavepacket simula tion of methane scattering including one or more rotational degrees of free dom and the vibra- tional stretch modes will be a good starting model to study th e direct and indirect effects, since most of the kinetic energy changes ar e observed in the stretch modes and so the bending and umbrella modes are only r elevant with accurate PESs. Eventually dissociation paths can be introd uced in the PES along one or more bonds. Beside of our descriptions of the possible isotopes effect fo r the dissociation extracted of the scatter simulations we have to keep in mind t hat also a tunnel- ing mechanism can be highly responsible for the higher obser ved isotope effect in the experiment and that a different dissociation barrier i n the simulations can enhance this effect of tunneling. 4 Conclusions The scattering is in all cases predominantly elastic. Howev er, the observed in- elastic scattering is higher for CD 4compared with previous simulation on CH 4 for the PES with an elongated equilibrium bond length close t o the surface. When the molecule hits the surface, we observe in general a hi gher vibra- tional excitation for CD 4than CH 4. The PES with an elongated equilibrium 13bond length close to the surface gives for both isotopes almo st the same defor- mations, although we observe a somewhat smaller bond length ening for CD 4 in the orientation with three bonds pointing towards the sur face. The other model PESs show differences in the bond angle deformations an d in the distri- bution of the excitation probabilities of CD 4and CH 4, especially for the PES with only an anharmonic intramolecular potential. Energy distribution analysis contributes new information on the scattering dynamics. A higher transfer of translational energy toward s vibrational kinetic energy at the surface results in higher inelastic scatterin g. The highest increase of vibrational kinetic energy is found in the ν3asymmetrical stretch modes for all orientations and also in the ν1symmetrical stretch mode for the orientation with three bonds pointing towards the surface, when the PES h as an elongated equilibrium bond length close to the surface. Our simulations give an indication that the isotope effect in the methane dissociation is caused mostly by the difference in the scatte ring behaviour of the molecule in the orientation with three bonds pointing to wards the surface. At least multiple vibrational stretch modes should be inclu ded for a reasonable description of isotope effect in the methane dissociation re action. Acknowledgments This research has been financially supported by the Council f or Chemical Sciences of the Netherlands Organization for Scientific Res earch (CW-NWO). This work has been performed under the auspices of NIOK, the N etherlands Institute for Catalysis Research, Lab Report No. TUE-99-5- 02. 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The a1vibrational normal modes in the C 3vsymmetry; ν1symmetrical stretch (X 2),ν3asymmetrical stretch (X 4), and ν4umbrella (X 3). 17Table 1 Overview of the relations between the mass-weighted coordi nates Xi; the force con- stants ki(in atomic units) for CD 4, the designation, and the symmetry in T d, C3v and C 2v. i k i designation T dC3vC2v 1 translation t2a1a1 2 8.897·10−5ν1; symmetrical stretch a1a1a1 3 2.008·10−5ν4; umbrella t2a1a1 4 1.060·10−4ν3; asymmetrical stretch t2a1a1 5 2.447·10−5ν2; bending e e a 1 6 2.447·10−5ν2; bending e e a 2 7 2.008·10−5ν4; umbrella t2e b 1 8 2.008·10−5ν4; umbrella t2e b 2 9 1.060·10−4ν3; asymmetrical stretch t2e b 1 10 1 .060·10−4ν3; asymmetrical stretch t2e b 2 Table 2 αandβvalues (in atomic units) of Vsurffor CD 4with one, two or three deuteriums pointing towards the surface [see Eqs. (4), (5) and (6)]. one two three α15.617·10−35.617·10−35.617·10−3 α28.882·10−35.128·10−32.960·10−3 α34.703·10−3−4.614·10−3−7.720·10−3 α4−1.353·10−2−5.103·10−3−2.295·10−3 α5 −7.252·10−3 β1 4.187·10−3 β2 7.252·10−3 β3 4.659·10−32.196·10−3 β4 3.804·10−3 β5 4.212·10−32.295·10−3 β6 3.439·10−3 18Table 3 γvalues (in atomic units) of VMorsefor CD 4with one and three deuteriums pointing towards the surface [see Eq. (8)]. one three value γ12,γ22,γ32,γ42 γ12,γ22,γ32,γ42 7.629·10−3 γ13,−3γ23,−3γ33,−3γ43 −γ13,3γ23,3γ33,3γ43 1.397·10−3 γ14,−3γ24,−3γ34,−3γ44 −γ14,3γ24,3γ34,3γ44 −1.454·10−2 γ17,γ18,γ19,γ1,10,γ28,γ2,10γ17,γ18,γ19,γ1,10,γ28,γ2,10 0.0 γ27,−2γ37,−2γ47 −γ27,2γ37,2γ47 1.318·10−3 γ38,−γ48 γ38,−γ48 −1.114·10−3 γ29,−2γ39,−2γ49 −γ29,2γ39,2γ49 −1.371·10−2 γ3,10,−γ4,10 −γ3,10,γ4,10 1.187·10−2 Table 4 γvalues (in atomic units) of VMorsefor CD 4with two deuteriums pointing towards the surface [see Eq. (8)]. two value γ12,γ22,γ32,γ42 7.629·10−3 γ13,γ23,−γ33,−γ43,γ17,−γ27,γ37,−γ47,γ18,−γ28,−γ38,γ48 −8.070·10−4 γ14,γ24,−γ34,−γ44,γ19,−γ29,γ39,−γ49,γ1,10,−γ2,10,−γ3,10,γ4,10 8.396·10−3 Table 5 Excitation probabilities at the surface, at an initial tran slational energy of 96 kJ/mol and all initial vibrational states in the ground state, for t he intramolecular PES with elongation of the C–D bonds [see Eq. (12)] in the a1modes of the C 3vand C 2v symmetry, with one, two or three deuteriums pointing toward s the surface. These modes are a ν1(a1) symmetrical stretch, a ν2(e) bending, a ν3(t2) asymmetrical stretch, and a ν4(t2) umbrella. In parenthesis is the irreducible representati on in T d symmetry. orientation ν1(a1) stretch ν2(e) bending ν3(t2) stretch ν4(t2) umbrella one 0.460 0.910 0.174 two 0.792 0.092 0.830 0.495 three 0.868 0.756 0.387 19Table 6 Expectation values of the kinetic energy per mode in mHartre e for CH 4and CD 4, at an initial translational energy of 96 kJ/mol and all initi al vibrational states in the ground state, for the intramolecular PES with elongatio n of the C–H/D bonds [see Eq. (12)] in the a1modes of the C 3vand C 2vsymmetry, with one, two or three deuteriums pointing towards the surface. These modes are a ν1(a1) symmetrical stretch, a ν2(e) bending , a ν3(t2) asymmetrical stretch, and a ν4(t2) umbrella. In parenthesis is the irreducible representation in T dsymmetry. isotope orientation translation ν1(a1) stretch ν2(e) bending ν3(t2) stretch ν4(t2) umbrella CH4 initial 36.57 3.30 1.75 3.39 1.50 one 16.76 3.56 4.53 1.51 two 14.59 3.50 1.79 4.67 1.57 three 20.53 5.32 4.39 1.58 CD4 initial 36.57 2.33 1.24 2.52 1.12 one 16.17 2.61 4.09 1.18 two 14.00 2.78 1.27 4.05 1.27 three 20.06 4.37 3.80 1.28 Table 7 Expectation values of the potential energy terms in mHartre e for CH 4and CD 4, at an initial translational energy of 96 kJ/mol and all initial vibrational states in the ground state, for the intramolecular PES with elongation of the C–H/D bonds [see Eq. (12)]. Vsurfis the total surface hydrogen repulsion; Vharm(ν2) and Vharm(ν4) are the harmonic terms of the intramolecular PES of the a1modes in the C 3vand C 2v symmetry corresponding to a ν2(e) bending and ν4(t2) umbrella modes respectively in the T dsymmetry. Vbond(Rup) and Vbond(Rdown) are the potential energy in a single C–H/D bond pointing respectively towards and away fr om the surface. isotope orientation Vsurf Vharm(ν2)Vharm(ν4)Vbond(Rup)Vbond(Rdown) CH4 initial 0.00 1.75 1.50 3.39 3.39 one 18.20 1.87 3.25 3.85 two 18.55 2.18 4.01 2.75 3.45 three 9.22 2.94 3.00 3.74 CD4 initial 0.00 1.24 1.12 2.48 2.48 one 17.94 1.89 2.43 3.44 two 18.45 1.68 4.52 2.29 2.74 three 8.71 3.49 2.28 3.21 20

arXiv:physics/0003033v1 [physics.chem-ph] 15 Mar 2000Bond breaking in vibrationally excited methane on transiti on metal catalysts R. Milot and A. P. J. Jansen Schuit Institute of Catalysis, ST/SKA, Eindhoven Universi ty of Technology P.O. Box 513, NL-5600 MB Eindhoven, The Netherlands (December 22, 2013) The role of vibrational excitation of a single mode in the sca ttering of methane is studied by wave packet simulations of oriented CH 4and CD 4molecules from a flat surface. All nine internal vibrations are included. In the translational energy range from 32 up to 128 kJ/mol we find that initial vibrational excitations enhance the transfer of tr anslational energy towards vibrational energy and increase the accessibility of the entrance channel for d issociation. Our simulations predict that initial vibrational excitations of the asymmetrical stret ch (ν3) and especially the symmetrical stretch (ν1) modes will give the highest enhancement of the dissociatio n probability of methane. The dissociative adsorption of methane on transition metal s is an important reaction in catalysis; it is the rate limiting step in steam reforming to produce syngas, and it is prototypical for catalytic C–H activation. Although the reaction mechanism has been studied intensively, it is n ot been fully understood yet. A number of molecular beam experiments in which the dissociation energy was measu red as a function of translational energy have observed that vibrationally hot CH 4dissociates more readily than cold CH 4, with the energy in the internal vibrations being about as effective as the translational energy in inducing di ssociation.1–7Two independent bulb gas experiment with laser excitation of the ν3asymmetrical stretch and 2 ν4umbrella modes on the Rh(111) surface,8and laser excitation of the ν3and 2 ν3modes on thin films of rhodium9did not reveal any noticeable enhancement in the reactivity of CH4. A recent molecular beam experiment with laser excitation o f theν3mode did succeed in measuring a strong enhancement of the dissociation on a Ni(100) surface. Howev er, this enhancement was still much too low to account for the vibrational activation observed in previous studies an d indicated that other vibrationally excited modes contrib ute significantly to the reactivity of thermal samples.10 Wave packet simulations of the methane dissociation reacti on on transition metals have treated the methane molecule always as a diatomic up to now.11–16Apart from one C–H bond (a pseudo ν3stretch mode) and the molecule surface distance, either (multiple) rotations or some lattice motion were included. None of them have looked at the role of the other internal vibrations, so there is no mo del that describes which vibrationally excited mode might be responsible for the experimental observed vibrational a ctivation. In previous papers we have reported on wave packet simulations to determine which and to what extent int ernal vibrations are important for the dissociation of CH 4 in the vibrational ground state,17and the isotope effect of CD 4.18We were not able yet to simulate the dissociation including all internal vibrations. Instead we simulated th e scattering of methane, for which all internal vibrations c an be included, and used the results to deduce consequences for the dissociation. These simulations indicate that for methane to dissociate the interaction of the molecule with t he surface should lead to an elongated equilibrium C–H bond length close to the surface. In this letter we report on n ew wave packet simulations of the role of vibrational excitations for the scattering of CH 4and CD 4molecules with all nine internal vibrations. The dynamical features of these simulations give new insight into the initial steps of the dissociation process. The conventional explanation is that vibrations help dissociation by adding energy needed t o overcome the dissociation barrier. We find that two other new explanations play also a role. One of them is the enhanced transfer of translational energy into the dissociation channel by initial vibrational excitations. The other more important explanation is the increased accessibility of th e entrance channel for dissociation. We have used the multi-configurational time-dependent Hart ree (MCTDH) method for our wave packet simulation.19,20This method can deal with a large number of degrees of freedom and with large grids. (See Ref. 21 for a recent review.) Initial translational energy has been chosen in the range of 32 to 128 kJ/mol. The initial state has been written as a product state of ten functions; one for t he normally incident translational coordinate, and one for each internal vibration. All vibrations were taken to be in the ground state except one which was put in the first excited state. The orientation of the CH 4/CD4was fixed, and the vibrationally excited state had a1symmetry in the symmetry group of the molecule plus surface (C 3vwhen one or three H/D atoms point towards the surface, and C2vwhen two point towards the surface.) The potential-energy s urface is characterised by an elongation of the C–H bonds when the molecule approaches the surfaces, no surface corrugation, and a molecule-surface part appropriate for Ni(111). It has been shown to give reasonable results, an d is described in Refs. 17 and 18. These articles give also the computational details about the configurational basis a nd number of grid points, and contain illustrations of the orientations and the important vibrational modes. We can obtain a good idea about the overall activation of a mod e by looking at the kinetic energy expectation 1values /angbracketleftΨ(t)|Tj|Ψ(t)/angbracketrightfor each mode j. During the scattering process the change in the translatio nal kinetic energy is the largest. It is plotted in Fig. 1 as a function of time for CH 4in the orientation with three bond pointing towards the surface with an initial kinetic energy of 96 kJ/mol and di fferent initial vibrational states. When the molecule approaches the surface the kinetic energy falls down to a min imum value. This minimum value varies only slightly with the initial vibrational states of the molecule. The tot al loss of translational kinetic energy varies substantial ly, however. The initial translational kinetic energy is not co nserved. This means that the vibrational excitation enhanc es inelastic scattering. Especially an excitation of the ν1symmetrical stretch and to a lesser extend the ν3asymmetrical stretch mode result in an increased transfer of kinetic ener gy towards the intramolecular vibrational energy. The inelastic scatter component (the initial minus the final tra nslational energy) for both isotopes in the orientation with three bonds pointing towards the surface, shows the fol lowing trend for the initial vibrational excitations of the modes; ν1> ν3> ν4>ground state. CH 4scatters more inelastic than CD 4over the whole calculated range of translational kinetic energies, if the molecule has an in itial excitation of the ν3stretch mode. CH 4scatters also more inelastically than CD 4in the ν1symmetrical stretch mode at higher energies , but at lower en ergies it scatters slightly less inelastically. For the molecules with the non -excited state or an excitation in the ν4umbrella mode CD 4 has a higher inelastic scattering component than CH 4. At an initial translational kinetic energy of 128 kJ/mol th e excitation of the ν4umbrella mode results in a strong enhancement of the inelast ic scattering component. For CD 4 the inelastic scattering component for the initial excited ν4umbrella mode can become even larger than for the initial excited ν3stretch mode. For the orientation with two bonds pointing to wards the surface we observe the same trends for the relation between the inelastic scatter components a nd the excited initial vibrational modes, but the inelastic scatter component are less than half of the values for the ori entation with three bonds pointing towards the surface. Also the excitation of the ν3asymmetrical stretch modes results now in a higher inelasti c scattering component for CD4than for CH 4. Excitation of the ν2bending mode gives a little higher inelastic scatter compon ent than the vibrational ground state. For the orientation with one bond pointing towards the surface we observe an even lower inelastic scattering component. At an initial kinetic ener gy of 128 kJ/mol we find that both the ν1andν3stretch modes have on inelastic component of around 6.5 kJ/mol for CD 4and 4.0 kJ/mol for CH 4. At an initial translational energy of 32 kJ/mol we observe for both isotopes in all orient ations a very small increase of translational kinetic energy for the vibrational excited molecule, which means th at there is a net transfer from intramolecular vibrational energy through the surface repulsion into the translationa l coordinate. There seem to be two groups of vibrations with different quali tative behavior with respect to (de)excitation when the molecule hits the surface. The first group, let’s call it t he “stretch” group, consist of the ν3asymmetric stretch in any orientation and the ν1symmetric stretch in the orientation with three hydrogen/d euterium atoms pointing to the surface. The second, let’s call it the “bending” group , consists of all bending vibrations and the ν1in other orientations. When the molecule is initially in the vibrati onal ground state the kinetic energy in the vibrations increases, reaches a maximum at the turn-around point, and t hen drops back almost to the initial level except for a small contribution due to the inelastic scattering compone nt. The vibrations within a group have very similar amounts of kinetic energy, but the “stretch” group has clearly a larg er inelastic component than the “bending” group, and also the kinetic energy at the turn-around point is larger. When t he molecule has initially an excitation of a vibration of the “stretch” group then the kinetic energy of that vibrat ion increases, reaches a maximum at the turn-around point, and drops to a level lower than it was initially. For an excitation of a vibration of the “bending” group there is no maximum, but its kinetic energy simply drops to a lower l evel. We see that in all cases there is not only a transfer of energy from the translation to vibrations, but a lso an energy flow from the initially excited vibration to other vibrations. However, the total energy of the vibratio nal kinetic energy and the intramolecular potential energy increases, because it has to absorb the inelastic scatterin g component. Figure 2 shows the (repulsive) interaction with the surface during the scattering process of CH 4at an initial kinetic energy of 96 kJ/mol and different initial vibrational excita tions for the orientation with three hydrogens pointing towards the surface. Since this is a repulsive term with a exp onential fall-off changes in the repulsion indicate the motion of the part of the wave packet closest to the surface. A t the beginning of the simulation the curves are almost linear in a logarithmic plot, because the repulsion hardly c hanges the velocity of the molecule. After some time the molecule enters into a region with a higher surface repulsio n and the slopes of the curves drop. This results in a maximum at the turn-around point, where most of the initial t ranslational kinetic energy is transfered into potential energy of the surface repulsion. For a classical simulation it would have meant no translational kinetic energy, but it corresponds with the minimum kinetic energy for our wave p acket simulations. Past the maximum, a part of the wave packet will accelerate away from the surface, and the sl ope becomes negative. The expectation value of the translational kinetic energy (see Fig.1) increases at the s ame time. The slope of the curves in Fig. 2 becomes less negative towards the end of the simulation, although the exp ectation value of the translational kinetic energy in this time region is almost constant. The reason for this is that a p art of the wave packet with less translational kinetic energy is still in a region close to the surface. We see also th at the height of the plateaus for the different initial vibrational excitations is again in the order; ν1> ν3> ν4>ground state. This again indicates that a larger part of 2the wave packet is inelastically scattered when ν1is excited than when ν3is excited, etc. At lower initial translational kinetic energies the platea us have a lower position and the main gap exist between the plateaus of the ν1andν3stretch modes and the lower positioned plateaus of the ν4umbrella and the ground state. At an initial translational kinetic energies of 128 kJ/mol t he positions of the plateaus are higher and the differences between the initial vibrational excitations are also small er. The plateau of the ν3stretch mode is even around the same position as the ν4umbrella mode for CD 4in the orientation with three bonds pointing towards the sur face at this initial energy. The orientation with two bonds pointin g towards the surface shows the same trends. The plateaus of the initial excited ν2bending mode are located slightly above the ground state for both isotopes. For the orientation with one bond pointing towards the surface the relative posi tions of the plateaus of the different initial excitations are the same as at low energies in the orientation with three b onds pointing towards the surface. Even though we did not try to describe the dissociation itsel f, the scattering simulation do yield indications for the role of vibrational excitations on the dissociation of m ethane, and compare these with experimental observations. The dissociation of methane occurs over a late barrier, beca use it is enhanced by vibrational energy.22Conventionally, the role of vibrational excitation on the enhancement of dis sociation probability was discussed as an effect of the availability of the extra (vibrational kinetic) energy for overcoming the dissociation barrier. Our simulations show that such a process might play a role, but they show also that t wo other processes occur through vibrational excitation. Firstly, an initial vibrational excitation increases tran slational kinetic energy transfer towards the intramolecu lar vibrational energy. The simulations show that this inelast ic scatter component can be seen in an large enhancement of the vibrational kinetic energy in the stretch modes at the tu rn-around point. This increase is larger for higher initial translational kinetic energy and is most effective in the ori entation with three bonds pointing towards the surface. If the dissociation of methane occurs primarily in this orient ation, then we would expect, based on the total available vibrational energy after hitting the surface, that excitat ion of the ν1symmetrical stretch mode is the most effective for enhancing the dissociation probability. The ν3asymmetrical stretch mode appears to be less so. An explanat ion of the enhanced inelastic scatter compound by vibrational exc itation is that through excitation the bonds are weakened, which will ease excitation in the initial non-excited modes . Other excitations than the ν2,ν3, orν4witha1symmetry for a particular orientation can possibly result in higher e nergy transfers, but we think that the difference with ν1 (which has always a1symmetry) would be still large. Secondly, the accessibility of the dissociation channel en hances also the dissociation probability. We have conclude d previously that our potential mimics reasonably the entran ce channel for dissociation.17In this letter we find that a part of the wave packet has a longer residence time at the surf ace. It is this part of the wave packet that accesses the dissociation channel, and it is also this part that is able to come near to the transition state for dissociation. From Figs. 1 and 2 we conclude that the ν1stretch mode will enhance the dissociation probability the most. The enhanced accessibility by vibrational excitation is explained by th e spread of the wave packet along a C–H bond, which gives a higher probability for the system to be atop the dissociatio n barrier. The molecular beam experiment with excitations of the ν3asymmetrical stretch mode of CH 4of Ref. 10 shows that a single excitation of the ν3asymmetrical stretch mode enhances dissociation, but the m easured reactivity of the ν3 stretch mode is too low to account for the total vibrational a ctivation observed in the molecular beam study of Ref. 5. It means that excitation of another mode than the ν3stretch will be more effective for dissociation. Our simulat ions show that indeed excitation of ν3stretch will enhance dissociation, but predict that excita tion of the ν1symmetrical stretch mode will be more effective if the dissociation occur s primary in the orientation with multiple bonds pointing towards the surface. The contribution of the ν1symmetrical stretch mode cannot be measured directly, beca use it has no infra-red activity. However, the contribution of the ν1mode can be estimated using a molecular beam study as follows. The contribution of the ν3stretch has already been determined.10Similarly the contribution of the ν4 umbrella mode can be determined. The contribution of the ν2bending can be estimated from our simulations to be somewhat lower than the ν4umbrella contribution. The total contribution of all vibra tions is known from Ref. 5, and a simple subtraction will give us then the contribution of the ν1stretch. At high translational energies the accessibility of the dissociation channel for molecules with an excited ν4umbrella mode is near to that of the molecules with excited stretch modes, and for CD 4the inelastic scattering is also enhanced. So the excitatio n of the ν4umbrella mode can still contribute significantly to the vibrational a ctivation, because it has also higher Boltzmann population in the molecular beam than the stretch modes. This research has been financially supported by the Council f or Chemical Sciences of the Netherlands Organization for Scientific Research (CW-NWO), and has been performed und er the auspices of the Netherlands Institute for Catalysis Research (NIOK). 31C. T. Rettner, H. E. Pfn¨ ur, and D. J. Auerbach, Phys. Rev. Let t.54, 2716 (1985). 2C. T. Rettner, H. E. Pfn¨ ur, and D. J. Auerbach, J. Chem. Phys. 84, 4163 (1986). 3M. B. Lee, Q. Y. Yang, and S. T. Ceyer, J. Chem. Phys. 87, 2724 (1987). 4A. C. Luntz and D. S. Bethune, J. Chem. Phys. 90, 1274 (1989). 5P. M. Holmbad, J. Wambach, and I. Chorkendorff, J. Chem. Phys. 102, 8255 (1995). 6J. H. Larsen, P. M. Holmblad, and I. Chorkendorff, J. Chem. Phy s.110, 2637 (1999). 7A. V. Walker and D. A. King, Phys. Rev. Lett. 82, 5156 (1999). 8J. T. Yates, Jr., J. J. Zinck, S. Sheard, and W. H. Weinberg, J. Chem. Phys. 70, 2266 (1979). 9S. B. Brass, D. A. Reed, and G. Ehrlich, J. Chem. Phys. 70, 5244 (1979). 10L. B. F. Juurlink, P. R. McCabe, R. R. Smith, C. L. DiCologero, and A. L. Utz, Phys. Rev. Lett. 83, 868 (1999). 11J. Harris, J. Simon, A. C. Luntz, C. B. Mullins, and C. T. Rettn er, Phys. Rev. Lett. 67, 652 (1991). 12A. C. Luntz and J. Harris, Surf. Sci. 258, 397 (1991). 13A. C. Luntz and J. Harris, J. Vac. Sci. A 10, 2292 (1992). 14A. C. Luntz, J. Chem. Phys. 102, 8264 (1995). 15A. P. J. Jansen and H. Burghgraef, Surf. Sci. 344, 149 (1995). 16M.-N. Carr´ e and B. Jackson, J. Chem. Phys. 108, 3722 (1998). 17R. Milot and A. P. J. Jansen, J. Chem. Phys. 109, 1966 (1998). 18R. Milot and A. P. J. Jansen, Surf. Sci. (2000), to be publishe d,arXiv:physics.chem-ph/0003031 . 19U. Manthe, H.-D. Meyer, and L. S. Cederbaum, J. Chem. Phys. 97, 3199 (1992). 20A. P. J. Jansen, J. Chem. Phys. 99, 4055 (1993). 21M. H. Beck, A. J¨ ackle, G. A. Worth, and H.-D. Meyer, Phys. Rep .324, 1 (2000). 22R. D. Levine and R. B. Bernstein, Molecular Reaction Dynamics and Chemical Reactivity (Oxford University Press, Oxford, 1987). 5060708090100 0 100 200 300 400 500Translational kinetic energy (kJ/mol)ground state umbrella asym. stretch sym. stretch time (fs)4 ν1ν3ν FIG. 1. Translational kinetic energy versus time for a CH 4molecule with three bonds pointing towards the surface. The initial translational kinetic energy is 96 kJ/mol. 0.0010.010.1110100 0 100 200 300 400 500Surface repulsion (kJ/mol) time (fs)ground statesym. stretchν asym. stretchν umbrellaν431 FIG. 2. Surface repulsion versus time during the scattering dynamics of CH 4at an initial translational energy of 96 kJ/mol in the orientation with three bonds pointing towards the sur face. 4

arXiv:physics/0003034v1 [physics.class-ph] 16 Mar 2000Excitation of resonators by electron beams Yukio Shibata†, Satoshi Sasaki†, Kimihiro Ishi†, Mikihiko Ikezawa†, †Research Institute for Scientific Measurements, Tohoku Uni versity, Japan E.G.Bessonov†† ††Lebedev Physical Institute AS, Moscow, Russia Abstract In this paper the main consequences of the vector theory of ex citation of resonators by particle beams are presented. Some features of excitatio n of broadband radiation in longitudinal modes of the enclosed and open resonators are d iscussed. 1 Introduction The excitation of resonators is described by Maxwell equati ons in vacuum [1] - [3] div/vectorE= 4πρ(a)rot/vectorH=4π c/vectorJ+1 c∂/vectorE ∂t(b), rot/vectorE=−1 c∂/vectorH ∂t(c), div /vectorH= 0 ( d). (1) These equations are a set of two vector and two scalar equatio ns for vectors of electric /vectorE(/vector r,t) and magnetic /vectorH(/vector r,t) field strengths or eight equations for six independent comp onents of the electric and magnetic fields. We suppose that the charg e density ρ(/vector r,t) and current density /vectorJ(/vector r,t) are given values. It means that only four components of the e lectromagnetic field strengths are independent. The solution of these equations includes transverse electr omagnetic field strengths of free elec- tromagnetic waves /vectorEtr,/vectorHtrand accompanied longitudinal electric field strengths /vectorElof Coulomb fields of the beam crossing the resonator. Transverse electr omagnetic field strengths excited by the beam in the resonator comply the condition div/vectorEtr=div/vectorHtr=div/vectorH= 0. Longitudinal electric field strength comply the condition rot/vectorEl= 0,div/vectorEl= 4πρ[2] - [5]1. Free electro- magnetic fields in resonators are solutions of homogeneous M axwell equations ( /vector =ρ= 0) with corresponding boundary conditions. These solutions a re a sum of eigenmodes of the res- onator which include a discrete set of eigenfrequences ωλand corresponding to them functions /vectorEλ(/vector r,t),/vectorHλ(/vector r,t) for the electric and magnetic field strengths (further we wi ll omit the super- scripts trandlin the fields). The subscript λincludes three numbers ( m,n,q ) corresponding to transverse and longitudinal directions of the resonator ax is. In the case of open resonators the transverse electromagnetic TEM mnqmodes are excited. When the number qis very high then 1In general case transverse fields are not only free electroma gnetic waves. Both a static magnetic field, a magnetic field accompanying a homogeneously moving particl e and arbitrary time depended magnetic field are transverse one. A part of the Coulomb electrical field accomp anying a relativistic particle is transverse one. The most simple example of the transverse electric field strengt h is the electric field strength of the homogeneously moving relativistic particle /vectorEtr=/vectorE−/vectorEl, where /vectorE=e/vector r/γ2r∗3,/vectorEl=e/vector r/r3,/vector ris the radius vector directed from the particle to the observation point, γ=/radicalbig 1−β2relativistic factor of the particle, R∗= (x−vt)2+(1−β2)(x2+y2), β=v/c,vthe velocity of the particle [1], [2]. After a particle beam c ross a resonator then only transverse free electromagnetic waves stay at the resonator. 1this number is omitted. Usually in the open resonators many l ongitudinal modes are excited even in the case of free-electron lasers emitting rather mon ochromatic radiation. The solution of the problem of excitation of resonators is si mplified by introduction of a transverse vector potential /vectorA(/vector r,t) =/summationtext λ/vectorAλ(/vector r,t) of free electromagnetic fields in Coulomb gauge div/vectorA= 0, where scalar potential ϕ= 0 when ρ= 0 (here we omitted the superscripts trand lin the vectors /vectorAtr). The corresponding wave equation for this vector can be sol ved by the method of separation of variables when we suppose /vectorAλ(/vector r,t) =qλ(t)·/vectorAλ(/vector r), where qλ(t) is the amplitude of the vector potential and /vectorAλ(/vector r) is the eigenfunction of the resonator normalized by the condition/integraltext|/vectorAλ(/vector r)|2dV= 1. In this case the total free electromagnetic field in the re sonator is described by the expression /vectorA(/vector r,t) =/summationtext λ/vector qλ(t)/vectorAλ(/vector r). The electric and magnetic field strengths of the transverse f ree fields in resonators can be expressed through the vector potential in the form /vectorEλ(/vector r,t) =−d/vectorAλ(/vector r,t)/dct=−˙qλ(t)·/vectorAλ(/vector r)/c, /vectorHλ(/vector r,t) =rot/vectorAλ(/vector r,t) =qλ(t)·rot/vectorAλ(/vector r), where ˙ qλ(t) =dqλ(t)/dt. When the charge and current densities are in the resonator then a scalar ϕσand a longitudinal vector potential /vectorAl(rot/vectorAl= 0) determine Coulomb fields of the beam in the resonator. We are n ot interesting them in this paper. When active and diffractive losses in the open resonator are a bsent then the vector potential of a free electromagnetic field in the resonator excited by th e beam can be presented in the form /vectorA(/vector r,t) =/summationdisplay λqmλ/vectorAλ(/vector r)eiωλt, (2) where the coefficient qmλis the amplitude of the excited eigenmode. The electromagnetic fields excited by the electromagnetic b eam are determined by the non- homogeneous Maxwell equations or the corresponding equati on for the vector potential ∆/vectorA(/vector r,t)−1 c2∂2/vectorA(/vector r,t) ∂t2=−4π c/vectorJ(/vector r,t). (3) The solution of the Eq(3) can be found in the form /vectorA(/vector r,t) =/summationtext λqλ(t)/vectorAtr λ(/vector r)+/summationtext σqσ(t)/vectorAl σ(/vector r) (here we stayed superscripts trandland used the conditions /vectorAλ,σ(/vector r,t) =qλ,σ(t)·/vectorAλ,σ(/vector r)). If we will substitute this expression into equation (3), int egrate over the volume of the res- onator, use the condition of normalization/integraltext|/vectorAλ(/vector r)|2dV=/integraltext|/vectorAσ(/vector r)|2dV= 1,/integraltext/vectorAλ(/vector r)/vectorAλ′(/vector r)dV=/integraltext/vectorAλ(/vector r)/vectorAσ(/vector r)dV=/integraltext/vectorAσ(/vector r)/vectorAσ′(/vector r)dV= 0 and take into account that the vector /vectorAλcomply with the condition ∆ /vectorAλ=−(ωλ/c)2/vectorAλthen we will receive the equation for change of the amplitude of the eigenmode qλfor free fields in the resonator ¨qλ+ω2 λqλ=4π c/integraldisplay V/vectorJ(/vector r,t)/vectorAλ(/vector r)dV. (4) The expression (4) is the equation of the oscillator of unit m ass excited by a force f(t) = (4π/c)/integraltext V/vectorJ(/vector r,t)/vectorAλ(/vector r)dV. It describes the excitation of both enclosed and open reson ators [3] - [5]. The same expression for force determine the excitation of waveguides [2]. The eigenmodes of the rectangular resonators (cavities) we re discovered by J.Jeans in 1905 when he studied the low of thermal emission. The equations (4 ) was used later for quantization of the electromagnetic field in quantum electrodynamics [3] . 2 Emission of electromagnetic radiation by electron beams i n open resonators The equation (4) does not take into account the energy losses of the emitted radiation in the resonator. These losses can be introduced through the quali ty of the resonator Qλ 2¨qλ+ωr Qλ˙qλ+ω2 λqλ=4π c/integraldisplay V/vectorJ(/vector r,t)/vectorAλ(/vector r)dV, (5) where in the case of the open resonator ωr= 2π/T,T= 2L/cis the period of oscillations of the light wavepacket between the resonator mirrors when it pass es along the axis of the resonator (notice that in general case the frequencies ωλ=ωmnqdepend on m,n,q and slightly differ from frequencies ωrq). Here we have introduced a version of a definition of a resona tor quality connected with the frequency ωr. Another version of a quality is usually connected with the frequency ωλ. Our definition is more convenient for the case of free-elect ron lasers using open resonators. Using (5) we can derive the expression for the energy balance in the resonator. For this purpose we can multiply this equation by ˙ qλand integrate over the volume of the resonator. Then we receive the equation 1 2d dt[ ˙q2 λ+ω2 λq2 λ] + (ωr Qλ)2˙q2 λ= 4π/integraldisplay V/vectorJ(/vector r,t)/vectorEλ(/vector r,t)dV. (6) If we take into account that /vectorEλ(/vector r,t) =−˙qλ(t)·/vectorAλ(/vector r)/c,/vectorHλ(/vector r,t) =rot/vectorAλ(/vector r),rot/vectorAλ= ωλ/vectorAλ/c,/integraltext|/vectorAλ(/vector r)|2dV= 1 then the energy of the free electromagnetic field in the res onator can be presented in the form εem λ=/integraltext[(|/vectorEλ|2+|/vectorHλ|2)/8π]dV= [ ˙q2 λ+ω2 λq2 λ]/8πc2and the equation (7) can be presented in the another form ˙εem λ+ (ωr/Qλ)εem λ=/integraldisplay V/vectorJ(/vector r,t)/vectorEλ(/vector r,t)dV. (7) The equation (5) is the pendulum equation with a friction. It determine the time evolution of the electromagnetic field stored at the resonator, when th e time dependence of the beam current /vectorJ(/vector r,t) is given. The amplitude qλ(/vector r,t) according to (5) is determined by the coefficient of expansion of the given current into series of eigenfuncti ons of the resonator. Notice that the value /vectorAλ[/vector re(t)] depends on tonly through /vector re(t) and the value /vectorEλ[t,/vector re(t)] =−˙qλ(t)·/vectorAλ[/vector re(t)]/c depends on tdirectly through qλ(t) and through /vector re(t). In the case of one particle of a charge ”e” the beam current den sity/vectorJ(/vector r,t) =e/vector v(t)δ[/vector r−/vector re(t)]. In this case the force f(t) =e/vector v[/vector re(t)]/vectorAλ[/vector re(t)] and the power transferred from the electron beam to the resonator wave mode λexcited in the resonator Pλ(t) =/integraltext V/vectorJ(/vector r,t)/vectorEλ(/vector r,t)dV= e/summationtext i/vector ve i(t)/vectorEλ[(/vector re,ı(t),t)]. Using these expressions of force and power for all electr ons ”i” of the beam we can present the equations (5), (7) in the form ¨qλ+ω2 λqλ=4πe c/summationdisplay i/vector ve i(t)/vectorAλ[/vector re i(t)], (8) ˙εem λ+ (ωr/Qλ)εem λ=e/summationdisplay i/vector ve i(t)/vectorEλ[(/vector re i(t),t)]. (9) It follows from (5), (7) and (8), (9) that transverse resonat or modes are excited only in the case when the force f(t)/negationslash= 0 and the power Pλ(t)/negationslash= 0 that is when the particle trajectory passes through the regions where the corresponding resonat or modes have large intensities and when the particle velocity has transverse and/or longitudi nal components directed along the direction of the electric field strength. Open resonators on the level with enclosed ones have modes with longitudinal components of electric field streng th (see Appendix). It means that open resonators can be excited even in the case when the parti cle trajectory have no transverse 3components and its velocity is directed along the axis of the resonator2. Using external fields of a single bending magnet can increase the power of the generate d radiation. Both in the case of lack of a banding magnet and presence of one bending magnet the bro adband radiation is emitted. The experiment confirms this observations [7]. Using extern al fields of undulators and beams bunched at frequencies of the emitted radiation can lead to e mission of rather monochromatic radiation. In the simplest case when the beam current density /vectorJ(/vector r,t) is a periodic function of time then the force can be expanded in the series f(t) =/integraltext V/vectorJ(/vector r,t)/vectorAλ(/vector r)dV=/summationtext∞ ν=−∞fλνexp[i(νωbt− ϕλν)], where ωb= 2π/TbandTbare a period and frequency of the current density oscillatio n accordingly, fλν= (1/Tb)/integraltextTb/2 −Tb/2f(t)exp(iνωbt)dt, are the known coefficients, ϕλνphase. The value fλ−ν=f∗ λ ν, where f∗ λνis the complex conjugate of fλν. The solution of the equation (5) for the case of the established oscillations ( t≫QλTb) is qλ(t) =∞/summationdisplay ν=1Aλνexp[i(νωbt−θλν)], (10) where Aλν=fλν/radicalBig (ω2 λ−ν2ω2 b)2+ (νωrωb/Qλ)2, θλν=ϕλν+arctgνωrωb Qλ(ω2 λ−ν2ω2 b). It follows from the equation (10) that the maximum of the ampl itude of the vector potential Aλν=Qλfλν/ωrω2 λtakes place at resonance νωb=ωλ=ωmnq≃ωrq. Notice that all modes λ are excited at the same frequency ωbof the oscillator. In general case ωb/negationslash=ωλ. The equation (10) is the first order linear equation of the ene rgy change in the resonator excited by the electron beam. It follows from this equation t hat after switching off the beam current at some moment t0(/vectorJ(/vector r,t)|t>t0= 0) the energy in the resonator will be changed by the law εem λ=εem λ,/0exp[−(t−t0)/τ], where τ=Qλ/ωr,εem λ0=εem λ|t=t0. On the contrary after switching on the beam current at some moment t0the energy in the resonator will be changed by the law εem λ=εem λm(1−exp[−(t−t0))/τ], where the energy of the electromagnetic field in the resonator εem λmis determined by the parameters of the resonator and the beam . The considered example describes the emission of an oscilla tor or a system of oscillators which are in phase with the excited mode and have zero average velocity (trajectory has a form re=re0+/vector ıa0cosω0t, where /vector ıis the unite vector directed along the axis x). More complicated examples of trajectories of particles using for excitation of resonators by electron beams can be considered (the arc of circle, sine- or helical-like trajec tories in bending magnets and undulators). 3 Vector TEM modes of open resonators The theory of high quality open resonators does not differ fro m enclosed ones. But eigenmodes of open resonators have some unique features. The spectrum o f the open resonators is rarefied, the operating mode spectrum has maximum selectivity. The di mensions of open resonators are much higher then the excited wavelengths and the dimensions of the enclosed resonators are of the order of excited wavelengths. The quality of open resona tors at the same wavelengths is higher then enclosed ones. 2In this case the transition radiation is emitted by particle s when they pass the walls of a resonator. The electromagnetic radiation will be emitted in the form of thi n spherical layers at the first and second resonator mirrors [6]. It will be reflected then repeatedly by resonato r mirrors. The expansion of the electromagnetic fields of the spherical layers will be described by the series (2). 4There are some methods of calculation of TEM modes in open res onators. Usually scalar wave equations are investigated [8], [9]. There is a small in formation in technical publications about distribution of vectors of the electric and magnetic fi eld strengths in such resonators. In this section we search some distributions. In the Appendix t he foundations of the excitation of resonators by electron beams are presented . We will present the result for the Cartesian coordinates. In this case the solution of the scalar wave equation (24) (see Appendix) has a form [10] Vmn(x,y,z) =C/radicalBig wx(z)wy(z)Hm/parenleftBigg√ 2x wx(z)/parenrightBigg Hn/parenleftBigg√ 2y wy(z)/parenrightBigg · exp/braceleftBigg ik 2/parenleftBigg x2 qx(z)+y2 qy(z)/parenrightBigg −i(m+1 2)arctgλz πw2 0x−i(n+1 2)arctgλz πw2 0y/bracerightBigg (11) and for the cylindrical coordinates V(r,φ,z) =C/parenleftbiggr w(z)/parenrightbiggm/parenleftBigg sinmφ cosmφ/parenrightBigg Lm n/parenleftBigg 2r2 w2(z)/parenrightBigg exp/braceleftBigg ikr2 q(z)−i(m+ 2n+ 1)arctgλz πw2 0/bracerightBigg w(z)−1, (12) where Hm,Hnare the Hermittian polynomials, Lm nthe Lagerian polynomials, λ= 2πc/ω is the wavelength, C = constant, 1 q(z)=1 R(z)+iλ πw2(z), R (z) =z 1 +/parenleftBigg πw2 0 λz/parenrightBigg2 , w2(z) =w2 0/bracketleftBigg 1 +/parenleftbiggλz πw2 0/parenrightbigg2/bracketrightBigg . In (11), (12) R(z) is the radius of the wave front of Gaussian beam, w(z) the radius of the beam, w0(z) the radius of the waist of the beam. Atm=n= 0 we have the main mode of the Gaussian beam. If w0x=w0y=w0then the main modes for the Cartesian and cylindrical coordinates ar e the same U(x,y,z) =C w(z)exp/braceleftBigg −x2+y2 w2(z)+ik 2x2+y2 R(z)−iarctgλz πw2 0/bracerightBigg expi(kz−ωt). (13) We have the solutions (11), (12) of the scalar wave equation ( 24) for the space limited beam. Now we can find vectors of the electric and magnetic field stren gths using the expressions (23) and possible ways of construction of Hertz vectors. Let us su ppose the next compositions with the electric Hertz vector assuming that magnetic Hertz vect or is zero: 1) Πe x=U(x,y,z), Πe y= Πe z= 0. 2) Πe x= 0, Πe y=U(x,y,z), Πe z= 0. 3) Πe x= 0, Πe y= 0, Πe z=U(x,y,z). In the first case div/vectorΠ =∂Πx/∂x=∂V/∂x exp[i(kz−ωt)], (rot/vectorΠ)x= 0, (rot/vectorΠ)y= (∂V/∂z +ikV)exp[i(kz−ωt)], (rot/vectorΠ)z=−(∂V/∂y )exp[i(kz−ωt)] 5and E1 x=∂2V/∂x2+k2V,E1 y=∂2V/∂x∂y , E1 z=∂2V/∂x∂z +ik∂V/∂x ,H1 x= 0,H1 y=ik∂V/∂z −k2V,H1 z=ik∂V/∂y . The upper superscript shows the first composition of the Hert z vector. A common multiple exp[i(kz−ωt)] for all field components is omitted. The values ∂2V/∂x i∂xk≪k∂V/∂x i≪k2V. That is why in this case E1 x≫E1 y,E1 z, H1 y≫H1 z. The second case does not differ from the first one. It is necessa ry to substitute variable xby yand vise versa. In the third case div/vectorΠ =∂Πx/∂z=∂V/∂x exp[i(kz−ωt)], (rot/vectorΠ)x=∂V/∂y exp[i(kz−ωt)], (rot/vectorΠ)y=−∂V/∂x exp[i(kz−ωt)], (rot/vectorΠ)z= 0 and E3 x=∂2V/∂x∂z +ik∂V/∂x ,E3 y=∂2V/∂z∂y +∂V/∂y , E3 z= 2ik∂V/∂z ,H3 x=ik∂V/∂y ,H3 y=−ik∂V/∂x ,H3 z= 0. It follows that in the case of the main mode the electric and ma gnetic field strengths corre- sponding to the electric Hertz vector have components: E1 x=k2U(x,y,z), E1 y≃0, E1 z= 2ikx/bracketleftbigg1 w2(z)+ik R(z)/bracketrightbigg U(x,y,z), H1 x≃0, H1 y=−k2U(x,y,z), H1 z= 2iky/bracketleftbigg1 w2(z)+ik R(z)/bracketrightbigg U(x,y,z) E2 x=≃0, E2 y=k2U(x,y,z), E2 z= 2iky/bracketleftbigg1 w2(z)+ik R(z)/bracketrightbigg U(x,y,z), H2 x=−k2U(x,y,z), H2 y≃0, H2 z= 2ikx/bracketleftbigg1 w2(z)+ik R(z)/bracketrightbigg U(x,y,z) E3 x= 2ikx/bracketleftbigg1 w2(z)+ik R(z)/bracketrightbigg U(x,y,z), E3 y=−2iky/bracketleftbigg1 w2(z)+ik R(z)/bracketrightbigg U(x,y,z), (14) E3 z= 2ik  4λ(x2+y2)z (πw0)2w3(z)+ik(x2+y2) 2R2(z) 1−/parenleftBigg πw2 0 λz/parenrightBigg2 −iλ πw2 0/bracketleftbigg 1 +/parenleftBig λz πw2 0/parenrightBig2/bracketrightbigg  U(x,y,z), H3 x= 2iky/bracketleftbigg1 w2(z)+ik R(z)/bracketrightbigg U(x,y,z), H3 y=−2iky/bracketleftbigg1 w2(z)+ik R(z)/bracketrightbigg U(x,y,z), H3 z= 0. The electric and magnetic field strengths received from magn etic Hertz vector can be received from the fields (14) as well. For this purpose we can take the ve ctor of the electric field strength 6received from magnetic Hertz vector equal to the negative va lue of the magnetic field strength received from the electric Hertz vector /vectorE′→ −/vectorHand by analogy we can take /vectorH′→/vectorE. The general solution for the electromagnetic field strength of the main mode of Gaussian beam TEM 00can be presented in the form /vectorE=c1/vectorE1+c2/vectorE2+c3/vectorE3−c4/vectorH1−c5/vectorH2−c6/vectorH3, /vectorH=c1/vectorH1+c2/vectorH2+c3/vectorH3+c4/vectorE1+c5/vectorE2+c6/vectorE3, (15) where ciare the arbitrary coefficients determined by the conditions o f excitation of the mode by the electron beam. Waves determined by the only coefficient ci(when another ones are equal to zero) can be excited independently. Higher modes in the open resonator will be described by the ex pressions (15) and by the expressions similar to (14) for the electromagnetic field st rengths of the main mode. They will form orthogonal and full set of fundamental waves. The arbit rary wave may be expanded into these waves. Of cause, real electric and magnetic field stren gths are determined by the real part of the expression (15). In the open resonators the same Gaussian beams are excited. T hey propagate between mirrors both in zand in −zdirections. However the resonators will be excited on discr ete set of eigenfrequences (wavelengths) [10]. We can see that according to (14) all considered waves /vectorEi,/vectorHiare transverse. At the same time they have longitudinal components. This is the general property of the convergent and divergent waves [4], [10]. Such waves have longitudinal com ponents which permit the lines of the electric and magnetic field strengths to be closed. The fields /vectorE1,/vectorH1describe an electromagnetic wave with one direction of pola rization and the fields /vectorE2,/vectorH2with another one. They have high transverse components of th e electric and magnetic field strengths and zero longitudinal components o n the axis z. Electromagnetic fields /vectorE3,/vectorH3are a new kind of fields. They have zero transverse compo- nents of the electric and magnetic field strengths and high va lue longitudinal component of the electric field strength at the axis z(similar to the wave E01at the axial region of the cylindrical waveguide). It means that in this case the lines of the electr ic and magnetic field strengths are closed in the directions both at the central part of the beam p ropagation that is near to the axis zand far from the axis that is near to the region of theirs envel opes (caustics)3. Usually the scalar functions V(x,y,z) orU(x,y,z) =V(x,y,z)exp[i(kz−ωt)] are used when the modes in open resonators are investigated [4], [8], [9]. It was supposed that the waves are transverse ones and the values of the electromagnetic field s trengths are distributed near the same way as the values of the scalar functions. At that some fe atures like the existence of the wave /vectorE3,/vectorH3were hidden. Such waves have longitudinal components of the electric field strength and hence can be excited through the transition rad iation emitted on the inner sides of the resonator walls by an electron homogeneously moving a long the axis z. Such excitation was observed in the experiments published in [7]. 4 Conclusion Open resonators permit an effective generation of broadband radiation at the main and/or other transverse modes under conditions when many longitudinal m odes are excited. The longitudi- 3Notice that usually the divergent waves with high directivi ty emitted by antennas are described and drawn by the lines of the electric and magnetic field strengths whic h are closed in the directions far from the axis of the beam propagation near to the region of theirs envelops. 7nal modes are limited in the longwavelength region by the diff raction losses and in the short wavelength region by the longitudinal electron beam dimens ions (coherence conditions). Open resonators can be excited in the case when the external fields in the resonator are absent and the particle trajectory is directed along the axis of the resona tor. Using external fields of a single bending magnet can increase the power of the generated radia tion [7]. Appendix Generation and propagation of electromagnetic waves in vac uum is described by Maxwell equations (1). We noticed above that these equations are a se t of eight equations for six inde- pendent components of the electric and magnetic fields. Only four components of the electro- magnetic field are independent. These equations added with i nitial and boundary conditions describe all processes in electrodynamics. There is no general solution of the system of Maxwell equatio ns with boundary conditions similar to the Lienard-Viechert solution for the fields prod uced by charged particles moving along some trajectories at a given low in free space. It means that p rivate problems must be solved separately for every concrete case. At that when the boundar y conditions exist, interactions of particles with surrounding media and intrabeam interactio ns of particles are essential then the beam density and beam current can not be given and the dynamic al Lorentz equations must be added. Below we will consider the case when the beam densit y and the density of the beam current (particle trajectories) are given. One of the possible simplifications of the solution of the Max well equations is to transform linear Maxwell equations to the equations of the second orde r relative to the field strengths or potentials. First of all the Maxwell equations can be transformed to the e quations separately for the electric and magnetic fields. For this purpose we can differen tiate equation (1.b) with respect tot, use equation (1.c) and employ the vector identity rot rot /vectorF=graddiv /vectorF−∆/vectorF, where ∆ is the Laplacian operator. Such a way we will receive the equati on for the electric field strength and then by analogy we will receive the equation for the magne tic field strength. They are ✷/vectorE=4π c2˙/vectorJ+ 4π gradρ, (a)✷/vectorH=−4π crot/vectorJ.(b) (16) where ✷= ∆−∂2/c2∂t2is the d’Alembertian operator,˙/vectorJ=∂/vectorJ/∂t. The equations (16) are the nonhomogeneous linear equations of the second order. We must add the equations (1.a), (1.d) to the system of the equations (16). It means that we have again a system of two vector and two scalar equations (in component s they are eight equations) for six unknown components of the electric and magnetic field str engths Ei,Hi. The divergence of the equation (16.a) leads to a more general continuity equation ( ∂/∂t)(∂ρ/ ∂t+div/vectorJ) = 0 which is valid when the continuity equation ∂ρ/∂t +div/vectorJ= 0 is valid. The solution of the Maxwell equations will be the solution of these second order equations. The second order equations are another equations. Strictly speaking they are not equivalent to Maxwell equations. We must check theirs solutions by subs tituting these solutions into the linear Maxwell equations to reject unnecessary solutions. This is very difficult problem even for simple cases. A way out can be found by introducing of elec tromagnetic field potentials. The vector potential /vectorAand scalar potential ϕare introduced by the equations /vectorH=rot/vectorA, /vectorE=−gradϕ −(1/c)(∂/vectorA/∂t). In this case both from Maxwell equations and from the equat ions (16) it follows the equations for vector and scalar potentia ls 8✷/vectorA=−4π c/vectorJ(a),✷ϕ=−4πρ(b) (17) and additional condition coupling the potentials (Lorentz gauge) div/vectorA=−1 c∂ϕ ∂t. (18) It is convenient to use the electric and magnetic Hertz vecto rs as well. They permit to simplify the solutions of the problem of propagation of wave s in resonators and free space which is described by the homogeneous wave equations ( ρ= 0,/vectorJ= 0). Both the electric and magnetic Hertz vectors /vectorΠe,/vectorΠmare introduced by the same expressions /vectorA=1 c∂/vectorΠe/m ∂t; ϕ=−div/vectorΠe/m. (19) Such way defined potentials /vectorAandϕwill satisfy the equation (11) simultaneously. Different superscripts e/min this case are used on the stage of introduction of the conne ction between electric and magnetic field strengths through Hertz vectors. The electric field strength can be expressed through the electric and magnetic Hertz vec tor by the equations /vectorE=graddiv /vectorΠe−1 c2∂2/vectorΠe ∂t2; /vectorH=1 c∂ ∂trot/vectorΠe, (20) /vectorE=−1 c∂ ∂trot/vectorΠm, /vectorH=graddiv /vectorΠm−1 c2∂2/vectorΠm ∂t2. (21) These manipulations are valid because of both definitions (2 0) and (21) satisfy Maxwell equa- tions (1) and equations (16). This is because of homogeneous wave equations for electromagnetic fields ✷/vectorF= 0 (22) are symmetric relative to fields /vectorE,/vectorH(/vectorF=/vectorE,/vectorH). If /vectorEand/vectorHare some solutions of the homogeneous Maxwell equations (1b), (1c) then vectors /vectorE′=−/vectorHand/vectorH′=/vectorEwill satisfy these and another Maxwell equations as well. In general case the problem may be reduced to solving of wave e quation if potentials /vectorΠe,/vectorΠm will be introduced simultaneously in the form [10] /vectorE=graddiv Πe−1 c2∂2/vectorΠe ∂t2−1 c∂ ∂trot/vectorΠm, /vectorH=1 c∂ ∂trot/vectorΠe+graddiv /vectorΠm−1 c2∂2/vectorΠm ∂t2.(23) We can be convinced that /vectorEand/vectorHdescribed by (23) fulfil to Maxwell equations at ρ=/vectorJ= 0 when vectors /vectorΠeand/vectorΠmfulfil the wave equation (22) with replaced /vectorF→on/vectorΠeand/vectorΠm. Equation (22) is valid for each component of vectors /vectorΠeand/vectorΠm. That is why it is possible to use scalar wave equation ✷U= 0 (24) and identify its solution Uwith one of components of vectors /vectorΠeor/vectorΠmand the rest components of these vectors equate to zero (say we can take /vectorΠe=/vector ex·0+/vector ey·0+/vector ez·U,/vectorΠm= 0). Substituting the constructed such a way vector with one component in (16) w e will find the electromagnetic field strengths /vectorE,/vectorHwhich satisfy the Maxwell and wave equations. Then we can ide ntify the same solution with another component of the Hertz vector, eq uate the rest components to zero 9and calculate another electromagnetic field strengths /vectorE,/vectorHwhich satisfy the Maxwell and wave equations as well. After we will go through all compositions with components then we will have a set of six different solutions for field strengths /vectorE,/vectorH. These solutions will be six electromagnetic waves with different structures. Sum of these solutions with some coefficients will be a solution of the Maxwell equations as well. This will be algorithm of el ectromagnetic field determination through Hertz vector. Equation (24) has many different solutions. We must find such s olutions which will corre- spond to the problem under consideration to a considerable e xtent. Below we will deal with monochromatic light beams of the limited diameter related w ith resonator modes. In general case such beams can be written in the form U(x,y,z) =V(x,y,z)ei(kz−ωt)(25) where V(x,y,z) is a function of coordinate slowly varying in comparison wi th exp i(kz−ωt). A complex form of values will be used for computations and then we will proceed to a real part of the form. Substituting (25) into (24) and taking into account the slow variation of V(x,y,z) compared with exp i(kz−ωt) that is the condition |∂2V/∂z2| ≪2k|∂V/∂z |and the condition k=ω/cwe will receive the equation i∂V ∂z+1 2k(∂2V ∂x2+∂2V ∂y2) = 0 (26) which describes a space limited beam. In the general case the limited in the transverse direction w ave propagating in free space or in a resonator have rather complicated structure. That is why it is desirable to find full, orthogonal set of fundamental waves with the well known feat ure of propagation. Then an arbitrary wave may be expanded into series of these waves. Di fferent series of fundamental waves can be found for this problem and the arbitrary wave can be expanded into one or another series. The method of separation of variables is use d to solve the wave equation. For example, in the Cartesian coordinates V(x,y,z) =X(x,y,z)·Y(x,y,z) and in the cylindrical coordinates V(x,y,z) =G(u)Φ(ϕ)exp[ikr2/2q(z)]·exp[iS(z)], where randϕare cylindrical coordinates on a plane transverse to z,u=r/w(z). These solutions are considered in [10]. References [1] Landau, L. D., and E. M. Lifshitz, The Classical Theory of Fields, 3rd Reversed English edition, Pergamon, Oksford and Addison-Wesley, Reading, M ass. (1971). [2] J.D.Jackson, Classical electrodynamics , John Wiley & Sons, New York, 1975. [3] W.Heitler, The quantum theory of radiation , Oxford at the Clarendon Press, 1954. [4] L.A.Vainshtein, Electromagnetic waves , Sovetskoe Radio, Moscow, 1957; Open resonators and open waveguides , Sovetskoe radio, Moscow, 1966; L.A.Vainshtein, V.A.Soln tsev, Lektsii po sverhvysokochastotnoi electronike, Moscow, Sovetskoe Radio, 1973. [5] V.M.Lopuhin, Vozbuzhdenie electromagnitnych kolebanii i voln electron nymi potokami , GITTL, Moscow, 1953. [6] Yu.Shibata, E.G.Bessonov, Preprint FIAN No 35, Moscow, 1996; e-print: http://xxx.lanl.gov/abs/physics/9708023. 10[7] Yukio Shibata, Kimihiro Ishi, Shuichi Ono, Phys. Rev. Le tt., 1997, v.78, No 14, pp. 2740- 2743; Yu.Shibata, Kimihiro Ishi Ono, Yuta Inone, Nucl. Inst r., Meth. B 145 (1998), pp. 49-53. [8] A.Maitland and M.H.Dunn, Laser Physics , North-Holland Publishing Company Amsterdam - London, 1969. [9] Orazio Svelto, Principles of lasers , Plenum Press - NY and London, 1972. [10] A.N.Oraevskiy, Gaussian Beams and Optical resonators , Nova Science Publishers, Inc. Commack, 1995 (Proc. Lebedev Phys. Inst., v.222). 11

arXiv:physics/0003035v1 [physics.pop-ph] 16 Mar 2000Will the Population of Humanity in the Future be Stabilized? L.Ya.Kobelev, L.L.Nugaeva Department of Physics, Urals State University Lenina Ave., 51, Ekaterinburg 620083, Russia E-mail: leonid.kobelev@usu.ru A phenomenological theory of growth of the population of hum ankind is proposed. The theory based on the assumption about a multifractal nature of the se t of number of people in temporal axis and contains control parameters. In particular cases t he theory coincide with known Kapitza , Foerster, Hoerner phenomenological theories . 01.30.Tt, 05.45, 64.60.A; 00.89.98.02.90.+p. I. INTRODUCTION The problem of mankind population growth is the one of the global problems concerning the development of the mankind and its future. Will the demographic ex- plosion existing now at the mankind population of the whole world stop as already it has stopped in the most developed countries? Will the population of the Earth will be stabilized as it follows from S.Kapitza’s theory (see [1])near 12 billions, or it will continue its growth at slower rate? What are the driving parameters that govern the development of mankind (such as presents in non-linear open system (see [2])? Are these param- eters genetically predetermined or can they be changed and controlled by means of human activity? Stabiliza- tion of the mankind population of earth in the future is a sad prospect for mankind, because the absence of the numerical increasing of any biological population almost always leads, early or late, to cessation of any develop- ment (the examples are many species of insects, e.g. ter- mites, frozen in development for millions years). Hence, the appearance of more active biological species becomes in that case quite probable. These active form of life will dominate the mankind and may force it out from its present ecological niche. The aim of the present paper is to introduce a new parameter in the phenomenologi- cal theories of humankind growth describing the devel- opment of mankind population if regard humankind as a large non-linear multifractal system. Namely, the frac- tal dimension of the whole mankind number in arbitrary moment of time - the local fractional dimension d(t) (the fractional dimension for whole population of mankind). The introducing of this parameter allows to receive, as a special cases, the results of theories (see [1]), [3]- [4]) , and several new scenarios of the mankind’s future as well. Alongside with probable increasing of the mankind pop- ulation, the scenario of ruin of the present civilization - diminution of its number down to zero - is shown to be possible. We note, that correct analysis of dynamics of multifractal sets (see [5]) requires the introduction of th e mathematical concept of fractional derivatives (see [6]- [7]), which allow to take into account partly the memoryof system about the past (in this case it is the memory, that includes genetic memory of mankind about its past development). II. FRACTIONAL DIMENSION QUANTITY OF MANKIND ON TIME AXIS We assume that the dynamics of the human population can be described within the framework of fractal geom- etry concepts and mathematical formalism of fractional differential equations (see [6]- [7]). For this purpose let us consider the set of all people as a multifractal set N(t) (0≤t≤ ∞) consisting of N(t) of elements at the given timet. Following the Kapitza theory (see [1]), assume for a certain interval of time the existence of a self-similarit y of this set characterized by its fractal dimension and in- troduce a local fractional dimension (LFD) d(t) which de- scribes the fractal properties of the set at the time t. This local fractional dimension is determined by those vari- ables and their functions (these variables will be defined later) , that are treated as the driving (control) parame- ters of the human community development. Among such parameters there can be parameters of genetic origin (for example, probably, density of the population in cities per unit urban area, etc.) and ”external” parameters (e.g., a possibility of supplying the mankind population of earth by necessary amounts of food, water and energy or quick climate changes or pollution of the environment and the atmosphere, etc.). We shall characterize an alteration of the mankind population N(t) over a small time in- terval by the generalized fractional Riemann - Liouville derivative [7] (which coincides with the usual Riemann - Liouville derivative if d=const.) D1+ν(t) +,tN(t) =∂α ∂tαt/integraldisplay 0N(t′)dt′ Γ(α−d(t′))(t−t′)d(t′)−α+1,(1) d(t) = 1 + ν(t)>0 1ν(t)≡ν(X1(t), X2(t)...Xi(t)), i= 1,2, ..., α ={d}+ 1 (2) In (1) ν(t) is the fractional quantity and defines the differences between the derivatives of integer order and fractional derivatives (1) thus being the driving parame- ter for the growth of mankind as a whole, {d}is equals to the integer part of d(t)>0 (α−1≤d(t)< α) ,α= 0 ford <0 , and the set Xiare the control parameters determining all external and internal influences on the mankind population growth . The explicit information about the function νand, hence, about LFD can be obtained only after a care- ful investigations and processing the statistical data of different events, circumstances and situations impact on the development of a human population. The fractional derivative using, as defined in (1), instead of the integer first derivative allows to introduce and take into account an obvious thing - a certain kind of mankind’s memory of the past and memory about the development rates over the past years (integration with a weight function over all times till tbeginning from a fixed moment). It gives a way of considering of different parameters Xi that influence at the mankind’s development by means of LFD’s dependence upon them. The present theory, as well as ( [1]), is a phenomenological theory and the ex- act definition of function ν(t) form is beyond its scope. We note that it has sense to consider ν(t)/ne}ationslash= 1, ap- parently, only for times greater than T′ 1because before a certain time, introduced in ( [1], [3]- [4]) (the time of demographic transition T=T′ 1), the theories men- tioned describe the empirical data about the number and progress of the mankind population quite well. More- over, we shall restrict ourselves in this paper to analyzing growth of the mankind population for three special cases ofν(t) :ν(t) = 0, ν(t) = 1 and ν(t) =−1. III. FOERSTER,HOERNER, KAPITZA THEORIES It is known, that growth of the population of earth from ancient times untill now is well approximated by an empirical relation suggested by Foerster Von H. (see [3]) and improved by H. von Xoerner (see [4]) N(t) =C T1−T′, (3) d(N(t)) dt=C (T1−T)2(4) C= 2.1011, T1= 2025 The reasonable generalization (3),(4) for future time (suitable for T > T′ 1that not gives as result an infinite value N(t) for values T=T1) was given in the phe- nomenological theory of the population S.Kapitza (see[1]) with the help of introducing of the mean people’s lifetime τ(τ=42 of years). In this theory the relation forN(T) (with C’=1,86.1011, T′ 1= 2007) takes place ∂N(t) dt=C′ (T′ 1−T)2+τ2(5) From (5) the restriction of the mankind population of earth by quantity 14.109follows. Unfortunately, the theory of the population S.Kapitza does not take into account neither exterior nor interior control parameters (even in simple form) basing only at the self-similarity of growth of a population of the people. IV. GENERALIZATION OF FOERSTER, HOERNER AND KAPITZA THEORIES FOR MULTIFRACTAL SET OF A QUANTITY OF MANKIND N(T) Let us assume the hypothesis about a fractal nature of setN(t) (maintaining assumption about selfsimilar the setsN(t)). In that case derivative on time∂ ∂tin equation (5) should be substitute for fractional derivative D1+ν(t) +,t. This operation take into account the memory of mankind about the past development. The right part (5) must be changed too for including in (5) an influence of the fractal dimension. So, instead of (5), obtain an equation D1+ν(t) +,tN(t) =C′ |T′ 1−T|2+ν(t)+2+ν(t) 2τ2+ν(t)(6) The equation (6) can be considered as the basic equa- tion of the phenomenological theory of development mankind’s population offered in this paper. The selection of different functions of fractional corrections for ν(t) al- lows to estimate character of changing N(t) as functions of time. V. FORECASTS OF THE FUTURE DEVELOPMENT OF MANKIND POPULATION FOR SPECIAL CASES OF THE FRACTIONAL DIMENSION CHOICE Some simple special cases forecasting of growth of the mankind’s population of earth are considered below at the basis of the equation (6). The meanings a fractal dimension, for simplicity , chosen as integer. a. Case ν(t) = 0 Atν(t) = 0 the fractional derivative Dcoincides with ∂ ∂tderivative and the equation (6) coincides with equa- tion (5) (hence, (6) includes S.Kapitza theory (see [1]) as a special case). It corresponds, probably, a compensa- tion the positive and the negative control parameters Xi (including exterior and ”interior” parameters), driving population of mankind. b. Case ν(t)→ −1 2The selection ν(t)→ −1 corresponds to a dominance of the negative tendencies in the future development of mankind and it is stipulated, for example, occurrence of irreversible changes in molecules DNA (owing to epi- demic AIDS, etc.), irreversible cosmic cataclysmic (for example, drop on earth of meteorites of huge mass), im- possibility for mankind to cope with negative factors of biogeozinos results in by effects of development of a tech- nical civilization. In this case the equation (6) reduce to equation N(t) =2C′ 2|T′ 1−T|+τ(7) From (7) the presence of the maximum of number of mankind follows at T=T′(i.e. in 2007) and it is equal 8,86.109(ifτis not changed). After transition through a maximum N(t) the number of mankind decreases (if the scenario will not vary) and in 2107 year N(t) will be equal 1 ,54.109. By the year 3007 the number of mankind will decrease down to 182.106, i.e. the mankind popula- tion of all earth will be equal to number of the people is occupying a dozen of large modern cities. The com- plete degenerating of mankind, as a result of decreasing of mankind population in the considered scenario and sub- sequent leaving mankind from biological arena may be expected through millions years ( not large time from bi- ological point of view). By then population the mankind of earth will decrease up to several thousand. c. Case ν(t) =−2. The equation (6) for this case is transformed into an integral equation at maintenance of the general tendency to accumulation of the negative factors resulting in to negative value d(t) . At negative values d(t) the integral equation gives the prompt diminution of mankind popu- lation and extinction of existence of mankind as a species is follows. So, at d(t)→ −1 equations (6) transforms to a T/integraldisplay T′N(t)dt=C′(8) supposing the absence the mankind at the earth (in that case equation (8) has no solution for N(t)/ne}ationslash= 0 thou for ν >−2 solution of equation (6) is exist (so N(t)→0 for ν→ −2). The time interval (necessary for disappearance mankind) is determined in this case by time for which d(t) will transfer from value equal to unity (status of mankind now) to value equal to nega- tive unity. This time interval can be very short: from several years (cosmic cataclysmic) to about several cen- turies (virus pandemia with lethal change of a heredity, increasing of the mean temperature of earth on some de- grees owing to throw out of carbonic gas etc.). It is nec- essary to pay attention on possibility of change of the negative scenario of mankind development stipulated by appearance and dominance of the positive control factorsXi(including those factors that due to conscious activity of mankind). In that case the inevitability of diminution of mankind population and destruction of mankind are not inevitable. d. case ν= 1. We shall consider the optimistic scenario of change of mankind population. It relevant to a dominance of pos- itive driving parameters: d(t)>1.So, e.g. for d(t) = 2 (ν= 1) from (6) the equation follows ∂2 ∂t2N(t) =C′ |T′−T|3+ 1,5τ3(9) The precise solution of an initial value problem of this equation is unwieldy, so we shall note , that at T >> τ the quantity N(t) will increase faster then first degree of time ( N(t)∼(T−T1)). The mankind population is increase and it characterizes in this case by following form (if there will not be includes appearance of a factors of conscious mankind activity which change the scenario and restricts unlimited growth of population) N(t)|t→∞∼C′(T−T′ 1) 2,29τ2ln(T−T′ 1) τ(10) So, at 3000 years, at the rate of increasing of the popu- lation defined by (10) (with the allowances that the cor- rections to N(t) of value ln[(T−T′ 1) τ] are dropped) popula- tion of mankind will increase up to 150 billions. That is improbable large, though, but it is may be not unreason- able because of future technical possibilities of mankind (probably, this number is the upper number for existence mankind population occupying the earth). For fractional values LFD d(t), increasing or decreasing of a population of the mankind will be characterized by intermediate de- pendencies between the received for the whole values d(t). In case of a periodic dependencies d(t) from time the pop- ulation of the world will change periodically depending on a concrete choice of dandN(t) and will not be a monotonous function of time. The examples are considered allow to determine the interval of change of fractional dimension d(t) in rea- son boundary for number of quantity mankind in future: −1< d < 2. The boundary values ( d=−1, d= 2) are result in or to ruin of mankind, or to so large mankind population that Earth may not endure. The last case must result in to change of scenario and it consist in the change in correlation between positive and negative com- ponents of control parameters Xitowards increasing of influence of negative parameters. Let the basic equation (6) is replaced by generalization of the S.Kapitza equation (5). Basic equation in that case reads ∂1+ν(t) ∂t1+ν(t)n(t) =Ksin2n(t) K=1 K, (11) n=N(t) K, t=T−T′ 1 τ, K=√ C′τ−1 3It gives for N(t) the qualitative effects analogous to results obtained from the equation (6). So, in this con- nection, a selection for describing the future increasing the population of mankind by equation (6), as the basis, or equation (11), containing, as well as the equation (6), driving parameters Xiin fractional dimension d(t), is not simple. As one of advantages of use of the equations (6) or (11) for describing demographic problems (with some of them the mankind already has confronted now) we shall stress an opportunity of insert and account in the theory many factors (such as , incurable illnesses, natural cataclysmic etc.) defining future of mankind as a result of influences the control parameters Xi(included in the dimension d(t) of fractal set for number’s distribution of the people in the time axis). VI. CONCLUSIONS 1.The main purpose of this paper was to analyze pos- sibility of introducing mankind’s population driving pa- rameters Xiin the phenomenological theories of the mankind population of the earth ( [1], [3], [4]) by method of attributing to set of the people N(t) the fractional di- mension d(t), depending from these parameters. At a choice d(t) = 1 the numbered theories are a special case of this phenomenological theory. The consideration of examples with integer meanings of d(t) is caused only by their mathematical simplicity and gives a reasonable meaning of fractal dimensions ( −1< d < 2) for describ- ing the time dependence of population of mankind. 2. In case when the interpretation the fractal proper- ties of set of the homosapience given in this paper cor- responds to a reality, ( more real cases correspond to fractional meanings d(t)) the future of mankind is not so mournful as in the case of the S.Kapitza theory (see [1]) and the exposition of change of number of mankind within the framework of the phenomenological theories of the population can be reduced to a selection of control parameters and filling them by the concrete contents. 3. The change of number of mankind (described in the framework a phenomenological theories of the pop- ulation) can be adjusted by such choice of control pa- rameters (and filling the fractal dimension d(t) by the concrete contents of dependencies of them) at which the population of mankind will grow so slowly, that overpop- ulation and the problems connected with it will do not arise in the foreseen future. Last will allow the theory be more realistic for predicting and menaging the future growth population of mankind as one of biological species occupying our world. 4. The growth of mankind population regulation (in- cluded in the parameters Xi) will allow to avoid degen- erating of mankind and to keep as much as long time the main ecological niche at Earth occupied by mankind. Last will give time for more realistic forecasting of the future mankind as one of biological kinds occupying ourworld. [1] Kapitza S.P., Uspehi Fizicheskih Nauk (Russia), 1966, vol.166, No.1, pp.63-79; Kapitza S.P., How Many People Lived, Live and are to Live in the World. An essay on the theory of growth of humankind , Moscow, Inst. Phys. Problem RAS, 1999,238p. [2] Klimontovith Yu.L., Statistical Theory of Open Systems. Vol.1, Moscow, Yanus, 1995, 686p. (in Russian); Kluwer Academic Publishers, Dordrecht, 1995; Klimontovich Yu.L., Statistical Physics of Open systems. Vol.2 , Moscow, Yanus, 1999, 450p. (in Rusian). [3] Foerster, Von H. et al., Science, v.132, p.1291 (1960) [4] Hoerner, von SJ., British Interplanetary Society, v.28 , 691, (1975) [5] Mandelbrot B., Fractal Geometry of Nature , W.H.Freeman, San Francisco, 1982 [6] Samko S.G, Kilbas A.A., Marithev I.I., Integrals and Derivatives of the Fractional Order and Their Applica- tions, (Gordon and Breach, New York, 1993). [7] Kobelev L.Ya, Fractal Theory of Time and Space , Ekater- inburg, Konros, 1999, 136p. (in Russian);Kobelev L.Ya., What Dimensions Do the Time and Space Have: In- teger or Fractional? arXiv:physics/0001035; Kobelev L.Ya., Can a Particle’s Velocity Exceeds the Speed of Light in the Empty Space? arXiv:gr-qc/0001042; Kobelev L.Ya.,Physical Consequences of Moving Faster than Light in Empty Space, arXiv:gr-qc /0001043 ;Kobelev L.Ya., Multifractality of Time and Space, Covariant Derivatives and Gauge Invariance,arXiv:hep-th/ 0002005; Kobelev L.Ya.,Generalized Riemann -Liouville Fractional Deriva- tives for Multifractal Sets, arXiv:math. CA/0002008,; Ko- belev L.Ya., The Multifractal Time and Irreversibility in Dynamic Systems, arXiv:physics/0002002; Kobelev L.Ya., Is it Possible to Transfer an Information with the Veloc- ities Exceeding Speed of Light in Empty Space?,arXiv: physics/ 0002003; Kobelev L.Ya., Maxwell Equation, Shroedinger Equation, Dirac Equation, Einstein Equation Defined on the Multi fractal Sets of the Time and the Space, arXiv:gr-qc/0002003 4

arXiv:physics/0003036v1 [physics.gen-ph] 16 Mar 2000Are the Laws of Thermodynamics Consequences of a Fractal Pro perties of Universe? Kobelev L.Ya. Department of Physics, Urals State University Lenina Ave., 51, Ekaterinburg 620083, Russia E-mail: leonid.kobelev@usu.ru Why in our Universe the laws of thermodynamics are valid? In t he paper is demonstrated the reason of it: if the time and the space are multifractal and th e Universe is in an equilibrium state the laws of the thermodynamics are consequences of it’s mult ifractal structure. 01.30.Tt, 05.45, 64.60.A; 00.89.98.02.90.+p. I. INTRODUCTION It is well known, that the multifractal sets have the characteristics very similar to the characteristics of a physical quantities (a free energy, an entropy, a tempera- ture etc) with which these characteristics can be formally compared. The connections between the characteristics of multifractal set and characteristics of physical quanti - ties formally correspond to connections between the ther- modynamic quantities. This surprising correspondence till now is completely inexplicable. In the present paper the multifractal analysis advanced in by Mandelbrot [1], [2], Renyi [3], Halsy [4], etc. (see for example, Rudolf [5]) was used for a substantiation of thermodynamics laws on the base of the supposition that the space and the time are multifractal sets. If our Universe state is the state of equilibrium (or the state nearly equilibrium) the connec- tions between the global characteristics of Universe as a whole (multifractal time and space) and their local frac- tal characteristics will be the same that thermodynamic laws (it is shown on the basis of the fractal theory of time and space (see [6] - [8]). From the point of view of the fractal theory of time and space the thermodynamics relations (as well as thermodynamics in whole) are con- sequence of the multifractal structure (structure of time and space) of our Universe. In the theory [6] - [8] the time and the space are treated as a real physical fields. These fields consist of small mul- tifractal subsets of time and space (”elements” of time and space), in turn, approximately treated as ”points”. Multifractal sets of time and space defined on a set of the carrier of a measure Rn, and contain all characteristics of the real world by reflected it’s in their fractional dimen- sions. The fractal dimensions dtr(ordtanddr) in small neighborhood of points t, r,for these ”points” (belonging the sets of t) are global dimensions. At the same time for all space - time continuum these dimensions are local fractal dimensions (Gelder exponents). The purpose of the paper is the establishment of connection between the global and the local characteristics of multifractal space and time on the basis of the multifractal analysis. We suppose that the state of the Universe (consisting from multifractal time and spatial sets) may be described asthe state close to a thermodynamic equilibrium. The es- tablishment of such connections enables on to view the new reason of origin of thermodynamic relations exist- ing in our world, reducing it to presence of the fractal properties at time and space. We shall show, that the thermodynamic relations used in physics are a natural consequence of known mathematical connections between the multifractal characteristics of the Universe (Univers e is considered as multifractal space - time set described within the framework of the fractal theory of time and space [6] - [8]). Thus thermodynamics can be considered as a natural consequence of multifractal characteristics of time and space of the world in which we live. II. CONNECTION BETWEEN THE PHYSICAL AND MULTIFRACTAL CHARACTERISTICS IN THE MULTIFRACTAL UNIVERSE Let’s consider the Universe as a dynamic system at the state that close to a thermodynamic equilibrium (at the present stage of it’s development), defined on a multifrac- tal set X. Let the state of the Universe is characterized by fractal dimensions of a space - time continuum and by mean values of an internal energy, a free energy, an entropy, a temperature. If the state of Universe is close to the thermodynamic equilibrium, it’s characteristics is possible to describe by a free energy F, entropy S, inter- nal energy Eand temperature T. Let the Universe has a multifractal nature stipulated by fractional dimensions of time and space (according to the fractal theory of time and space [6] - [8]) and is characterized by multifractal setX(r, t) it’s space - time points. The multifractal set Xis defined on the carrier of a measure (set Rnwith topological dimensions), i.e. X isthesubsetofRn. Let the set of the carrier of a measure is characterized by the temperature T0, by the internal energy E0=T0(in the system in which the Boltzmann constant is equal to unity), by a free energy F0. Let’s define a measure µ on the set Xand consider connection between invariant scaling characteristics of the multifractal Universe with a measure µon the basis of the theory [6] - [8] and hy- pothesis about the origin of the Universe as a result of 1explosion (big bang). Because of multifractality of space - time sets, the scaling transformations (at measuring volume of the Universe with the help of covering by four- dimension orbs (or cubes) with radius δ), for example, for mean values of probability of the casual mass distribu- tion< pq>(or the random distributions of densities of energy of physical fields), will look like (see, for example, [4] - [5]) < pq>∼δτ(q+1)(1) where qis scale factor bound with q-dimensions Renyi dimq B(X) by relation dimq B(X) =τ(q) q−1(2) The dimension Renyi characterizes global scailing char- acteristics of the Universe. For definition of it’s physical sense we shall consider local properties of the Universe near to the point r, t. The local fractal dimensions in this point (Gelder’s exponent) according to [6] - [8] looks like α(x)≡dt,r(/vector r(t), t) = 4 +/summationdisplay iβiLi,t/vector r(/vector r(t), t) (3) where Li,t/vector rare densities of energy of physical fields in this point and characterized by the densities of La- grangians. The quantity piin a neighborhood of a point (r, t) is transformed as pi∼δdtr(/vector r(t),t)(4) From definition of q-dimensions Renyi dimq B(/vector r, t) =1 q−1limδ→0log/summationtext ipq i logδ(5) follows dimq B(/vector r, t) =qd(/vector r(t), t q−1(6) The fractal dimensions d(r, t) in (3) are the dimension- less internal energies (after multiplication on E0the rela- tion (3) and correspond an internal energies of Universe in a point with coordinates ( r, t)) and so, for q >> 1, follows dimq B(/vector r, t)≈d(/vector r(t), t) (7) Therefore the dimq B(X) should has sense of an ener- gies. For describing of a thermodynamic equilibrium of the Universe there are only two energies (internal Eand freeF) and E0is bound with dtr(r, t), therefore the di- mensions Renyi there corresponds to a free energy of the Universe ( Fdivided by E0) in the q- state. Let’s define nowq- state. From (2) follows, as the Universe cools down and also it’s temperature is decrease and it’s vol- ume grows, that qmust depends on temperature and willincrease with Universe cooling. The simplest dimension- less function satisfying to this requirement is the functio n T=T0/T (8) Now it is necessary to define a function of state of the Universe - the entropy S. Let’s consider subsets S′(α) (of the set X) with identical Gelder’s exponents drt=α (in our case it corresponds to a selection of an isoener- getic sets of the ”internal energy” of the Universe). In this case joining of subsets S′(α), stratifying original set X, will coincide with the original set. Let’s introduce a spectrum of fractal dimensions f(α).The joining of all such subsets makes set X. Let’s the fractal dimensions of set S(α(q)) (obtained as a result of such stratifying) is f(α(q)) (spectrum of singularities). For each value of q the state of the Universe is determined as a single-valued state and at alteration q(that is decreasing of energy of the Universe because of decreasing of it’s tempera- ture) and expansion of the Universe) function f(α(q)) , describing scaling properties of set S(α(q)), will grows. Such behavior corresponds to behavior of an entropy (a q- entropy) which we shall designate by S. Hence, to the every state of Universe there corresponds a spectrum of singularities f(α(q)) equal to an q- entropy S. III. CONNECTION OF A FREE ENERGY AND AN ENTROPY AS A CONSEQUENCE OF A MULTIFRACTAL NATURE OF THE UNIVERSE We use now the known relation of the multifractal anal- ysis between q-dimensions Renyi, spectrum of a singu- larity f(α(q)) and local fractal dimensions q, (see, for example, [5]) (q−1)dimq B(X) =qα(q)−f(α(q)) (9) ForT << T 0, substituting in (9) instead of dimen- sions Renyi the spectrum of singularities f(α(q)), the lo- cal fractal dimensions α(q) and the scaling factor qtheir physical values (that we have received earlier) the rela- tion reads F=E−TS (10) The relation (10) is the basic relation of the thermody- namics. As relation (10) is fulfilled for the Universe as a whole, it will be fulfilled and for it’s parts with the state of thermodynamically equilibrium. Therefore in the Uni- verse with the multifractal time and space the realization of the laws of thermodynamics is a simple consequence of it’s structure. The analysis of connections of global dimensions and local fractal characteristics of the fractal space - time ca r- ried out above allows to make the following statements, that are true for a case of equilibrium (or nearly so by equilibrium) of the state of the Universe: 2a)The free energy of the Universe Fcan be viewed as fractal q- dimensions Renyi ( q=T0/T) of space - time setXthat consist the Universe dimT0/T B(/vector r, t) =T T0−Tlimlog(/summationtext iµT0/T i) logδ=F(11) where µia measure of i-th of four-dimensional element of space - time; b) The inverse temperature of the Universe T0/Tcor- responds to the q-characteristics of the scaling transfor- mation of multifractal space - time; c) The entropy of Universe Scorresponds the spectrum of fractal dimensions f(dtr(T0/T)), defined by dependen- cies of space-time of dimensions Renyi dimT0/T B(/vector r, t) , mean temperature T0/Tand local fractal dimensions of space - time sets with identical energy dtr(T0/T); d) The knowledge of the fractal spectrum and dimen- sions dtr(q) allows to find dimensions Renyi from (11). If the dimensions Renyi is known, the differentiation (****6.10) with respect to qgives in the equation dtr(q) =d dq[(q−1)dimq B(/vector r, t) (12) It is possible to find, using (9), the entropy (i,e. the spectrum of fractal dimensions f(T0/T) =f(q)); e) The thermodynamics in viewed model is a conse- quence of the multifractality of space - time continuum. IV. CONCLUSION The problem of a substantiation of the thermodynam- ics within the frame- work of the fractal theory of time and space presented in this paper, (as well as a substan- tiation of irreversibility of time and spatial events (see [9]) is reduced to a postulating of multifractal properties of space and time. If model of fractal time and spaces [6] - [8] is correct, the Universe is open system and exchange it’s energy with the carrier of a measure Rn(or with the alien Universe of inflationary model [9] or model [10]). [1] Mandelbrot B.B., The fractal geometry of nature (W.H. Freeman, New York, (1982)) [2] B.B.Mandelbrot, in ”Random Fluctuations and Pattern Growth”, vol.157 of NATO Advanced Study Institute, Series E: Applied Science, ed. by N.E. Stanley and D. Ostrowsky (Kluwer, Dortrecht, 1988) [3] A.Renyi, Introduction to information theory , Appendix in:Probability theory (North Holland, Amsterdam,1988) [4] Halsey T.C., Jensen M.N.,Kadanoff L.P. Procaccia J, Shraiman B., Phys.Rev.A.33, 1986, p.1141-1151[5] Rudolph O. Fortshritte der Physik, v.43 (1995), No5, p.349-450 [6] Kobelev L.Ya. Fractal Theory of Time and Space , Ed. L.Kobelev (Konross, Ekaterinburg,1999) [7] Kobelev L.Ya. What Dimensions Do the Space and Time Have: Integer or Fractional? arXiv:physics/0001035 [8] Kobelev L.Ya. Can a Particle’s Velocity Exceeds the Speed of Light in the Empty Space?. arXiv:gr- qc/0001042 [9] Kobelev L.Ya.The Multifractal Time and Irreversibilit y in Dynamic Systems. arXiv:physics/0002002 [10] Guth A.H. Phys.Rev.(1981), v.108 B, p.347; Linde A.D. Phys.Letter. 1982, v.108 B, p.389 [11] Kobelev L.Ya. Redshift Yielded by Dependence of Rate of Changing of Time Upon the Time Passes Since Galaxy’s Birth. Ural State Univ., Ekaterinburg, 1996. Dep v VINITI, No.3142-B96. 28.10.96 (in Russian) 3

arXiv:physics/0003037v1 [physics.bio-ph] 16 Mar 2000Crystalizing the Genetic Code L. Frappata, A. Sciarrinob,a, P. Sorbaa aLaboratoire d’Annecy-le-Vieux de Physique Th´ eorique LAP TH CNRS, UMR 5108, associ´ ee ` a l’Universit´ e de Savoie BP 110, F-74941 Annecy-le-Vieux Cedex, France bPermanent adress: Dipartimento di Scienze Fisiche, Univer sit` a di Napoli “Federico II” and I.N.F.N., Sezione di Napoli Complesso di Monte S. Angelo, Via Cintia, I-80126 Napoli, It aly Abstract New developments are presented in the framework of the model introduced by the authors in refs. [1, 2] and in which nucleotides as well as cod ons are classified in crystal bases of the quantum group Uq(sl(2)⊕sl(2)) in the limit q→0. An operator which gives the correspondence between the amino-acids and the co dons is now obtained for any known genetic code. The free energy released by base pairing of dinucleotides as well as the relative hydrophilicity and hydrophobicity of the dinu cleosides are also computed. For the vertebrate series, a universal behaviour in the ratios o f codon usage frequencies is put in evidence and is shown to fit nicely in our model. Then a first a ttempt to represent the mutations relative to the deletion of a pyrimidine by action of a suitable crystal spinor operator is proposed. Finally recent theoretical descript ions are reviewed and compared with our model. PACS number: 87.10.+e, 02.10.-v LAPTH-787/00 DSF-TH-9/00 physics/0003037 March 20001 Introduction Among the numerous and important questions offered to the phy sicist by the sciences of life, the ones relative to the genetic code present a particular in terest. Indeed, in addition to the fundamental importance of this domain, the DNA structur e on the one hand and the mechanism of polypeptid fixation from codons on the other han d possess appealing aspects for the theorist. Let us, in a brief summary, select some esse ntial features [3]. First, as well known, the DNA macromolecule is constituted by two linear ch ains of nucleotides in a double helix shape. There are four different nucleotides, characte rized by their bases: adenine (A) and guanine (G) deriving from purine, and cytosine (C) and thymi ne (T) coming from pyrimidine. Note also that an A (resp. T) base in one strand is connected wi th two hydrogen bonds to a T (resp. A) base in the other strand, while a C (resp. G) base is related to a G (resp. C) base with three hydrogen bonds. The genetic information is t ransmitted to the cytoplasm via the messenger ribonucleic acid or mRNA. During this operati on, called transcription, the A, G, C, T bases in the DNA are associated respectively to the U, C , G, A bases, U denoting the uracile base. Then it will be through a ribosome that a triple t of nucleotides or codon will be related to an amino-acid. More precisely, a codon is defined a s an ordered sequence of three nucleotides, e.g. AAG, ACG, etc., and one enumerates in this way 4×4×4 = 64 different codons. Following the universal eukariotic code (see Table 4), 61 of such triplets can be connected in an unambiguous way to the amino-acids, except the three foll owing triplets UAA, UAG and UGA, which are called non-sense or stop-codons, the role of w hich is to stop the biosynthesis. Indeed, the genetic code is the association between codons a nd amino-acids. But since one distinguishes only 20 amino-acids1related to the 61 codons, it follows that the genetic code is degenerated. Still considering the standard eukariotic co de, one observes sextets, quadruplets, triplets, doublets and singlets of codons, each multiplet c orresponding to a specific amino-acid. Such a picture naturally suggests to look for an underlying s ymmetry able to describe the observed structure in multiplets, in the spirit of dynamica l symmetry scheme which has proven so powerful in atomic, molecular and nuclear physics. We rev iew at the end of this paper these recent approaches. In refs. [1, 2] we have proposed a mathematical framework in w hich the codons appear as composite states of nucleotides. The four nucleotides be ing assigned to the fundamental irreducible representation of the quantum group Uq(sl(2)⊕sl(2)) in the limit q→0, the codons are obtained as tensor product of nucleotides. Indee d, the properties of quantum group representations in the limit q→0, or crystal basis, are well adapted to take into account the nucleotide ordering. Then properties of this model have bee n considered. We will generalize some of them in the following and also propose new developmen ts. 1Alanine (Ala), Arginine (Arg), Asparagine (Asn), Aspartic acid (Asp), Cysteine (Cys), Glutamine (Gln), Glutamic acid (Glu), Glycine (Gly), Histidine (His), Isole ucine (Ile), Leucine (Leu), Lysine (Lys), Methionine (Met), Phenylalanine (Phe), Proline (Pro), Serine (Ser), T hreonine (Thr), Tryptophane (Trp), Tyrosine (Tyr), Valine (Val). 1The paper is organized as follows. We start in sect. 2 by recal ling the main aspects of the model. In sect. 3 we build out of the generators of Uq→0(sl(2)⊕sl(2)) a reading operator, which gives the correct correspondence between codons and amino- acids for each of the 12 presently known genetic codes. This construction generalizes in a syn thetical way the one started in [1] for the eukariotic and vertebrate mitochondrial codes, the different reading operators acting on codons and providing the same eigenvalue for a given amino -acid whatever the considered code. In sect. 4 some physical properties of dinucleotide st ates are fitted. In sect. 5, we analyze ratios of codon usage frequency for several biological spec ies belonging to the vertebrate class and put in evidence a universal behaviour, which fits natural ly in our model. In sect. 6, making use of the general crystal basis mathematical framework, we represent the mutation induced by the deletion of a pyrimidine by the action of a suitable cry stal spinor operator. In sect. 7 we review and compare with our model the recent symmetry appr oaches to the genetic code. Finally in sect. 8 we give a few conclusions and discuss some d irections of future developments. 2 The Model We consider the four nucleotides as basic states of the (1 2,1 2) representation of the Uq(sl(2)⊕ sl(2)) quantum enveloping algebra in the limit q→0. A triplet of nucleotides will then be obtained by constructing the tensor product of three such fo ur-dimensional representations. Actually, this approach mimicks the group theoretical clas sification of baryons made out from three quarks in elementary particles physics, the building blocks being here the A, C, G, T/U nucleotides. The main and essential difference stands in the property of a codon to be an ordered set of three nucleotides, which is not the case for a baryon. Constructing such pure states is made possible in the framew ork of any algebra Uq→0(G) with Gbeing any (semi)-simple classical Lie algebra owing to the e xistence of a special basis, called crystal basis, in any (finite dimensional) representation o fG. The algebraG=sl(2)⊕sl(2) appears the most natural for our purpose. The complementary rule in the DNA–mRNA tran- scription may suggest to assign a quantum number with opposite values to the couples (A,T/U) and (C,G). The distinction between the purine bases (A,G) an d the pyrimidine ones (C,T/U) can be algebraically represented in an analogous way. Thus c onsidering the fundamental repre- sentation (1 2,1 2) ofsl(2)⊕sl(2) and denoting±the basis vector corresponding to the eigenvalues ±1 2of the J3generator in any of the two sl(2) corresponding algebras, we will assume the fol- lowing “biological” spin structure: sl(2)H C≡(+,+)←→ U≡(−,+) sl(2)V/arrowbothv /arrowbothv sl(2)V (1) G≡(+,−)←→ A≡(−,−) sl(2)H 2(2) the subscripts H(:= horizontal) and V(:= vertical) being just added to specify the algebra. Now, we consider the representations of Uq(sl(2)) and more specifically the crystal bases obtained when q→0. Introducing in Uq→0(sl(2)) the operators J+andJ−after modification of the corresponding simple root vectors of Uq(sl(2)), a particular kind of basis in a Uq(sl(2))- module can be defined. Such a basis is called a crystal basis an d carries the property to undergo in a specially simple way the action of the J+andJ−operators: as an example, for any couple of vectors u, vin the crystal basis B, one gets u=J+vif and only if v=J−u. More interesting for our purpose is the crystal basis in the tensorial product of two representations. Then the following theorem holds [4] (written here in the case of sl(2)): Theorem 1 (Kashiwara) LetB1andB2be the crystal bases of the M1andM2Uq→0(sl(2))- modules respectively. Then for u∈B1andv∈B2, we have: J−(u⊗v) =/braceleftbiggJ−u⊗v∃n≥1such that Jn −u/negationslash= 0andJ+v= 0 u⊗J−votherwise(3) J+(u⊗v) =/braceleftbiggu⊗J+v∃n≥1such that Jn +v/negationslash= 0andJ−u= 0 J+u⊗votherwise(4) Note that the tensor product of two representations in the cr ystal basis is not commutative. However, in the case of our model, we only need to construct th en-fold tensor product of the fundamental representation (1 2,1 2) ofUq→0(sl(2)⊕sl(2)) by itself, thus preserving commutativity and associativity. Let us insist on the choice of the crystal basis, which exists only in the limit q→0. In a codon the order of the nucleotides is of fundamental importa nce (e.g. CCU→Pro, CUC→ Leu, UCC→Ser). If we want to consider the codons as composite states of the (elementary) nucleotides, this surely cannot be done in the framework of L ie (super)algebras. Indeed in the Lie theory, the composite states are obtained by performing tensor products of the fundamental irreducible representations. They appear as linear combin ations of the elementary states, with symmetry properties determined from the tensor product (i. e. for sl(n), by the structure of the corresponding Young tableaux). On the contrary the crys tal basis provides us with the mathematical structure to build composite states as purestates, characterized by the order of the constituents. In order to dispose of such a basis, we need to consider the limit q→0. Note that in this limit we do not deal anymore either with a Lie algebra or with an universal deformed enveloping algebra. To represent a codon, we have to perform the tensor product of three (1 2,1 2) representations ofUq→0(sl(2)⊕sl(2)). However, it is well-known (see Tables 4) that in a multi plet of codons relative to a specific amino-acid, the two first bases constit uent of a codon are “relatively stable”, the degeneracy being mainly generated by the third nucleotide. We consider first the tensor product: (1 2,1 2)⊗(1 2,1 2) = (1 ,1)⊕(1,0)⊕(0,1)⊕(0,0) (5) 3where inside the parenthesis, j= 0,1 2,1 is put in place of the 2 j+ 1 = 1 ,2,3 respectively dimensional sl(2) representation. We get, using Theorem 1, the following t ableau: →su(2)H (0,0) (CA) (1 ,0) ( CG UG UA ) ↓ su(2)V (0,1) CU GU GA  (1,1) CC UC UU GC AC AU GG AG AA  From Table 4, the dinucleotide states formed by the first two n ucleotides in a codon can be put in correspondence with quadruplets, doublets or singlets o f codons relative to an amino-acid. Note that the sextets (resp. triplets) are viewed as the sum o f a quadruplet and a doublet (resp. a doublet and a singlet). Let us define the “charge” Qof a dinucleotide state by Q=J(1) H,3+J(2) H,3+J(2) V,3 (6) where the superscript (1) or (2) denotes the position of a cod on in the dinucleotide state. The dinucleotide states are then split into two octets with r espect to the charge Q: the eight strong dinucleotides associated to the quadruplets (as well as tho se included in the sextets) of codons satisfy Q >0, while the eight weakdinucleotides associated to the doublets (as well as those included in the triplets) and eventually to the single ts of codons satisfy Q <0. Let us remark that by the change C↔AandU↔G, which is equivalent to the change of the sign ofJ3,αor to reflexion with respect to the diagonals of the eq.(2), th e 8 strong dinucleotides are transformed into weak ones and vice-versa. If we consider the three-fold tensor product, the content in to irreducible representations of Uq→0(sl(2)⊕sl(2)) is given by: (1 2,1 2)⊗(1 2,1 2)⊗(1 2,1 2) = (3 2,3 2)⊕2 (3 2,1 2)⊕2 (1 2,3 2)⊕4 (1 2,1 2) (7) The structure of the irreducible representations of the r.h .s. of Eq. (7) is (the upper labels denote different irreducible representations): (3 2,3 2)≡ CCC UCC UUC UUU GCC ACC AUC AUU GGC AGC AAC AAU GGG AGG AAG AAA  (3 2,1 2)1≡/parenleftbiggCCG UCG UUG UUA GCG ACG AUG AUA/parenrightbigg (3 2,1 2)2≡/parenleftbiggCGC UGC UAC UAU CGG UGG UAG UAA/parenrightbigg (1 2,3 2)1≡ CCU UCU GCU ACU GGU AGU GGA AGA (1 2,3 2)2≡ CUC CUU GUC GUU GAC GAU GAG GAA  4(1 2,1 2)1≡/parenleftbigg CCA UCA GCA ACA/parenrightbigg (1 2,1 2)2≡/parenleftbigg CGU UGU CGA UGA/parenrightbigg (1 2,1 2)3≡/parenleftbiggCUG CUA GUG GUA/parenrightbigg (1 2,1 2)4≡/parenleftbiggCAC CAU CAG CAA/parenrightbigg The correspondence with the amino-acids is given in Table 10 (for the eukariotic code). Let us close this section by drawing the reader’s attention t o Fig. 1 where is specified for each codon its position in the appropriate representati on. The diagram of states for each representation is supposed to lie in a separate parallel pla ne. Thick lines connect codons associated to the same amino-acid. One remarks that each seg ment relates a couple of codons belonging to the same representation or to two different repr esentations. This last case occurs for quadruplets or sextets of codons associated to the same a mino-acid. 3 The Reading (or Ribosome) operator R 3.1 General structure of the reading operator As expected from formula (7), our model does not gather codon s associated to one particular amino-acid in the same irreducible multiplet. However, it i s possible to construct an operator Rout of the algebra Uq→0(sl(2)⊕sl(2)), acting on the codons, that will describe the various genetic codes in the following way: Two codons have the same eigenvalue under Rif and only if they are associated to the same amino-acid. This operator Rwill be called the reading operator. It is a remarkable fact that the various genetic codes share t he same basic structure. As we mentioned above, the dinucleotides can be split into “stron g” dinucleotides CC, GC, UC, AC, CU, GU, CG and GG that lead to quartets and “weak” ones UU, AU, U G, AG, CA, GA, UA, AA that lead to doublets. Let us construct a prototype of the r eading operator that reproduces this structure. The first part of the reading operator Ris responsible for the structure in quadruplets given essentially by the dinucleotide content. It is given by (the ciare arbitrary coefficients) 4 3c1CH+4 3c2CV−4c1PHJH,3−4c2PVJV,3. (8) The operators Jα,3(α=H, V) are the third components of the total spin generators of the algebraUq→0(sl(2)⊕sl(2)). The operator Cαis a Casimir operator of Uq→0(sl(2)α) in the crystal basis. It commutes with Jα±andJα,3and its eigenvalues on any vector basis of an irreducible representation of highest weight JisJ(J+ 1), that is the same as the undeformed standard second degree Casimir operator of sl(2). Its explicit expression is Cα= (Jα,3)2+1 2/summationdisplay n∈Z+n/summationdisplay k=0(Jα−)n−k(Jα+)n(Jα−)k. (9) 5Note that for sl(2)q→0the Casimir operator is an infinite series of powers of Jα±. However in any finite irreducible representation only a finite number of terms gives a non-vanishing contribution. PHandPVare projectors given by the following expressions: PH=Jd H+Jd H− andPV=Jd V+Jd V−. (10) The second part of Rgives rise to the splitting of the quadruplets into doublets . It reads −2PDc3JV,3 (11) where the projector PDis given by PD= (1−Jd V+Jd V−)(Jd H+Jd H−)(Jd H−Jd H+) + (1−Jd H+Jd H−)(1−Jd V+Jd V−) + (1−Jd H+Jd H−)(Jd V+Jd V−)(Jd H−Jd H+). (12) The third part of Rallows to reproduce the sextets viewed as quartets plus doub lets. It is −2PSc4JV,3 (13) where the projector PSis given by PS= (Jd H−Jd H+) [(Jd H+Jd H−)(1−Jd V+Jd V−) + (Jd V+Jd V−)(Jd V−Jd V+)(1−Jd H+Jd H−)].(14) At this point, one obtains the eigenvalues of the reading ope ratorRfor the 64 codons, where Y = C,U (pyrimidines), R = G,A (purines) and N = C,U,G,A: CCN =−c1−c2 GCN =−c1+ 3c2 UCN = 3 c1−c2 ACN = 3 c1+ 3c2 CUN = c1−c2 GUN = c1+ 3c2 CGN =−c1+c2 GGN =−c1+ 5c2 UUY = 5 c1−c2−3c3 UUR = 5 c1−c2−c3 AUY = 5 c1+ 3c2−c3−c4 AUR = 5 c1+ 3c2+c3+c4 UGY = 3 c1+c2−c3−c4 UGR = 3 c1+c2+c3+c4 AGY = 3 c1+ 5c2+c3+c4 AGR = 3 c1+ 5c2+ 3c3+ 3c4 CAY = c1+c2−c3 CAR = c1+c2+c3 GAY = c1+ 5c2+c3 GAR = c1+ 5c2+ 3c3 UAY = 5 c1+c2−c3 UAR = 5 c1+c2+c3 AAY = 5 c1+ 5c2+c3 AAR = 5 c1+ 5c2+ 3c3(15) The coefficients c3andc4are fixed as follows. The coefficient c3is set to the value c3= 4c1by requiring that the quartet CUN and the doublet UUR, associat ed to the amino-acid Leu, lead to the sameR-eigenvalue. It remains to reproduce the Sersextet. This is achieved by taking for the coefficient c4the value c4=−4c1−6c2, such that the final eigenvalues for the codons 6are the following: CCN =−c1−c2 GCN =−c1+ 3c2 UCN = 3 c1−c2 ACN = 3 c1+ 3c2 CUN = c1−c2 GUN = c1+ 3c2 CGN = c1+c2 GGN =−c1+ 5c2 UUY =−7c1−c2 UUR = c1−c2 AUY = 5 c1+ 9c2 AUR = 5 c1−3c2 UGY = 3 c1+ 7c2 UGR = 3 c1−5c2 AGY = 3 c1−c2 AGR = 3 c1−13c2 CAY =−3c1+c2 CAR = 5 c1+c2 GAY = 5 c1+ 5c2 GAR = 13 c1+ 5c2 UAY = c1+c2 UAR = 9 c1+c2 AAY = 9 c1+ 5c2 AAR = 17 c1+ 5c2 (16) The prototype of the reading operator Rtakes finally the form: R=4 3c1CH+4 3c2CV−4c1PHJH,3−4c2PVJV,3+ (−8c1PD+ (8c1+ 12c2)PS)JV,3(17) and the correspondence codons/amino-acids is given as foll ows: CCN→Pro UCN→Ser GCN→Ala ACN→Thr CUN→Leu GUN→Val CGN→Arg GGN→Gly UUY→Phe AUY→Ile UGY→Cys AGY→Ser UUR→Leu AUR→Met UGR→Trp AGR→unassigned (X) CAY→His UAY→Tyr GAY→Gln AAY→Asn CAR→Gln UAR→Ter GAR→Glu AAR→Lys(18) 3.2 The various genetic codes In this section, we will determine the reading operators for the following genetic codes: – the Eukariotic Code (EC), – the Vertebral Mitochondrial Code (VMC), – the Yeast Mitochondrial Code (YMC), – the Invertebrate Mitochondrial Code (IMC), – the Protozoan Mitochondrial and Mycoplasma Code (PMC), – the Echinoderm Mitochondrial Code (EMC), – the Ascidian Mitochondrial Code (AMC), – the Flatworm Mitochondrial Code (FMC), – the Ciliate Nuclear Code (CNC), – the Blepharisma Nuclear Code (BNC), – the Euplotid Nuclear Code (ENC), – the Alternative Yeast Nuclear Code (alt. YNC), Let us emphasize that each of these codes is very close to the a ssignment (18). The main differences between the biological codes and the prototype c ode (18) are the following: •assignment of the doublet AGR either to Arg(codes EC, YMC, PMC, CNC, BNC, ENC, aYNC),Ser(codes IMC, EMC, FMC), Gly(code AMC) or the stop signal Ter(code VMC). Such an assignment is done by the following term in the readin g operator: c5PAG/parenleftig 1 2−J(3) V,3/parenrightig (19) 7The operators J(3) α,3are the third components corresponding to the third nucleot ide of a codon. Of course, these last two operators can be replaced by J(3) α,3=Jα,3−Jd α,3. The projectorPAGis given by PAG= (Jd H+Jd H−)(Jd H−Jd H+)(1−Jd V+Jd V−)(Jd V−Jd V+) (20) and the coefficient c5by forArg c5=−4c1+ 14c2 forSer c5= 12c2 forGly c5=−4c1+ 18c2 forTer c5= 6c1+ 14c2(21) •splitting of some doublets into singlets (one element of the singlet combining to another doublet to form a triplet): Met→Met+Ilefor the EC, PMC, EMC, FMC, CNC, BNC, ENC, aYNC codes; Lys→Lys+Asnfor the FMC and EMC codes; Trp→Trp+Terfor the EC, CNC, BNC, aYNC codes; Trp→Trp+Cysfor the ENC code; Ter→Tyr+Terfor the FMC code; Such an assignment is done through the following term in the r eading operator: c6PXY/parenleftig 1 2−J(3) V,3/parenrightig/parenleftig 1 2−J(3) H,3/parenrightig (22) where we use the projector PAUfor the splitting of the Metdoublet,PAAfor theLys doublet,PUGfor theTrpdoublet, andPUAfor theTerdoublet. These projectors are given by PAU= (1−Jd H+Jd H−)(Jd H−Jd H+)(Jd V+Jd V−)(Jd V−Jd V+) (23) PAA= (1−Jd H+Jd H−)(Jd H−Jd H+)(1−Jd V+Jd V−)(Jd V−Jd V+) (24) PUG= (Jd H+Jd H−)(Jd H−Jd H+)(1−Jd V+Jd V−)(1−Jd V−Jd V+) (25) PUA= (1−Jd H+Jd H−)(Jd H−Jd H+)(1−Jd V+Jd V−)(1−Jd V−Jd V+) (26) The coefficient c6takes the following values: forMet→Met+Ile c6= 12c2 forLys→Lys+Asn c6=−8c1 forTrp→Trp+Cys c6= 12c2 forTrp→Trp+Ter c6= 6c1+ 6c2 forTer→Ter+Tyr c6=−8c1(27) •in the case of the CNC and BNC codes, the Terdoublet is changed in Glnas follows: Ter→Glnfor the CNC code by the term −4c1PUA/parenleftig 1 2−J(3) V,3/parenrightig (28) Ter→Ter+Glnfor the BNC code by the term −4c1PUA/parenleftig 1 2−J(3) V,3/parenrightig/parenleftig 1 2+J(3) H,3/parenrightig (29) 8•in the case of the alternative YNC code, the last quartet Leuis split into a triplet Leu coded by (CUC,CUU,CUA) and a doublet Sercoded by (CUG). The corresponding term in the reading operator is 2c1PCU/parenleftig 1 2−J(3) V,3/parenrightig/parenleftig 1 2+J(3) H,3/parenrightig (30) where the projector PCUis given by PCU= (1−Jd H+Jd H−)(1−Jd H−Jd H+)(Jd V+Jd V−)(1−Jd V−Jd V+) (31) •in the case of the Yeast Mitochondrial Code, the quartet CUN c odes the amino-acid Thrrather than Leu. This change is achieved by multiplying the quartets term (8 ) by (1 + 2PCU) for the horizontal part and by (1 −4PCU) for the vertical part. 3.2.1 The Eukariotic Code (EC) The Eukariotic Code is the most important one and is often ref erred to as the universal code. The differences between the Eukariotic Code and the prototyp e code are the following: prototype code EC prototype code EC AUG Met Met AUA Met Ile AGG X Arg AGA X Arg UGG Trp Trp UGA Trp Ter Hence from (19), (21), (22) and (27), the reading operator fo r the Eukariotic Code is REC=4 3c1CH+4 3c2CV−4c1PHJH,3−4c2PVJV,3+ (−8c1PD+ (8c1+ 12c2)PS)JV,3 +(−4c1+ 14c2)PAG/parenleftig 1 2−J(3) V,3/parenrightig +/bracketleftig 12c2PAU+ (6c1+ 6c2)PUG/bracketrightig/parenleftig 1 2−J(3) V,3/parenrightig/parenleftig 1 2−J(3) H,3/parenrightig (32) 3.2.2 The Vertebral Mitochondrial Code (VMC) The Vertebral Mitochondrial Code is used in the mitochondri ae of vertebrata. The differences between the Vertebral Mitochondrial Code and the prototype code are the following: prototype code VMC prototype code VMC AGG X Ter AGA X Ter Hence from (19) and (21), the reading operator for the Verteb ral Mitochondrial Code is RV MC =4 3c1CH+4 3c2CV−4c1PHJH,3−4c2PVJV,3+ (−8c1PD+ (8c1+ 12c2)PS)JV,3 +(6c1+ 14c2)PAG/parenleftig 1 2−J(3) V,3/parenrightig (33) 93.2.3 The Yeast Mitochondrial Code (YMC) The Yeast Mitochondrial Code is used in the mitochondriae of yeast such as Saccharomyces, Candida, etc. The differences between the Yeast Mitochondri al Code and the prototype code are the following: prototype code YMC prototype code YMC CUC Leu Thr CUU Leu Thr CUG Leu Thr CUA Leu Thr AGG X Arg AGA X Arg Hence from (19) and (21), the reading operator for the Yeast M itochondrial Code is RY MC = (4 3c1CH−4c1PHJH,3)(1 + 2PCU) + (4 3c2CV−4c2PVJV,3)(1−4PCU) +(−8c1PD+ (8c1+ 12c2)PS)JV,3+ (−4c1+ 14c2)PAG/parenleftig 1 2−J(3) V,3/parenrightig (34) 3.2.4 The Invertebrate Mitochondrial Code (IMC) The Invertebrate Mitochondrial Code is used in the mitochon driae of some arthopoda, mollusca, nematoda and insecta. The differences between the Invertebr ate Mitochondrial Code and the prototype code are the following: prototype code IMC prototype code IMC AGG X Ser AGA X Ser Hence from (19) and (21), the reading operator for the Invert ebrate Mitochondrial Code is RIMC =4 3c1CH+4 3c2CV−4c1PHJH,3−4c2PVJV,3+ (−8c1PD+ (8c1+ 12c2)PS)JV,3 +12c2PAG/parenleftig 1 2−J(3) V,3/parenrightig (35) 3.2.5 The Protozoan Mitochondrial and Mycoplasma Code (PMC ) The Protozoan Mitochondrial and Mycoplasma Code is used in t he mitochondriae of some protozoa (leishmania, paramecia, trypanosoma, etc.) and f or many fungi. The differences between the Protozoan Mitochondrial and Mycoplasma Code an d the prototype code are the following: prototype code PMC prototype code PMC AUG Met Met AUA Met Ile AGG X Arg AGA X Arg Hence from (19), (21), (22) and (27), the reading operator fo r the Protozoan Mitochondrial Code is RPMC =4 3c1CH+4 3c2CV−4c1PHJH,3−4c2PVJV,3+ (−8c1PD+ (8c1+ 12c2)PS)JV,3 +(−4c1+ 14c2)PAG/parenleftig 1 2−J(3) V,3/parenrightig + 12c2PAU/parenleftig 1 2−J(3) V,3/parenrightig/parenleftig 1 2−J(3) H,3/parenrightig (36) 103.2.6 The Echinoderm Mitochondrial Code (EMC) The Echinoderm Mitochondrial Code is used in the mitochondr iae of some asterozoa and echi- nozoa. The differences between the Echinoderm Mitochondria l Code and the prototype code are the following: prototype code EMC prototype code EMC AUG Met Met AUA Met Ile AGG X Ser AGA X Ser AAG Lys Lys AAA Lys Asn Hence from (19), (21), (22) and (27), the reading operator fo r the Echinoderm Mitochondrial Code is REMC =4 3c1CH+4 3c2CV−4c1PHJH,3−4c2PVJV,3+ (−8c1PD+ (8c1+ 12c2)PS)JV,3 +12c2PAG/parenleftig 1 2−J(3) V,3/parenrightig +/bracketleftig 12c2PAU−8c1PAA/bracketrightig/parenleftig 1 2−J(3) V,3/parenrightig/parenleftig 1 2−J(3) H,3/parenrightig (37) 3.2.7 The Ascidian Mitochondrial Code (AMC) The Ascidian Mitochondrial Code is used in the mitochondria e of some ascidiacea. The differ- ences between the Ascidian Mitochondrial Code and the proto type code are the following: prototype code AMC prototype code AMC AGG X Gly AGA X Gly Hence from (19) and (21), the reading operator for the Ascidi an Mitochondrial Code is RAMC =4 3c1CH+4 3c2CV−4c1PHJH,3−4c2PVJV,3+ (−8c1PD+ (8c1+ 12c2)PS)JV,3 +(−4c1+ 18c2)PAG/parenleftig 1 2−J(3) V,3/parenrightig (38) 3.2.8 The Flatworm Mitochondrial Code (FMC) The Flatworm Mitochondrial Code is used in the mitochondria e of the flatworms. The differ- ences between the Flatworm Mitochondrial Code and the proto type code are the following: prototype code FMC prototype code FMC UAG Ter Ter UAA Ter Tyr AUG Met Met AUA Met Ile AGG X Ser AGA X Ser AAG Lys Lys AAA Lys Asn Hence from (19), (21), (22) and (27), the reading operator fo r the Flatworm Mitochondrial Code is RFMC =4 3c1CH+4 3c2CV−4c1PHJH,3−4c2PVJV,3+ (−8c1PD+ (8c1+ 12c2)PS)JV,3 +12c2PAG/parenleftig 1 2−J(3) V,3/parenrightig +/bracketleftig 12c2PAU−8c1PAA−8c1PUA/bracketrightig/parenleftig 1 2−J(3) V,3/parenrightig/parenleftig 1 2−J(3) H,3/parenrightig (39) 113.2.9 The Ciliate Nuclear Code (CNC) The Ciliate Nuclear Code is used in the nuclei of some ciliata , dasyclasaceae and diplomonadida. The differences between the Ciliate Nuclear Code and the prot otype code are the following: prototype code CNC prototype code CNC UGG Trp Trp UGA Trp Ter UAG Ter Gln UAA Ter Gln AUG Met Met AUA Met Ile AGG X Arg AGA X Arg Hence from (19), (21), (22), (27) and (28), the reading opera tor for the Ciliate Nuclear Code is RCNC =4 3c1CH+4 3c2CV−4c1PHJH,3−4c2PVJV,3+ (−8c1PD+ (8c1+ 12c2)PS)JV,3 +/bracketleftig (−4c1+ 14c2)PAG−4c1PUA/bracketrightig/parenleftig 1 2−J(3) V,3/parenrightig +/bracketleftig 12c2PAU+ (6c1+ 6c2)PUG/bracketrightig/parenleftig 1 2−J(3) V,3/parenrightig/parenleftig 1 2−J(3) H,3/parenrightig (40) 3.2.10 The Blepharisma Nuclear Code (BNC) The Blepharisma Nuclear Code is used in the nuclei of the blep harisma (ciliata) (note that this code is very close to the CNC which is used for the ciliata ). The differences between the Blepharisma Nuclear Code and the prototype code are the foll owing: prototype code BNC prototype code BNC UGG Trp Trp UGA Trp Ter UAG Ter Gln UAA Ter Ter AUG Met Met AUA Met Ile AGG X Arg AGA X Arg Hence from (19), (21), (22), (27) and (29), the reading opera tor for the Blepharisma Nuclear Code is RBNC =4 3c1CH+4 3c2CV−4c1PHJH,3−4c2PVJV,3+ (−8c1PD+ (8c1+ 12c2)PS)JV,3 +(−4c1+ 14c2)PAG/parenleftig 1 2−J(3) V,3/parenrightig −4c1PUA/parenleftig 1 2−J(3) V,3/parenrightig/parenleftig 1 2+J(3) H,3/parenrightig +/bracketleftig 12c2PAU+ (6c1+ 6c2)PUG/bracketrightig/parenleftig 1 2−J(3) V,3/parenrightig/parenleftig 1 2−J(3) H,3/parenrightig (41) 3.2.11 The Euplotid Nuclear Code (ENC) The Euplotid Nuclear Code is used in the nuclei of the euploti dae (ciliata). The differences between the Euplotid Nuclear Code and the prototype code are the following: prototype code ENC prototype code ENC UGG Trp Trp UGA Trp Cys AUG Met Met AUA Met Ile AGG X Arg AGA X Arg Hence from (19), (21), (22) and (27), the reading operator fo r the Euplotid Nuclear Code is RENC =4 3c1CH+4 3c2CV−4c1PHJH,3−4c2PVJV,3+ (−8c1PD+ (8c1+ 12c2)PS)JV,3 +(−4c1+ 14c2)PAG/parenleftig 1 2−J(3) V,3/parenrightig + 12c2(PAU+PUG)/parenleftig 1 2−J(3) V,3/parenrightig/parenleftig 1 2−J(3) H,3/parenrightig (42) 123.2.12 The alternative Yeast Nuclear Code (alt. YNC) The alternative Yeast Nuclear Code is used in the nuclei of so me yeast (essentially many candidae). The differences between the alternative Yeast Nu clear Code and the prototype code are the following: prototype code alt. YNC prototype code alt. YNC CUG Leu Ser CUA Leu Leu UGG Trp Trp UGA Trp Ter AUG Met Met AUA Met Ile AGG X Arg AGA X Arg Hence from (19), (21), (22), (27) and (30), the reading opera tor for the alternative Yeast Nuclear Code is RaY NC =4 3c1CH+4 3c2CV−4c1PHJH,3−4c2PVJV,3+ (−8c1PD+ (8c1+ 12c2)PS)JV,3 +(−4c1+ 14c2)PAG/parenleftig 1 2−J(3) V,3/parenrightig + 2c1PCU/parenleftig 1 2−J(3) V,3/parenrightig/parenleftig 1 2+J(3) H,3/parenrightig +/bracketleftig (6c1+ 6c2)PUG+ 12c2PAU/bracketrightig/parenleftig 1 2−J(3) V,3/parenrightig/parenleftig 1 2−J(3) H,3/parenrightig (43) 3.3 Reading values for the amino-acids We have therefore constructed reading operators for the gen etic codes specified above, starting from a prototype code that emphasizes the quartet/doublet s tructure of the different codes. The different reading operators are such that they give the sa me value for a given amino-acid, whatever the code under consideration. Finally, we get the f ollowing eigenvalues of the reading operators for the amino-acids (after a rescaling, setting c≡c1/c2): a.a. value ofR a.a. value ofR a.a. value ofR Ala−c+ 3 Gly−c+ 5 Pro−c−1 Arg−c+ 1 His−3c+ 1 Ser 3c−1 Asn 9c+ 5 Ile 5c+ 9 Thr 3c+ 3 Asp 5c+ 5 Leu c−1 Trp 3c−5 Cys 3c+ 7 Lys 17c+ 5 Tyr c+ 1 Gln 5c+ 1 Met 5c−3 Val c+ 3 Glu 13c+ 5 Phe−7c−1 Ter 9c+ 1(44) Remark that the reading operators R(c) can be used for any real value of c, except those conferring the same eigenvalue to codons relative to two diff erent amino-acids. These forbidden values are the following: −7,−5,−4,−3,−5 2,−7 3,−2,−5 3,−3 2,−4 3,−1,−5 6,−4 5,−3 4,−5 7,−2 3, −3 5,−1 2,−3 7,−2 5,−3 8,−1 3,−3 10,−2 7,−1 4,−2 9,−1 5,−1 6,−1 7,−1 8,−1 9, 0,1 7,1 6,1 5,1 4,1 3,2 5,1 2,2 3, 1, 4 3,3 2, 2,5 2, 3, 4, 5. At this point, let us emphasize the specific properties of our model. To each nucleotide are assigned specific quantum numbers characterizing its purin e/pyrimidine origin and involving 13the complementary rule. Then ordered sequences of bases can be constructed and character- ized in this framework. Ordered sequences of three bases hav e been just above examined and the correspondence codon/amino-acid represented by the re ading operatorR. Finally let us remark that the coefficients ci, which above have been taken as constants, can more gener- ally be considered as functions of some external variables ( biological, physical and chemical environment, time, etc.). In this way it is possible to expla in the observed discrepancy in the correspondence codons/amino-acid in biological species u nder stress conditions (in vitro). In this scheme the evolution process of genetic code can also be discussed. However, we believe that a better understanding of the reasons of the evolution, i.e. which kind of optimization process takes place, has still to be acquired. 4 Physical properties of the dinucleotides The model we have at hand, with nucleotides characterized by quantum numbers, is well adapted to elaborate formulae expressing biophysical prop erties. A particularly interesting quantity is the free energy released by base pairing in doubl e stranded RNA. The data are not provided for a doublet of nucleotides, with one item in each s trand, but for a pair of nucleotides, for ex. CG, lying on one strand and coupled with another pair, i.e. GC on the second strand ; note also that the direction on a strand being perfectly defi ned, the release of energy for the doublet sequence CG on the first strand running from 5′to 3′related to the doublet GC on the complementary strand running from 3′to 5′, will be different to the one related to the doublet GC, itself associated to CG. It appears clear that such quant ities involve pairs of nucleotides, and that naturally ordered crystal bases obtained from tens or product of two representations are adapted for such a calculation. We will also consider two other quantities involving again p airs of nucleotides, namely the relative hydrophilicity Rfand hydrophobicity Rxof dinucleosides. Before presenting our results, let us mention that fits for th e same biophysical properties can be found in a recent preprint [5] where polynomials in 4 or 6 coordinates in the 64 codon space are constructed. In their approach, the authors assoc iate two coordinates ( d, m) to each nucleotide of any codon, as follows: A= (−1,0),C= (0,−1),G= (0,1),U= (1,0), labelling in this way each codon with 6 numbers. The above labelling of t he nucleotides is related to our labels Eq. (2) in the following way: d m CJV,3−JH,3−(JV,3+JH,3) UJV,3−JH,3−(JV,3+JH,3) GJV,3+JH,3JV,3−JH,3 AJV,3+JH,3JV,3−JH,3 Therefore the labels ( d, m) just correspond up to a sign for the pyrimidine (resp. purin e) to the antidiagonal and diagonal (resp. diagonal and antidiag onal)Uq→0(sl(2)). In the following we compare our results with those of [5]. 14Free energy In [1] we have fitted the experimental data with a four-parame ter operator. Here we fit the more recent data [6] with a two-parameter operator obtained from the one used in [1] by setting two parameters to zero: ∆G0 37=α0+α1(CH+CV)Jd 3H (45) Using a least-squares fit, one finds for the coefficients αi: α0=−2.14, α1=−0.295 (46) The standard deviation of the two-parameter fit (46) is found to be equal to 0 .149, which is to be compared to the standard deviation 0 .16 of the four-parameter fit of ref. [5]. The experimental and fitted values of the free energies ∆ G0 37of the dinucleotides are displayed in Table 1. /bracketleftig CA−2.1 −2.14/bracketrightig /bracketleftig CG−2.4 −2.73UG−2.1 −2.14UA−1.3 −1.55/bracketrightig  CU−2.1 −2.14 GU−2.2 −2.14 GA−2.4 −2.14  CC−3.3 −3.32UC−2.4 −2.14UU−0.9 −0.96 GC−3.4 −3.32AC−2.2 −2.14AU−1.1 −0.96 GG−3.3 −3.32AG−2.1 −2.14AA−0.9 −0.96  Table 1: Dinucleotides free energies ∆ G0 37. The upper (resp. lower) values are the experimental (resp. fi tted) values. Hydrophilicity We fit the values of the relative hydrophilicity Rfof the 16 dinucleoside monophosphates [7] with the following four-parameter operator: Rf=α0+α1CV+α2Jd 3V+α3/summationdisplay i=1,2(Ji 3H+Ji 3V)(Ji 3H+Ji 3V−1) (47) (the last term in α3is equal to 4 for AA, to 2 for CA, GA, UA and zero for the other din u- cleotides). Using a least-squares fit, one finds for the coefficients αi: α0= 0.135, α1= 0.036, α2= 0.147, α3=−0.016 (48) The standard deviation of the four-parameter fit (48) is foun d to be equal to 0 .027, which is to be compared to the standard deviation 0 .033 of the six-parameter fit of ref. [5]. The experimental and fitted values of the hydrophilicity Rfof the dinucleosides are displayed in Table 2. 15/bracketleftig CA0.083 0.103/bracketrightig /bracketleftig CG0.146 0.135UG0.160 0.135UA0.090 0.103/bracketrightig  CU0.359 0.354 GU0.224 0.207 GA0.035 0.028  CC0.349 0.354UC0.378 0.354UU0.389 0.354 GC0.193 0.207AC0.118 0.175AU0.112 0.175 GG0.065 0.060AG0.048 0.028AA0.023 -0.004  Table 2: Dinucleosides relative hydrophilicities Rf. The upper (resp. lower) values are the experimental (resp. fi tted) values. Hydrophobicity We fit the values of the relative hydrophobicity Rxof the 16 dinucleoside monophosphates as reported in [8] with the following four-parameter operator : Rx=α0+α1Jd 3V+α2Jd 3H+α3[(J1 3H+J1 3V)2+ (J2 3H+J2 3V)2] (49) (the last term in α3is equal to 2 for AA, AC, CA and CC, to 2 for AU, AG, UA, UC, GC, GA, CU and CG and zero for UU, UG, GU, GG). Using a least-squares fit, one finds for the coefficients αi: α0= 0.294, α1=−0.240, α2=−0.105, α3= 0.136 (50) Using a least-squares fit without the dinucleoside AA, one fin ds new coefficients αi, which lead to better values of Rxfor the remaining dinucleosides: α0= 0.309, α1=−0.203, α2=−0.068, α3= 0.099 (51) The standard deviation of the four-parameter fit (50) is equa l to 0.049, which is the same of /bracketleftig CA0.494 0.507/bracketrightig /bracketleftig CG0.326 0.340UG0.291 0.309UA0.441 0.476/bracketrightig  CU0.218 0.205 GU0.291 0.309 GA0.660 0.611  CC0.244 0.236UC0.218 0.205UU0.194 0.174 GC0.326 0.340AC0.494 0.507AU0.441 0.476 GG0.436 0.444AG0.660 0.611AA1 0.778  Table 3: Dinucleosides relative hydrophobicities Rx. The upper (resp. lower) values are the experimental (resp. fi tted) values. the four-parameter fit of ref. [5]. Using the fit (51), the stan dard deviation becomes 0 .074 16(including the value for AA) or 0 .024 (excluding the value for AA). For this last case, the standard deviation of ref. [5] is still equal to 0 .031. The experimental and fitted values (second fit) of the relative hydrophobicity Rxof the dinucleosides are displayed in Table 3. 5 Universal behaviour of ratios of codon usage frequency In the following the labels X, J, Z, K represent any of the 4 bases C, U, G, A . Let XJZ be a codon in a given multiplet, say mi, encoding an a.a., say Ai. We define the probability of usage of the codon XJZ as the ratio between the frequency of usage nZof the codon XJZ in the biosynthesis of Aiand the total number nof synthesized Ai, i.e. as the relative codon frequency, in the limit of very large n. It is natural to assume that the usage frequency of a codon in a multiplet is connected to its probability of usage P(XJZ→Ai). We define [2] the branching ratio BZKas BZK=P(XJZ→Ai) P(XJK→Ai)(52) where XJK is another codon belonging to the same multiplet mi. It is reasonable to argue that in the limit of very large number of codons, for a fixed bio logical species and amino-acid, the branching ratio depends essentially on the properties o f the codon. In our model this means that in this limit BZKis a function, depending on the type of the multiplet, on the quantum numbers of the codons XJZ andXJK, i.e. on the labels Jα, Jα,3, where α=HorV, and on an other set of quantum labels leaving out the degeneracy on Jα; in Table 4 different irreducible representations with the same values of Jαare distinguished by an upper label. We have put in evidence a correlation in the codon usage frequ ency for the quartets and the quartet subpart of the sextets, i.e. the codons in a sexte t differing only for the third codon, for the vertebrates in [2] and for biological species belong ing to the vertebrates, invertebrates, plants and fungi in [9], and we have shown that these correlat ions fit well in our model with the assumed dependence on BZK. Here we remark that for thirteen biological species belong ing to the vertebrate class, with a statistics of codons larger tha n 95,000 (see Table 5), the ratio of BAG BUC=BAU BGC=P(XJA→Ai) P(XJG→Ai)P(XJC→Ai) P(XJU→Ai)(53) for quartets and the quartet subpart of the sextets has a beha viour independent of the specific biological species. Moreover, for the same amino-acids for which we have remarked correlations, the values of the ratio BAG/BUCare almost the same (see Table 8). We show that these behaviour and correlations find a nice explanation in our mod el. In Tables 6 and 7, we report respectively the values of the branching ratios BAGandBUCas computed from the database [10] (release of February 2000) and in Table 8 the ratio of the se quantities. The average values/angbracketleftBAG/BUC/angbracketright, the standard deviations σand the ratios σ//angbracketleftBAG/BUC/angbracketrightare displayed in 17the following table: Pro AlaThr SerGly ValLeu Arg /angbracketleftBAG/BUC/angbracketright 2.502.843.302.672.210.330.261.32 σ 0.460.530.560.350.300.040.030.14 σ//angbracketleftBAG/BUC/angbracketright0.190.190.170.130.140.130.100.11 The above behaviour can be easily understood considering a d ependence on BZKnot only on the irreducible representations to which the codons XJZ andXJK appearing in the numerator and the denominator belong, but also on the specific states de noting these codons, and refining the factorized form of [2] as BZK=FZK(IR(XJZ);IR(XJK))GH(b.s.;JH,3(XJZ))GV(b.s.;JV,3(XJZ)) GH(b.s.;JH,3(XJK))GV(b.s.;JH,3(XJK))(54) where we have denoted by b.s.the biological species, by IR(XJZ) and Jα,3(XJZ) the irre- ducible representation to which the codon XJZ belong (see Table 4), and the value of the third component of the α-spin of the state XJZ. Note that we have still neglected the dependence on the type of the biosynthetized amino-acid. The ratio BAG/BUCusing Eq. (54), is no more depending on the biological species but only on the value of t he irreducible representations of the codons . Moreover, for Pro,Ala,Thr,Ser, (resp.ValandLeu), the irreducible representations appearing in the Ffunctions are the same as can be seen from Table 9, so we expect the same value for the ratio, which is indeed the case (see above Table ), the value of BAG/BUCfor the first four amino-acids (resp. for the last two amino-acids) l ying in the range 2 .90±15% (resp. 0.30±15%). These values should be compared with the value 1.32 for Argand 2.21 for Gly. Let us end this section by the following remark. From the abov e table, one might be tempted to consider the value of the ratio BAG/BUCforGlyof the same order of magnitude as the ones forPro,Ala,Thr,Ser. Then one distinguishes, following this ratio, three group s of codons quartets: the one associated to the five just mentioned amino -acids, another one relative to Val andLeu, and a last one with Arg. Now, let us look at the dinucleotide pairs constituting the first two nucleotides in a codon in the light of our results of s ect. 2: the pairs CC, GC, AC, UC and GG relative to Pro,Ala,Thr,Ser, andGlyrespectively belong to the representation (1,1) ofUq→0(sl(2)⊕sl(2)); the states GU and CU relative to ValandLeurespectively belong to the representation (0,1); finally CG relative to Argalso lies in a different representation (1,0). 6 Mutations in the genetic code In this section, we present a mathematical framework to desc ribe the single-base deletions in the genetic code. In [11] starting from the observation that the single-base deletions in DNA, which occur far more frequently that single base additions, take place in the opposite site to a purine R, (R = G, A ) i.e. a pyrimidine Y(Y = C, U/T ) is deleted, arguments have been presented to explain why the Stop codons have the structure t hey have, see Table 4. We refer to the paper for more details and for references to the biolog ical literature on the subject and 18we recall here just the main ideas and conclusions of [11]. Th e starting point is the observed fact that deletions occur more frequently in the following s equences: YR,TTR,YTG and TR. In ref. [11] all these sequences have been refined as YTRV , (V = C, A, G ). Starting from the structure of this dangerous sequence and using the c omplementarity property, an analysis shows that four codons – TAA,TAG,TTA,CTA – are both potential deletion site codons and reverse-complementary potential site codons. A s a mutation at the end of a protein chain just implies the addition of further peptides, the aut hors conclude that the assignment of codons TAA andTAG as Stop codons minimizes the possible deleterious effects of deletion. Indeed the codon usage frequency of the dangerous codon CTA, as it can be seen from fig. (5) of [2] and from fig. (2) of [9], is very low. An analysis of the co don usage frequency exhibits an analogous behaviour for the codon TTA. The mechanism by which the above specified sequences are pref erred in the deletion process is unclear. In the following we will present a mathematical s cheme in which these properties can be settled. Let us recall that the Wigner-Eckart theorem, ha s been extended to the quantum algebra Uq(sl(n)), and recently in [12] to the case of Uq→0(sl(2)). In [12] ( q→0)-tensor operators have been introduced, called crystal t ensor operators, which transform as J3(τj m)≡mτj mJ±(τj m)≡τj m±1 (55) Clearly, if|m|> jthenτj mhas to be considered vanishing. The (q→0)-Wigner-Eckart theorem can be written ( j1≥j) τj m|j1m1/angbracketright= (−1)2j2j/summationdisplay α=0/angbracketleftj1+j−α/bardblτj/bardblj1/angbracketright|j1+j−α, m 1+m/angbracketright (δm1,j1−α+δ−m,j−α−δm1,j1−αδ−m,j−α) (56) The (q→0)-Wigner-Eckart theorem has the peculiar feature that the selection rules do not depend only on the rank of the tensor operator and on the in itial state, but in a crucial way from the specific component of the tensor in consideratio n. The tensor product of two irreducible representations in the crystal basis is not com mutative (see sect. 2), therefore one has to specify which is the first representation. In the follo wing, as in [12], the crystal tensor operator has to be considered as the first one. Let us also remark the following peculiar property of crysta l basis which will be used in the following. We specify it only for the case we are interested i n, but it is a completely general property. An ordered sequence, or chain, of nnucleotides is a state belonging to an irreducible repre- sentation of Uq→0((sl(2)⊕sl(2)) appearing in the n-fold product of the fundamental irreducible representation (1 /2,1/2). Moreover the same property holds for any subsequence of m(m < n ) nucleotides. We can mimick the deletion of a Nnucleotide in a generic position of a coding sequence by a local annihilation operator of the Nnucleotide. In order to take into account the observed fact that the deletion of the nucleotide depend s on the nature of the neighboring 19nucleotides, we require the annihilation operator to behav e as a defined crystal tensor opera- tor under Uq→0(sl(2))VorUq→0(sl(2))Hor both. In our mathematical description we have to specify the action of the annihilation operator on a chain of nucleotides. If we assume that the annihilation of the Nnucleotide behaves e.g. as a spinor crystal operator for the Uq→0(sl(2))V, we have to require that the deletion of the Nnucleotide from the initial chain of Knucleotides, described by the state |Ji, Mi; Ωi/angbracketright, leading to the final chain of K−1 nucleotides, described by the state|Jf, Mf; Ωf/angbracketright, is compatible with the ( q→0)-Wigner-Eckart theorem prescription for the action of the definite crystal spinor operator betwee n the initial state |Ji, Mi; Ωi/angbracketrightand the final state|Jf, Mf; Ωf/angbracketright, where we have denoted by Ω the set of all the labels necessary to identify completely the state. As we shall see, this is far from being trivial and will put constraints on the type of nucleotides surrounding the nucl eotide N. We have to specify which chain has to be considered in order to study the action of the c rystal tensor operator. It seems reasonable to take into account chains formed by K= 2 and 3 nucleotides starting from Nin the sense of the reading of the codon sequence. So we are defini ng on the chain the action of a “matrioska” crystal tensor operator. We assume: Assumption : The biological mechanism responsible for the deletion of a pyrimidine C(resp. U) in a sequence can be schematized by a local crystal tensor op erator τ1/2 −1/2forUq→0(sl(2)V) andτ1/2 −1/2(resp. τ1/2 1/2) forUq→0(sl(2)H), which transforms the state YX(resp. YXZ) into the stateX(resp. XZ),X,Zbeing any nucleotide. By “local crystal tensor operator” we mean an operator which , in the sequence of RNA, acts on the K-chain ( K= 2,3) starting with Y, deleting the pyrimidine, according to the selection rules imposed by the assumed type of the crystal tensor. Let us point out that, differently to ref. [11], where the DNA s equence was analyzed, we consider the transcripted RNA sequence and the deletion in t he trascription of a Y. There are 8 possible cases (we denote the initial and final sta tes with the notation of sect. 2 and by A (resp. F) the allowed (resp. forbidden) transition ). We analyze the deletion of a C (on the left) and of an U(on the right). Action of τ1/2 −1/2,H⊕τ1/2 −1/2,V (1,1)→(1 2,1 2) CC C F–F (0,1)→(1 2,1 2) CU U A–F (1,0)→(1 2,1 2) CG G F–A (0,0)→(1 2,1 2) CA A A–AAction of τ1/2 1/2,H⊕τ1/2 −1/2,V (1,1)→(1 2,1 2) UC C A–F UU U A–F (1,0)→(1 2,1 2) UG G A–A UA A A–A So for the transition for the state of dinucleotide to one nuc leotide state, from the assumed nature of the crystal tensor operator, it follows that a pyri midine can be deleted if followed by 20a purine. Now let us consider what happens if we consider the t ransition from a trinucleotide to a dinucletide state. Using the previous result we conside r only the state in which a purine is in second position so we have to consider 16 cases: Action of τ1/2 −1/2,H⊕τ1/2 −1/2,V (1 2,1 2)4→(1,1) CAC AC A–A CAU AU A–A CAA AA A–A CAG AG A–A (3 2,1 2)2→(1,1) CGC GC F–A CGG GG F–A (1 2,1 2)2→(0,1) CGU GU F–A CGA GA F–AAction of τ1/2 1/2,H⊕τ1/2 −1/2,V (3 2,1 2)2→(1,1) UAC AC A–A UAU AU A–A UAA AA A–A UAG AG A–A (3 2,1 2)2→(1,1) UGC GC A–A UGG GG A–A (1 2,1 2)2→(0,1) UGU GU A–A UGA GA A–A So, from the assumed nature of the crystal tensor operator, t he transition from a trinucleotide to a dinucleotide state is horizontally forbidden for the de letion of a Cif the second nucleotide is aG. Let us note that we have made the simplified assuption that the transitions depend only on the values of Jα, Jα,3of the initial and final state. Moreover, both to take into account the data of [11] and to che ck that the results are not very sensible to the choice of the initial state, we consider the deletion of a purine in second position in a four-nucleotide state and impose that the proc ess may take place only if the initial and final state can be connected by a spinor crystal operator τ1/2 −1/2,H⊕τ1/2 −1/2,Vfor the deletion ofCorτ1/2 1/2,H⊕τ1/2 −1/2,Vfor the deletion of U. As the two pyrimidines differ by their value of JH,3, the constraints imposed by the tensor operator τ1/2 ±1/2,Hare weaker than those imposed by the tensor operator τ1/2 −1/2,V. In Appendix (in sect. 2) we have reported all the irreducible representations arising by the 4-fold (3-fold) tensor product of the fundamental represen tation. A detailed analysis shows that only the following deletions may happen (we report all t he transitions that are allowed at 21least once): Action of τ1/2 −1/2,H⊕τ1/2 −1/2,V (2,1)3→(3 2,3 2) GCGC GGC F–A ACGC AGC F–A GCGG GGG F–A ACGG AGG F–A (2,0)2→(3 2,1 2)2 CCGG CGG F–A UCGG UGG F–A (1,1)7→(1 2,3 2)1 GCGU GGU F–A ACGU AGU F–A GCGA GGA F–A ACGA AGA F–A (1,0)4→(1 2,1 2)2 CCGA CGA F–A UCGA UGA F–A (1,1)9→(1 2,3 2)2 GCAC GAC F–A GCAG GAG F–A (1,0)6→(1 2,1 2)4 CCAG CAG F–A (1,1)9→(3 2,3 2) ACAC AAC A–A ACAU AAU A–A ACAG AAG A–A ACAA AAA A–A (1,0)6→(3 2,1 2)2 UCAG UAG A–A UCAA UAA A–AAction of τ1/2 −1/2,H⊕τ1/2 −1/2,V (1,2)3→(3 2,3 2) UCUC UUC A–F UCUU UUU A–F ACUC AUC A–F ACUU AUU A–F (0,2)2→(1 2,3 2)2 CCUU CUU A–F GCUU GUU A–F (1,1)8→(3 2,1 2)1 UCUG UUG A–F UCUA UUA A–F ACUG AUG A–F ACUA AUA A–F (0,1)5→(1 2,1 2)3 CCUA CUA A–F GCUA GUA A–F (1,1)9→(3 2,1 2)2 UCAC UAC A–F UCAU UAU A–F (0,1)6→(1 2,1 2)4 CCAU CAU A–F (0,0)4→(1 2,1 2)4 CCAA CAA A–A (0,1)6→(1 2,3 2)2 GCAU GAU A–A GCAA GAA A–A So we remark: •The deletion of C, allowed or horizontally forbidden, may happen only if it is followed by a purine. In the allowed cases, it must be followed by the nu cleotide A, in agreement with the observed data. •A nucleotide Abefore the deleted nucleotide Cappears only in the transition (1 ,1)9→ (3 2,3 2). This feature is present in the observed data with a very low occurrence, which in our language would mean that the matrix element of τbetween these two irreducible representations is small. 22Now we consider the case of deletion of U. A detailed analysis shows that only the following deletions may happen: Action of τ1/2 1/2,H⊕τ1/2 −1/2,V (1,2)1→(1 2,3 2)2 CUUC CUC A–F CUUU CUU A–F GUUC GUC A–F GUUU GUU A–F (1,2)2→(3 2,3 2) CUCC CCC A–F GUCC GCC A–F (1,1)3→(1 2,1 2)4 CUAC CAC A–F CUAU CAU A–F (1,1)6→(1 2,1 2)3 UUCA UCA A–F AUCA ACA A–F (2,1)2→(3 2,1 2)1 UUCG UCG A–F UUUG UUG A–F UUUA UUA A–F AUCG ACG A–F AUUG AUG A–F AUUA AUA A–F (2,1)3→(3 2,1 2)2 UUGC UGC A–F UUAC UAC A–F UUAU UAU A–F (1,1)7→(1 2,1 2)3 UUGU UGU A–F (0,1)4→(1 2,1 2)2 CUGU CGU A–FAction of τ1/2 1/2,H⊕τ1/2 −1/2,V (1,1)2→(1 2,1 2)3 CUUG CUG A–F GUUG GUG A–F CUUA CUA A–F GUUA GUA A–F (1,1)2→(3 2,1 2)1 CUCG CCG A–F GUCG GCG A–F (1,2)2→(1 2,3 2)1 UUCU UCU A–F AUCU ACU A–F (0,2)1→(1 2,3 2)1 CUCU CCU A–F GUCU GCU A–F (2,2)→(3 2,3 2) UUCC UCC A–F UUUC UUC A–F UUUU UUU A–F AUCC ACC A–F AUUC AUC A–F AUUU AUU A–F (0,1)3→(1 2,1 2)1 CUCA CCA A–F GUCA GCA A–F (1,1)3→(3 2,1 2)2 CUGC CGC A–F 23Action of τ1/2 1/2,H⊕τ1/2 −1/2,V (1,1)7→(1 2,3 2)1 AUGU AGU A–A AUGA AGA A–A (0,1)4→(1 2,3 2)1 GUGU GGU A–A GUGA GGA A–A (1,1)3→(1 2,3 2)2 GUAC GAC A–A GUAU GAU A–A GUAG GAG A–A GUAA GAA A–A (2,0)2→(3 2,1 2)2 UUGG UGG A–A UUAG UAG A–A UUAA UAA A–A (1,0)2→(3 2,1 2)2 CUGG CGG A–AAction of τ1/2 1/2,H⊕τ1/2 −1/2,V (1,1)3→(3 2,3 2) GUGC GGC A–A GUGG GGG A–A (1,0)2→(1 2,1 2)4 CUAG CAG A–A CUAA CAA A–A (2,1)3→(3 2,3 2) AUGC AGC A–A AUAC AAC A–A AUAU AAU A–A AUGG AGG A–A AUAG AAG A–A AUAA AAA A–A (0,0)2→(1 2,1 2)2 CUGA CGA A–A So we remark: •The deletion of Umay happen only if it is followed by Aor by G. In the observed data only Ais considered; however in [11] the reported deletion of Uare about 1/4 with respect to the reported deletion of C. So our modelisation just foresees a different environment for the deletion of UandC. •The last nucleotide in the four-nucleotide sequence in whic h the deletion occurs may be any nucleotide, but the case in which it is a purine seems more frequent than the case in which it is a pyrimidine. •There are no transition which are only horizontally forbidd en. In conclusion, both from considering the transitions on the K-chains ( K= 2,3) to the (K−1)-chains or the transition from the four-nucleotide state s to the three-nucleotide states under the action of the crystal tensor operators, we deduce t hat the deletion of a pyrimidine may happen if it is followed by a purine. In particular, for th e deletion of C the preferred purine is the adenine A, whilst for the deletion of U also the guanine G may appear. This makes a difference between the two cases and it would be extremely int eresting to see if more accurate data may confirm this asymmetry. Moreover the next following nucleotide may be of any type but there is indication that a purine is preferred. So our mat hematical scheme explains the main features of the observed data [11]. A more quantitative analysis should require higher statistics in the experimental data. 247 Recent theoretical approaches: a comparison The use of continuous symmetries in the genetic code has been considered by different teams these recent years2. It appears of some importance to summarize each of these app roaches, and to make clear how the model we propose differ from them. In 1993, an underlying symmetry based on a continuous group h as been proposed [13]. More precisely, considering the eukaryotic code, the authors tr ied to answer the following question: is it possible to determine a Lie algebra Gcarrying a 64-dimensional irreducible representation Rand admitting a subalgebra Hsuch that the decomposition of Rinto irreducible multiplets underHgives exactly the 21 different multiplets, the different codo ns in each of the first 20 multiplets being associated to the same amino-acid, the l ast multiplet containing the stop codons ? They proposed as starting symmetry the symplectic a lgebra sp(6), which indeed admits an irreducible representation of dimension 64, equa l to the number of different codons, with the successive breakings: sp(6)⊃sp(4)⊕su(2)⊃su(2)⊕su(2)⊕su(2)⊃su(2)⊕U(1)⊕su(2)⊃su(2)⊕U(1)⊕U(1) (57) Such a chain of symmetry breaking could be considered as refle cting the evolution of the genetic code, the six amino-acids relative to the codons in the irred ucible representations obtained after the first breaking (in which 64 = 16 + 4 + 20 + 10 + 12 + 2) appearing as primordial amino- acids in their approach. However, the authors were obliged, in order to reproduce the actual multiplet pattern, to assume in the final breaking, a partial breaking or a “freezing” in the sense that the breaking of the last su(2) into U(1) does not occur for all the multiplets. As an example, such a freezing has to be imposed to the sextets co rresponding to Leu and Ser, which otherwise would decompose into three doublets. In the same way, freezing will forbid the doublets related to Lys and Cys to split into singlets. In a second further paper, dated 1997 [14], a refinement of thi s approach has been considered, with the use of Lie groups instead of Lie algebras: then, glob al properties, for example non connexity of O(2) = U(1)×Z2, can be exploited. In this context, the authors proposed ano ther chain of breaking starting with the exceptional group G2, which also allows a 64 dimensional irreducible representation. But here again, the freezing p athology cannot be avoided. One can also mention the work of [15] where the unifying algeb ra before breaking is so(14). Meantime (1997), interpreting the double origin of the nucl eotides, each arising either from purine or from pyrimidine, as a Z2-grading a supersymmetric model was proposed [16], involvi ng superalgebras for such a program. The Z2-grading specific of a simple superalgebra is there used to separate purine and pyrimidine: indeed, by putting t he four nucleotids in the the 4 dimensional representation of su(2/1) one can confer to the A and G purines (R) an even grading, and to the C and U pyrimidines (Y) an odd grading; not e that the R states are then in thesu(2) doublet and the Y ones su(2) singlets. The notion of polarity spin is also introduced , 2See section “Symmetry techniques in Biological Systems” in Proc. XXII Int. Coll. on Group Theoretical Methods in Physics, pp. 142-165. 25allowing to distinguish the C and G nucleotides with two loca lly polarized sites, from the A and U ones with three polarized sites: the C and G (resp. A and U ) will be assigned in a doublet (resp. in singlets) of another su(2). Then the authors consider the sum of algebras: su(2)⊕su(2)⊕su(2|1) with the first (second) su(2) acting as polarity spin on the first (second) nucleotid of a codon, and the su(2|1) acting on the third nucleotid only. Moreover the two su(2) would act in an alternating way on the first and second posi tion, that is as 1 /2,−1/2 and −1/2, 1/2. This sum of algebras can be embedded in the superalgebra su(6|1), which admits a 64 dimensional irreducible representation, and could be al so used for a superalgebraic approach to the genetic code evolution, with the chain of symmetry bre aking: su(6|1)⊃su(2)⊕su(3|1)⊃su(2)⊕su(2)⊕su(2|1)⊃U(1)⊕U(1)⊕su(2|1) ⊃U(1)⊕U(1)⊕gl(1|1) (58) Again the problem of freezing, that is the last breaking appl ies to some but not all the multiplets, is present with this choice of (super)algebras. It seems necessary to remark that in this proposal which impl ies (super)algebras acting in the same time on nucleotides and on codons – one must say in a ra ther complicated way – the nucleotides cannot appear as building blocks from which one algebraically constructs the codons, by performing tensorial products of representatio ns, as is the case of our model. In fact, the problem of ordering the nucleotides inside a codon forbi ds this natural way of proceeding as long as only usual (super)algebras are involved. Note that i t is the limit of quantum algebras that we use in our approach: then, we have at hand the so-calle d crystal bases, which exactly solve the ordering problem. In a last month preprint, two authors of the same team [5] prop osed to fit biophysical properties of nucleic acids by constructing polynomials in 6 coordinates in the 64 dimensional codon space. As already mentioned in sect. 4, the two coordin ates they associate to each nucleotide is direcetly related to the nucleotide eigenval ues of our model. The authors present their computations as independent of a particular choice of algebra or superalgebra as long as the underlying algebra is of rank 6 – which is in particular the dimension of the Cartan subalgebra of su(6|1) – and admits a 64 dimensional irreducible representation . We note that our model does allow to calculate the biophysical quantitie s considered in ref. [5] without the constraint on representations, but more importantly, with only a two rank algebra. A detailed and systematic study of superalgebras and supera lgebra breaking chains has been performed by the authors of [17]: it is the orthosymplectic osp(5|2) superalgebra which emerges from their algebraic analysis. Finally, it is amazing to remark that, just a few years after t he the concept of genetic code was formulated, an attempt to give a mathematical descripti on of its properties was started by the russian physicist Yu. B. Rumer [18]. Indeed he remarke d that the 16 roots, i.e. the combinations of the first two codons, divide in a strong octet which form quartets ou sub-part of sextets and a weak octet which form doublets, triplets and singlets, attempting to g ive a 26systematic description of the genetic code. A few years afte r, with B.G. Konopel’chenko [19] they formulated the strong assumption that with respect to a ny property of the codons the 16 roots can be gathered into two octets with opposite “charge” , whose positive (negative) value respectively characterizes the strong and weak roots. This description comes out naturally in our model, such a charge Qbeing defined in Eq. (5) of sect. 2. 8 Conclusion Our model is based on the algebra Uq→0(sl(2)⊕sl(2)) that we have chosen for two main characteristics. First it encodes the stereochemical prop erty of a base, and also reflects the complementarity rule, by conferring quantum numbers to eac h nucleotide. Secondly, it admits representation spaces or crystal bases in which an ordered s equence of nucleotides or codon can be suitably characterized. Let us emphasize that Uq→0(sl(2)⊕sl(2)) is really neither a Lie algebra nor an enveloping deformed algebra. We still use in a loose sense the word algebra, just to emphasize the fact that we use largely the mathematical to ols of representation space, tensor operators etc. which are typical of the algebraic structure s. Let us add that it is a remarkable property of a quantum algebra in the limit q→0 to admit representations, obtained from the tensorial product of basic ones, in which each state appears as a unique sequence of ordered basic elements. In this framework, the correspondence codon/amino-acid is realized by the operator Rc, constructed out of the symmetry algebra, and acting on codon s: the eigenvalues provided by Rcon two codons will be equal or different depending on whether t he two codons are associated to the same or to two different amino-acids. It is remarkable t hat this correspondence can be obtained for all the genetic codes and that the reading opera tors have a bulk common to the various genetic codes (the prototype reading operator) and differ only for a few additive terms, analogous to perturbative terms present in most Hamiltonia ns describing complex physical systems. Moreover they depend on parameters, presently ass umed as constants, which in principle can be considered as functions of suitable variab les. These feature may be of some interest in the study of the evolution of the genetic code, pr oblem which has not yet been tackled in our model. Then, restricting to the case of states made of two nucleotid es, the experimental values of the free energy, released by base pairing in the formation of double stranded nucleic acids, of the hydrophibicity and of the hydrophilicity have been fit ted with expressions depending respectively on 2, 4 and 4 parameters and constructed out of t he generators ofUq→0(sl(2)⊕ sl(2)). The model does not necessarily assign the codons in a multipl et (in particular the quartets, sextets and triplet) to the same irreducible representatio n. Let us remark that the assignments of the codons to the different irreducible representations i s a straightforward consequence of the tensor product, once assigned the nucleotides to the fun damental irreducible representation. This feature is relevant, since it can explain the correlati on between the branching ratios of the 27codon usage of different codons coding the same amino-acid as discussed in [2] and [9]. Here we have shown that the universal pattern (inside the class of ve rtebrates) of BAG/BUCcan simply be reproduced in our model. Moreover our mathematical description of the genetic code a llows a modelisation of some biological process. A first step in this direction has been pr esented in sect. 6, where we have shown that the observed data related to the a pyrimidine dele tion can be simulated by introduc- ing the concept of q→0 – or crystal – tensor operator. Finally let us mention some d irections for future development of our model. Going further in the ana lysis of the branching ratios, we want to refine our analysis and make a more detailed study taki ng into account the dependence on the family of biological species. Indeed preliminary ana lysis on plants, invertebrates and bacteriae shows that, even if the pattern of the correlation is still approximatively present, large deviations appear which presumably exhibit evidence that the dependence on subclass or family of biological species cannot any more be neglected, d ifferently to the case of vertebrates. A further investigation of the possibility of mathematical ly modelising or simulating biological processes, in particular mutations, by crystal tensor oper ators, is in progress. Other questions are still to be investigated: in particular how could the gen etic code evolution be reproduced in our model ? References [1] L. Frappat, A. Sciarrino, P. Sorba, A crystal base for the genetic code, Phys. Lett. A 250 (1998) 214 and physics/9801027 . [2] L. Frappat, A. Sciarrino, P. Sorba, Symmetry and codon usage correlations in the genetic code, Phys. Lett. A 259(1999) 339 and physics/9812041 . [3] M. Singer, P. Berg, Genes and Genomes, Editions Vigot, Paris (1992). [4] M. Kashiwara, Crystallizing the q-analogue of universal enveloping algebras, Commun. Math. Phys. 133(1990) 249. [5] J.D. Bashford, P.D. Jarvis, The genetic code as a periodic table: algebraic aspects, physics/0001066 . [6] D.H. Mathews, J. Sabina, M. Zucker, D.H. Turner, J. Mol. B iol.288(1999) 911. [7] A.L. Weber, J.C. Lacey, Genetic code correlations: amino-acids and their anticodo n nu- cleotides, J. Mol. Evol. 11(1978) 199. [8] J.R. Jemcyck, The genetic code as a periodic table, J. Mol. Evol. 11(1978) 211. [9] M.L. Chiusano, L. Frappat, A. Sciarrino, P. Sorba, Codon usage correlations and Crystal Basis model of the genetic code, preprint LAPTH-736/99, DSF-Th-17/99. 28[10] Y. Nakamura, T. Gojobori, and T. Ikemura, Nucleic Acids Research 26(1998) 334. [11] J.L. Jestin, A. Kempf, Chain termination codons and polymerase-induced frameshi ft mu- tations, FEBS Letters 419(1997) 153. [12] V. Marotta, A. Sciarrino, Tensor operator and Wigner-Eckart theorem for Uq→0(sl(2)), preprint DSF-T-43/98, math.QA/9811143 , submitted to J. Math. Phys. [13] J.E. Hornos, Y. Hornos, Algebraic model for the evolution of the genetic code, Phys. Rev. Lett.71(1993) 4401. [14] M. Forger, Y. Hornos, J.E. Hornos, Global aspects in the algebraic approach in the genetic code, Phys. Rev. E 56(1997) 7078. [15] R.D. Kent, M. Schlesinger, B.G. Wybourne, On algebraic approach to the genetic code, Proc. XXII Int. Coll. on Group Theoretical Methods in Physic s, Eds. B.P. Corney, R. Delbourgo and P.D. Jarvis, International Press (Boston, 19 99), pp. 152. [16] J.D. Bashford, I. Tsohantjis, P.D. Jarvis, Codon and nucleotide assignments in a super- symmetric model of the genetic code, Phys. Lett. A 233(1997) 481 and A supersymmetric model for the evolution of the genetic code, Proc. Nat. Acad. Sci. USA 95(1998) 987. [17] M. Forger and S. Sachse, Lie Superalgebras and the Multiplet Structure of the Ge- netic Code I: Codons Representations, II: Branching Scheme s,math-ph/9808001 and math-ph/9905017 . [18] Yu.B. Rumer, Systematization of the Codons of Genetic Code, Translated from Doklady Akademi Nauk SSSR 167, N.6, (1966) 1393-1394 and Systematization of the Codons of Genetic Code, Translated from Doklady Akademi Nauk SSSR 187, N.4, (1969) 937-938. [19] B.G. Konopel’chenko, Yu.B. Rumer, Classification of Codons in the Genetic Code, Trans- lated from Doklady Akademi Nauk SSSR 223, N.2, (1975) 471-474. 29Figure 1: Classification of the codons in the different crysta l bases. t t t tQQQQQQQQQQQQQQQQQQQQQQQQ t t t tQQQQQQQQQQQQQQQQQQQQQQQQ t t t tQQQQQQQQQQQQQQQQQQQQQQQQ t t t tCCC UCC UUC UUU GCC ACC AUC AUU GGC AGC AAC AAU GGG AGG AAG AAA t t t tQQQQQQQQQQQQQQQQQQt t t tQQQQQQQQQQQQQQQQQQCCU UCU GCU ACU GGU AGU GGA AGAQQQQQQQQQQQQ QQQQQQQQQQQQ t t t tQQQQQQQQQQQQQQQQQQQQQQQQ t t t tCCG UCG UUG UUA GCG ACG AUG AUA t tQQQQQQQQQQQQ t tCCA UCA GCA ACAGlySer2 Arg2Phe Ile Asn Lys Pro AlaSer4 Thr Leu2(3/2,3/2) (1/2,3/2)1 (3/2,1/2)1 (1/2,1/2)1 30Figure 1 (continued) t t t tQQQQQQQQQQQQQQQQQQt t t tQQQQQQQQQQQQQQQQQQCUC CUU GUC GUU GAC GAU GAG GAA t tQQQQQQQQQQQQ t tCUG CUA GUG GUALeu4 ValAsp Glu t t t tQQQQQQQQQQQQQQQQQQQQQQQQ t t t tCGC UGC UAC UAU CGG UGG UAG UAAQQQQQQ QQQQQQQQQQQQ QQQQQQt tQQQQQQ t tCGU UGU CGA UGAArg4 CysTyr Ter Trp t tQQQQQQQQQQQQ t tCAC CAU CAG CAAHis Gln(1/2,3/2)2 (1/2,1/2)3 (3/2,1/2)2 (1/2,1/2)2 (1/2,1/2)4 31Table 4: The eukariotic code. The upper label denotes differe nt irreducible representations. codon a.a. JH JVcodon a.a. JH JV CCC Pro 3/2 3/2 UCC Ser 3/2 3/2 CCU Pro (1/2 3/2)1UCU Ser (1/2 3/2)1 CCG Pro (3/2 1/2)1UCG Ser (3/2 1/2)1 CCA Pro (1/2 1/2)1UCA Ser (1/2 1/2)1 CUC Leu (1/2 3/2)2UUC Phe 3/2 3/2 CUU Leu (1/2 3/2)2UUU Phe 3/2 3/2 CUG Leu (1/2 1/2)3UUG Leu (3/2 1/2)1 CUA Leu (1/2 1/2)3UUA Leu (3/2 1/2)1 CGC Arg (3/2 1/2)2UGC Cys (3/2 1/2)2 CGU Arg (1/2 1/2)2UGU Cys (1/2 1/2)2 CGG Arg (3/2 1/2)2UGG Trp (3/2 1/2)2 CGA Arg (1/2 1/2)2UGA Ter (1/2 1/2)2 CAC His (1/2 1/2)4UAC Tyr (3/2 1/2)2 CAU His (1/2 1/2)4UAU Tyr (3/2 1/2)2 CAG Gln (1/2 1/2)4UAG Ter (3/2 1/2)2 CAA Gln (1/2 1/2)4UAA Ter (3/2 1/2)2 GCC Ala 3/2 3/2 ACC Thr 3/2 3/2 GCU Ala (1/2 3/2)1ACU Thr (1/2 3/2)1 GCG Ala (3/2 1/2)1ACG Thr (3/2 1/2)1 GCA Ala (1/2 1/2)1ACA Thr (1/2 1/2)1 GUC Val (1/2 3/2)2AUC Ile 3/2 3/2 GUU Val (1/2 3/2)2AUU Ile 3/2 3/2 GUG Val (1/2 1/2)3AUG Met (3/2 1/2)1 GUA Val (1/2 1/2)3AUA Ile (3/2 1/2)1 GGC Gly 3/2 3/2 AGC Ser 3/2 3/2 GGU Gly (1/2 3/2)1AGU Ser (1/2 3/2)1 GGG Gly 3/2 3/2 AGG Arg 3/2 3/2 GGA Gly (1/2 3/2)1AGA Arg (1/2 3/2)1 GAC Asp (1/2 3/2)2AAC Asn 3/2 3/2 GAU Asp (1/2 3/2)2AAU Asn 3/2 3/2 GAG Glu (1/2 3/2)2AAG Lys 3/2 3/2 GAA Glu (1/2 3/2)2AAA Lys 3/2 3/2 32Table 5: Biological species sample used in analysis of sect. 5 Species Number Number of sequences of codons 1 Homo sapiens 17625 8707603 2 Rattus norvegicus 4907 2469130 3 Gallus gallus 1592 763008 4 Xenopus laevis 1433 646214 5 Bos taurus 1382 614602 6 Oryctolagus cuniculus 713 358447 7 Sus scrofa 658 275045 8 Danio rerio 500 213258 9 Rattus rattus 342 153049 10 Canis familiaris 317 142944 11 Rattus sp. 299 112039 12 Ovis aries 327 101591 13 Fugu rubripes 157 95979 Table 6: BAGratios for the quartets Pro Ala Thr Ser Val Leu Arg Gly 12.34 2.03 2.29 2.51 0.23 0.17 0.53 0.99 22.40 2.17 2.33 2.35 0.22 0.17 0.61 1.03 31.77 1.90 1.96 1.93 0.25 0.14 0.52 1.02 44.10 4.23 4.08 3.45 0.48 0.32 1.00 1.67 52.02 1.80 1.94 2.32 0.21 0.14 0.56 1.01 61.45 1.45 1.30 1.45 0.15 0.10 0.44 0.88 71.60 1.60 1.52 1.69 0.16 0.12 0.46 0.89 81.39 1.47 1.71 1.68 0.22 0.18 0.89 1.94 92.28 1.97 2.19 2.26 0.21 0.17 0.66 1.03 102.09 1.72 1.81 1.90 0.21 0.15 0.49 1.01 112.22 2.15 2.27 2.24 0.21 0.16 0.62 1.07 122.15 1.60 1.76 1.99 0.15 0.13 0.60 1.08 131.60 1.40 1.28 1.42 0.17 0.12 0.73 1.23 33Table 7: BUCratios for the quartets Pro Ala Thr Ser Val Leu Arg Gly 10.85 0.64 0.64 0.82 0.72 0.64 0.43 0.47 20.91 0.69 0.61 0.78 0.59 0.57 0.48 0.49 30.75 0.80 0.69 0.77 0.84 0.64 0.45 0.51 41.27 1.15 1.05 1.17 1.26 1.24 0.98 0.87 50.78 0.61 0.57 0.79 0.65 0.57 0.41 0.47 60.62 0.47 0.46 0.54 0.51 0.43 0.29 0.34 70.68 0.54 0.49 0.65 0.50 0.47 0.33 0.38 81.02 0.88 0.69 0.83 0.82 0.64 0.60 0.68 90.88 0.71 0.59 0.76 0.59 0.57 0.50 0.51 100.76 0.61 0.57 0.76 0.56 0.55 0.37 0.53 110.94 0.69 0.58 0.83 0.55 0.55 0.47 0.49 120.70 0.53 0.45 0.73 0.50 0.46 0.41 0.43 130.77 0.68 0.55 0.71 0.60 0.49 0.57 0.64 Table 8: BAG/BUCratios for the quartets Pro Ala Thr Ser Val Leu Arg Gly 12.75 3.15 3.57 3.05 0.32 0.26 1.25 2.11 22.63 3.15 3.81 3.02 0.38 0.30 1.28 2.10 32.38 2.38 2.83 2.50 0.30 0.21 1.14 2.00 43.22 3.69 3.89 2.96 0.38 0.25 1.02 1.92 52.60 2.96 3.40 2.92 0.32 0.25 1.36 2.17 62.33 3.08 2.80 2.67 0.29 0.24 1.55 2.60 72.34 2.97 3.11 2.60 0.32 0.25 1.38 2.36 81.36 1.68 2.48 2.03 0.27 0.27 1.48 2.87 92.58 2.78 3.68 2.98 0.36 0.31 1.32 2.00 102.74 2.82 3.17 2.51 0.38 0.28 1.32 1.91 112.36 3.14 3.93 2.71 0.38 0.29 1.34 2.18 123.08 3.03 3.92 2.72 0.30 0.28 1.45 2.52 132.09 2.06 2.31 2.01 0.27 0.24 1.28 1.92 Table 9: Ffunctions appearing in the BAG/BUCratios Pro Ala Thr Ser FAG/parenleftig (1 2,1 2)1; (3 2,1 2)1/parenrightig FUC/parenleftig (1 2,3 2)1; (3 2,3 2)/parenrightigFAG/parenleftig (1 2,1 2)1; (3 2,1 2)1/parenrightig FUC/parenleftig (1 2,3 2)1; (3 2,3 2)/parenrightigFAG/parenleftig (1 2,1 2)1; (3 2,1 2)1/parenrightig FUC/parenleftig (1 2,3 2)1; (3 2,3 2)/parenrightigFAG/parenleftig (1 2,1 2)1; (3 2,1 2)1/parenrightig FUC/parenleftig (1 2,3 2)1; (3 2,3 2)/parenrightig Val Leu Arg Gly FAG/parenleftig (1 2,1 2)3; (1 2,1 2)3/parenrightig FUC/parenleftig (1 2,3 2)2; (1 2,3 2)2/parenrightigFAG/parenleftig (1 2,1 2)3; (1 2,1 2)3/parenrightig FUC/parenleftig (1 2,3 2)2; (1 2,3 2)2/parenrightigFAG((1 2,1 2)2; (3 2,1 2)2/parenrightig FUC/parenleftig (1 2,1 2)2; (3 2,1 2)2/parenrightigFAG/parenleftig (1 2,3 2)1; (3 2,3 2)/parenrightig FUC/parenleftig (1 2,3 2)2; (3 2,3 2)/parenrightig 34Table 10: Amino-acid content of the ⊗3(1 2,1 2) representations (3 2,3 2)≡ P−Pro S−Ser F−Phe F−Phe A−Ala T−Thr I−Ile I−Ile G−Gly S−Ser N−Asn N−Asn G−Gly R−Arg K−Lys K−Lys  (3 2,1 2)1≡/parenleftbigg P−Pro S−Ser L−Leu L−Leu A−Ala T−Thr M−Met I−Ile/parenrightbigg (3 2,1 2)2≡/parenleftbiggR−Arg C−Cys Y−Tyr Y−Tyr R−Arg W−Trp Ter Ter/parenrightbigg (1 2,3 2)1≡ P−Pro S−Ser A−Ala T−Thr G−Gly S−Ser G−Gly R−Arg  (1 2,3 2)2≡ L−Leu L−Leu V−Val V−Val D−Asp D−Asp E−Glu E−Glu  (1 2,1 2)1≡/parenleftbigg P−Pro S−Ser A−Ala T−Thr/parenrightbigg (1 2,1 2)2≡/parenleftbigg R−Arg C−Cys R−Arg Ter/parenrightbigg (1 2,1 2)3≡/parenleftbigg L−Leu L−Leu V−Val V−Val/parenrightbigg (1 2,1 2)4≡/parenleftbigg H−His H−His Q−Gln Q−Gln/parenrightbigg 35Table 11: Four-fold tensor product of the (1 2,1 2) representation of Uq→0(sl(2)⊕sl(2)) (1 2,1 2)⊗(1 2,1 2)⊗(1 2,1 2)⊗(1 2,1 2) = (1 2,1 2)⊗/bracketleftig (3 2,3 2)⊕2 (3 2,1 2)⊕2 (1 2,3 2)⊕4 (1 2,1 2)/bracketrightig = (2,2)⊕3 (2,1)⊕3 (1,2)⊕9 (1,1)⊕2 (2,0) ⊕2 (0,2)⊕6 (1,0)⊕6 (0,1)⊕4 (0,0) One has (The upper label denotes different irreducible repre sentations): (1 2,1 2)⊗(3 2,3 2) = (2 ,2)⊕(2,1)1⊕(1,2)1⊕(1,1)1 where (2,2) = (1 ,2)1= CCCC UCCC UUCC UUUC UUUU GCCC ACCC AUCC AUUC AUUU GGCC AGCC AACC AAUC AAUU GGGC AGGC AAGC AAAC AAAU GGGG AGGG AAGG AAAG AAAA  CUCC CUUC CUUU GUCC GUUC GUUU GACC GAUC GAUU GAGC GAAC GAAU GAGG GAAG GAAA  (2,1)1= (1 ,1)1= CGCC UGCC UACC UAUC UAUU CGGC UGGC UAGC UAAC UAAU CGGG UGGG UAGG UAAG UAAA  CACC CAUC CAUU CAGC CAAC CAAU CAGG CAAG CAAA  (1 2,1 2)⊗(3 2,1 2)1= (2,1)2⊕(2,0)1⊕(1,1)2⊕(1,0)1 where (2,1)2= (1 ,1)2= CCCG UCCG UUCG UUUG UUUA GCCG ACCG AUCG AUUG AUUA GGCG AGCG AACG AAUG AAUA  CUCG CUUG CUUA GUCG GUUG GUUA GACG GAUG GAUA  (2,0)1= (1 ,0)1=/parenleftbigCGCG UGCG UACG UAUG UAUA/parenrightbig /parenleftbigCACG CAUG CAUA/parenrightbig (1 2,1 2)⊗(3 2,1 2)2= (2,1)3⊕(2,0)2⊕(1,1)3⊕(1,0)2 where (2,1)3= (1 ,1)3= CCGC UCGC UUGC UUAC UUAU GCGC ACGC AUGC AUAC AUAU GCGG ACGG AUGG AUAG AUAA  CUGC CUAC CUAU GUGC GUAC GUAU GUGG GUAG GUAA  (2,0)2= (1 ,0)2=/parenleftbig CCGG UCGG UUGG UUAG UUAA/parenrightbig /parenleftbig CUGG CUAG CUAA/parenrightbig 36(1 2,1 2)⊗(1 2,3 2)1= (1,2)2⊕(0,2)1⊕(1,1)4⊕(0,1)1 where (1,2)2= CCCU UCCU UUCU GCCU ACCU AUCU GGCU AGCU AACU GGGU AGGU AAGU GGGA AGGA AAGA (0,2)1= CUCU GUCU GACU GAGU GAGA  (1,1)4= CGCU UGCU UACU CGGU UGGU UAGU CGGA UGGA UAGA  (0,1)1= CACU CAGU CAGA  (1 2,1 2)⊗(1 2,3 2)2= (1,2)3⊕(0,2)2⊕(1,1)5⊕(0,1)2 where (1,2)3= CCUC UCUC UCUU GCUC ACUC ACUU GGUC AGUC AGUU GGAC AGAC AGAU GGAG AGAG AGAA (0,2)2= CCUU GCUU GGUU GGAU GGAA  (1,1)5= CGUC UGUC UGUU CGAC UGAC UGAU CGAG UGAG UGAA  (0,1)2= CGUU CGAU CGAA  (1 2,1 2)⊗(1 2,1 2)1= (1,1)6⊕(1,0)3⊕(0,1)3⊕(0,0)1 where (1,1)6= CCCA UCCA UUCA GCCA ACCA AUCA GGCA AGCA AACA  (0,1)3= CUCA GUCA GACA  (1,0)3=/parenleftbig CGCA UGCA UACA/parenrightbig (0,0)1=/parenleftbig CACA/parenrightbig (1 2,1 2)⊗(1 2,1 2)2= (1,1)7⊕(1,0)4⊕(0,1)4⊕(0,0)2 where (1,1)7= CCGU UCGU UUGU GCGU ACGU AUGU GCGA ACGA AUGA  (0,1)4= CUGU GUGU GUGA  (1,0)4=/parenleftbig CCGA UCGA UUGA/parenrightbig (0,0)2=/parenleftbig CUGA/parenrightbig 37(1 2,1 2)⊗(1 2,1 2)3= (1,1)8⊕(1,0)5⊕(0,1)5⊕(0,0)3 where (1,1)8= CCUG UCUG UCUA GCUG ACUG ACUA GGUG AGUG AGUA  (0,1)5= CCUA GCUA GGUA  (1,0)5=/parenleftbigCGUG UGUG UGUA/parenrightbig (0,0)3=/parenleftbigCGUA/parenrightbig (1 2,1 2)⊗(1 2,1 2)4= (1,1)9⊕(1,0)6⊕(0,1)6⊕(0,0)4 where (1,1)9= CCAC UCAC UCAU GCAC ACAC ACAU GCAG ACAG ACAA  (0,1)6= CCAU GCAU GCAA  (1,0)6=/parenleftbig CCAG UCAG UCAA/parenrightbig (0,0)4=/parenleftbig CCAA/parenrightbig 38

arXiv:physics/0003038v1 [physics.bio-ph] 16 Mar 2000The Force-Velocity Relation for Growing Biopolymers A. E. Carlsson Department of Physics Washington University St. Louis, Missouri 63130-4899 (Submitted to Physical Review,July 24, 2013) The process of force generation by the growth of biopolymers is simulated via a Langevin-dynamics approach. The interac tion forces are taken to have simple forms that favor the growth of straight fibers from solution. The force-velocity relation is obtained from the simulations for two versions of the monome r-monomer force field. It is found that the growth rate drops off more rapidly with applied force than expected from the sim plest theories based on thermal motion of the obstacle. The discrepancies amount to a factor of three or more when the app lied force exceeds 2 .5kT/a, where ais the step size for the polymer growth. These results are explained on the basis of r estricted diffusion of monomers near the fiber tip. It is also f ound that the mobility of the obstacle has little effect on the grow th rate, over a broad range. PACS numbers: 87.15.Rn, 87.16.Ac, 87.17.Jj I. INTRODUCTION The growth of biopolymers is a key ingredient in the crawling motion and internal transport processes of al- most all eukaryotic cells. They crawl among each other and over substrates by motion of the cytoplasm into pro- trusions known as lamellipodia, filopodia, or microspikes according to their shapes. The forces driving the exten- sion of these protusions are believed to comes from the growth of a collection of fibers assembled from monomers of the protein actin. The actin fibers are approximately 7 nm in diameter. With no opposing force, they can grow at velocities [1] of over 1 µm/sec at physiological actin concentrations [2,3] of 10–50 µM; the velocities of the cell protrusions are typically [4,5] in the range of 0 .1µm/sec. Actin fiber growth also can power the motion of bacte- ria and viruses through the cell cytoplasm. The veloci- ties usually range from 0 .02 to 0 .2µm/sec, but velocities up to 1 .5µm/sec have been observed. As they move, they leave behind “comet tails” made up of actin fibers [6,7]. Recent experiments have studied the minimal in- gredients necessary for such propulsion. For example, Ref. [8] shows that polystyrene beads coated with a cat- alytic agent for actin polymerization spontaneously move in cell extracts at velocities of 0 .01 to 0 .05µm/sec, form- ing comet tails similar to those caused by bacteria and viruses. It has also been shown recently that Listeria andShigella bacteria can move in a medium much sim- pler than a cell extract, containing in addition to actin monomers only the proteins Arp2/3 complex, actin de- polymerizing factor, and capping protein. In particular, myosin-type motors are not necessary for motion driven by actin polymerization. The minimal ingredients lead to velocities of 0 .01 to 0 .02µm/sec; supplementation of this mix with other ingredients including profilin, α- actinin, and the VASP protein increases the velocitiesup to 0 .05µm/sec. To our knowledge, there have been no measurements of the force-velocity relation for grow- ing actin filaments. However, recent measurements of the actin fiber density [9] and Young’s modulus [10] at the leading edge of lamellipodia would suggest forces on the order of 1 pN per fiber if all fibers are contributing equally; this is roughly equal to the basic force unit for fiber growth, kT/a, where kis Boltzmann’s constant, T is temperature, and a= 2.7 nm is the incremental fiber length per added monomer. Microtubules, which are thicker fibers (22 nm) assem- bled from tubulin subunits, also exert forces when they grow. Microtubule assembly and disassembly is crucial in intracellular processes such as mitosis, the formation of cilia and flagella, and the transport of nutrients across the cell. Recent measurements [11] on microtubules in vitrohave yielded explicit force-velocity curves. At zero force, the velocity is about 0 .02µm/sec; with increasing force, the velocity drops off roughly exponentially. It is clear that growth of the fiber against a force results in a lowering of the system’s free energy if the opposing force is sufficiently small, since the exothermic contribu- tion from the attachment of monomers at the end of the polymer will outweigh the work done to move the ob- stacle against the external force. The critical force at which polymerization stops is determined by the balance of these two contributions. However, it is not yet under- stood in detail what factors determine the rate of growth and the maximum force at which a useful speed can be obtained. The basic difficulty of the polymer’s growth process is that when the obstacle impinges directly on the fiber tip, there is not enough room for a new monomer to move in. Thus the rate of growth must be connected to the fluctuations of either the obstacle or the filament tip, which create temporary gaps between the tip and the obstacle. This effect has been treated explicitly in 1the “thermal ratchet” model [12]. In this model, one as- sumes that the obstacle must be a critical distance afrom the tip for growth to occur. The fiber is assumed to be rigid. The growth rate is obtained by solution of a drift- diffusion type equation. For conditions of slow growth, i.e. in which the time to add a monomer is much longer than the time it takes the obstacle to diffuse a distance a, this equation can be solved analytically. The forward growth rate is proportional to the probability that the obstacle-tip separation exceeds a. If depolymerization is sufficiently slow to be ignored, this yields the following dependence of the velocity von the opposing force F: v∝exp (−Fa/kT ) (1) where kis Boltzmann’s constant and Tis the tempera- ture. This result is equivalent to application of the prin- ciple of detailed balance [13], on the assumption that the depolymerization rate is independent of the oppos- ing force. This work has been extended to flexible fibers at non-perpendicular incidence [14,15], and to interactin g systems of fibers [16]. For flexible fibers, it is again found that the velocity is proportional to the probability form- ing of a gap large enough to admit the next monomer. It is the purpose of this paper to evaluate the force- velocity relation for growing fibers using a model more realistic than those used previously. The model used to derive Eq. (1) does not explicitly treat the diffusion of monomers to the filament tip, but treats only the diffu- sive behavior of the variable describing the distance be- tween the obstacle and the tip. It is assumed that once this distance exceeds a, that the monomers can enter with a fixed probability independent of the tip-obstacle distance. This assumption needs to be evaluated by ex- plicit treatment of the diffusion in the monomers. In addition, although the form of Eq. (1) is confirmed by the force-velocity relation for microtubules [11] the de- cay rate of the velocity with applied force was about twice as large as expected from Eq. (1). One possible expla- nation of this, suggested by Mogilner and Oster [16], is subsidy effects between the thirteen fibers comprising a microtubule “protofilament”. We intend to investigate the extent to which other mechanisms can account for such discrepancies. II. MODEL Our model system contains a fiber of protein monomers growing perpendicular to a flat rigid obstacle in two di- mensions. We will be mainly interested in the actin sys- tem, but the basic physics of our results is relevant to any fiber growing against an obstacle. Our choice of two dimensions is dictated mainly by computational practi- cality: the simulations took over two weeks of CPU time on a Compaq 21264 processor and our preliminary stud- ies indicate that the three-dimensional simulations take about 30 times longer. The fundamental units of thesimulation are the monomers; their internal and rota- tional degrees of freedom are assumed to be included in our effective interaction energies. The motions of the monomers and the obstacle are treated via Langevin dy- namics. The z-direction is taken as the growth axis, with the obstacle parallel to the x-direction. The coordinates of the monomer centers-of-mass are given by /vector ri, and the z-coordinate of the obstacle is called Z. The Langevin equations for this system are µ−1 id/vector ri/dt=−/vectorFi+/vectorfi(t) for the monomers and µ−1 OdZO/dt=−FO+fO(t) for the obstacle, where the µ’s are mobilities, Fdenotes deter- ministic interaction forces, and /vectorfiand and fOare random forces satisfying ∝angbracketleftfx i(t)fx j(t′)∝angbracketright=∝angbracketleftfz i(t)fz j(t′)∝angbracketright= 2µ−1 ikTδijδ(t−t′),(2) ∝angbracketleftfx i(t)fz j(t′)∝angbracketright= 0, (3) and ∝angbracketleftfO(t)fO(t′)∝angbracketright= 2µ−1 OkTδ(t−t′). (4) The Langevin equations are implemented with a finite time step ∆ tfollowing the procedure of Ref. [17]: /vector ri(t+ ∆t) =/vector ri(t) + ∆tµi/vectorFi(t) +/vector g(t)/radicalbig kTµ i, (5) andZ(t+ ∆t) =Z(t) + ∆tµO/vectorFO(t) +h(t)/radicalbig kTµ O,(6) where /vector g(t) and h(t) are random functions with zero time average, satisfying ∝angbracketleftgx(t)gx(t′)∝angbracketright=∝angbracketleftgz(t)gz(t′)∝angbracketright=∝angbracketlefth(t)h(t′)∝angbracketright= 2∆tδtt′. (7) To implement the last set of correlations, at each time step we choose gx,gz, and hfrom a uniform random distribution random from −√ 6∆tto√ 6∆t. A. Force Laws The obstacle experiences an external force of magni- tudeFextin the −zdirection. In the absence of reliable force fields for the monomer- monomer interactions, we use a simple model form for the interactions which has a linear filament as the lowest- energy structure. This form contains two-body and three-body interactions. The two-body interactions are repulsive and have the form V2(rij) =Vrepexp [−κrep(rij−a)] ; (8) the three-body interaction energy has the form V3(/vector rij,/vector rik) =Vattexp[−κatt(rij−a)] exp [−κatt(rik−a)](α+ cos θij),(9) with 0 < α < 1. It is attractive for θij>cos−1(−α). The monomer-obstacle interactions have only a two-body repulsive component, and have the form W2(zi) =Vobstexp (−κobst|zi−(Z−a)|).(10) 2The forces are obtained as gradients of these energy terms. The energies are modified by subtraction of ap- propriate constants to force the interaction energy to go to zero at a cutoff distance rmax(in the case of the three- body terms this means that the energy vanishes if either rijorrikbecomes greater than rmax). We use two parameter sets, whose values are given in Table I. These two parameter sets are chosen mainly to sample different shapes of the “basin of attraction” for the addition of a monomer, and by no means exhaustively sample the range of possible model force fields. The first corresponds to a narrow basin of attraction. The large value of αmeans that the three-body terms are positive only for a small range of angles. This is partly compen- sated for by the choice of prefactors to avoid the binding energy becoming too small. We will call this the “hard” force field. The corresponding energy contours are shown in Figure 1a. The width of the basin of attraction, or the region over which the force pulls the next monomer into its minimum-energy position, is about a tenth the size of a monomer, which would correspond to a few ˚A for actin monomers. Figure 1b shows the energy contours for the parameters corresponding to a wider basin of attraction, which is about a half the size of a monomer. We call this the “soft” force field. For both of the force fields, the binding energies are very large compared to kT, so that monomer subtraction from the fiber does not occur in the simulations. This is a reasonable approximation; from the measured on and off constants in Ref. [1], the ratio of on to off rates at physiological actin monomer concentrations would be less than 0 .01. With regard to the mobilities, the only physically rel- evant factor is the ratio of the obstacle mobility to the monomer mobility, since multiplicative changes in all the mobilities simply serve to scale up the fiber growth ve- locities; these will thus factor out of our velocity results , which are scaled by 1 /µkT. For most of our simula- tions, we use a mobility of the obstacle equal to that of the monomers for simplicity. This would correspond to identifying the obstacle with a part of a fluctuating membrane, rather than an entire rigid particle. We have varied the obstacle mobility in a few cases, with results to be discussed below. TABLE I. Parameters used in simulations. Energies are given in units of kT,κ-parameters in units of a−1, and rmax in units of a, the equilibrium monomer spacing. Force Field VrepVattVobstκrepκattκobstα r max Hard 141.3 6510 19.14 8.267 4.960 4.960 0.940 1.412 Soft 257.5 2151 27.44 7.666 4.600 4.600 0.770 1.522−7 −6 −5 −4 −3 −2 −1 0 1 2−5−4−3−2−1012345 −8−7−6−5−4−3−2−1 012−5−4−3−2−1012345 FIG. 1. Energy contours for monomer approaching fiber tip with hard (a) and soft (b) force fields. Contour heights corre - spond to integer multiples of kT, with lighter corresponding to lower energies. The length units are nm, assuming a monomer step size of 2.7 nm as for actin. B. Filament-Growth Procedure A typical physiological concentration of actin (10 µM) is low in the sense that the average spacing between actin monomers is about 60 nm, roughly 10 times the monomer size. This means that the probability that two free monomers are near enough to interact with each other is very small. For this reason we adopt a growth procedure in which only one free monomer at at time interacts with the tip. This is accomplished as follows. We start with a fiber of six monomers pointing in the z- direction, at their equilibrium spacing. A free monomer is then added at a point on a circle of radius Rcentered on the next attachment site [18] (defined as one monomer spacing beyond the monomer at the fiber tip). We choose R= 2.5a, which places the added monomer well beyond the interaction range of the monomer at the tip. The rela- tive probabilities of monomer addition at different points on the circle are proportional to exp [ −W2(z)/kT]. This weighting is accomplished by choosing a random number for each potential addition point; if this random number is less than exp [ −W2(z)/kT], then this point is rejected and another one is chosen. A new point is also chosen if the monomer overlaps the fiber (i.e., its distance to the fiber is less than rmax). The system is then stepped for- ward in time according to the procedure described above, until one of two possible termination events occur: 3•1. The monomer diffuses outside of the R-circle. In this case it is restarted on the circle as above. If the obstacle abuts the fiber, the position of the monomer is constrained to be out of the interaction range of the obstacle. •2. The monomer attaches to the tip. In this case, another monomer is started on the R-circle. In this way, the CPU time that is used in the simulation is focused on the time that the monomers spend close to the tip. A typical snapshot of a simulation configuration is shown in Figure 2. FIG. 2. Typical fiber-obstacle configuration during simula- tions. One can use the computed growth rates to predict growth rates for low concentrations c, by simply mul- tiplying the computed rates by the probability P(c) of finding a monomer inside the R-circle. We obtain this probability numerically as P(c) =1 c/integraldisplay r<Rexp(−U(/vector r)/kT)d2r , (11) where U(/vector r) is the energy (from both fiber and obstacle) associated with placing a monomer at the point /vector r, and the coordinates are given with respect to the next attachment point [19]. We plot our force-velocity relations in terms of the force acting between the obstacle and the fiber tip. This exceeds the external force applied on the obstacle by an amount corresponding to the viscous drag on the obstacle as it moves through the medium. The total force is thus given as F=Fext+v/µO. III. RESULTS Our simulations involve 10 runs, each of which involves the addition of 30 monomers to the fiber tip. This cor- responds to a statistical uncertainty of/radicalbig 1/300 = 6% in the growth velocities. Typical results for the fiber length as a function of time are shown in Figure 3. Note that there are no backwards steps, because the parameters that are used in the force field result in an exothermic enthalpy for monomer addition that exceeds kTby at least a factor of 20. We use a concentration correspond- ing to one monomer per square of side 20 a; in our model,velocities at other concentrations would be given by a linear proportionality. 0 20 40 60 80 NUMBER OF TIME STEPS (millions)010203040NUMBER OF MONOMERS FIG. 3. Representative plot of number of monomers in fiber vs. number of time steps. Obtained for Fa/kT = 1.5 and hard force field. A. Force-Velocity Relation Figure 4a shows growth velocity (solid circles) vs. ap- plied force, for the “hard” force field ( cf.Figure 1a). For comparison, a curve proportional to exp ( −Fa/kT ) is shown. The simulation results give noticeably lower velocities at finite applied forces than the exponential prediction. The discrepancy is about 65% at Fa/kT = 1, and 85% at Fa/kT = 2.5. The results can be roughly fitted to different exponential curve, of the form exp(−1.7Fa/kT ). Thus the growth velocity is much more sensitive to force than the thermal-ratchet model would predict. Figure 4b shows similar results for the soft force field ( cf.Figure 1b). The free-fiber growth velocity is about twice that for the “hard” force field, because the attraction basin is larger. The discrepan- cies between the simulation results and the analytic the- ory are comparable to those seen for the “hard” force field, but somewhat less pronounced. The discrepancy atFa/kT = 2.5 is 70%, and the exponential fit curve is exp(−1.5Fa/kT ). The open diamonds in Fig. 4b corre- spond to the results of varying the mobility µO; for the leftmost one the mobility is doubled, and for the right- most one it is reduced by a factor of ten. The effects of these variations are very minor, as predicted by the “thermal-ratchet” model [12]. 40 1 2 3 Fa/kT0123 V 0 1 2 3 Fa/kT012345 V FIG. 4. Growth rates (solid circles) for hard (a) and soft (b) force fields vs. total force F. Rates given in units of µkTc, where µis the monomer mobility and cis the concentration. Force given in units of a/kT. Solid line corresponds to expo- nential decay ( cf.Eq. (1)). Diamonds in (a) correspond to mobility enhanced by factor of 2 (left) and reduced by factor of 10 (right). Dashed curves correspond to theory of Eq. (12) . B. Interpretation We believe that the discrepancies seen in Fig. 4 re- sult from the restriction of monomer diffusion to the fiber tip by the impinging obstacle. Such restriction will occur even when the obstacle is elevated by a dis- tance aor more. Figure 5 shows energy contours for a monomer approaching the tip, when the obstacle is ele- vated a distance 1 .25arelative to its equilibrium position forFa/kT = 1.0. The contours at at integer multiples ofkT. The easily accessible paths corresponding to en- ergies less than kTare confined to a narrow band by the presence of the obstacle. This is expected to slow the diffusion to the tip. Effectively, the monomers must travel through a tunnel in order to get to the basin of attraction near the tip. Another possible explanation for the observed effect would be that even in the region with energy less than kT, there is a finite energy from the interaction with the obstacle. However, this energyis proportional to the the length scale of the interaction between the obstacle and the monomers. In a few cases, we have made this length scale five times smaller, and the velocities are unchanged to within a few percent. There- fore, this monomer-obstacle interaction energy does not seem to be the major factor, but rather the blocking ef- fects of the obstacle. −7−6−5−4−3−2−1 012−5−4−3−2−1012345 FIG. 5. Energy contours for monomer approaching fiber tip with hard force field, in presence of obstacle. Contours are as in Fig. 1. To make this physical picture more precise, we have calculated the velocities for model fiber configurations in which the obstacle is held at a fixed distance from the fiber tip. The results are shown in Figs. 6a and b, for the “hard” and “soft” force fields respectively. The distance Zis measured in units of the monomer size, and the edge of the obstacle is defined as the point where the monomer-obstacle interaction energy is equal to kT. Thus when Z= 0, the interaction energy of the last monomer in the fiber with the obstacle is kT. In both cases, the velocity at Z/a= 1 is nearly zero. Only for Z/a > 2 is the velocity within 20% of the free-growth velocity. The appropriate generalization of Eq. (1) is then the following: v(F) =/integraldisplay∞ 0v(Z)P(Z, F)dZ , (12) where Fis the applied force, Zis the obstacle position, andP(Z, F) = (const)exp( −E/kT) is the probability of a certain value of Z. Here the obstacle-fiber inter- action energy is E=W2(z−Z) +FZ, where zis the z-coordinate of the last monomer in the fiber. Equa- tion (12) reduces to Eq. (1) if v(Z) has the form of a step function beginning at a Z=a, and W2is sufficiently short-ranged. The dashed lines in Figs. 4a and 4b cor- respond to a numerical evaluation of Eq. (12). For both force fields, the agreement with the simulation results is quite close, with only about 20% discrepancies occurring for small but non-zero forces. Thus the gradual rise of the velocity seen in Fig. 6, as opposed to an abrupt jump, is at the heart of the observed effect. 50 1 2 3 4 5 Z/a0123 V 0 1 2 3 4 5 Z/a012345 V FIG. 6. Fiber growth velocity with fixed tip-fiber spacing, for hard (a) and soft (b) force fields. Zis measured relative to point at which tip-fiber interaction energy is kT. IV. CONCLUSIONS The physics underlying the above results is general enough that in most systems involving fiber growth against an obstacle, one should expect a decay of velocity with applied force more rapid than the simple exponen- tial form (1). This may explain some of the discrepan- cies pointed out in connection with the measured force- velocity relation of Ref. [11]. However, our application to microtubule growth is not quantitative enough to de- termined whether the present effect exceeds the subsidy effects discussed in Ref. [16]. The results obtained here should be useful in explaining the basic physics of motion based on actin polymerization. For example, in a recent study van Oudenaarden and Theriot [20] have simulated the propulsion of plastic beads in cell extracts with a model based on a number of fibers exerting forces on the beads. In their simulations, an assumed form is taken for the probability of monomer addition to a fiber in terms of the time-averaged position of the fiber relative to the bead, or equivalently the force acting between the two. A better knowledge of the relationship between the force and the monomer addition rate can help pin down thevalidity of the assumptions underlying such simulations. Because the bead-motion simulations include only the thermal energy required to achieve a certain tip-obstacle spacing, it is likely that the addition rate will drop off more rapidly with increasing force than is assumed in Ref. [20]. The results obtained here are also expected to have no- ticeable results on the structure of membranes that are being pushed forward by collections of actin fibers. As a result of random fluctuations, some fibers will eventu- ally get ahead of others, and these will be exerting larger forces on the membrane. If the velocity drops off rapidly with the force, then these fibers will be slowed down sig- nificantly. This will result in the membrane surface being smoother than otherwise expected. Future work should treat such many-fiber effects, and also explore the effects of fiber growth angle and branching. ACKNOWLEDGMENTS I am grateful to John Cooper for stimulating my in- terest in this project, and to Jonathan Katz and Elliot Elson for useful conversations. This research was sup- ported by the National Institutes of Health under Grant Number GM38542-12. [1] T. Pollard, J. Cell Biol. 103, 2747 (1986). [2] J. A. Cooper, Ann. Rev. Physiol. 53, 585 (1991). [3] J.-B. Marchand et al., J. Cell Biol. 130, 331 (1995). [4] V. Argiro, M. Bunge, and J. Johnson, J. Neurosci. Res. 13, 149 (1985). [5] S. Felder and E. L. Elson, J. Cell Biol. 111, 2513 (1990). [6] L. G. Tilney and D. A. Portnoy, J. Cell Biol. 109, 1597 (1989). [7] A. S. Sechi, J. Wehland, and J. V. Small, J. Cell Biol. 137, 155 (1997). [8] L. A. Cameron, M. J. Footer, A. van Oudenaarden, and J. A. Theriot, Proc. Natl. Acad. Sci. 96, 4908 (1999). [9] V. C. Abraham, V. Krisnamurthi, D. L. Taylor, and F. Lanni, Biophys. J. 77, 1721 (1999). [10] C. Rotsch, K. Jacobson, and M. Radmacher, Proc. Natl. Acad. Sci. USA 96, 921 (1999). [11] M. Dogterom and B. Yurke, Science 278, 856 (1997). [12] C. S. Peskin, G. M. Odell, and G. F. Oster, Biophysical Journal 65, 316 (1993). [13] T. L. Hill, Linear Aggregation Theory in Cell Biology (Springer-Verlag, New York, 1987), Chap. 2. [14] A. Mogilner and G. Oster, Biophys. J. 71, 3030 (1996). [15] A. Mogilner and G. Oster, Eur. Biophys. J. 25, 47 (1996). [16] A. Mogilner and G. Oster, Eur. Biophys. J. 28, 235 (1999). [17] M. Doi and S. F. Edwards, The Theory of Polymer Dy- namics (Clarendon Press, Oxford, 1998), Chap. 3. 6[18] In order to avoid an excessive number of excursions back and forth across the circle radius, the added monomer is placed a small distance inside R. [19] In two dimensions, the time for a particle to diffuse to ca p- ture is not strictly proportional to the area of the region in which it diffuses, but contains logarithmic corrections. Therefore the calculated velocities are not strictly indep en- dent of R. I have verified by use of a few test cases with larger values of Rthat the predicted logarithmic scaling is observed. Extrapolating to a value of Rcorresponding to a physiological interparticle spacing for actin monomer s would modify the calculated velocities by a constant factor of about two, which we have not included because our fo- cus is the force-dependence of the velocity rather than its absolute magnitude. [20] A. van Oudenaarden and J. A. Theriot, Nature Cell. Biol. 1, 493 (1999). 7

arXiv:physics/0003039v1 [physics.chem-ph] 16 Mar 2000A fully ab initio potential curve of near-spectroscopic quality for OH−ion: importance of connected quadruple excitations and scalar relativistic effects Jan M.L. Martin* Department of Organic Chemistry, Kimmelman Building, Room 262, Weizmann Institute of Science, IL-76100 Reh .ovot, Israel. E-mail:comartin@wicc.weizmann.ac.il (Special issue of Spectrochimica Acta A : Received March 6, 2000; In final form March 16, 2000) Abstract A benchmark study has been carried out on the ground-state po tential curve of the hydroxyl anion, OH−, including detailed calibration of both the 1- particle and n-particle basis sets. The CCSD(T) basis set li mit overestimates ωeby about 10 cm−1, which is only remedied by inclusion of connected quadru- ple excitations in the coupled cluster expansion — or, equiv alently, the inclu- sion of the 2 πorbitals in the active space of a multireference calculatio n. Upon inclusion of scalar relativistic effects (-3 cm−1onωe), a potential curve of spectroscopic quality (sub-cm−1accuracy) is obtained. Our best computed EA(OH), 1.828 eV, agrees to three decimal places with the bes t available ex- perimental value. Our best computed dissociation energies ,D0(OH−)=4.7796 eV and D0(OH)=4.4124 eV, suggest that the experimental D0(OH)=4.392 eV may possibly be about 0.02 eV too low. I. INTRODUCTION Molecular anions play an important role in the chemistry of t he interstellar medium [1], of carbon stars [2], and the Earth’s ionosphere [3]. As point ed out in Ref. [4], the presence of 1anions in the interstellar medium may have profound consequ ences for our understanding of the interstellar processing of the biogenic elements (see e .g. Ref. [5] and references therein). Yet as judged from the number of entries in the compilations o f Huber and Herzberg [6] (for diatomics) and of Jacox [7] (for polyatomics), high- or even medium-resolution spec- troscopic data for anions are relatively scarce compared to the amount of data available for neutral or even cationic species: in the 1992 review of Hirot a [8] on spectroscopy of ions, only 13 molecular anions were listed in Table VII, compared t o 4 1/2 pages worth of entries for cations. (Early reviews of anion spectroscopy are found in Refs. [9,10], while ab initio studies of structure and spectroscopy of anions were review ed fairly recently by Botschwina and coworkers [11].) Some of the reasons for this paucity are discussed in the introductions to Refs. [12,4]. One such species is the hydroxyl anion, OH−. By means of velocity modulation spec- troscopy [13], high-resolution fundamentals were obtaine d [14,15] for three isotopomers, namely16OH−,16OD−, and18OH−; in addition, some pure rotational transitions have been observed [16]. Lineberger and coworkers [17] earlier obtai ned some rotational data in the course of an electron photodetachment study, and obtained p recise electron affinities (EAs) of 14741.03(17) and 14723.92(30) cm−1, respectively, for OH and OD. Very recently, the same group re-measured [18] EA(OH) and obtained essentiall y the same value but with a higher precision, 14741.02(3) cm−1. The spectroscopic constants of OH−were previously the subject of ab initio studies, notably by Werner et al. [19] using multireference configura tion interaction (MRCI) methods, and recently by Lee and Dateo (LD) [12] using coupled cluster theory with basis sets as large as [7s6p5d4f3g2h/6s5p4d3f2g]. The LD paper is particularly relevant here. The CCSD(T) (cou pled cluster with all single and double substitutions [20] and a quasiperturbati ve treatment for triple excitations [21]) method, in combination with basis sets of at least spdfg quality and including an account for inner-shell correlation, can routinely predic t vibrational band origins of small polyatomic molecules with a mean absolute error on the order of a few cm−1(e.g. for 2C2H2[22], SO 2[23]). Yet while LD found very good agreement between their c omputed CCSD(T)/[6s5p4d3f2g/5s4p3d2f] spectroscopic constants and available experimental data, consideration of further basis set expansion and of inner-s hell correlation effects leads to a predicted fundamental νat the CCSD(T) basis set limit of 3566.2 ±1 cm−1, about 11 cm−1 higher than the experimental results [14] of 3555.6057(22) cm−1, where the uncertainty in parentheses represents two standard deviations. In a recent benchmark study [24] on the ground-state potenti al curves of the first-row diatomic hydrides using both CCSD(T) and FCI (full configura tion interaction) methods, the author found that CCSD(T) has a systematic tendency to overe stimate harmonic frequencies of A–H stretching frequencies by on the order of 6 cm−1. Even so, the discrepancy seen by LD is a bit out of the ordinary, and the question arises as to wh at level of theory is required to obtain ‘the right result for the right reason’ in this case . In the present work, we shall show that the discrepancy betwe en the CCSD(T) basis set limit and Nature is mostly due to two factors: (a) neglect of the effect of connected quadruple excitations, and (b) neglect of scalar relativis tic effects. When these are properly accounted for, the available vibrational transitions can b e reproduced to within a fraction of a cm−1from the computed potential curve. In the context of the pres ent Special Issue, this will also serve as an illustrative example of the type of accu racy that can be achieved for small systems with the present state of the art. Predicted ba nd origins for higher vibrational levels (and ‘hot bands’) may assist future experimental wor k on this system. Finally, as by-products of our analysis, we will show that the electron a ffinity of OH can be reproduced to very high accuracy, and tentatively propose a slight upwa rd revision of the dissociation energy of neutral hydroxyl radical, OH. II. COMPUTATIONAL METHODS The coupled cluster, multireference averaged coupled pair functional (ACPF) [25], and full CI calculations were carried out using MOLPRO 98.1 [26] running on DEC/Compaq Al- pha workstations in our laboratory, and on the SGI Origin 200 0 of the Faculty of Chemistry. 3Full CCSDT (coupled cluster theory with all connected singl e, double and triple excitations [27]) and CCSD(TQ) (CCSD with quasiperturbative correctio ns for triple and quadruple ex- citations [28]) calculations were carried out using ACES II [29] on a DEC Alpha workstation. Correlation consistent basis sets due to Dunning and cowork ers [30,31] were used through- out. Since the system under consideration is anionic, the re gular cc-pV nZ (correlation consis- tent polarized valence n-tuple zeta, or V nZ for short) basis sets will be inadequate. We have considered both the aug-cc-pV nZ (augmented correlation consistent, or AV nZ for short) basis sets [32] in which one low-exponent function of each angular momentum is added to both the oxygen and hydrogen basis sets, as well as the aug′-cc-pV nZ basis sets [33] in which the ad- dition is not made to the hydrogen basis set. In addition we co nsider both uncontracted ver- sions of the same basis sets (denoted by the suffix ”uc”) and the aug-cc-pCV nZ basis sets [34] (ACV nZ) which include added core-valence correlation functions . The largest basis sets con- sidered in this work, aug-cc-pV6Z and aug-cc-pCV5Z, are of [ 8s7p6d5f4g3h2i/7s6p5d4f3g2h] and [11s10p8d6f4g2h/6s5p4d3f2g] quality, respectively. The multireference ACPF calculations were carried out from a CASSCF (complete ac- tive space SCF) reference wave function with an active space consisting of the valence (2σ)(3σ)(1π)(4σ) orbitals as well as the (2 π) Rydberg orbitals: this is denoted CAS(8/7)- ACPF (i,e, 8 electrons in 7 orbitals). While the inclusion of the (2 π) orbitals is essential (see below), the inclusion of the (5 σ) Rydberg orbital (i.e., CAS(8/8)-ACPF) was considered and found to affect computed properties negligibly. In addition , some exploratory CAS-AQCC (averaged quadratic coupled cluster [35]) calculations we re also carried out. Scalar relativistic effects were computed as expectation va lues of the one-electron Darwin and mass-velocity operators [36,37] for the ACPF wave funct ions. The energy was evaluated at 21 points around re, with a spacing of 0.01 ˚A. (All energies were converged to 10−12hartree, or wherever possible to 10−13hartree.) A polynomial in (r−re)/reof degree 8 or 9 (the latter if an F-test revealed an acceptabl e statistical significance for the nonic term) was fitted to the energies. Us ing the procedure detailed in Ref. [24], the Dunham series [38] thus obtained was transfor med by derivative matching into 4a variable-beta Morse (VBM) potential [39] Vc=De/parenleftBig 1−exp[−z(1 +b1z+b2z2+. . .+b6z6)]/parenrightBig2(1) in which z≡β(r−re)/re,Deis the (computed or observed) dissociation energy, and βis an adjustable parameter related to that in the Morse functio n. Analysis of this function was then carried out in two different manners: (a) analytic di fferentiation with respect to (r−re)/reup to the 12th derivative followed by a 12th-order Dunham ana lysis using an adaptation of the ACET program of Ogilvie [40]; and (b) numer ical integration of the one- dimensional Schr¨ odinger equation using the algorithm of B alint-Kurti et al. [41], on a grid of 512 points over the interval 0.5 a0—5a0. As expected, differences between vibrational energies obtained using both methods are negligible up to the seventh vibrational quantum, and still no larger than 0.4 cm−1for the tenth vibrational quantum. III. RESULTS AND DISCUSSION A.n-particle calibration The largest basis set in which we were able to obtain a full CI p otential curve was cc- pVDZ+sp(O), which means the standard cc-pVDZ basis set with the diffuse sandpfunction from aug-cc-pVDZ added to oxygen. A comparison of computed p roperties for OH−with different electron correlation methods is given in Table I, w hile their errors in the total energy relative to full CI are plotted in Figure 1. It is immediately seen that CCSD(T) exaggerates the curvatu re of the potential surface, overestimating ωeby 10 cm−1. In addition, it underestimates the bond length by about 0.0006 ˚A. These are slightly more pronounced variations on trends p reviously seen [24] for the OH radical. The problem does not reside in CCSD(T)’s quasiperturbative treatment of triple excita- tions: performing a full CCSDT calculation instead lowers ωeby only 1.7 cm−1and lengthens the bond by less than 0.0001 ˚A. Quasiperturbative inclusion of connected quadruple exc ita- tions, however, using the CCSD(TQ) method, lowers ωeby 8.5 cm−1relative to CCSD(T), 5and slightly lengthens the bond, by 0.00025 ˚A. (Essentially the same result was obtained by means of the CCSD+TQ* method [42], which differs from CCSD(TQ ) in a small sixth-order termE6TT.) No CCSDT(Q) code was available to the author: approximati ng the CCSDT(Q) energy by the expression E[CCSDT (Q)]≈E[CCSDT ]+E[CCSD (TQ)]−E[CC5SD(T)] = E[CCSDT ] +E5QQ+E5QT, we obtain a potential curve in fairly good agreement with fu ll CI. What is the source of the importance of connected quadruple e xcitations in this case? Analysis of the FCI wave function reveals prominent contrib utions to the wave function from (1 π)4(2π)0→(1π)2(2π)2double excitations; while the (2 π) orbitals are LUMO+2 and LUMO+3 rather than LUMO, a large portion of them sits in th e same spatial region as the occupied (1 π) orbitals. In any proper multireference treatment, the afo rementioned excitations would be in the zero-order wave function: obvio usly, the space of all double excitations therefrom would also entail quadruple excitat ions with respect to the Hartree- Fock reference, including a connected component. Since the basis set sizes for which we can hope to perform CCSD T(Q) or similar calcu- lations on this system are quite limited, we considered mult ireference methods, specifically ACPF from a [(2 σ)(3σ)(4σ)(1π)(2π)]8reference space (denoted ACPF(8/7) further on). As might be expected, the computed properties are in very close agreement with FCI, except forωebeing 1.5 cm−1too high. AQCC(8/7) does not appear to represent a further im prove- ment, and adding the (5 σ) orbital to the ACPF reference space (i.e. ACPF(8/8)) affect s properties only marginally. B. 1-particle basis set calibration All relevant results are collected in Table II. Basis set con vergence in this system was previously studied in some detail by LD at the CCSD(T) level. Among other things, they noted that ωestill changes by 4 cm−1upon expanding the basis set from aug-cc-pVQZ to aug- cc-pV5Z. They suggested that ωethen should be converged to about 1 cm−1; this statement 6is corroborated by the CCSD(T)/aug-cc-pV6Z results. Since the negative charge resides almost exclusively on the oxygen, the temptation exists to use aug′-cc-pV nZ basis sets, i.e. to apply aug-cc-pV nZ only to the oxygen atom but use a regular cc-pV nZ basis set on hydrogen. For n=T, this results in fact in a difference of 10 cm−1onωe, but the gap narrows as nincreases. Yet extrapolation suggests convergence of the computed fundamental to a value about 1 cm−1higher than the aug-cc-pV nZ curve. For the AV nZ and A’V nZ basis sets ( n=T,Q), the CAS(8/7)-ACPF approach systemat- ically lowers harmonic frequencies by about 8 cm−1compared to CCSD(T); for the funda- mental the difference is even slightly larger (9.5 cm−1). Interestingly, this difference decreases forn=5. It was noted previously [24] that the higher anharmonicity c onstants exhibit rather greater basis set dependence than one might reasonably have expected, and that this sensi- tivity is greatly reduced if uncontracted basis sets are emp loyed (which have greater radial flexibility). The same phenomenon is seen here. In agreement with previous observations by LD, inner-shell correlation reduces the bond lengthen slightly, and increases ωeby 5–6 cm−1. This occurs both at the CCSD(T) and the CAS(8/7)-ACPF levels. C. Additional corrections and best estimate At our highest level of theory so far, namely CAS(8/7)-ACPF( all)/ACV5Z, νis pre- dicted to be 3559.3 cm−1, still several cm−1higher than experiment. The effects of fur- ther basis set improvement can be gauged from the difference b etween CCSD(T)/AV6Z and CCSD(T)/AV5Z results: one notices an increase of +1.0 cm−1inωeand a decrease of 0.00006 ˚A inre. We also performed some calculations with a doubly augmente d cc-pV5Z basis set (i.e. d-AV5Z), and found the results to be essentially indis tinguishable from those with the singly augmented basis set. Residual imperfections in the e lectron correlation method can be gauged from the CAS(8/7)-ACPF −FCI difference with our smallest basis set, and appear to consist principally of a contraction of reby 0.00004 ˚A and a decrease in ωeby 1.5 cm−1. 7Adding the two sets of differences to obtain a ‘best nonrelati vistic’ set of spectroscopic con- stants, we obtain ν=3558.6 cm−1, still 3 cm−1above experiment. In both cases, changes in the anharmonicity constants from the best directly compute d results are essentially nil. Scalar relativistic corrections were computed at the CAS(8 /7)-ACPF level with and without the (1 s)-like electrons correlated, and with a variety of basis set s. All re- sults are fairly consistent with those obtained at the highe st level considered, CAS(8/7)- ACPF(all)/ACVQZ, namely an expansion of reby about 0.0001 ˚A and — most importantly for our purposes — a decrease of ωeby about 3 cm−1. Effects on the anharmonicity constants are essentially nonexistent. Upon adding these corrections to our best nonrelativistic s pectroscopic constants, we obtain our final best estimates. These lead to ν=3555.44 cm−1for16OH−, in excellent agreement with the experimental result [14] 3555.6057(22) cm−1. The discrepancy between computed (3544.30 cm−1) and observed [14] (3544.4551(28) cm−1) values for18OH−is quite similar. For16OD−, we obtain ν=2625.31 cm−1, which agrees to better than 0.1 cm−1with the experimental value [15] 2625.332(3) cm−1. Our computed bond length is slightly shorter than the observed one [14] for OH−, but within the error bar of that for OD−[15]. If we assume an inverse mass dependence for the experimental di abatic bond distance and extrapolate to infinite mass, we obtain an experimentally de rived Born-Oppenheimer bond distance of 0.96416(16) cm−1, in perfect agreement with our calculations. While until recently it was generally assumed that scalar re lativistic corrections are not important for first-and second-row systems, it has now been s hown repeatedly (e.g. [43–45]) that for kJ/mol accuracy on computed bonding energies, scal ar relativistic corrections are indispensable. Very recently, Csaszar et al. [46] consider ed the effect of scalar relativistic corrections on the ab initio water surface, and found correc tions on the same order of mag- nitude as seen for the hydroxyl anion here. Finally, Bauschl icher [47] compared first-order Darwin and mass-velocity corrections to energetics (for si ngle-reference ACPF wave func- tions) with more rigorous relativistic methods (specifical ly, Douglas-Kroll [48]), and found that for first-and second-row systems, the two approaches yi eld essentially identical results, 8lending additional credence to the results of both Csaszar e t al. and from the present work. (The same author found [49] more significant deviations for t hird-row main group systems.) Is the relativistic effect seen here in OH−unique to it, or does it occur in the neutral first- row diatomic hydrides as well? Some results obtained for BH, CH, NH, OH, and HF in their respective ground states, and using the same method as for OH−, are collected in Table III. In general, ωeis slightly lowered, and revery slightly stretched — these tendencies becoming more pronounced as one moves from left to right in the Periodi c Table. The effect for OH− appears to be stronger than for the isoelectronic neutral hy dride HF, and definitely compared to neutral OH. The excellent agreement ( ±1 cm−1on vibrational quanta) previously seen [24] for the first-row diatomic hydrides between experiment and CCSD(T)/ACV5Z potential curves with an FCI correction is at least in part due to a cance llation between the effects of further basis set extension on the one hand, and scalar relat ivistic effects (neglected in Ref. [24]) on the other hand. The shape of the relativistic contri bution to the potential curve is easily understood qualitatively: on average, electrons ar e somewhat further away from the nucleus in a molecule than in the separated atoms (hence the s calar relativistic contribution to the total energy will be slightly smaller in absolute valu e atrethan in the dissociation limit): as one approaches the united atom limit, however, th e contribution will obviously increase again. The final result is a slight reduction in both the dissociation energy and on ωe. In order to assist future experimental studies on OH−and its isomers, predicted vibra- tional quanta G(n)−G(n−1) are given in Table V for various isotopic species, togethe r with some key spectroscopic constants. The VBM parameters of the potential are given in Table IV. The VBM expansion generally converges quite rapidly [39 ] and, as found previously for OH, parameters b5andb6are found to be statistically not significant and were omitte d. The VBM expansion requires the insertion of a dissociation e nergy: we have opted, rather than an experimental value, to use our best calculated value (see next paragraph). Agreement between computed and observed fundamental frequ encies speaks for itself, as does that between computed and observed rotational constan ts. At first sight agreement 9for the rotation-vibration coupling constants αeis somewhat disappointing. However, for 16OH−and18OH−, the experimentally derived ‘ αe’ actually corresponds to B1−B0, i.e. to αe−2γe+. . .. If we compare the observed B1−B0with the computed αe−2γeinstead, excellent agreement is found. In the case of16OD−, the experimentally derived αegiven is actually extrapolated from neutral16OD: again, agreement between computed and observed B1−B0is rather more satisfying. We also note that our calculations validate the conclusion b y Lee and Dateo that the experimentally derived ωeandωexefor16OH should be revised upward. D. Dissociation energies of OH and OH−; electron affinity of OH This was obtained in the following manner, which is a variant on W2 theory [44]: (a) the CASSCF(8/7) dissociation energy using ACVTZ, ACVQZ, and AC V5Z basis sets was ex- trapolated geometrically using the geometric formula A+B/Cnfirst proposed by Feller [50]; (b) the dynamical correlation component (defined at CAS(8/7 )-ACPF(all) −CASSCF(8/7)) of the dissociation energy was extrapolated to infinite maxi mum angular momentum in the basis set, l→ ∞ from the ACVQZ ( l=4) and ACV5Z ( l=5) results using the formula [51]A+B/l3; (c) the scalar relativistic contribution obtained at the C AS(8/7)-ACPF level was added to the total, as was the spin-orbit splitting [52] f or O−(2P). Our final result, D0=4.7796 eV, is about 0.02 eV higher than the experimental one [6]; interestingly enough, the same is true for the OH radical (computed D0=4.4124 eV, observed 4.392 eV). In com- bination with either the experimental electron affinity of ox ygen atom, EA(O)=1.461122(3) eV [53] or the best computed EA(O)=1.46075 eV [54], this lead s to electron affinities of OH, EA(OH)=1.8283 eV and 1.8280 eV, respectively, which agr ee to three decimal places with the experimental value [18] 1.827611(4) eV. We note tha t the experimental De(OH−) is derived from De(OH)+EA(OH) −EA(O), and that a previous calibration study on the atomization energies of the first-row hydrides [55] suggest ed that the experimental De(OH) may be too low. While a systematic error in the electronic str ucture treatment that cancels almost exactly between OH and OH−cannot entirely be ruled out, the excellent agreement 10obtained for the electron affinity does lend support to the com puted Devalues. IV. CONCLUSIONS We have been able to obtain a fully ab initio radial function o f spectroscopic quality for the hydroxyl anion. In order to obtain accurate results for this system, inclusion of connected quadruple excitations (in a coupled cluster expansion) is i mperative, as is an account for scalar relativistic effects. Basis set expansion effects bey ondspdfgh take a distant third place in importance. While consideration of connected quad ruple excitation effects and of basis set expansion effects beyond spdfgh would at present be prohibitively expensive for studies of larger anions, no such impediment would appear to exist for inclusion of the scalar relativistic effects (at least for one-electron Darwin and m ass-velocity terms). Our best computed EA(OH), 1.828 eV, agrees to three decimal p laces with the best available experimental value. Our best computed dissociat ion energies, D0(OH−)=4.7796 eV and D0(OH)=4.4124 eV, suggest that the experimental D0(OH)=4.392 eV (from which the experimental D0(OH−) was derived by a thermodynamic cycle) may possibly be about 0.02 eV too low. 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Martin, F. De Proft, and P. Geerli ngs, Phys. Rev. A 60(1999)1034 [55] J. M. L. Martin, Chem. Phys. Lett. 273(1997)98 15TABLES TABLE I. Computed total energy (hartree), bond distance ( ˚A), harmonic frequency (cm−1) and anharmonicity constants (cm−1) of16OH−using the cc-pVDZ+sp(O) basis set as a function of the electron correlation method Ee re ωe ωexe ωeye ωeze FCI -75.623457 0.97503 3701.7 96.65 0.454 -0.024 CCSD -75.616478 0.97209 3747.1 95.28 0.537 -0.010 CCSD(T) -75.622380 0.97442 3711.6 96.45 0.401 -0.031 CC5SD(T) -75.621379 0.97428 3709.5 97.74 0.367 -0.025 CCSDT -75.622656 0.97449 3709.9 96.37 0.465 -0.023 CCSD(TQ) -75.621660 0.97467 3703.1 98.17 0.352 -0.024 CCSD+TQ* -75.621473 0.97463 3702.8 98.48 0.337 -0.023 approx. CCSDT(Q) -75.622937 0.97488 3703.5 96.78 0.452 -0. 022 approx. CCSDT+Q* -75.622750 0.97484 3703.2 97.10 0.438 -0. 020 CAS(8/7)-ACPF -75.623089 0.97499 3703.2 96.60 0.455 -0.02 3 CAS(8/7)-AQCC -75.622147 0.97500 3702.9 96.54 0.456 -0.02 9 CAS(8/8)-ACPF -75.623084 0.97501 3703.0 96.66 0.444 -0.02 4 CAS(8/8)-AQCC -75.622669 0.97493 3704.2 96.59 0.443 -0.02 4 16TABLE II. Computed bond distance, harmonic frequency, anha rmonicity constants, and Dun- ham correction to harmonic frequency for16OH−as a function of basis set and electron correlation method. All data in cm−1except re(˚A) Corr. method basis set 1s corr? re ωe ωexe ωeye ωeze Y10−ωe ν CAS(8/7)-ACPF aug’-cc-pVTZ no 0.96776 3725.01 92.738 0.36 23 -0.0566 -0.37 3540.07 CAS(8/7)-ACPF aug’-cc-pVQZ no 0.96517 3742.24 93.610 0.38 55 -0.0068 -0.24 3556.00 CAS(8/7)-ACPF aug’-cc-pVQZ no+REL 0.96528 3739.00 93.564 0.3881 -0.0066 -0.24 3552.86 CAS(8/7)-ACPF aug’-cc-pV5Z no 0.96476 3745.58 93.856 0.49 68 -0.0192 -0.14 3559.24 CCSD(T) aug’-cc-pVTZ no 0.96741 3733.55 91.987 0.3284 -0.0 524 -0.40 3549.99 CCSD(T) aug’-cc-pVQZ no 0.96486 3750.37 92.948 0.3474 -0.0 121 -0.27 3565.28 CCSD(T) aug’-cc-pV5Z no 0.96456 3751.56 93.183 0.4643 -0.0 227 -0.17 3566.42 CAS(8/7)-ACPF AVTZ no 0.96809 3716.44 92.083 0.2144 -0.013 3 -0.42 3532.49 CAS(8/7)-ACPF AVQZ no 0.96551 3737.30 93.868 0.4277 -0.003 4 -0.19 3550.75 CAS(8/7)-ACPF AV5Z no 0.96488 3744.47 93.816 0.5236 -0.015 7 -0.13 3558.33 CCSD(T) AVTZ no 0.96781 3723.56 91.345 0.1745 -0.0188 -0.46 3540.88 CCSD(T) AVQZ no 0.96520 3745.61 93.159 0.3900 -0.0107 -0.22 3560.29 CCSD(T) AV5Z no 0.96472 3749.39 93.193 0.4966 -0.0291 -0.15 3564.32 CCSD(T) d-AV5Z no 0.96476 3749.31 93.079 0.4900 -0.0283 -0. 16 3564.45 CCSD(T) AV6Z no 0.96466 3750.41 93.237 0.4839 -0.0214 -0.14 3565.26 CCSD(T) AVTZuc no 0.96734 3724.84 92.600 0.4875 -0.0734 -0. 39 3540.46 CCSD(T) AVQZuc no 0.96522 3744.72 93.044 0.4081 -0.0219 -0. 27 3559.58 CCSD(T) AV5Zuc no 0.96473 3749.21 93.243 0.4435 -0.0103 -0. 16 3563.95 CAS(8/7)-ACPF ACVTZ no 0.96789 3713.45 91.642 0.2137 0.000 0 -0.41 3530.45 CAS(8/7)-ACPF ACVQZ no 0.96558 3735.72 93.894 0.4219 -0.01 30 -0.23 3549.01 CAS(8/7)-ACPF ACV5Z no 0.96501 3740.66 94.081 0.4691 -0.00 05 -0.14 3553.87 CCSD(T) ACVTZ no 0.96768 3718.89 91.145 0.1639 -0.0044 -0.4 5 3536.66 CCSD(T) ACVQZ no 0.96525 3744.90 93.038 0.3867 -0.0191 -0.2 6 3559.73 CCSD(T) ACV5Z no 0.96472 3749.22 93.225 0.4361 -0.0101 -0.1 7 3563.96 CAS(8/7)-ACPF ACVTZ yes 0.96725 3714.74 92.017 0.1855 -0.0 035 -0.43 3530.86 CAS(8/7)-ACPF ACVQZ yes 0.96468 3741.86 94.110 0.4205 -0.0 129 -0.23 3554.71 CAS(8/7)-ACPF ACV5Z yes 0.96410 3746.51 94.317 0.4682 0.00 09 -0.14 3559.26 CCSD(T) ACVTZ yes 0.96688 3725.04 91.122 0.1509 -0.0022 -0. 46 3542.81 CCSD(T) ACVQZ yes 0.96435 3751.76 93.151 0.3929 -0.0202 -0. 26 3566.37 CCSD(T) ACV5Z yes 0.96378 3756.27 93.347 0.4427 -0.0088 -0. 17 3570.80 CAS(8/7)-ACPF ACVQZ all yes+REL 0.96478 3738.69 94.098 0.4 193 -0.0102 -0.24 3551.57 ∆REL 0.00010 -3.17 -0.012 -0.0012 0.0027 -0.01 -3.14 best calc. 0.96417 3742.87 94.404 0.4527 0.0100 -0.14 3555.44 The suffix “+REL” indicates inclusion of scalar relativistic (Darwin and mass-velocity) ef- fects obtained as expectation values for the wave function i ndicated. 17TABLE III. Effect of scalar relativistic contributions on th e bond lengths ( ˚A) and harmonic frequencies (cm−1) of the AH (A=B–F) diatomics. All calculations were carried out at the CAS(2 σ3σ4σ1π)-ACPF/ACVQZ level with all electrons correlated ∆re ∆ωe BH -0.00001 -0.57 CH +0.00001 -1.08 NH +0.00003 -1.77 OH +0.00004 -2.35 HF +0.00005 -2.80 OH−+0.00010 -3.14 Effects on the anharmonicity constants are negligible. TABLE IV. Parameters for the VBM representation, eq. (1), ob tained from our best potential. De,reare in cm−1and˚A, respectively; the remaining parameters are dimensionle ss De 40398.7079 re 0.964172 β 2.128977 b1 -0.047181 b2 0.022371 b3 -0.0070906 b4 0.0018429 18TABLE V. Spectroscopic constants and band origins (in cm−1) of different isotopomers of the hydroxyl anion obtained from our best potential 16OH− 16OD− 18OH− 18OD− calc obsdacalc obsdbcalc obsdacalc Y00 2.38 1.26 2.36 1.25 Y10≈ωe 3742.72 3738.44(99)c2724.79 2723.5(10) 3730.35 2707.77 −Y20≈ωexe 94.298 91.42(49)c49.979 49.72(50) 93.676 49.357 Y30≈ωeye 0.4686 0.1808 0.38(15) 0.4639 0.1774 Y01≈Be 19.126021 19.12087(37) 10.136936 10.13599(30) 18.999788 18.99518(49) 10.010698 −Y11≈αe 0.779874 0.77167(13) 0.300914 0.3043(5) 0.772165 0.76409 (16) 0.295310 Y21≈γe 0.003913 0.001099 0.003861 0.001072 αe-2γe 0.772048 0.77167(13) 0.298716 0.2984(3) 0.764443 0.76409 (16) 0.293166 −Y02≈De 0.001998 0.001995(6) 0.000561 0.000559(2)d0.001972 0.000031(2) 0.000547 Y12≈βe 0.000032 0.000032(2) 0.000006 0.000008(2) 0.000031 0.000 031(2) 0.000006 ZPVE 1850.23 1351.19 1844.18 1342.81 G(1)-G(0) 3555.63 3555.6057(22) 2625.42 2625.332(3) 3544 .49 3444.4551(28) 2609.63 G(2)-G(1) 3371.17 2527.06 3361.24 2512.49 G(3)-G(2) 3189.42 2429.75 3180.66 2416.38 G(4)-G(3) 3010.39 2333.49 3002.78 2321.29 G(5)-G(4) 2834.11 2238.28 2827.63 2227.23 G(6)-G(5) 2660.70 2144.12 2655.31 2134.21 G(7)-G(6) 2490.31 2051.03 2485.97 2042.24 The Dunham constants Ymninclude higher-order corrections to the mechanical spectr oscopic constants (like ωe, ωexe) as obtained from the potential function. (a) Ref. [14]. Uncertainties in parentheses correspond to t wo standard deviations. (b) Ref. [15]. Uncertainties in parentheses correspond to t hree standard deviations. (c) LD proposed ωe=3741.0(14) and ωexe=93.81(93) cm−1, obtained by mass scaling of the 16OD−results, as more reliable. (d) From observed D0andD1in Ref. [15]. 19FIGURES FIG. 1. Deviation from the FCI potential curve of OH−for different electron correlation methods 00.00050.0010.00150.0020.0025 0.87 0.92 0.97 1.02 1.07 r[O-H] (Å)E-E[FCI] (hartree)CCSD(T) CCSDT CCSD+TQ CCSD+TQ* CC5SD(T) CAS(8/8)-ACPF CAS(8/8)-AQCC approx. CCSDT+Q* approx. CCSDT+Q 20

arXiv:physics/0003040v1 [physics.optics] 16 Mar 2000Pulsed pump in optical displacement transducer for experim ents with probe bodies Victor V. Kulagin Sternberg Astronomical Institute, Moscow State Universit y, Universitetsky prospect 13, 119899, Moscow, Russia, e-mai l: kul@sai.msu.ru Abstract The sensitivity of the displacement transducer pumped with a train of high-intensity laser pulses is estimated. Due to the multicomponent charac ter of the pump a consid- eration of transformations of the signal and the noises betw een optical modes plays an important role in estimation of the potential sensitivity. An expression for the mini- mal detectable external classical force resembles those fo r the continuous wave pumping with substitution of the laser power by a time averaged power of pulsed laser. Possible scheme for back action noise compensation for such transduc ers is considered. For full suppression of back action noise the field of local oscillato r has to be pulsed with the same time dependence as the pump field. PACS: 03.65.Bz; 42.50.Dv; 42.50.Lc 1 Introduction The longbase laser interferometric gravitational wave det ectors are under construction at present time [1-3]. Their sensitivity to metric perturbati on will be about h≈10−21that cor- responds to the classical regime of operation. However for f uture installations with projected sensitivity 10−22÷10−23the quantum features of the measurement process can play a si gnif- icant role. At the same time there are no limits of principle o n the accuracy of measurement of external classical force. Therefore the methods and sche mes which give the possibility to overcome the quantum measurement limitations (or the so cal led standard quantum limit, SQL) is of vital importance for future generation of gravita tional wave experiments. There are several procedures which allow to achieve the sens itivity larger than the SQL [4,5]. For example in [5] an optimal filtration procedure for the simplest variant of the optical sensor - a mirror attached to a mechanical resonator and illuminated with a coherent pump field - was considered. An external force acting on the me chanical oscillator displaces its equilibrium position and thus changes the phase of the re flected field. The vacuum fluctuations of the input light act on the oscillator through the radiation pressure effect and constitute the back action noise of the measuring appara tus. For such system two quadratures of reflected wave are correlated. Using correla tion (phase sensitive) processing of two quadratures one can increase a signal-to-noise ratio and overcome the SQL. However the gain in sensitivity for the schemes overcoming t he SQL is usually propor- tional to the square root of the ratio of laser power used for p umping the interferometer and an optimal power that corresponds to the point where the sens itivity of the interferometer achieve the SQL [4-6]. Unfortunately the optimal power is im practicably large, about several dozens of kilowatts that restrains the experimental implem entation of the technique. 1The pumping with the ultrashort periodic laser pulses can be technically advantageous over a continuous wave pumping for practical realization of the schemes overcoming the SQL. Actually for a large power a problem of generating a train of s hort high-intensity laser pulses can be technically easier than a problem of cw light generati on (when the averaged powers for two cases are equal) because in the first case the energy in laser resonator is spread over the large frequency band (and different spatial longitudina l modes) and high intensities can be produced relatively easily. At the same time the amplitud e and frequency stability of the pulsed pump in the case of a mode locked laser can be at the s ame level as for the monochromatic pump [7,8]. For example in [8] the stability o f intermode beats for the mode locked laser output was estimated as 5 ·10−12in 10 s. Another consideration is that the perspectives of squeezed states generation with high nonclassicality seem more realistic for the case of short hi gh-intensity laser pulses allowing the use of squeezed pulsed pump in displacement transducers [9]. Finally an analog to digital conversion is usually used in mo dern experiment during the processing of the output. Therefore it seems natural to take the pulsed pump at once so that the output will comprise a set of the values for appropri ate variable at definite times. The goals of this article are to consider a displacement tran sducer consisting of a mirror attached to a mechanical oscillator and illuminated with a t rain of high-intensity laser pulses, to reveal the algorithm of optimal signal processing for suc h transducer and to estimate the sensitivity of the scheme to a measurement of classical exte rnal force. The model of displacement transducer and basic equations of motion is considered in section 2. The sensitivities for traditional measurement s cheme and for correlative processing of the output quadratures in the case of time independent pum p are estimated in sections 3 and 4 correspondingly. The pulsed pump for the displacemen t transducer is considered in section 5. The conclusions are in section 6. 2 Model for displacement transducer and transforma- tion of quadrature components Let consider the most simple case of optical displacement tr ansducer - a mirror attached to a mass of a mechanical oscillator and illuminated with a trai n of high-intensity laser pulses. An external force displaces an equilibrium position of mech anical oscillator changing the phase of reflected wave. The variation of the reflected field ph ase is measured by a readout system. This model is easy to calculate and it contains at the same time all features of displacement transducers with pulsed pump. For the inciden tEiand reflected Erwaves one can use the quasimonochromatic approximation Ei= (A(t−x/c) +a1)·cosωp(t−x/c)−a2·sinωp(t−x/c) Er= (B(t+x/c) +b1)·cosωp(t+x/c)−b2·sinωp(t+x/c) (1) where A(t−x/c) and ωpare an amplitude (mean value) and a frequency of the pump wave , a1anda2are the operators of the quadrature components (fluctuation s) of the pump wave (vacuum for coherent state) , B(t+x/c) is an amplitude (mean value) of the reflected wave, b1 2andb2are the operators of the quadrature components (fluctuation s) of the reflected wave. The periodic envelope function A(t−x/c) consists of a train of equally spaced pulses and the duration of each pulse is considerably larger than the perio d of light wave but considerably smaller than the period of the mechanical oscillator. To obtain the equation coupling the amplitudes of the incide nt and reflected waves for the moving mirror one can use a transformation of electromag netic field for moving reference frame [10]. For a constant velocity of the mirror Vone has Er=−[(1−V/c)/(1 +V/c)]·Eiexp(−2iωpX/c) (2) where for simplicity the reflection coefficient of the mirror i s taken to be r≈ −1 and Xis the position of the mirror. Let suppose that this expression is v alid also for the slowly varying velocity V(t) and position X(t) of the mirror and |V(t)|≪c(the validity of equation (2) has been proved for the mirror consisting of free electrons f or the general case of relativistic velocity V(t) in [11]). Then in linear approximation in V/cone can obtain from equation (2) the following expression Er=−(1−2V(t)/c−2iωpX(t)/c)·Ei (3) The first term in (3) is an amplitude modulation of the reflecte d wave due to the mirror movement and the second is a phase modulation. For slow motio n of the mirror V≈ωµX (ωµis a frequency of mechanical oscillator) and the second term in brackets is considerably smaller than the third term. Therefore for the transformati on of the quadrature components of the field one can obtain b1(t) = −a1(t) b2(t) = −a2(t) + 2ωpA(t)X(t)/c (4) For the equation of mirror motion one has ¨X(t) + 2δµ˙X(t) +ω2 µX(t) =M−1(Fs(t) +Fp(t) +Fth(t)) (5) where Mandδµare the mass and the damping coefficient of mechanical oscilla tor,Fs(t) is a signal force, Fp(t) is radiation pressure force and Fth(t) is a force associated with the damping of the oscillator. Let suppose for simplisity that δµtends to zero. Then the displacement X(t) of the mirror will consist of two parts - a signal displaceme ntXs(t) and a radiation pressure displacement Xp(t). For Fp(t) one has Fp(t) =SA(t)·a1(t)/(4π) (6) where Sis a cross section of the laser beam. Therefore the equations of motion for the displacement transducer are b1(t) = −a1(t) b2(t) = −a2(t) + 2ωpA(t)X(t)/c (7) ¨X(t) + 2 δµ˙X(t) +ω2 µX(t) =M−1(Fs(t) +SA(t)·a1(t)/(4π)) 33 Sensitivity for a traditional measurement scheme For traditional measurement scheme [4,6] the amplitude of t he pump is constant. There- fore one can easily obtain the transformation relations for the quadratures b1andb2from equations (7) b1(ω) = −a1(ω) b2(ω) = −a2(ω) +λξ(ω)A2a1(ω) +Aξ(ω)Fs(ω) (8) where ξ(ω) = 2ωpG(ω)/c,G(ω) =/bracketleftBig M(−ω2−2δµiω+ω2 µ)/bracketrightBig−1is mechanical oscillator trans- fer function and λ=S/(4π). Only quadrature b2contains the signal and it is this quadrature that is measure d in traditional measurement scheme [4,6]. This corresponds to the measurement of the phase of the reflected wave. The first term in the right hand side of equa tion (8) for b2can be treated as an additive noise and the second term as a back action noise . For small pump amplitudes the sensitivity is increasing with the increase of Abecause the signal is proportional to A. However for large pump amplitudes the second term in r.h.s. o f (8) becomes dominant and the sensitivity is decreasing with the increase of A. Therefore there is an optimal value of the pump amplitude and the sensitivity to external force at t his pump amplitude is just the SQL [6]. 4 Correlative processing of quadratures for time inde- pendent amplitude of the pump Two quadratures of the reflected field according to equation ( 8) have the dependence on the amplitude fluctuations of the incident field a1. Therefore one can expect that the sensitivity can be increased for the correlative processing of the outpu t [5,12]. Actually if one combine with appropriate weight coefficients the quadratures b1andb2of the output wave then in this combination the noise term depending on a1can be cancelled. This weighting can be done by a homodyne detector with appropriate choise of a loca l oscillator phase φ. Let the field of the local oscillator have the form EL(t) =ALcos(ωpt+φ) (9) Then the photodetector output is proportional to the follow ing expression according to (1), (9) Ipd∝AL(b1cosφ+b2sinφ) (10) and at certain frequency ωfone can obtain Ipd∝AL[a1(ωf)(−cosφ+λξ(ωf)A2sinφ)−a2(ωf) sinφ+Aξ(ωf)Fs(ωf) sinφ] (11) Therefore choosing the phase φaccording to the equation ( ξ(ωf) is real for δµ= 0) 4−cosφ+λξ(ωf)A2sinφ= 0 (12) one can make the photocurrent insensitive to the amplitude fl uctuations a1of the input field at certain frequency ωfof the signal. In this case the increase of the pump amplitude A results in the relative increase of the output signal at freq uency ωfaccording to the equation (11) with respect to the noise level defined by a2. For compensation of the back action noise inside definite fre quency band one has to use the time dependent local oscillator phase φ(t) [12,13]. In this case the optimal dependence of φontis defined by the displacement transducer transfer function ξ(ωf) and by the spectrum of the external force Fs(ω) [12]. So a signal-to-noise ratio is proportional to A2(there is no optimal power) and in principle there is no sensitivity limitation by the SQL. In real experi ment when the pump power gets larger the output signal and noises get smaller according to equation (11) if the condition (12) is kept valid therefore when Abecomes greater than a certain value then the noises of photodetector electronics can limit the sensitivity. Ho wever this noises have technical character and will be neglected in the following. Another sensitivity restriction can arise due to the dampin g in mechanical oscillator (mirror) [14,15]. This problem is general for all supersens itive measurements. At the same time an intrinsic dissipation obtained in modern experimen ts for mechanical oscillator is far larger (by several orders of magnitude) than the value expec ted from the first principles [16] therefore it can be treated also as a technical problem now an d will not be adressed below. It is worth to mention that the increase in sensitivity over t he usual measurement scheme occurs here due to utilization of the internal squeezing (se lf-squeezing) of the reflected beam because of the nonlinear (quadratic) interaction of the inc ident light and the mirror [17,18]. Actually two quadratures of the reflected beam are correlate d and it is this fact that allow to use the correlative processing of the output. On the other hand the correlation of the quadratures according to equations (8) means the squeezing of the beam and the larger the correlation coefficient λξ(ω)A2the larger the internal squeezing [17]. 5 Sensitivity for the pulsed pump Let consider the periodic envelope A(t) which consists of a train of equally spaced pulses with duration τand period T. The spectrum of this pump has also the form of a train of pulses in frequency domain with the distance between neighb our pulses ωq= 2πT−1(13) For the amplitude of the pump A(t) one can use now the expansion into the Fourier series A(t) =∞/summationdisplay n=−∞gnexp(−inωqt) (14) and the particular form of A(t) is defined by the set of Fourier amplitudes gn. The response of the displacement transducer now have many fr equency components at ω=nωq, n= 0,1. . .according to the equations (4) and each frequency component contains 5the signal part besides the radiation pressure force Fp(t) have also wide spectrum (cf. (6)). So there are two problems: how to collect the signal parts fro m the whole spectral band of the output and how to achieve the compensation of the radiation p ressure noise in the output. It is clear that the monochromatic local oscillator is inappro priate for the homodyning because quadrature b1(t) of the output signal contains in this case the quadrature a1(t) of the input noises only from one frequency and the radiation pressure fo rceFp(t) in expression for b2(t) (cf. (7)) contains a1(t) from all frequencies nωqtherefore the full compensation is impossible. Fortunately two problems can be overcome by the use of the pul sed local oscillator with the amplitude time dependence resembling that for the pump. For the radiation pressure displacement Xpof the mechanical oscillator one has from equations (5), (6) and (14) the following expression Xp(ω) =G(ω)Fp(ω) =λG(ω)∞/summationdisplay n=−∞gna1(ω−nωq) (15) For the quadrature transformation one can obtain instead of (8) the following equations from (4) and (14) b1(ω) = −a1(ω) b2(ω) = −a2(ω) + 2ωpc−1∞/summationdisplay k=−∞gk(Xp(ω−kωq) +Xs(ω−kωq)) (16) Let suppose the local oscillator field in the form of EL(t) =AL(t) cos(ωpt+φ) (17) where the dependence of the amplitude AL(t) ontis much slower than cos ωpt. Then for the envelope of the local oscillator field AL(t) the Fourier expansion similar to (14) is valid AL(t) =∞/summationdisplay n=−∞enexp(−inωqt) (18) The photodetector current has now the following form Ipd∝AL(t)(b1(t) cosφ+b2(t) sinφ) (19) and in the frequency domain one has Ipd(ω)∝cosφ·∞/summationdisplay n=−∞enb1(ω−nωq) + sin φ·∞/summationdisplay n=−∞enb2(ω−nωq) (20) Let consider different parts in the photodetector output. Th e first term in equation (20) depends only on the amplitude fluctuations of the input field a ccording to (16) cosφ·∞/summationdisplay n=−∞enb1(ω−nωq) =−cosφ·∞/summationdisplay n=−∞ena1(ω−nωq) (21) 6The second term in equation (20) contains the signal and the n oise parts. The noise part consists of the additive noise and the back action noise and h as the following expression according to (15) and (16) [sinφ·∞/summationdisplay n=−∞enb2(ω−nωq)]noise=−sinφ·∞/summationdisplay n=−∞ena2(ω−nωq) + 2ωpc−1sinφ·λ· ∞/summationdisplay n=−∞∞/summationdisplay k=−∞engkG(ω−kωq−nωq){∞/summationdisplay m=−∞gma1(ω−kωq−nωq−mωq)} (22) Let consider only the photocurrent at small frequencies ω≈ωµ. Then the main input into the photocurrent will be given by the resonant terms for which k+n= 0. With this supposition one has from equation (22) [sinφ·∞/summationdisplay n=−∞enb2(ω−nωq)]noise=−sinφ·∞/summationdisplay n=−∞ena2(ω−nωq) + sinφ·λξ(ω)·∞/summationdisplay m=−∞emg−m·∞/summationdisplay n=−∞gna1(ω−nωq)} (23) Comparing equations (21) and (23) one can conclude that full compensation of back action noise is possible only for en=αgn (24) where αis the same for all numbers nso the forms of pump and local oscillator fields have to be the same (apart from the scale factor α). Let now consider the signal part of the second term in the r.h. s. of equation (20). From equations (7), (16) and (20) one has [sinφ·∞/summationdisplay n=−∞enb2(ω−nωq)]signal= sinφ·∞/summationdisplay n=−∞en{∞/summationdisplay k=−∞gkξ(ω−kωq−nωq)Fs(ω−kωq−nωq)} (25) Evaluation of this expression for the condition k+n= 0 gives [sinφ·∞/summationdisplay n=−∞enb2(ω−nωq)]signal= sinφ·ξ(ω)Fs(ω)∞/summationdisplay n=−∞eng−n (26) Combining equations (20), (23), (24) and (26) and supposing that the back action noise is compensated in the output of the photodetector one can obtai n for the spectral density of noises in the photocurrent the following expression N(ω)∝sinφ·N0·∞/summationdisplay n=−∞gng−n= sinφ·N0P (27) 7where it is supposed that fluctuations at frequencies ω−nωq, n= 0,1. . .are uncorrelated and have the same spectral density N0(this assumption is valid for not very small duration of pump pulses), Pis proportional to the time averaged power of the pulsed pump . Then for the signal-to-noise ratio µone has from equations (26) and (27) the following expressio n µ∝N−1 0P/integraldisplay∞ −∞|ξ(ω)Fs(ω)|2dω (28) This value is just equal to the signal-to-noise ratio for con tinuous wave pump with a power Pand correlative processing of the output (cf. equation (11) ). Note that the sensitivity here is not limited by the SQL like in the case of correlative proce ssing of quadratures for the monochromatic pump and is increasing with the increase of P. It is worth to mention that the condition for the back action n oise compensation for the pulsed pump is just the same as for the monochromatic pump (cf. equation (12)) with substitution of the A2with the time averaged value P. Therefore the compensation of the back action noises for the finite frequency band can be possib le for the time varying phase of the local oscillator [12, 13]. 6 Conclusion The pumping of the displacement transducer with a train of th e short high-intensity laser pulses is considered. The algorithm of optimal signal proce ssing for such transducer is re- vealed. It consists of the correlative processing of the out put using the pulsed local oscillator with the same envelope as for the pump field (apart from the sca le factor). In this case the back action noise due to the radiation pressure force can be fully compensated and the sensitivity of the scheme to a detection of a classical exter nal force is not limited by the SQL (as for the case of correlative quadrature processing and mo nochromatic pump field). The pulsed pump can be advantageous over the single frequenc y pumping when the non- linear optical elements are used unside the system. Thus con siderable increase in sensitivity can be achieved for a gravitational interferometric Fabry- Perot type detector with a non- linear optical element placed in a waist of the beam [19]. The use of the phase-conjugate mirrors in a gravitational detector of the Michelson type al lows to construct the system with the parallel arms [20]. For such systems an efficiency depends on the instant power of the light beam and can be high for the short intensive pulses. In this article only the problem of the force detection with k nown spectrum is considered. The reconstruction of unknown external force acting on the d isplacement transducer with the pulsed pump below the standard quantum limit will be cons idered elsewhere. References [1] Thorne K. S. 1987 300 Years of Gravitation eds. S. W. Hawking and W. Israel (Cam- bridge Univ. Press, Cambridge) p. 330. [2] Abramovici A. et al 1992 Science 256325. 8[3] Brillet A. et al 1991 The Detection of Gravitational Waves ed. D. G. Blair (Cambridge Univ. Press, Cambridge) p. 369. [4] Caves C. M., Thorne K. S., Drever R. W. P., Sandberg V. D., Z immerman M. 1980 Rev. Mod. Phys. 52341. [5] Gusev A. V., Kulagin V. V. 1997 Appl. Phys. B64137. [6] Caves C. M. 1981 Phys. Rev. D231693. [7] Spence D. E., Dudley J. M., Lamb K., Sleat W. E., Sibbett W. 1994Opt. Letters 19 481. [8] Bagaev S. N., Denisov V. I., Korel I. I. et al. 1998 Abstracts IX Int. Conf. on Laser Optics (St. Petersburg, Russia, June 1998) p. 79. [9] Kulagin V. V. 1998 Proc. 5th Int. Wigner Symp. ed. P. Kasperkovitz, D. Grau (Singa- pore: World Scient. Publ. Co.) p. 509 [10] Landau L. D., Lifshitz E. M. 1988 Classical Theory of the Field (N. Y. - L.: Pergamon Press). [11] Il’in A. S., Kulagin V. V., Cherepenin V. A. 1999 J. of Commun. Technology and Electronics 421158. [12] Gusev A. V., Kulagin V. V. 1996 Proc. 4th Int. Conf. on Squeezed States and Un- certainty Relations (Taiyuan, Shanxi, P. R. China) NASA Conference Publ. 3322, p. 427. [13] Vyatchanin S. P., E.A. Zubova E.A. 1994 Optics Communications 111303. [14] Gusev A. V., Kulagin V. V. 1997 Quantum Communication, Computing and Measure- ment, ed. O. Hirota, A. S. Holevo, C. M. Caves (Plenum Press, N.Y.- L) p. 491 [15] Jaekel M. T., Reynaud S. 1990 Europhys. Lett. 13301 [16] Braginsky V. B., Mitrofanov V. P., Okhrimenko O. A. 1993 Phys. Lett. A 17582 [17] Kulagin V. V. 1998 Gravitational Wave Experiments , ed. E. Coccia, G. Pizzella, F. Ronga (World Scientific Publ. Co., Singapore) 2p. 428 [18] Heidmann A., Hadjar Y., Pinard M. 1997 Appl. Phys. B64173. [19] Kulagin V. V., Rudenko V. N. 1990 Phys. Lett. A ,143p. 353 [20] Grishchuk L. P., Kulagin V. V., Rudenko V. N. 1989 Proc. 5th Marcel Grossman Meeting (Perth, Australia, 8-13 August) p. 345 9

arXiv:physics/0003041v1 [physics.class-ph] 16 Mar 2000Cavity equipartition revisited V Guruprasad∗ T J Watson Research Center, Yorktown Heights, NY 10598 Using hindsight from Landauer’s principle and the FPU probl em, I show that Planck’s law implies a classical equipartition of antinodal lobes instead of who le modes, which makes sense because the information of excitation is inherently replicated in each antinode of a given mode. I show that the law could have been derived by considering merely the classi cal thermalisation due to wall jitter, which is not considered in the quantum derivations. As a resu lt,hemerges as the spectral equivalent ofkBin this context. I. INTRODUCTION Ever since Planck’s discovery a hundred years ago, it has bee n thought that “classical theory is absolutely incapable of describing the distribution of light from a blackbody” [1 , I-41-2]. Applying classical rules of equipartition to the wall oscillators had led to the ultra-violet divergence in R ayleigh’s law, and the correct law was achieved only after the wall interactions were assumed to be quantised [2]. Quan tisation of the radiation itself [1, III-4-5] is of course th e more precise picture we have today, but why the procedure sho uld at all work remains a mystery. Furthermore, the Fermi-Pasta-Ulam problem [3] [4, §5.5.1] [5] makes it possible that thermal equilibrium might not be established by time-symmetric micro-dynamics alone, so the existing deri vations cannot be relied on as the complete picture. I show below that at least in the case of radiation, the necessity of quantum mechanics happens to be purely the result of faulty thermodynamics, of not considering the involvement of wall thermal motion. More particularly, I obtain Planck’s law by attributing the thermalisation to wall jitter, which overcomes the FPU problem by indirectly involving the rest of the universe, an d more importantly, affects the radiation by constantly changing the cavity dimensions, i.e. without depending on nonlinearity of wall reflections or photonic ab sorption and emission. Thermal equilibrium is thus guaranteed within th e cavity and with its environment even at frequencies well below the atomic spectra, and the wall jitter continually ch anges the stationary modes of the cavity, but always in whole numbers of antinodal lobes, assuming that the jitter m otion is generally slow compared to the electromagnetic transit times within the cavity. This is sufficient basis for a pplying equiparition over the antinodal lobes, rather than whole modes, and directly leads to Planck’s law. The present derivation fundamentally breaks from the past i n not assuming quantisation. The fact that Planck’s law emerges nonetheless implies that photonic transitions , which have been basis of all past work, are not essential in the problem. Instead, the very notion of stationarity guara ntees that any such transitions must likewise involve whole numbers of antinodes, maintaining consistency with the cla ssical quantisation. It will be shown that the condition of equilibrium suffices to ensure the consistency of the trans ition probabilities with those of the classical exchange of energies between the modes by Doppler spreading and uneven r eflective scattering due to the wall jitter. Although the derivation does not suffice to explain other aspe cts of quantum physics, such as entanglement and wave-particle duality, for example, there is some new insig ht to be gained relating thermodynamics and quantum theory, specifically that Planck’s constant hnow emerges as the equivalent of kB, the Boltzmann constant, for the spectral domain. Furthermore, we arrive at an unsuspected degree of laxity, a s the antinodal quantisation is only relevant under the stipulations of stationarity and equilibrium, so that phot onic transitions must intimately involve thermal equilibr ium as well. This is entirely consistent with the observation th at the very detection of photons necessarily depends on an irreversible change in the observing system , since every act of measurement or learning must change the m acroscopic physical state of the observer representing its state of kno wledge. Correspondingly, the laxity is consistent with the causality of undisturbed quantum systems, which is the prem ise of Schr¨ odinger’s equation [6, §27]. By our reasoning, at absolute zero temperature, there would be no wall jitter a nd therefore no mechanism to mix energies across the modes, so that the radiation Hamiltonian becomes not only ca usal but also deterministic, being governed entirely by Maxwell’s equations. Photonic interactions at the walls co uld violate this classical determinism, but the concern is purely pendantic in absence of observations. Each observat ion would still incur photonic events, but the probabilitie s involved are now clearly those arising from the thermal irre versibility of the observer’s changing state of knowledge. ∗Electronic address: prasad@watson.ibm.com2 At ordinary temperatures, of course, the wall jitter suffices for randomising the energy distribution. We are thus able to fundamentally attribute the statistical nature of quant um mechanics entirely to thermal randomisation. This is not a contradiction of prior theory as the inherent irrevers ibility of learning was only discovered in 1961 [7]. II. PHYSICAL INFORMATION Presumably, as will become clear from the following section s, the present derivation is largely a reverse-engineering of the Planck and the Bose-Einstein derivations. The differenc e, as stated, is that we no longer depend on the assumption of quantisation as in the past, but are able to deduce it on the basis of classical mechanics, viz by recognising that wall jitter continually changes the cavity dimensions, forcing stationary modes to change by whole numbers of antinodal lobes, and thus thermalises the radiation. The quantisatio n thus follows from the very notion of stationary modes, but we need to be able to relate it to sound principles of measu rement and thermodynamic information, as follows. At least at microwave frequencies, a precise determination of the amplitude and instantaneous phase in a given cavity mode involves measuring the induced current in a suffic iently thin probe at the mode frequency. Depending on the location of such a probe, the induced current jwill vary as j(x, t) = [aE0sinφ+bωB0cosφ]eiωt, φ ≡2πx/λ, (1) where EandBare the maximum electric and magnetic field intensities, i.e . at the antinodal points φ=nπ+π/2, along each coordinate x, provided that we are not too close to the walls. Eq. (1) also d escribes the fields acting on the wall atoms as the latter are, like the microwave probes, much smaller than the wavelengths of interest. Importantly, we need to measure EandBat only one location, in order to determine E0andB0everywhere else in the cavity, so theinformation of excitation is independently available at almost every point within the cavity. This also means that the energy of the mode must be thermodynamically active at th ese points, i.e. that these non-nodal points, rather than whole modes as in the Rayleigh-Jeans theory, must be con sidered as the potential candidates for the correct classical equipartition. This at first poses a measure-theo retic difficulty, as the points constitute a continuum except for the countable set of nodes, but as the notion of spectral d istribution specifically concerns stationary modes, we only need to consider whole antinodal lobes. The notion that the antinodal lobes are thermodynamically significant independently, is evident in the fact that the energy of a lob e, u=1 2/integraldisplayπ 0/bracketleftbig ǫ0(E0sinφ)2+µ−1 0(B0cosφ)2/bracketrightbig dφ =1 4[ǫ0|E0|2+µ0|B0|2], where φ≡2πx/λ,(2) isindependent of the frequency . Per our ideas, we must now apply classical, i.e. Boltzmann, equipartition rules to the antinodal lobes, obtaining p≡p(U) =e−U/k BT(3) for the probability of excitation of a given lobe to energy U. As the number of antinodal lobes in a mode is proportional to its frequency, their frequency-independence leads dire ctly to Planck’s quantisation rule Um(f) =Uf, (4) Udenoting the energy of any one antinodal lobe, correspondin g, as will be shown, to h. The corresponding probability of modal excitation is, accordingly, pm(f) =pf=e−Uf/k BT. (5) Eqs. (3-5) are identically applicable to fixed fractions or m ultiples of the antinodal lobes, for which the frequency- independence property clearly holds as well, prompting our interpretation of has the spectral analogue of kB. III. DYNAMIC STATIONARITY A set of modes differing by an exact number of antinodal lobes c onstitutes a harmonic family of the form {f, 2f, 3f, ...}. The energy sum of the harmonic family of fis then expected to be Uh(f) =U(f·pf+ 2f·p2f+ 3f·p3f+...) =Uf·pf (1−pf)2,(6)3 which is spread over a total of nh(f) = 1 + pf+p2f+p3f+... =1 1−pf(7) lobes, yielding for the energy expectation of the mode /hatwideUm(f) =Uf p−f−1=Uf eUf/k BT−1. (8) This is identical to Planck’s law, but with the lobe energy Ureplacing h, showing that the formof the law does not depend on quantum assumptions and is simply indicative of eq uipartition over antinodes. The question to be examined, naturally, is why we had needed h armonic oscillators in the first place. The oscillator energy levels are exactly the same as the member frequencies of the harmonic families, which seems to imply that, correspondingly, a given family may have only one member fre quency active at a time. Since eqs. (6-8) do not depend on this restriction, we need to understand why this was neede d in the quantum picture. Recall that in the quantum derivations, the cavity was assum ed to have a fixed set of stationary modes, which is impossible given the thermal motion at the walls, and conv ersely, the premise tends to make the very occurrence of thermalisation classically unlikely if not impossible. More importantly, however, notice that in our classical wal l jitter picture, the modes themselves are constantly changi ng. In fact, in absence of photonic transitions, the only way energy can be removed from one mode and given to another is whe n the first mode itself disappears and the second gets created. Since both the initial and the final modes contain a whole numb er of antinodal lobes by definition, the change involved is always integral in the number of such l obes. Since the stationary modes are fixed in Planck’s theory, their lobes are fixed as well, making it necessary to r estrict the energy exchanges separately. The harmonic oscillator indeed reproduces our notion of mode changes ind irectly, because when an oscillator loses or gains energy, it can no longer operate at its original frequency, and, as no ted above, its eigenfrequencies do constitute a harmonic family. The oscillator concept clearly is a kludge, however , and our use of the families is purely as a mathematical tool in evaluating the modal expectation Um(f). IV. DETAILED BALANCE Since we did not consider the precise effects of the Doppler sp reading and non-uniform scattering due to the wall jitter, let alone the nonlinearity of the wall atoms and thei r quantum transitions, we need to ensure that the modal energy expection Um(f) would indeed be consistent with all such mechanisms. This g uarantee is provided, rather ingeniously in hindsight, by the Bose-Einstein derivation [1, III-4-5]. Consider a pair of modes containing mfandnf antinodal lobes, whose probabilities of excitation would b e related by eq. (5) as Pr[n] Pr[m]=e−(n−m)U/k BT. (9) By eq. (4), their energies would be m·Ufandn·Uf, respectively. Regardless of the mechanisms responsible f or the thermalisation, the condition of equilibrium demands that thepower flow between the modes would be balanced, so that Pr[m]·m·Uf=Pr[n]·n·Uf, (10) which, on combining with eq. (9), yields m n=e−(n−m)Uf/k BT. (11) Setting n−m= 1 and solving for nproduces n=1 eUf/k BT−1, (12) so that the energy expectation of the n-lobe mode turns out to be /hatwideUm(f) =n·Uf=Uf eUf/k BT−1, (13) the same as eq. (8). We have thus proved that the conditions of stationarity and equilibrium indeed suffice to insulate the spectral distribution from the precise mechanisms resp onsible for the exchange of energy between the modes. ✷4 V. CONSTANCY BY TRANSITIVITY It remains to be shown that Uis not specific to a given cavity and its total energy, but a uni versal constant identifiable as h. The reasoning for this is partly contained in Landauer’s pr inciple, as the physical states of the observing system representative of data are necessarily st ationary as well. Since Fourier theory defines a spectral component to extend over infinite time, the observation of sp ectral lines necessarily requires enough time for the establishment of equilibrium between the source and the obs erving system. Both our conditions of stationarity and equilibrium are thus applicable to every observer, and betw een every pair of cavities compared by a given observer. If the cavities were to be observed at different times, or with di fferent instruments, the value of Uin each case would be necessarily identical so long as the cavities are maintaine d at the same temperature; in the first case, it is a necessary premise that the observer’s data bearing state is maintaine d intact between the observations, and in the second, the implicit act of calibration against a common referent ensur es the transitivity. Key to the transitivity argument is the fact that Uappears in the exponent term in eqs. (8) and (13), so that Uf/k BTmust be a pure number, which makes Ucomputable from the shape of the spectrum and the temperatur eT. Eqs. (8) and (13) thus “expose” hto direct measurement in much the same way as the mean square t ravel in Brownian motion, ∝an}bracketle{tR2∝an}bracketri}ht ∝kB[1, I-41-10], exposes the Boltzmann constant. We can thus co nclude that U(h) is inherently a thermodynamic constant specific to the spectral domain anal ogous to kBin the positional one. Further support comes from Dirac’s demonstration [6, §21] that given any anti-commutating relation [ ., .] and four dynamical variables u1,v1,u2andv2, we would obtain, with no assumption of inter-dependence, [u1, v1](u2v2−v2u2) = (u1v1−v1u1)[u2, v2] (14) so that we must have u1v1−v1u1=K[u1, v1] andu2v2−v2u2=K[u2, v2].(15) The constancy of Kfollows by transivity to arbitrary sets of variables. We may , for instance, choose EandBas u1andv1, and for u2andv2, select the dynamical variables governing a given spectros cope; we would then get the same value of Kfor its internal structure and dynamics as for the cavity rad iation. We have thus established the universality of K≡i¯h∼iU, and therefore of U∼h.✷ VI. CONCLUSION I have shown that the historical failure of classical ideas t o arrive at the blackbody distribution law is not really indicative of an intrinsic failure of classical mechanics, as hitherto believed, but of inadequate treatment of radiat ion modes and their thermalisation, since the known premises of stationarity and equilibrium do suffice, with due attention to the involvement of thermal wall jitter, for arriving at th e correct form of the spectral law. The reason that wall jitter was completely ignored in the previous consideratio ns seems to be that the precise mechanism of thermalisation was not considered important, as the principle of equiparti tion had apparently worked in kinetic theory without one. The traditional intuition in the latter case, nonlinearity of the molecular interactions, is turning out to be inadequa te for the purpose, as remarked in the introduction. In their rush to apply equipartition as a universal principl e, Rayleigh and others overlooked what might in hindsight seem a most obvious, and eminently classical, cause. Had the y instead analysed the picture more carefully, they would have discovered the frequency-independence of antinodal e nergy (eqs. 2-5), and thence the correct classical spectral law, eq. (8). From the appearance of the law ( §V), they would have been forced to interpret Uas a universal constant, and the transitivity argument would have followed as ration alisation. The mystery of quantum mechanics would have been avoided, and perhaps Landauer’s principle discov ered much earlier as well, in consequence. Conversely, it should now be clear that the established quantum derivation s are imperfect precisely because they assume perfectly rigid walls when stipulating a fixed set of stationary modes. There seems to be no way to take the wall motion into account without also rendering the usual premises of photon ic interactions and wall nonlinearities unnecessary for th e thermalisation and the spectral law. It is particularly interesting that the precise cause of the rmalisation, which could not be isolated by kinetic theory, is now clearly identifiable, as explained in the introductio n and §V, with the necessary involvement of the rest of the universe via the irreversibility of learning, at the instan ts of observation. This conclusion, which in effect identifie s quantum randomness with the thermal, is at variance with the current notion of a priori randomness, for instance in the jittery motion of particles ( zitterbewegung ) historically introduced for explaining the apparent emer gence of cas the instantaneous speed [8]. However, the mathematical s olution of Dirac’s equation actually describes strictly5 sinusoidal motion [6, eq.(29), p.263], in keeping with the p remise of causality, and the indicated speed does not necessarily contradict special relativity, as apparently now assumed [9], because it only refers to unobservable motion. This was in fact explained by Dirac [6, p.262], so that the mot ion is akin to phase velocity in that respect. The further rationalisation that the frequency, O(2mc2/¯h), would be too high for precise measurement, is also unneces sary, and specious, considering that it is quite finite, being only abo ut 1020Hz for the electron. The more general case of quantum fluctuations might not present a difficulty either, as they are not intrinsically associable with finite, confined systems like cavity radiation at 0 K, for which causality alm ost amounts to determinism. However, since the Fourier relation of coordinates and momenta, as in eq. (1), is also di rectly responsible for the Uncertainty Principle, consist ency of with the latter does need to be demonstrated. I believe we h ave achieved this recently, but as the reasoning is considerably involved, it will have to be presented separat ely. Acknowledgments Many thanks are owed to R Landauer, C H Bennett and B M Terhal of IBM Research for valuable discussions in the context. Note this does not imply endorsement of the idea s presented here by them or any other individual. [1] R P Feynman, R Leighton, and M Sands. The Feynman Lectures on Physics . Addison-Wesley, 1964. [2] R Resnick and D Halliday. Fundamentals of Physics . 2 edition. [3] E Fermi, J Pasta, and S Ulam. In Collected papers of Enrico Fermi , volume 2, page 977. Univ of Chicago, 1965. Los Alamos Rpt LA-1940 (1955). [4] M Toda, R Kubo, and N Saito. Statistical Physics I: Equilibrium Statistical Mechanics . Springer-Verlag, 1992. [5] A Fillipov et al. Energy transport between two attractor s...J of Phys A , 31:7719–7728, 1998. [6] P A M Dirac. The principles of quantum mechanics . Cambridge Univ, 4 edition, 1953. [7] R Landauer. Irreversibility and Heat Generation in the C omputing Process. IBM Journal , Jul 1961. [8] E Schr¨ odinger. Sitzungsb. d. Berlin Akad. , page 418, 1930. [9] L Brillouin. Science and information theory . Acad Press, 1962.

arXiv:physics/0003042v1 [physics.atm-clus] 17 Mar 2000Electron correlation in C 4N+2carbon rings: aromatic vs. dimerized structures Tommaso Torelli1and Lubos Mitas2 1Department of Physics, University of Illinois at Urbana-Ch ampaign, Urbana, Illinois 61801 2National Center for Supercomputing Applications, Univers ity of Illinois at Urbana-Champaign, Urbana, Illinois 6180 1 (February 21, 2014) The electronic structure of C 4N+2carbon rings exhibits competing many-body effects of H¨ ucke l aromaticity, second-order Jahn-Teller and Peierls instab ility at large sizes. This leads to possible ground state structures with aromatic, bond angle or bond le ngth alternated geometry. Highly accurate quantum Monte Carlo results indicate the existenc e of a crossover between C 10and C 14 from bond angle to bond length alternation. The aromatic iso mer is always a transition state. The driving mechanism is the second-order Jahn-Teller effect wh ich keeps the gap open at all sizes. The discovery of carbon fullerenes and nanotubes has opened a new materials research area with a vast poten- tial for practical applications. Unfortunately, our under - standing of the rich variety of structural and electronic properties of carbon nanostructures is far from satisfac- tory. For example, experiments [1] indicate that quasi- one-dimensional structures such as chains and rings are among the primary precursors in the formation process of fullerenes and nanotubes. However, our insights into their properties and behavior are incomplete due to the complicated many-body effects involved. In particular, recent studies [2,3] have demonstrated a profound im- pact of the electron correlation on stability and other properties of such all-carbon structures. An important example of such nanostructures is the system of planar monocyclic carbon rings C nwithn= 4N+2, where N is a natural number. These closed-shell molecules mani- fest an intriguing competition between conjugated aro- maticity, second-order Jahn-Teller and, at large sizes, Peierls instability effects. Consequently, this leads to di f- ferent stabilization mechanisms that tend to favor one of the following structures: a cumulenic ring ( A), with full Dnhsymmetry, with all bond angles and bond lengths equal; or either of two distorted ring structures, of lower Dn 2hsymmetry, with alternating bond angles ( B) or bond lengths ( C). Further structural details are given in Fig. 1. Accurate studies for the smallest sizes (C 6and C 10) find isomer Bto be the most stable. However, for larger sizes the results from commonly used methods are contradic- tory and available experiments [4] are unable to clearly specify the lowest energy structures. In order to identify the most stable isomers and to elu- cidate the impact of many-body effects, we carried out an extensive study of electronic structure and geometries of C4N+2rings of intermediate sizes up to 22 atoms (with some methods up to 90 atoms). We employed a number of electronic structure methods including the highly ac- curate quantum Monte Carlo (QMC) method which has been proven very effective in investigations of C 20[2] and larger carbon clusters [3], as confirmed also by an inde- pendent study by Murphy and Friesner [5]. Our QMC re- sults reveal that the C 4N+2ground state structures have alternated geometries at all sizes while cumulenic isomerAis a structural transition state. The results also pro- vide valuable insights into the shortcomings of the den- sity functional approaches such as inaccurate balance be- tween exchange and correlation in commonly used func- tionals. In addition, the letter presents a first evaluation of interatomic forces in large systems within the QMC framework. According to the H¨ uckel rule, the 4 N+2 stoichiometry implies the existence of a double conjugated π-electron system (in- and out-of-plane). Combined with the ring planarity, this suggests a strong aromatic stabilization i n favor of isomer A. Although the highest occupied and the lowest unoccupied molecular orbitals (HOMO and LUMO) are separated by a gap of several eV, a double degeneracy in the HOMO and LUMO states opens the possibility for a second-order Jahn-Teller distortion [6] , resulting in either cumulenic Bor acetylenic Cstructure. Such distortion lowers the symmetry and splits the de- generacy by a fraction of an eV, with an overall energy gain. Moreover, as N→ ∞, the picture is complicated further by the fact that the system becomes a semimetal- lic polymer with two half-filled πbands. As first pointed out by Peierls [7], such a one-dimensional system is intrin- sically unstable and undergoes a spontaneous distortion which lowers the symmetry. The symmetry breaking en- ables the formation of a gap, in analogy to the elusive case of trans-polyacetylene [8]. FIG. 1. The most stable isomers of C 4N+2rings (shown for C 10). The parameters needed to specify the geometries are: average bond length ¯ r= (r1+r2)/2, and bond angle ¯α= (α1+α2)/2; relative bond length and Ar= (r1−r2)/¯r bond angle alternation Aα= (α1−α2)/¯α It is very instructive to see how the commonly used 1computational methods deal with such many-body ef- fects. Density functional theory (DFT) methods tend to favor a “homogenized” electronic structure with delocal- ized electrons. In fact, for sizes larger than C 10, there is no indication of any stable alternation up to the largest sizes we have investigated (C 90). Calculations performed within the local density approximation (LDA) and gen- eralized gradient approximations (GGA, with BPW91 functional) consistently converge to the aromatic struc- tureA, in agreement with other studies [9]. Only by extrapolation to the infinite-chain limit, Bylaska, Weare et al. [10] claim to observe a very small, yet stable, bond alternation within LDA. A very different picture arises from the Hartree-Fock (HF) method, which shows a pro- nounced dimerization for C 10and larger. This agrees with the HF tendency to render structures less homoge- neous in order to increase the impact of exchange effects. We also verified that using GGA functionals with an ad- mixture of the exact HF exchange (B3PW91) recovers qualitatively the HF results for large sizes ( >C46), as al- ready observed by others [11]. Obviously, this problem calls for much more accurate treatments. High-level post-HF methods, such as multi- configuration self-consistent field (MCSCF) and coupled cluster (CC), indeed provide answers for the smallest ring sizes (C 6[12] and C 10[13,14]). In particular, Martin and Taylor [14] have carried out a detailed CC study demonstrating that both C 6and C 10have angle alter- nated ground state structures, although for C 10the en- ergy of the aromatic isomer Ais found to be extremely close (1 kcal/mol). In addition, we have performed lim- ited CCSD calculations of C 14and have found the dimer- ized isomer to be stable by ≃6 kcal/mol. Unfortunately, these methods are impractical for larger cases or more extensive basis sets [11]. The quantum Monte Carlo (QMC) method was used to overcome these limitations. This method possesses the unique ability to describe the electron correlation ex- plicitly and its favorable scaling in the number of parti- cles enables us to apply it to larger systems [15]. In the variational Monte Carlo (VMC) method we construct an optimized correlated many-body trial wavefunction Ψ T, given by the product of a linear combination of Slater determinants and a correlation factor ΨT=/summationdisplay ndnD↑ n{ϕα}D↓ n{ϕβ}exp/summationdisplay I,i<ju(riI, rjI, rij) (1) where ϕare one-particle orbitals, i, jdenote the elec- trons, Ithe ions and riI, rjI, rijare the corresponding distances. The correlation part, u, includes two-body (electron-electron) and three-body (electron-electron- ion) correlation terms and its expansion coefficients are optimized variationally. Most of the variational bias is subsequently removed by the diffusion Monte Carlo (DMC) method, based on the action of the projection−120−100−80−60−40−20020Energy [kcal/mol]C10 C14 C18 C22HF 12.9% 14.6% 14.3%14.3% 0 3.5 7 10.5 14 17.5 21 24.5 Ar[%] − bond length alternation−40−20020 MCSCF 10.9%13.7%0.0% FIG. 2. HF and MCSCF energy as a function of the degree of dimerization Ar. Least-squares estimates of the positions of the minima are indicated by arrows. operator exp( −τH); in the limit of τ→ ∞, this projec- tor recovers the lowest eigenstate from an arbitrary trial function of the same symmetry and nonzero overlap. The fermion antisymmetry (or sign) problem is circumvented by the fixed-node approximation. More details about the method are given elsewhere [15]. DFT, HF and MCSCF calculations have been carried out using standard quantum chemistry packages [16]. All calculations employed an accurate basis set, consisting of 10s11p2dGaussians contracted to 3 s3p2d, and smooth ef- fective core potentials [17] to replace the chemically iner t core electrons. The geometries of smaller rings with 6 and 10 atoms have already been established from previous calcula- tions [12–14]. We have verified that the most reliable published structural parameters agree very well (within ≃0.002˚A and 1◦) with our own GGA values. However, since the dimerized isomer Cis unstable within DFT, we followed a different strategy. We began from HF ge- ometries, which show that the degree of bond length al- ternation saturates at Ar≈14% (Fig. 2). In order to correct for the HF bias favoring acetylenic structures, we performed limited MCSCF calculations (see below) for C10, C14, and C 18. The electron correlation has a pro- found effect on the geometry, to the extent of causing the dimerized isomer to be unstable for C 10, while for C14it decreases the dimerization to Ar≈10%. Clearly the limited MCSCF for C 14and C 18provides rather poor geometry improvement although one expects a larger cor- rection as more correlation energy is recovered. In order to verify this and to estimate the correct degree of dimer- ization for C 14, we carried out the evaluation of the Born- Oppenheimer forces by a finite-difference scheme using 2correlated sampling, in the VMC method [18,19]. The computation of interatomic forces is a new development in QMC methodology and, to our knowledge, this is the first application in this range of system sizes. We probed the tangential C-C stretching/shortening which leads to a change in the degree of dimerization, Ar. ForAr=7%, our force estimate is F=−dE/dθ = 0.010(2) a.u. (and a second derivative of H= 0.30(1) a.u.), suggesting proximity to the minimum. Moreover, at Ar= 10.5% we find a force of opposite sign: F=−0.013(3) a.u. (H= 0.33(1) a.u.). For C 18, we have instead performed two QMC single point calculations at Ar=7%,14% and found the first energy to be lower by ∆ E≃−12 kcal/mol. Finally, we assumed Ar=7% and ¯ r=1.286˚A as our best estimate for calculations of the acetylenic isomer with n >10. The crucial ingredient for very accurate QMC calcu- lations is a trial function with a small fixed-node error. The quality of the one-particle orbitals is of prime im- portance for decreasing such error. While HF or DFT orbitals are commonly used for construction of Slater de- terminants, our recent projects [20] have demonstrated that natural orbitals from limited correlated calculation s (e.g., MCSCF) lead to more consistent results. Inclu- sion of the electron correlation into the method used to generate the orbitals is indeed very relevant for ob- taining improved fermion nodes, especially for such sys- tems which exhibit strong non-dynamical correlation ef- fects [11,14]. Extensive tests confirmed that orbitals from MCSCF (with an active space consisting of 4 occupied and 4 virtual orbitals) yield the lowest energies and so they were used in all of our calculations. In addition, the inclusion of the most important excited configura- tions into Ψ T(about 20–30 determinants) provided fur- ther significant improvement of the total energies. In particular, the weights of single excitations were surpris- ingly large for the alternated geometries and comparable to the largest weights of configurations with double exci- tations. A more detailed analysis on the multi-reference nature of the wavefunction in these systems will be given elsewhere. Equipped with such high quality trial functions we have carried out QMC calculations from C 6to C 18. A plot of the energy differences, with comparison to other methods, is shown in Fig. 3. For the very compact C 6 ring, where the overlap between πin-plane orbitals is large, as observed by Raghavachari et al. [21], the angle alternated isomer Bis the most stable. The aromatic structure Ais instead a transition state leading to angle alternation (B 1umode), while the dimerized isomer Cis unstable in all methods. C10is the case which was studied extensively in the past. Our DMC results agree with calculations of Martin and Taylor [14]. We conclude that the angle alternated isomer is still the lowest energy structure, albeit C6 C10 C14 C18−8−6−4−20Ealt−Earom [kcal/mol per atom]LDAQMC MCSCF HF

arXiv:physics/0003043v1 [physics.atom-ph] 20 Mar 2000Ground and excited states of the hydrogen negative ion in str ong magnetic fields O.-A. Al-Hujaj and P. Schmelcher Theoretische Chemie, Institut f¨ ur Physikalische Chemie d er Universit¨ at Heidelberg, INF 229, 69120 Heidelberg, Ger many (July 24, 2013) The lowest bound states of the hydrogen negative ion and nega tive donor systems in a homo- geneous magnetic field are investigated theoretically via a full configuration interaction approach with an anisotropic Gaussian basis set. The broad magnetic fi eld regime γ= 8·10−4−4·103is covered. Nonrelativistic total energies, electron detach ment energies and transition wavelengths are presented assuming an infinite nuclear mass. The binding mec hanisms are discussed in detail. The accuracy for the energies is enhanced significantly compare d to previously published data. I. INTRODUCTION The term “strong field” characterizes a situation for which t he Lorentz force is of the order of magnitude or greater than the Coulomb binding force. For a hydrogen atom i n the ground state the corresponding field strength cannot be reached in the laboratory, but only in astrophysic al objects like white dwarfs ( B ≈102–105T) or neutron stars ( B ≈107–109T). Astrophysicists possess therefore a vivid interest in t he behavior and properties of matter in strong magnetic fields: theoretically calculated data of ma gnetized atoms can be used for the determination of the decomposition and magnetic field configuration of astrophys ical objects [1–4]. On the other hand the strong magnetic field regime is accessible in the laboratory if one considers highly excited Rydberg states of e.g. atoms [5,6]. In solid state physics donor states in semiconductors with p arabolic conduction bands are systems which possess a Hamiltonian equivalent to the one of hydrogen within an effe ctive mass approximation. Due to screening effects the Coulomb force is much weaker than in the case of hydrogen. The regime where the ground state of the system is dominated by magnetic forces can therefore be reached for ce rtain semiconductors in the laboratory. As an example we mention GaAs for which the effective mass is m∗= 0.067meand the static dielectric constant ǫs= 12.53ǫ0. Since the Hamiltonian of the atomic ion and the negative donor are c onnected through a scaling transformation the values for the energies given in the present work hold for both syste ms equally. The reader should however keep in mind that they are given in differently scaled units. Apart from the above atoms and molecules in strong magnetic fi elds are also of interest from a pure theoretical point of view. Due to the competition of the spherically symm etric Coulomb potential and the cylindrically symmetric magnetic field interaction we encounter a nonseparable, non integrable problem. Perturbation theory, which is possibl e in the weak and in the ultrastrong field regime, breaks down in the intermediate field regime. It is therefore necessary to develop new techniques to solve such problems. The neutra l hydrogen atom in a strong magnetic field is now understood to a high degree (see [5,7] and references therei n). Recently Kravchencko has published an “exact” solution which provides an infinite double sum for the eigenv alues [8]. With the presented method all energy values of bound states could in principle be calculated to arbitrar y precision. For two electron atoms the situation is significantly differe nt. The problems posed by the electron-electron interac- tion and the non-separability on the one-particle level hav e to be solved simultaneously, which is much harder. The H−ion provides an additional challenge since correlation pla ys an important role for its binding properties. With- out a field it possesses only one bound state [9]. In the presen ce of a magnetic field and for the assumption of an infinitely heavy nucleus it could be shown [10] that there exi sts an infinite number of bound states. For laboratory field strengths these states are, due to the binding mechanis m via a one dimensional projected polarization potential, very weakly bound [11]. Some finite nuclear mass effects can be included via scaling relations [7,12,14]. However, the influence of the center of mass motion has not been investigat ed in detail so far. In the present work we assume an infinitely heavy nucleus which represents a good approximat ion for the slow H−atomic ion in strong magnetic fields and describes simultaneously the situation of negatively c harged donors D−in the field. Relativistic corrections were neglected since they are assumed to be small compared to the e lectron detachment energy of the system. We will use in the following the spectroscopic notation2S+1Mfor the electronic states of the ion where MandSare the total magnetic and spin quantum numbers. Since states with n egative z-parity are not considered here we omit the corresponding label in our notation (see also section II A). Many authors have tackled the quantum mechanical problem of H−in a strong magnetic field. One of the first, who pursued a variational approach to this problem, were Hen ry et al. [15]. They give first qualitative insights into the weak and intermediate field regime. Mueller et al. [16] qu alitatively described the strong field ground state3(−1) and the10 state for high fields ( γ≈4 toγ≈20 000, where γ= 1 a. u. corresponds to 2 .3554 ·105T). 1Larsen has published a number of papers on this problem [17–1 9]. On the one hand he created very simple and physically motivated trial functions with only a small numb er of variational parameters. On the other hand his energies were “state of the art” in variational calculation s for a long time. In [17] he provides binding energies of the lowest10 state in the field regime γ= 0−5 and of the3(−1) state in the regime γ= 0−3. He also presents figures showing the binding energies of the singlet and triplet stat e forM=−2 and M=−3. Later [19] he presents total and electron detachment energies for the lowest10,3(−1) and3(−2) state in the high field regime. More specifically the regime γ= 20 −1 000 for the3(−1) state and γ= 20 −200 for the other states were investigated. Furthermore Park and Starace [20] provided upper and lower bounds for ene rgies and binding energies of the ground state10 for weak fields. In the nineties several authors [21–24] improved the accura cy of the binding energies and total energies by new techniques. Vincke and Baye [21] report total ionization en ergies for the lowest singlet and triplet states with M= 0,−1 and −2 for a few field strengths in the regime γ= 4−400. They are to our knowledge the first who reported that the1(−1) state becomes bound for sufficiently high field strengths an d realized that the1(−2) state is slightly stronger bound than the corresponding triplet state in the h igh field regime. Larsen and McCann present in [22] one-particle binding energies for the10 state in the broad magnetic field regime γ= 0−200. In [23] the same authors consider furthermore the singlet and triplet states of M=−1,−2. The triplet states are calculated for γ= 0.5−200, the1(−1) state in the field regime γ= 55 −2 000 and the1(−2) state is calculated for a few field strength in the range γ= 1−100. Blinowski and Szwacka [24] have subsequently used a Gau ssian basis set, similar to the one used in our calculation. They present results for the10 state, which are less accurate than those of ref. [22]. We also mention some Hartree–Fock calculations: very early Virtamo [25] has investigated the ground state energies fromγ≈20 to γ≈20 000. Thurner et al. [26] (results published in [7]) have ca lculated triplet states for M=−1,−2 and−3 for many field strength in the broad range γ= 2·10−4−2·103. However since they use spherical wave functions for weak fields and cylindrical ones for high fields, there rem ains a gap of inaccurate results in the intermediate field regime. In the present investigation we provide lower variational e nergies and higher one-particle binding energies for the atomic H−problem and respectively the negatively charged donor cent er D−problem in a strong magnetic field compared to all other published data sofar. An exception is t he field free situation: the calculation by Pekeris [27] gives −0.52775 a.u. for the ground state binding energy whereas we obt ain−0.5275488 a.u. Clearly the field-free situation is much better understood than the case of a strong field. The paper is organized as follows: in section II we consider t he symmetries of the Hamiltonian and the basis set we use in our calculations. In section III we will report on the s trategy we employed for the selection of basis functions in order to obtain accurate results. Section IV contains the discussion of our results and a comparison with the literature. II. HAMILTONIAN, SYMMETRIES AND BASIS SET A. Hamiltonian and Symmetries In the following we assume an infinite nuclear mass (fixed dono r). The magnetic field is chosen to point along the z-direction. The nonrelativistic Hamiltonian takes in ato mic units the form H=H1+H2+1 |r1−r2|(2.1) with Hi=1 2p2 i+1 2γlzi+γ2 8/parenleftbig x2 i+y2 i/parenrightbig −1 |ri|+γszi. (2.2) The Hamiltonian is splitted in its one-particle operators, where 1 /2γlziis the Zeeman term, γ2/8/parenleftbig x2 i+y2 i/parenrightbig is the diamagnetic term, −1/|ri|is the attractive Coulomb interaction with the nucleus (don or) and γszithe spin Zeeman term (we take the g-factor equal 2). The two-particle operat or 1/|r1−r2|represents the repulsive electron-electron interaction. The Hamiltonian (2.1) possesses four independent symmetri es and associated quantum numbers: the total spin S2, the total z-projection of the spin Sz, the z-component of the total angular momentum Mand the total z-parity Π z (parity is also conserved but not a further independent symm etry). 2B. One-particle basis set For our calculation we use an anisotropic Gaussian basis set , which has been put forward by Schmelcher and Cederbaum in ref. [28], for the purpose of investigating ato ms and molecules in strong magnetic fields. It has already successfully been applied to helium [12,13], H+ 2[29] and H 2[30]. Adapted to the problem discussed here this one-particle bas is set for the spatial part reads in the cylindrical coordinates as follows Φi(ρ, φ, z ) =ρnρiznzie−αiρ2−βiz2exp(imiφ). (2.3) These functions are eigenfunctions of the symmetry operati ons of the one-particle Hamiltonian Hi, i.e. eigenfunctions oflzandπz. The additional parameters nρiandnziobey the following restrictions: nρi=|mi|+ 2ki;ki= 0,1,2, . . . and mi=. . . ,−2,−1,0,1,2, . . . (2.4) nzi=πzi+ 2li; li= 0,1,2, . . . and πzi= 0,1. (2.5) The exponents αiandβiserve as positive, nonlinear variational parameters. Due t o these parameters, the one-particle functions are flexible enough to be adapted to the situation o f an arbitrary field strength: in the weak magnetic field regime a basis set with an almost isotropic choice of paramet ersαi≈βidescribes the slightly perturbed spherical symmetry. For very high magnetic fields it is appropriate to c hoose α=γ/4 since ρ|mi|exp(−γ/4ρ2) yields the ρ-dependence of the lowest Landau level for a given magnetic q uantum number. The βiwill be well tempered in a wide region. In the intermediate field regime the basis is com posed of functions with certain magnetic field dependent sets of {αi, βi}which mediate the extreme cases. The optimal choice is found by searching the set of {αi,βi}which yields the lowest eigenvalues of the one-particle Hamilton ian. The parameters {αi,βi}are successively optimized using the pattern search algorithm. In this manner we have optimiz ed up to five excited states in every symmetry subspace. The starting values for the parameters {αi,βi}have to be chosen very carefully to find a deep local or even the global minimum. Since the search in this high dimensional space is v ery time consuming, an optimal choice of the kiandli is crucial: for every new ki,liconfiguration a new optimization procedure has to be started . The resulting binding energies for the neutral hydrogen atom were identical to 7 – 9 digits with the one given in [8] for almost all field strengths for the ground state and 5 – 7 digits were recovered for states with higher magnetic quantum number |mi|. We point out that Blinowski and Szwacka [24] have used a simil ar basis set, but without the monomers ρ2kiand z2li. The additional monomers however decisively enhance the fle xibility and accuracy of the calculations. C. Two-particle configurations As a next step we build two-particle configurations from our o ptimized one-particle basis set and represent the Hamiltonian (2.1) in this configuration space. This is done f or each total symmetry ( S2,Πz, Lz) separately. The corresponding spectrum of H−is then obtained by diagonalizing the Hamiltonian matrix. W e hereby use all possible excited two-particle configurations constructed from our o ptimized one-particle basis set, i.e. our approach is a full configuration interaction method (full CI). The two-partic le functions are constructed from the one-particle functio ns by selecting combinations for mi+mj=Mandπzi+πzj= Π z. The spin part can be trivially separated. Due to the antisymmetrization of the spatial wave function the configu ration space of the triplet states is slightly smaller than that of the singlet states since for triplet configurations t here are no combinations with i=j. As our basis set is not orthogonal we have to solve a finite-dim ensional generalized real symmetric eigenvalue problem (H−ES)·c= 0 (2.6) where His the matrix representation of the Hamiltonian and Sthe overlap matrix. The resulting energies Eare strict upper bounds to the exact eigenvalues in the given sub space of symmetries. Some technical remarks concerning the calculation of the ma trix elements are in order. All matrix elements can be evaluated analytically. With the exception of the electr on-electron integrals all expressions can be calculated very rapidly. The electron-electron integrals, however, d eserve a special treatment: through a combination of trans- formation techniques as well as analytical continuation fo rmulae for the series of involved transcendental functions their representation has been simplified enormously (for de tails see ref. [12] and in particular [13]). It is due to this extremely efficient implementation of the electron-electro n integral that large basis sets of the order of 2500 −4000 could be used in the present work to perform CI calculations f or many field strengths. 3III. SELECTION OF THE BASIS FUNCTIONS Since the single bound state in the absence of the external fie ld is bound only due to correlation, and all the other states in the presence of the magnetic field are only wea kly bound, it is very important to include correlation by a proper choice of the one-particle basis functions build ing up the two-particle configurations. For the M= 0 singlet state this was achieved by selecting one-particle b asis functions not only with m1=m2= 0 but also with m1=−m2/ne}ationslash= 0. This allows one to describe the angular correlation whic h is particular important for the10 state. In general the enhanced binding properties of negative ions in the presence of a magnetic field are due to a balanced competition of the different interactions. On the one hand th e confinement due to the magnetic field raises the kinetic energy and the electrostatic repulsion due to the electron- electron interaction. These effects tend to lower the bindin g energy. On the other hand the confinement raises the nuclear a ttraction energy, the exchange energy and to some extent also the correlation energy which tend to enhance the binding energy. Of course one has to distinguish between, for example, the10 state whose binding properties are dominated by correlati on effects and the excited bound states with nonzero magnetic quantum numbers which possess a signi ficant contribution to their binding energy through exchange effects and due to the occupation of the series of tig htly bound hydrogenic orbitals 1 s, 2p−1, 3d−2,. . . etc. For the description of the lowest states with |M|>0 an effective one-particle picture can be employed [17]: the hydrogen negative ion consists of a tightly bound core elect ron with magnetic quantum number zero and a significantly less bound electron which carries the magnetic quantum numb er of the ion. The core electron is then described by one-particle basis functions with m1= 0. The outer electron is described by one-particle functio ns with m2=Min order to take into account the fact that it is weakly bound and thus spatially extended. In order to go beyond this effective one-particle picture we used, similar to the case M= 0 one-particle functions with other magnetic quantum numbers to obtain in particular the correlation behavior. The above picture is not valid for the tightly bound states in the high field regime: the number of functions with different magnetic quantum numbers can be reduced as we incre ase the field strength. This reduction in the number of basis functions is also suggested by the occurrence of lin ear dependencies for strong fields. The extent of this reduction can be seen from the fact that the number of two-par ticle basis functions drops from 4 000 for γ= 0 to less than 3 000 for γ= 4000 for the10 state. IV. RESULTS AND DISCUSSION As already mentioned the H−ion possesses only one bound state in the absence of the magne tic field [9]. Turning on the field it has been shown [10] that there exists (for infini te nuclear mass) for any nonzero field an infinite number of bound states. The corresponding proof [10] relies on the p hysical picture [11] that the external electron is for weak fields far from the neutral atomic core and experiences there fore to lowest order a polarization potential due to the induced dipole moment of the core. Perpendicular to the field the motion of the external electron is dominated by the field and it occupies approximately Landau orbitals wher eas parallel to the field it is weakly bound due to the projection of the mentioned polarization potential on the L andau orbitals which yields an one-dimensional binding along the field. For typical strong laboratory fields the corr esponding binding energies are of the order of 10−6eV for the hydrogen atom negative ion and significantly larger f or more electron atoms with a larger polarizability. To investigate theses states in the weak field regime goes clear ly beyond the feasibility of the present method. Instead we will investigate a number of states, starting from the value of the field strength for which they become significantly bound, which means that the outer electron is already relati vely close to the core and possesses a binding energy of at least a few meV. Clearly in that case the picture of the pola rization potential is no more valid since exchange and correlation effects rule the binding properties of the ion. W ithin our approach we could find one bound state for each negative magnetic quantum number of the ion considered ( −3≤M≤0) for both singlet and triplet states, except the 30 state, which is unbound. Their behavior has been studied fo r the complete range of field strengths 0 .01≤γ≤4000. The one bound state of the H−ion in the absence of the field represents, in the above sense, an exception since it is already significantly bound without the field. All these stat es possess positive z-parity and no bound states could be found for negative z-parity . A. Threshold energies The electron detachment energy is defined to be the energy we n eed to remove one electron from the atom without changing the quantum numbers of the total system. The corres ponding lowest threshold energy ETfor the H−ion can be expressed as: 4ET=γ 2(|M|+M+ 2 + geMs)−I(H) (4.1) where I(H) is the binding energy of the ground state of the neutral hy drogen atom in a magnetic field. The term γ/2(|M|+M+2) is the energy of an electron in the lowest Landau level wit h magnetic quantum number m=Mwhere the spin part is omitted. This means that the free electron ca rries the whole angular momentum of the state. For magnetic quantum numbers M≤0 the threshold energy ETis independent of the angular momentum M, i.e. there is only a singlet and a triplet threshold. The threshold energy is then ET=γ−I(H) for singlet states and ET=−I(H) for triplet states. We denote the electron detachment energ y byI(H−) which is given by I(H−) =ET−Etotwhere Etotis the total energy of the considered state of H−. B. Total, electron detachment and transition energies Before we discuss the individual states and their propertie s let us describe some general features of the states considered here. The total energy of the singlet states is mo notonically increasing with increasing field strength. Thi s fact is caused by the increase of the field-dependent kinetic energy. In contrast to this the total energy of the triplet states is monotonically decreasing with increasing field st rengths. This is a consequence of the additional spin Zeeman term (we consider here only the Sz=−1 component of the spin triplet states). The electron detachment energies are monotonically increasing with increasing field strengt h for all states considered here, i.e. both singlet and tripl et states. This has to be seen in view of the above-mentioned fact that th e zero-point kinetic (Landau) energy of the electrons is raised in the presence of the magnetic field and t herefore the threshold energy for loosing one electron is raised in the same way. For the10 state the total energy raises from −0.52754875 at γ= 0 to 3986 .49870 at γ= 4000. This state is the most tightly bound state for all field strengths. The detachm ent energy increases from 0 .027549 a.u. at γ= 0 to 2.29805 a.u. at γ= 4000. There are two reason which give rise to the fact, that t his state is the most tightly bound one. On the one hand the electrons are in this state much close r to the nucleus than in other states. This increases the binding due to the attractive nuclear potential energy. On the other hand correlation has an important impact on the binding energy. Both effects are reinforced with incre asing field strength as the electrons become more an more confined in the x-y plane perpendicular to the magnetic fi eld. These effects overcome the influence of the static electron-electron repulsion. The total energies and the de tachment energies of the10 state are presented in table I. It can be seen that the detachment energies for this most tigh tly bound state could be improved by 1-2% for all field strengths compared to the existing literature. This is not c orrect for a vanishing field, where much more efficient basis sets like the Hylleraas basis set are available. For numeric al reasons the relative accuracy for the detachment energie s is largest in the intermediate field regime. The30 state is not bound for all considered field strengths. This c an be understood in an effective particle picture as follows: for triplet states the spatial two-part icle wave function is antisymmetric with respect to particl e exchange and therefore the two particles have to occupy diffe rent spatial orbitals, i.e. we are exclusively dealing with excited configurations. For M/ne}ationslash= 0 it is (see later) possible to obtained tightly bound tripl et states in a strong magnetic field by occupying different orbitals of the hydroge nic series (1 s,2p−1,3d−2, . . .) which yields the one-particle excited configurations of the type 1 s2p−1,1s3d−2, . . .. For the case of the30 state, however, we have M= 0 and only configurations constructed from pairs of two orbitals w ith (m,−m) are allowed which are either of doubly excited character ( m/ne}ationslash= 0) or a singly excited configuration with m= 0. Therefore no magnetically tightly bound configurations are allowed for the30 state which illuminates its unbound character for any field strength. All singlet and triplet electron detachment energies of all the conside red bound states are presented also graphically: Figure 1 shows the singlet detachment energies and figure 2 the corres ponding energies for the triplet states. It is important to mention that the global ground state of the ion undergoes a crossover with respect to its symmetry with increasing field strength. For weak fields the10 state is the ground state of the system, whereas in strong fie lds the3(−1) state becomes the ground state which was first shown in ref. [15]. This is caused by the spin Zeeman term, which lowers the total energy of the triplet states. Th e crossover takes place at γc≈0.05 which corresponds to approximately 104T for the H−ion. The3(−1) state is very weakly bound when it becomes the ground state (atγcthe detachment energy is ≈3·10−4a.u.). This prevents us from localizing more exactly the fiel d strength at which the crossover takes place. The3(−1) state, being the ground state of the anion for γ > γ cnever becomes the most tightly bound state. At γ= 4 000 its electron detachment energy is 1 .25 a.u. and therefore much less than the detachment energy of the10 state. This is due to the fact that the tightly bound states a re formed by occupying the hydrogenic series 1 s,2p−1,3d−2, . . .(as mentioned above) and the10 states allows for the 1 s2configuration yielding the strongest binding although it represents an excited sta te for γ > γ cdue to its spin character. The singlet state1(−1) is not bound for weak fields. It becomes bound in the regime γ≈1−5 which is an unexpected behavior. The1(−1) state lies higher in the spectrum than the bound1(−2) and1(−3) states for the 5intermediate field region. In the high field region it however crosses both states. The crossing with the1(−3) takes place at γ≈300, the crossing with the1(−2) state is at γ/greaterorsimilar4 000. Unfortunately the accuracy of our method is not sufficient to provide a closer look at this crossing. The fa ct that the1(−1) state is not bound for weak fields but bound for strong fields is a consequence of the complicate d interplay of the different interactions. The Coulomb repulsion of the two electrons is much weaker for the spatial ly antisymmetric triplet states compared to the singlet states. The electron-electron repulsion is higher for the s tates with M=−1 compared to the states with M < −1. This pushes the |M|= 1 singlet states for weak fields beyond the threshold energy , i.e. makes them unbound. The total ionization and the detachment energies of the singlet and triplet states with M=−1 are presented in table II. The suppression of the binding for the singlet state can clea rly be seen from this table: the detachment energy of the singlet is 100 times lower than for the triplet at γ= 10, but at γ= 4000 the ratio is of the order 2. The comparison with the literature (see table II) shows that our detachment energies are variationally lower by several percent than the best available data. For the situation of weakly bound st ates the improvement is significantly larger. Let us now consider the energies for the states with M=−2 which are presented in table III. Focusing on the detachment energies we realize that for weak fields the tripl et state possesses a larger detachment energy than the singlet state, but for intermediate and high fields the singl et state is stronger bound than the triplet one, i.e. we encounter a crossover which is presented in figure 3. Compare d to the data of ref. [23], our method yields 5 −10% higher variational detachment energies for the triplet sta te and several times higher detachment energies for the singlet one. If we consider the singlet-triplet splitting w hich is the difference of the total energies between the singl et and the triplet state, where the spin-Zeeman shift is omitte d, it can be observed, that for all states this splitting behaves monotonically increasing with increasing field str ength in the weak field regime. The splitting for the states withM=−2 and M=−3 are shown in figure 4. The splitting for the M=−2 states increases in weak fields, but for high fields this splitting decreases and becomes nega tive above some critial field strength. It seems that the Coulomb repulsion, due to antisymmetrization of the wave fu nction is dominated by correlation effects. That the above observation is in fact a consequence of correlation is supported by Vincke and Baye [21]: the reversed order concerning the detachment energies (see figure 3) occurs if t hey include so-called transverse mixing, which simulates correlations in their approach. For states with M=−3 only a few published data are available. These states are on ly weakly bound, although they are stronger bound for γ/greaterorsimilar300 than the1(−1) state. The singlet state has for γ= 0.2 a detachment energy of 7.1 10−5a.u. and at γ= 1000 its detachment energy is 0 .19. The electron detachment energies of the triplet state are of the same order of magnitude and the absolute value of th e singlet triplet splitting is the lowest of the states considered here. As a consequence a careful convergence stu dy of the results (detachment energy) is indispensable. Our data are given in table IV. The wavelengths of the transitions of the singlet states are presented in figure 5. The wavelengths are monotonically decreasing with incresing field strength except for the tran sition from the1(−1) state to the1(−2) state. As mentioned above these states cross at γ/greaterorsimilar4 000. Therefore the corresponding wavelength for this tran sition diverges at the crossing field strength. The transition wavelengths for the triplet states shown in figure 6 are also monotonically decreasing with increasing field strength. Finally we comment on corrections due to the finite nuclear ma ss. There are two kinds of corrections, which are relevant here. One, which is special for ions in strong magne tic fields and which describes the coupling between the center of mass motion and the electronic motion. This coupli ng is due to a motional electric field of intrinsic dynamical origin seen by the moving ion in a magnetic field [31]. Second t here are corrections due to the replacement of the naked masses by reduced ones which can be easily included in o ur data by performing the corresponding shifts [12–14] . A full dynamical treatment of the atomic ion including the c ollective motion goes clearly beyond the scope of the present investigation. It is important to note that for the c ase of the fixed negative donors there naturally occur no such corrections. V. BRIEF SUMMARY We have investigated the H−ion, negative donors D−respectively, in a strong magnetic field via a fully correlat ed approach. The key ingredient is an anisotropic Gaussian bas is set, whose one-particle wave functions are nonlinearly optimized in order to obtain the spectrum of the one-particl e Hamiltonian. In contrast to other basis sets, which are appropriate either for the low field or for the high field re gime, our basis set is flexible enough to be adapted to the situation of arbitrary field strength and especially sui ted for the intermediate field regime. All calculations were performed in the infinite mass frame neglecting relativisti c corrections. We have investigated the low field ground state10, as well as singlet and triplet states for M=−1,−2,−3 for the broad field regime γ= 8·10−4−4·103. For all states and almost all field strengths we could reach a t least 61−2% higher binding energies, compared to all other published data. For some states our binding energies were larger by a factor up to two. The global ground state undergoe s a crossover with respect to its symmetry which is well-known in the literature [15]: for weak fields γ/lessorsimilar5·10−2the global ground state is the10 state, whereas for γ/greaterorsimilar5·10−2it is the3(−1) state, which is much weaker bound than the10 state for all field strengths. The1(−1) state becomes bound for γ/lessorsimilar5 and it crosses the1(−3) state at γ≈300 and the1(−2) state at γ/greaterorsimilar4000. We have also investigated the electronic states with M=−2 in detail. For γ/lessorsimilar1 the triplet state is stronger bound than the singlet, whereas for γ/greaterorsimilar1 the singlet is stronger bound than the triplet. Explanatio ns for the binding mechanisms of the considered states have been provided. The transition wavelengths for all allowed transitions as a function of the field strength are thereby obtained. No stationary trans itions which could be of relevance to the astrophysical observation in magnetized white dwarfs have been observed. Acknowledgments The Deutsche Forschungsgemeinschaft (OAA) is gratefully a cknowledged for financial support. We thank W. Becken for many fruitful discussions and for his help concerning co mputational aspects of the present work. [1] D. Wickramasinghe and L. Ferrario, Astrophys. J. 282, 222 (1984). [2] H. Ruder, G. Wunner, H. Herold, and F. Geyer, Atoms in strong magnetic fields (Springer Verlag, Berlin, 1994). [3] R. Henry and R. O’Connell, Astrophys. J. 282, L97 (1984). [4] S. Jordan, P. Schmelcher, W. Becken, and W. Schweizer, As tron. Astrophys. 336, L33 (1998). [5] H. Friedrich and D. Wintgen, Phys. Rep. 183, 37 (1989). [6]Atoms and Molecules in Strong External Fields , edited by P. Schmelcher and W. Schweizer (Plenum Press, New York, 1998). [7] H. Ruder, G. Wunner, H. Herold, and F. 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We also provide the results for the electron detachment energies given in the literature so far. γ Etot I(H−) I Lit(H−) 0 -0.52754875 0.02754875 0.2775a 8 10−4-0.52754430 0.02794446 1 10−3-0.52754053 0.02804078 0.02735b 2 10−3-0.52753777 0.02853877 0.02785b 5 10−3-0.52749800 0.03000425 0.0293b 8 10−3-0.52740873 0.03142473 0.01 -0.52734972 0.03237472 0.0317b 0.02 -0.52677018 0.03687014 0.0362b 0.05 -0.52314046 0.04876375 0.08 -0.51715770 0.05874669 0.1 -0.51223522 0.06470874 0.0634c 0.2 -0.47868356 0.08830200 0.08685c 0.5 -0.32804874 0.13083820 0.130d 0.8 -0.13939006 0.15710667 1.0 -0.00178881 0.17061922 0.1695d 2.0 0.75990486 0.21788123 0.2175d 5.0 3.3234387 0.2961625 0.2955d 8.0 6.0350437 0.3455713 10.0 7.8806402 0.3715626 0.371d 20.0 17.319887 0.464715 0.463d 50.0 46.359385 0.622747 0.618d 80.0 75.753770 0.721953 100.0 95.436219 0.773977 0.7665d 200.0 194.31374 0.95911 0.9385d 500.0 492.47687 1.26604 800.0 791.35985 1.45501 1000.0 990.78459 1.55299 2000.0 1988.8003 1.8949 4000.0 3986.4978 2.2981 aSee Pekeris [27]. bSee Park et al. [20]. cSee Larsen [17]. dSee Larsen [22]. 8TABLE II. Nonrelativistic total and electron detachment en ergies (atomic units) of singlet and triplet states with M=−1 as a function of the magnetic field strength γ. These states evolve from the1P−1and3P−1states for zero magnetic field. 1(−1)3(−1) γ Etot I(H−) I Lit(H−) E tot I(H−) I Lit(H−) 0.05 -0.52468218 0.00030547 0.00025a 0.08 -0.53959063 0.00117963 0.1 -0.54954554 0.00201906 0.0016a 0.2 -0.59861960 0.00823804 0.0072a 0.5 -0.72586763 0.02865709 0.027875b 0.8 -0.82643425 0.04415086 1.0 -0.88359474 0.05242585 0.0518b 2.0 -1.1036308 0.0814168 0.0805b 5.0 3.6194699 0.0001312 -1.5081497 0.1277508 0.1263b 8.0 6.3792043 0.0009946 -1.7751617 0.1557768 10.0 8.2504454 0.0017574 -1.9181202 0.1703230 0.168b 20.0 17.778209 0.006392 -2.436716 0.221318 0.2175b 50.0 46.961089 0.021050 -3.323515 0.305655 0.309b 80.0 76.441490 0.034306 -3.882658 0.358381 100.0 96.167783 0.042402 0.00281b-4.175890 0.386100 0.38015b 200.0 195.19141 0.07671 0.0407b-5.21214 0.48500 0.4771b 500.0 493.59274 0.15017 -6.90934 0.65226 800.0 792.61061 0.20426 -7.94296 0.75783 1000.0 992.10306 0.23452 0.1727b-8.47584 0.81343 2000.0 1990.3441 0.3511 0.2732b-10.3165 1.0117 4000.0 3988.2901 0.5057 -12.4576 1.2534 aSee Larsen [17] bSee Larsen and McCann [23]. TABLE III. Nonrelativistic total eigenenergies and electr on detachment energies (atomic units) of singlet and triple t states withM=−2 as a function of the magnetic field strength γ. These states evolve from the1D−2and3D−2states in the absence of a magnetic field. 1(−2)3(−2) γ Etot I(H−) I Lit(H−) E tot I(H−) I Lit(H−) 0.1 -0.44754846 2.20 10−5-0.54756898 4.25 10−5 0.2 -0.39082732 0.00044576 -0.59092467 0.00054311 0.5 -0.19993316 0.00272262 -0.70022629 0.00301575 0.0023a 0.8 0.01224680 0.00546981 -0.78790031 0.00561691 1.0 0.16158323 0.00724788 0.0015a-0.83835741 0.00718852 0.0064a 2.0 0.96305720 0.01472889 -1.03557327 0.01335936 0.0123a 5.0 3.5905353 0.0290659 -1.4049975 0.0245987 0.02290a 8.0 6.3421331 0.0384819 -1.6512176 0.0318326 10.0 8.2087000 0.0435028 0.006a-1.7838114 0.0360142 0.0335a 20.0 17.722831 0.061762 -2.2663464 0.0509479 0.047a 50.0 46.888223 0.093916 -3.096448 0.078587 0.0719a 80.0 76.360491 0.115232 -3.621917 0.097640 100.0 96.083393 0.126803 0.03a-3.897965 0.108128 0.09895a 200.0 195.10291 0.16995 -4.87531 0.14817 0.13535a 500.0 493.49377 0.24914 -6.48097 0.22388 800.0 792.51251 0.30235 -7.46090 0.27576 1000.0 992.00594 0.33164 -7.96662 0.30420 2000.0 1990.2556 0.4397 -9.7152 0.4105 4000.0 3988.2159 0.5799 -11.7535 0.5493 aSee Larsen and McCann [23]. 9TABLE IV. Nonrelativistic total and electron detachment en ergies (atomic units) of singlet and triplet states with M=−3 as a function of the magnetic field strength γ. These states evolve from the1F−3and3F−3states in the absence of a magnetic field. 1(−3)3(−3) γ Etot I(H−) E tot Etot(Lit.) I(H−) 0.2 -0.39045232 7.075 ·10−5-0.59046132 7.975 ·10−5 0.5 -0.19822524 0.00101470 -0.69827772 0.00106718 0.8 0.01561745 0.00209916 -0.38448370 0.00220031 1.0 0.16602786 0.00280324 -0.83410348 -0.7092618a0.00293459 2.0 0.97196637 0.00581972 -1.02829096 0.00607705 5.0 3.6077963 0.0118049 -1.3927524 0.0123535 8.0 6.3646091 0.0160059 -1.6361379 0.0167528 10.0 8.2339720 0.0182308 -1.7669036 0.0191064 20.0 17.757749 0.026853 -2.243571 0.028172 50.0 46.938916 0.043223 -3.063189 0.045328 80.0 76.420911 0.054812 -3.581678 0.057401 100.0 96.148914 0.061282 -3.853929 0.064125 200.0 195.18642 0.08643 -4.81732 0.09017 500.0 493.60717 0.13574 -6.39799 0.14091 800.0 792.64412 0.17074 -7.36188 0.17675 1000.0 992.14738 0.19020 -7.8591 -7.686295a0.19663 aSee Thurner et al. [7]. 10Figure Captions FIG. 1. Electron detachment energies of the singlet states i n atomic units. Note that the M=−1 singlet state is not bound for weak fields and is for 5 /lessorsimilarγ/lessorsimilar300 weaker bound than all other states considered. For γ≈300 and γ/greaterorsimilar4000 we encounter crossings of the detachment energies of the1(−1) with those of the1(−3) and1(−2) states, respectively. FIG. 2. Electron detachment energies of the triplet states i n atomic units. FIG. 3. Electron detachment energies of the singlet and trip let state of the M=−2 states in atomic units. Note that the singlet and triplet state reverse their order: For low field s trengths the triplet state is more bound whereas for high fiel ds (γ/greaterorsimilar1) the singlet state is more bound than the triplet state. FIG. 4. Singlet triplet splitting for the M=−2 and the M=−3 states in atomic units. The splitting due to the spin Zeeman term is omitted. Note that the splitting of the M=−2 states increases for low fields but decreases for high fields , whereas the splitting for the M=−3 states increases monotonically with increasing field stre ngth. FIG. 5. Singlet transition wavelengths between the conside redbound states in ˚Angstrøm as a function of the field strengths in atomic units on a logarithmic scale. FIG. 6. Triplet transition wavelengths between the conside red states in ˚Angstrøm as a function of the field strength in atomic units on a logarithmic scale. 1110−310−210−1100101102103 γ10−410−310−210−1100101I(H−)Figure 1 M=0 M=−1 M=−2 M=−3Singlet10−1100101102103 γ10−410−310−210−1100101I(H−)Figure 2 M=−1 M=−2 M=−3Triplet0.2 0.40.8 1.02.0 4.08.0 γ10−310−2I(H−)Figure 3 M=−2 singlet M=−2 triplet0.1 0.2 0.5 0.8 1.0 2.0 γ−0.00050.00.0005 −0.001Singlet−triplet splittingFigure 4 M=−2 M=−3100101102103104 γ102103104105106107WavelengthFigure 5 10−1(−1) 1(−1)−1(−2) 1(−2)−1(−3)100101102103 γ103104105106WavelengthFigure 6 3(−1)−3(−2) 3(−2)−3(−3)

arXiv:physics/0003044v1 [physics.gen-ph] 18 Mar 2000New Hierarchic Theory of Matter, general for liquids and solids: dynamics, thermodynamics and mesoscopic structure of water and ice Alex Kaivarainen University of Turku, JBL, FIN-20520 Turku, Finland H2o@karelia.ru URL: http://www.karelia.ru/˜alexk Summary A basically new hierarchic quantitative theory, general fo r solids and liquids, has been developed. It is assumed, that unharmo nic oscillations of particles in any condensed matter lead to em ergence of three-dimensional (3D) superposition of standing de Bro glie waves of molecules, electromagnetic and acoustic waves. Consequ ently, any condensed matter could be considered as a gas of 3D standing w aves of corresponding nature. Our approach unifies and develops s trongly the Einstein’s and Debye’s models. Collective excitations in form of coherent clusters, repre senting at certain conditions the mesoscopic molecular Bose condensa te, were analyzed, as a background of hierarchic model of condensed m atter. The most probable de Broglie wave (wave B) length is deter- mined by the ratio of Plank constant to the most probable impu lse of molecules, or by ratio of its most probable phase velocity to fre- quency. The waves B are related to molecular translations (t r) and librations (lb). As the quantum dynamics of condensed matter does not follow i n general case the classical Maxwell-Boltzmann distribution, the re al most probable de Broglie wave length can exceed the classical thermal de Brog lie wave length and the distance between centers of molecules many times. This makes possible the atomic and molecular partial Bose co n- densation in solids and liquids at temperatures, below boil ing point. It is one of the most important results of new theory, which we have confirmed by computer simulations on examples of water and ic e. Four strongly interrelated new types of quasiparticles (co llective excitations) were introduced in our hierarchic model: 1.Effectons (tr and lb) , existing in ”acoustic” (a) and ”optic” (b) states represent the coherent clusters in general case ; 2.Convertons , corresponding to interconversions between trandlbtypes of the effectons (flickering clusters); 13.Transitons are the intermediate [ a⇋b] transition states of the trandlb effectons; 4.Deformons are the 3D superposition of IR electromagnetic or acoustic waves, activated by transitons and [lb ⇋tr]convertons. Primary effectons (tr and lb) are formed by 3D superposition o f the most probable standing de Broglie waves of the oscillati ng ions, atoms or molecules. The volume of effectons (tr and lb) may con tain from less than one, to tens and even thousands of molecules. T he first condition means validity of classical approximation in des cription of the subsystems of the effectons. The second one points to quan tum properties of coherent clusters due to molecular Bose conde nsation. It leads from our computer simulations, that liquids are sem iclassical systems because their primary (tr) effectons contain less than one mo lecule and primary (lb) effectons - more than one molecule. The solids are quantu m systems to- tally because both kind of their primary effectons (tr and lb) are molecular Bose condensates. It is shown, that the 1st order [ gas→liquid ] transition is accompanied by strong decrease of librational (rotational) degrees of fre edom due to emergence of primary (lb) effectons. In turn, the [ liquid →solid] transition is followed by decreasing of translational degrees of freedom due to Bos e-condensation of primary (tr) effectons. In the general case the effecton can be approximated by par- allelepiped with edges corresponding to de Broglie waves le ngth in three selected directions (1, 2, 3), related to the symmetry of the molecular dynamics. In the case of isotropic molecular moti on the effectons’ shape may be approximated by cube. The edge-length of primary effectons (tr and lb) can be consid ered as the ”parameter of order”. The in-phase oscillations of molecules in the effectons corr espond to the effecton’s (a) - acoustic state and the counterphase oscillations correspond to their (b) - optic state. States (a) and (b) of the effectons differ in potential energy only, however, their kinetic energies, impulses and spatial dimensions - are the same. The b-state of the effectons has a common feature with Fr¨ olich’s polar mode. The(a→b)or(b→a)transition states of the primary effectons (tr and lb), defined as primary transitons, are accompanied b y a change in molecule polarizability and dipole moment withou t density fluctuation. In this case the transitions lead to absorption or radiation of IR photons, respectively. Superposition (interception) of three internal standing I R pho- tons, penetrating in different directions (1,2,3) - forms pr imary elec- tromagnetic deformons (tr and lb). On the other hand, the [lb ⇋tr]convertons andsecondary transitons are accompanied by the density fluctuations, leading to absorption or radiation of phonons . 2Superposition resulting from interception of standing phonons in three direc- tions (1,2,3), forms secondary acoustic deformons (tr and lb). Correlated collective excitations of primary and secondary effectons and deformons (tr and lb) ,localized in the volume of primary trandlb electromag- netic deformons ,lead to origination of macroeffectons, macrotransitons andmacrodeformons (tr and lb respectively) . Correlated simultaneous excitations of tr and lb macroeffec tonsin the vol- ume of superimposed trandlbelectromagnetic deformons lead to origination ofsupereffectons. In turn, the coherent excitation of both: tr andlb macrodeformons and macroconvertons in the same volume means creation of superdeformons. Su- perdeformons are the biggest (cavitational) fluctuations, leading to microbub- bles in liquids and to local defects in solids. Total number of quasiparticles of condensed matter equal to 4!=24, reflects all of possible combinations of the four basic ones [ 1-4], intro- duced above. This set of collective excitations in the form o f ”gas” of 3D standing waves of three types: de Broglie, acoustic and el ectro- magnetic - is shown to be able to explain virtually all the pro perties of all condensed matter. The important positive feature of our hierarchic model of ma tter is that it does not need the semi-empiric intermolecular potentials f or calculations, which are unavoidable in existing theories of many body systems. T he potential energy of intermolecular interaction is involved indirectly in di mensions and stability of quasiparticles, introduced in our model. The main formulae of theory are the same for liquids and solid s and include following experimental parameters, which take into ac- count their different properties: [1]- Positions of (tr) and (lb) bands in oscillatory spectra; [2]- Sound velocity; [3]- Density; [4]- Refraction index (extrapolated to the infinitive wave leng th of photon ). The knowledge of these four basic parameters at the same temp erature and pressure makes it possible using our computer program, to ev aluate more than 300 important characteristics of any condensed matter. Amo ng them are such as: total internal energy, kinetic and potential energies, heat-capacity and ther- mal conductivity, surface tension, vapor pressure, viscos ity, coefficient of self- diffusion, osmotic pressure, solvent activity, etc. Most of calculated parameters are hidden, i.e. inaccessible to direct experimental measu rement. The new interpretation and evaluation of Brillouin light sc attering and M¨ ossbauer effect parameters also have been done on the basis of hierarchic theory. Mesoscopic scenarios of turbulence, superconduct ivity and superfluity are elaborated. Some original aspects of water in organization and large-sc ale dynamics of biosystems - such as proteins, DNA, microtubules, membrane s and regulative 3role of water in cytoplasm, cancer development, quantum neu rodynamics, etc. have been analyzed in the framework of hierarchic theory. Computerized verification of our Hierarchic concept of matt er on examples of water and ice has been performed, using specia l computer program: Comprehensive Analyzer of Matter Proper ties [CAMP, copyright, 1997, Kaivarainen]. The new optoacousti c device (CAMP), based on this program, with possibilities much wide r, than that of IR, Raman and Brillouin spectrometers, has been prop osed (see URL: http://www.karelia.ru/˜alexk [CAMP & Innovatio ns]). This is the first theory able to predict all known experimenta l tem- perature anomalies for water and ice. The conformity betwee n theory and experiment is very good even without adjustable paramet ers. The hierarchic concept creates a bridge between micro- and m acro- phenomena, dynamics and thermodynamics, liquids and solid s in terms of quantum physics. I. INTRODUCTION A theory of liquid state, as well as a general theory of conden sed matter, is still absent. However, this fundamental problem becomes mo re and more crucial for different brunches of science and technology. The existing solid states theories did not allow to extrapol ate them success- fully to liquids. Widely used molecular dynamics method is based on classical approach and corresponding computer simulations. It cannot be consider ed as a general one. The understanding of hierarchic organization of matter and developing of gen- eral theory is essential, as a mesoscopic bridge between mic roscopic and macro- scopic physics, between liquids and solids. There are strong evidence obtained that at least part of mole cules of solids and liquids did not follow classical Maxwell-Bolt zmann dis- tribution. This means that only quantum approach is valid fo r elab- oration of general theory of condensed matter. Our theoretical study of water and aqueous systems was initiated in 1986. It was stimulated by necessity to explain the nontrivi al phenomena, ob- tained by different physical methods in our investigations o f water-protein solu- tions. For example, the Drost-Hansen temperature anomalie s in water [1], cor- relating with changes in large scale protein dynamics were f ound in our group by specially elaborated experimental approaches. In the pr ocess of study it becomes evident, that the water clusters and water hierarch ical cooperative properties are dominating factors in self-organization, f unction and evolution of biosystems. The living organisms are strongly dependent on water proper- ties, representing about 70% of the body mass. On the other hand, due to 4its complicity and numerous anomalies, water is an ideal sys tem for testing a new theories of condensed matter. One may be sure th at if the theory works well with respect to water and ice, then it mu st be valid for other liquids, glasses or crystals as well. For thi s reason we have made the quantitative verification of our hierarchic me soscopic concept [2, 3, 4] on examples of water and ice. Our theory proceeds from fact that two main types of molecula r heat mo- tion:translational (tr) andlibrational (lb) oscillations - are characterized with non Maxwell-Boltzmann distributions in three- dimensional (3D) impulse space. The most probable impulses (p) determine the most probable de Broglie wave length ( λB=h/p=vph/νB) and phase velocity ( vph). Conformational intramolecular dynamics is taken into acco unt indirectly, as far it has an influence on the intermolecular dynamics and the value of the most probable parameters of waves B in condensed matter. Solids and liquids are considered as a hierarchical system o f collective excita- tions - metastable quasiparticles of the four new type: effectons, transitons, convertons and deformons, strongly interrelated with each other. When the effecton sizes determined by standing waves B of mole cules exceed the distances between them, then the coherent molecu lar clus- ters appear as a result of high temperature molecular Bose-c ondensation. The possibility of molecular Bose-condensation in liquids and solids at ”normal” temperatures is one of the most important statem ents of our model, confirmed by computer simulations. Effectons represent the self-organization of matter on a mes oscopic (nanoscale) level. In contrast to Fr¨ olich’s single polar mode excitation [5], the co- herence of thermal oscillations in the effectons: three-dim ensional (3D) standing de Broglie waves (waves B) of molecules and ato ms is induced not only by Coulomb interaction, determined the in- phase oscillations of dipole moments of cluster of molecules in th e ”opti- cal” b-state, but mainly by the distant Van der Waals interac tions and equality of 3D waves B length of molecules, forming effect ons in both: (a) and (b) states. The values of standing wave B length, determined the dimensions of the effectons in selected directions (1,2,3) and may be consider ed as a mesoscopic parameter of order . The primary transitons andconvertons have common features with coherent dissipative structures introduced by Chatzidimitrov-Dreisman and Br¨ andas in 1988 [6]. Such structures were predicted on the background o f complex scaling method and Prigogine theory of star- unitary transformatio ns. Estimated from principle of uncertainty, the minimum boson ’s ”degrees of freedom”( nmin) in these spontaneous coherent structures are equal to: nmin≥τ(2πkBT/h) 5where τis relaxation time of coherent-dissipative structures. If, for example, τ≃10−12c, corresponding to excitation of a molecular sys- tem by infrared photon, then at T= 300 Kone get nmin≃250. It means that at least 250 degrees of freedom, i.e. [250 /6] molecules, able to translations and librations act coherently and produce a photon absorption/emission phe- nomenon. The traditional consideration of an oscillating i ndividual molecule as a source of photons is replaced by the notion of a correlation pattern in such a model. The interactions like Josephson’s junctions are possible b etween the effectons also. In associated liquids like water, all hydrogen bonds betwee n molecules of the primary effectons are saturated. This condition corre sponds to minimum of their thermal mobility and maximum of de Brogli e wave length. The quantum effects, related to Bose condensati on of molecules, forming primary effectons are responsible for st abilization of clusters in both: (a) and (b) states of the effectons. The Fr¨ olich’s polar mode reflects only private case of thermal coherent dy- namics, considered by our model, corresponding to optical ( b)-state of primary effectons. We have to point out that the interactions between atoms and m olecules in condensed matter is much stronger and thermal mobility/imp ulse much lesser, than in gas phase. It means that the temperature of Bose conde nsation, when condition (1) starts to work can be much higher in solids and l iquids than in the gas phase. The lesser is interaction between molecules or atoms the lower temperature is necessary for initiation of Bose conde nsation. This conclusion is confirmed in 1995 by Ketterle’s group in MI T and later by few other groups, showing the Bose-Einstein condensatio n in gas of neutral atoms, like sodium (MIT), rubidium (JILA) and lithium (Rice University) at very low temperatures, less than one Kelvin. However, at this conditions the number of atoms in primary ga s effectons (Bose condensate) was about 20,000 and the dimensions were a lmost macro- scopic: about 15 micrometers. For comparisons the number of water molecules in primary lib rational effec- ton at freezing point 2730K is only 280 and linear dimension about 20 ˚A (see Fig. 7). OurHierarchic concept of matter unites and extends strongly two earlier existing most general models of solid state [7]: a) the Einstein model of independent quantum oscillators; b) the Debye model, taking into account only collective phen omena - phonons in a solid body like in a continuous medium. Among earlier models of liquid state the model of flickering c lusters by Frank and Wen (1957) is closest of all to our model. In our days the qu antum field theoretical approach to description of biosystems with som e ideas, close to our ones has been developed intensively by Umesawa’s group [8,9 ] and Italian group [10,11]. 6Del Giudice and Preparata (1998) introduced the notion of Co herent Elec- trodynamic Condensate (CEC), which may occur in the volume o f 3D photon’s standing waves, radiated by atoms/molecules of matter. Thi s idea is close to our notions of mesoscopic collective excitations of condensed matter in the volumes of 3D IR photons (Kaivarainen, 1992, 1995), related to trans lations (transla- tional macroeffectons) and librations (librational macroe ffectons). The new physical ideas required a new terminology. It is a rea son why one can feel certain discomfort at the beginning of reading this work. To facilitate this process, we present below a description of a new quasipa rticles, notions and terms, introduced in our Hierarchic Theory of matter (see Ta ble 1). II. THE NEW NOTIONS AND DEFINITIONS, INTRODUCED IN HIERARCHIC CONCEPT OF MATTER The most probable de Broglie wave (wave B). In composition of condensed matter the dynamics of particle s could be char- acterized by the thermal unharmonic oscillations of two typ es:translational (tr) andlibrational (lb). The length of the most probable wave B of thermally activated molecule, atom or atoms group in condensed matter can be estimated by tw o following ways: (λ1,2,3=h/mv1,2,3 gr=v1,2,3 ph/ν1,2,3 B)tr,lb where the most probable impulse p1,2,3=mv1,2,3 gris equal to product of the particle mass (m) and most probable group velocity ( v1,2,3 gr). The wave B length also could be evaluated as the ratio of its most probable phas e velocity ( v1,2,3 ph) to most probable frequency ( ν1,2,3 B). The indices (1,2,3) correspond to selected directions of mo tion in 3D space, related to the main axes of the molecules symmetry and their t ensor of po- larizability. In the case when molecular motion is anisotro pic one, we have inequalities: λ1 B/ne}ationslash=λ2 B/ne}ationslash=λ3 B Due to unharmonicity of oscillations: ( mv2/2)<(kT/2) - the most proba- ble kinetic energy of molecules ( Tkin)tr,lbis lesser than potential one ( V)tr,lb. Consequently, the most probable wave B length: λ1,2,3 B> h/(mkBT)1/2. The most probable (primary) effectons (tr and lb). Such a new type of quasiparticles (excitations) is represented by 3D super position of three most probable standing waves B of molecules. The shape of primary effectons in a general case can be approximated by a parallelepiped, with t he length of its 3 ribs determined by 3 most probable waves B length. The volume of primary effectons is equal to: 7Vef= (9/4π)λ1λ2λ3. The number of molecules or atoms forming effectons is: nm= (Vef)/(V0/N0), where V 0andN0are molar volume and Avogadro number, correspondingly. Thenmincreases with temperature decreasing and may reach hundre ds or even thousands of molecules. In liquids, primary effectons may be checked as a clusters and in solids as domains or microcrystalline. The thermal oscillations in the volume of corresponding effe ctons are syn- chronized. It means the coherence of the most probable wave B of molecules and their wave functions. We consider the primary effectons as a result of partial Bose condensation of molecules of condensed matter. Primary effec- tons correspond to the main state of Bose-condensate with th e packing number np= 0, i.e. with the resulting impulse equal to zero. Primary effectons, as a coherent clusters, represent self-o rganization of con- densed matter on mesoscopic level. It is a quantum phenomeno n. For the other hand, if the volume of primary effectons is less than the volum e, occupied by one molecule (V0/N0), it points to the classical behavior of molecules. ”Acoustic” (a) and ”optical” (b) states of primary effectons . The ”acoustic” a-state of the effectons is such a dynamic state when molecules or other particles composing the effectons, oscillate in the same phase (i.e. with- out changing the distance between them). The ”optic” b-state of the effectons is such dynamic state when particles oscillate in the counterphase manner (i.e. with periodical change of the distance between particles). This state of primary effectons has a com mon features with Fr¨ olich’s mode [5]. The kinetic energies of ”acoustic” ( a) and ”optical” ( b) states are equal [Ta kin=Tb kin] in contrast to potential energies [ Va< Vb]. It means that the most probable impulses in ( a) and ( b) states and, consequently, the wave B length and spatial dimensions of the effectons in the both states are equal. The energy of intermolecular interaction (Van der Waals, Co ulomb, hydrogen bonds etc.) in a-state are bigger than that in b-state. Consequently, the molecular polarizability in a-state also could be bigger than in b-state. It means that dielectric properties of matter may change as a re sult of shift of the (a⇔b)tr,lbequilibrium of the effectons. Primary transitons (tr and lb). Primary transitons represent intermediate transition states between ( a) and (b) modes of primary effectons - translational and librational . Primary transi- tons (tr and lb) - are radiating or absorbing photons corresp onding to trans- lational and librational bands in oscillatory spectra. The volumes of primary transitons and primary effectons coincides (see Table 1). 8Primary electromagnetic and acoustic deformons (tr, lb). Electromagnetic primary deformons are a new type of quasipa rticles (excita- tions) representing a 3 Dsuperposition of three standing electromagnetic waves. The IR photons originate and annihilate as a result of ( a⇔b)tr,lbtransitions of primary effectons, i.e. corresponding primary transitons. Such quantum tran- sitions are not accompanied by the fluctuation of density but with the change of polarizability and dipole moment of molecules only. Electromagnetic deformons appear as a result of intercepti on - superposi- tion of 3 pairs of photons, penetrating in matter in different selected directions (1,2,3). We assume, that each of these 3 pairs of photons form a standing wave in the volume of condensed matter. The linear dimension of each of three ribs of primary deformo n is determined by the wave length of 3 intercepting standing photons: λ1,2,3= [(n˜ν)−1]1,2,3 tr,lb where: nis the refraction index and (˜ ν)tr,lb- the wave number of transla- tional or librational band. These quasiparticles as the big gest ones, are respon- sible for the long-range space-time correlation in liquids and solids. In the case when ( b→a)tr,lbtransitions of primary effectons are accom- panied by great fluctuation of density they may be followed by emission of phonons instead of photons. It happened when primary effecto ns are involved in the spatial organization of macro- and supereffectons (se e below). Primary acoustic deformons may originate or annihilate in such a way. But for p rimary effectons the probability of collective spontaneous emissi on of photons during (b→a)tr,lbtransition is much higher than that of phonons. The coherent electromagnetic radiation as a result of self- correlation of many dipole moments in composition of coherent cluster, lik e primary effectons, containing N≫1 molecules is already known as superradiance [12].The time of collective transition in the case of superradiance is les s than that of isolated molecule and intensity of superradiance (I∼N·hν/τ∼N2)is much bigger than that from the same number of independent molecules (I∼N·hν/T1∼N). The (b→a) transition time of the primary effectons has the reverse dep endence on the N(τ∼1/N).The relaxation time for isolated atoms or molecules ( T1) is independent on N. The main energy is radiated in the direction of most elongated volume, i.e. ends of tubes. In technic superradia nce is used for very powerful masers and lasers with ultra short optic impulses. The properties of water as an dipole laser were considered in [13]. Secondary effectons (tr and lb) . In contrast to primary effectons, this type of quasiparticle s isconventional . They are the result of averaging of the frequencies and energ ies of the ”acoustic” (a) and ”optical” ( b) states of effectons with packing numbers nP>0, having the resulting impulse more than zero. For averaging the energies of such states the Bose-Einstein distribution was used under the condition when T < T 0(T0is temperature of degeneration and, 9simultaneously, temperature of first order phase transitio n). Under this con- dition the chemical potential: µ= 0 and distribution has a form of Plank equation. Secondary transitons (tr and lb). Secondary transitons, like primary ones are intermediate t ransition state between ( ¯ a) and ( ¯b) states of secondary effecton - translational and libration al. Secondary transitons are responsible for radiation and abs orption of phonons. As well as secondary effectons, these quasiparticles are con ventional, i.e. a result of averaging. The effective volumes of secondary tran sitons and secondary effectons - coincides. Secondary ”acoustic” deformons (tr and lb). This type of quasiparticles is also conventional as a result of 3D superposi- tion of averaged thermal phonons. These conventional phono ns originate and annihilate in a process of (¯ a⇔¯b)1,2,3thermoactivated transitions of secondary conventional effectons (translational and librational tra nsitons). In the case of secondary transitons (¯ a⇔¯b)tr,lbtransitions are accompanied by the fluctuation of density. Convertons (tr⇔lb). These important excitations are introduced in our model as i nterconversions between translational and librational primary effectons. T he(acon) convertons correspond to transitions between the (a) states of these eff ectons and (bcon) convertons - to that between their (b)-states. As far the dimensions of t rans- lational primary effectons are much less than that of librati onal ones, the con- vertons could be considered as the dissociation and association of t he primary librational effectons (coherent clusters). Both of convert ons: (acon) and ( bcon) are accompanied by density fluctuation, inducing phonons wi th corresponding frequency in the surrounding medium. The ca- and cb- deformons, induced by convertons. Three-dimensional (3D) superposition of phonons, irradiated by t wo types of convertons: aconandbcon, represent in our model the acoustic ca- and cb- deformons. They have the properties, similar to that of secondary deformons, discussed above. The c-Macrotransitons (Macroconvertons) and c-Macrodefo rmons. Simultaneous excitation of the acon andbcontypes of convertons in the volume of primary librational effectons lead to origination of big fluctuation, like cavitational one, termed c-Macrotransitons or Macroconvertons. In turn, such fluctuations induce in surrounding medium high frequency thermal phonons. The 3D-superposition these phonons forms c- Macrodefor- mons. Macroeffectons (tr and lb). Macroeffectons (A and B) are collective simultaneous excitations of the pri- mary and secondary effectons in the [ A∼(a,¯a)]tr,lband[B∼(b,¯b)]tr,lbstates in the volume of primary electromagnetic translational and li brational deformons, 10respectively. This correlation of similar primary and seco ndary states results in significant deviations from thermal equilibrium. The A and B states of macroef- fectons (tr and lb) may be considered as the most probable vol ume-orchestrated thermal fluctuations of condensed matter. Macrodeformons or macrotransitons (tr and lb). This type of conventional quasiparticles is considered in o ur model as an intermediate transition state of macroeffectons. The ( A→B)tr,lband (B→ A)tr,lbtransitions are represented by the coherent transitions of primary and secondary effectons in the volume of primary electromagnetic deformons - trans- lational and librational. The ( A→B)tr,lbtransition of macroeffecton is accom- panied by simultaneous absorption of 3 pairs of photons and t hat of phonons in the form of electromagnetic deformons. If ( B→A)tr,lbtransition occur without emission of photons, then all the energy of the excited B-sta te is transmitted into the energy of fluctuation of density and entropy of Macroeffec ton as an isolated mesosystem. It is a dissipative process: transition from th e more ordered struc- ture of matter to the less one, termed Macrodeformons. The bi g fluctuations of density during ( A⇔B)tr,lbtransitions of macroeffectons, i.e. macrodefor- mons are responsible for the Raleigh central component in Br illouin spectra of light scattering (see Chapter 9 in [3]). Translational and l ibrational macrode- formons are also related to the corresponding types of visco sity (Chapter 12 in [3]). The volumes of macrotransitons/macrodeformons (tr o r lb) and macroef- fectons coincide with that of trorlb primary electromagnetic deformons correspondingly . Supereffectons. This mixed type of conventional quasiparticles is composed of translational and librational macroeffectons correlated in space and time in the volumes of superimposed electromagnetic primary deformons (transla tional and librational - simultaneously). Like macroeffectons, supereffectons may exist in the ground (A∗ S) and excited ( B∗ S) states representing strong deviations from thermal equi- librium state. Superdeformons or supertransitons. This collective excitations have the lowest probability as compared to other quasiparticles of our model. Like macrodeformons, superde formons represent the intermediate ( A∗ S⇔B∗ S) transition state of supereffectons. In the course of these transitions the translational and librational mac roeffectons undergo simultaneous [(A⇔B)trand(A⇔B)lb] transitions. The (A∗ S→B∗ S) transition of supereffecton may be accompanied by the absor p- tion of two electromagnetic deformons - translational and l ibrational simultane- ously. The reverse ( B∗ S→A∗ S) relaxation may occur without photon radiation. In this case the big cavitational fluctuation originates. Such a process plays an important role in the processes of sub limation, evap- oration and boiling. 11The equilibrium dissociation constant of the reaction: H2O⇋H++HO− should be related with equilibrium constant of supertransi tons: K B∗ S⇋A∗ S. The A∗ S→B∗ Scavitational fluctuation of supereffectons can be accompani ed by the activation of reversible dissociation of part of water mole cules. Supertransitons and macrotransitons have the properties o f dissipative sys- tems. In contrast to primary and secondary transitons and deformo ns, the notions of [macro- and supertransitons] and [macro- and superde- formons] coincide. Such types of transitons and deformons represent the dy- namic processes in the same volumes of corresponding primar y electromagnetic deformons. Considering the transitions of all types of translational deformons (primary, secondary and macrodeformons), one must keep in mind that th elibrational type of modes remains the same. And vice versa, in case of libr ational defor- mons, translational modes remain unchanged. Only the reali zation of a con- vertons and supereffectons are accompanied by the interconv ersions between the translational and librational modes, between translation al and librational effec- tons. Interrelation between quasiparticles forming solids and l iquids. Our model includes 24types of quasiparticles (Table. 1):  4 -Effectons 4 -Transitons 4 -Deformons translational and librational ,including primary and secondary(I)  2 -Convertons 2−C-deformons 1−Mc-transiton 1−Mc-deformon the set of interconvertions between translational and librational primary effectons(II) /bracketleftbigg2 -Macroeffectons 2 -Macrodeformons/bracketrightbiggtranslational and librational (spatially separated)(III) /bracketleftbigg1 -Supereffectons 1 -Superdeformons/bracketrightbiggtranslational ⇋librational (superposition of trandlbeffectons and deformons in the same volume)(IV) Each level in the hierarchy of quasiparticles (I - IV) introduced in our model is based on the principle of correlation of correspond ing type of a dynamic 12process in space and time. All of these quasiparticles are co nstructed on the same physical principles as 3D -superposition (intercepti on) of different types of standing waves. Such a system can be handled as an ideal gas of quasiparticles of 24 types. As far each of the effecton’s types: trandlb, macroeffectons [ tr+lb] and supereffectons [ tr/lb] has two states (acoustic and optic) the total number of excitations is equal to: Nex=31 Three types of standing waves are included in our model: - de Broglie waves of particles (waves B); - acoustic waves (phonons); - electromagnetic waves (IR photons). This classification reflects the duality of matter and field and represent their self-organization and interplay on mesoscopic and macrosc opic levels. Our hierarchical system includes a gradual transition from theOrder (pri- mary effectons, transitons and deformons) to the Chaos (macro- and superdefor- mons). It is important, however, that in accordance with the model proposed, this thermal Chaos is ”organized” by hierarchical superposition of definite types of the ordered quantum excitations. It means that the final dy namics condensed matter only ”looks” as chaotic one. Our approach makes it pos sible to take into account the Hidden Order of Condensed Matter. The long-distance correlation between quasiparticles is d etermined mainly by the biggest ones - an electromagnetic primary deformons , involving in its volume a huge number of primary and secondary effectons .The vol- ume of primary deformons [tr and lb] could be conventionally subdivided on two equal parts, within the nodes of 3D standing IR electroma gnetic waves. The big number of the effectons (primary and secondary) as wel l as secondary deformons in each of these parts is equal also. However the dy namics of the effectons is orchestrated in such a way, that when one half of t heir quantity in the volume of big primary deformon undergo ( a→b)tr,lbtransitions, the other half of the effectons undergo the opposite ( b→a)tr,lbtransition. These processes compensate each other due to equilibrium energy e xchange of IR pho- tons and phonons. Such ”internal” dynamic equilibrium make s it possible to consider macroeffectons and macrodeformons as the isolated systems. The sim- ilar orchestrated equilibrium dynamics is pertinent also f or supereffectons and superdeformons. The increasing or decreasing in the concentration of primar y deformons is directly related to the shift of ( a⇔b)tr,lbequilibrium of the primary effectons leftward or rightward, respectively. This shift, in turn, l eads also to correspond- ing changes in the energies and concentrations of secondary effectons, deformons and, consequently, to that of super- and macro-deformons. It means the ex- isting of feedback reaction between subsystems of the effect ons and deformons, necessary for long-range self-organization in macroscopic volumes of condensed matter. 13Table 1. Schematic representation of the 18 types of quasiparticles of condensed matter as a hierarchical dynamic system, based on the effectons, transitons and deformons. Total number of quasiparticles , introduced in Hierarchic concept is 24. Six collective exci tations, related to convertons - interconversions between primary librational and translational effectons and their derivatives are not re presented here for the end of simplicity. The situation is possible when spontaneous oscillations be tween the sub- systems of effectons and deformons are not accompanied by the change in the total internal energy due to compensation effect . In such a way a long-period macroscopic autooscillations in liquids, revealed experi mentally [5, 6], could be explained. Such kind of phenomena, related to equilibrium shift of two subsystems, could be responsible for long relaxation (m emory) of water containing systems after different perturbations ( like mag- netic treatment, ultra high dilution, etc.) . The instability of macrosystem 14arises from competition between discrete quantum andaveraged thermal equilib- rium types of energy distributions of coherent molecular clusters, as it leads from our theoretical calculations. The total internal energy of substance is determined by the c ontributions of all types of quasiparticles with due regard for their own ene rgy, concentration and probability of excitation. Contributions of super- andmacro effectons and corresponding super- andmacro deformons as well as polyeffectons and coherent superclusters to the internal energy of matter normally are small, due to th eir low probability of excitation, big volume and, consequentl y, low concentration. III. THE MAIN STATEMENTS AND BASIC FORMULAE OF HIERARCHIC MODEL As far the acoustic ( a) and optical ( b) thermal coherent modes of molecules in composition of elementary cells or bigger clusters of the condensed matter are unharmonic, the quantum a⇔btransitions (beats) with absorption and radiation of phonons or photons can exist. The number of acoustic and optical modes is the same and equal to three [3], if oscillation of all p-atoms in the basis are cohe rent in both optical and acoustic dynamic states. Remnant modes are degenerated. The states of system, minimizing the uncertainty relation, when: [∆p·∆x∼/planckover2pi1and∆x=L∼/planckover2pi1/∆p]1,2,3(3.1) are quantum coherent states . A system of effectons could be considered as a partially degen erate Bose-gas. The degree of the degenerateness is proportional to the numb er of molecules in the volume of primary effectons. Degeneration in liquids gro ws up at lowering temperature and make a jump up as a result of (liquid →solid) phase transition as it leads from our theory and computer calculations. It is known from the Bose-Einstein theory of condensation de veloped by London in 1938 [14] that if the degeneration factor: λ= exp( µ/kT) (3.2) is close to λ≃1 at a low chemical potential value: µ≪kT (3.2a) then the contribution of bosons with the resulting impulse Pef≃0 (like primary effectons) cannot be neglected, when calculating in ternal energy. We assume in our theory that for all types of primary and secon dary effectons of condensed matter (solids and liquids), the condition (3. 2a) is valid. 15Partial Bose-Einstein condensation leads to the coherence of the waves B of molecules and atoms forming primary effectons in the both: ac oustic (a) and optic (b) states. Primary effectons are described with wave f unctions coherent in the volume of an effecton. In non ideal Bose-gas, despite the partial Bose-condensati on, the quasiparti- cles exist with nonzero impulse, termed as secondary effectons. These effectons obeys the Bose-Einstein statistics. The sizes of primary effectons determine the mesoscopic scal e of the con- densed matter organization. According to our model, the domains, nods, crystallites, and clusters observed in solid bodies and in l iquid crys- tals, polymers and biopolymers - can be a consequence of prim ary translational or librational effectons. Stabilization of molecules, atoms or ions in composition of coherent clus- ters (effectons) and correlation between different effectons could be provided by distant Van der Waals interaction and new Resonant Vibro-Gravitational Interaction, introduced in Chapter 17 of book [3]. It leads from quantitative consequences of mesoscopic concept, that [gas →liquid] phase transition is related with appearance the con ditions for par- tial Bose-condensation, when the primary librational effec tons, containing more than one molecule emerge [3, 4]. At the same time it means the b eginning of degeneration when the chemical potential µ→0. At this condition wave B length, corresponding to librations, starts to exceed the m ean distances between molecules in the liquid phase. It means that the temperature, at which the phase transition [gas→liquid] occurs, coincides with the temperature of partial Bose- con densation ( Tc),[i.e. primary librational effectons formation] and degeneration temperature ( T0). The changes of quasiparticles volume and shape in three dimensional (3D) space are related to corresponding changes in the impul se space. The total macroscopic Bose-condensation, in accordance wi th our model, responds to conditions, when [ a⇔b] equilibrium of primary effectons strongly shifts to the main (a)- state and (b)- state becomes thermall y inaccessible. At the same time the wave B length tends to macroscopic value. For quantum systems at temperature (T) higher than degeneration temper ature T0(T > T 0), when chemical potential ( µi=∂Gi/∂ni)<0 has a negative value, the mean number of Bose-particles ( ni) in state (i) is determined by the Bose- Einstein distribution: ni={exp[(ǫi−µi)/kT]−1}−1(3.3) where ǫiis the energy of the particle in state (i). For ”normal” conde nsed matter ǫi≫µ≪kT. The Bose-Einstein statistics, in contrast to the Maxwell- B oltzmann statis- tics, is applied to the indistinguishable Bose- particles w ith zero or integer spin values. The Fermi-Dirac distribution is valid for systems o f indistinguishable particles with a semi-integer spin obeying the Pauli princi ple. 16In the case of condensed matter at the temperature: 0< T < [T0∼=Tc] N∗particles of Bose condensate have a zero impulse [3, 28]: N∗≃N[1−(T/T0)3/2] (3.4) where N is the total number of particles in a system. A. Parameters of individual de Broglie waves (waves B) The known de Broglie relation expressing Wave-Particle Dua lity, has a simple form: − →p=/planckover2pi1− →k=h/− →λB − →p=/planckover2pi1/− →LB=m− →vgr where /vectork= 2π//vectorλ= 1//vectorLBis the wave number of wave B with length /vectorλ= 2π/vectorLB, /vector pis the impulse of particle with mass (m) and group velocity ( vgr),/planckover2pi1= h/2πis the Plank constant. Each particle can be represented as wave packet with group ve locity: vgr=/parenleftbiggdωb dk/parenrightbigg 0 and phase velocity: vph=ωb k(3.4a) where: ωBis the angle frequency of wave B determining the total energy of the waveB: (EB=/planckover2pi1ωB). Total energy is equal to the sum of kinetic ( Tk) and potential ( VB) energies and is related to particle’s mass and product of phase and gro up velocities (vgrvph) as follows [15]: EB=/planckover2pi1ωB=Tk+VB=(/planckover2pi1k)2 2m+VB=mvgrvph (3.4b) where (m) is particle mass; (c) is light velocity. From 3.4a and 3.4b it is possible to get an important relation between phase and group velocities of wave B and its kinetic, potential and total energy: vph vgr=Tk+VB 2Tk=EB 2Tk(3.4c) 17B. Parameters of de Broglie waves of molecules of condensed m at- ter The formulae given below allow to calculate the frequencies of the corre- sponding primary waves B in the directions 1,2,3 in aandbstates of primary effectons (translational and librational) [2, 3, 4]: /bracketleftbig νa 1,2,3/bracketrightbig tr,lb=/bracketleftBigg ν1,2,3 p exp(hν1,2,3 p/kT)−1/bracketrightBigg tr,lb(3.5) /bracketleftbig νb 1,2,3/bracketrightbig tr,lb=/bracketleftbig νa 1,2,3+ν1,2,3 p/bracketrightbig tr,lb(3.6) The most probable frequencies of photons/bracketleftbig ν1,2,3 p/bracketrightbig tr,lbare related to the wave numbers of the maxima of corresponding bands (tr and lib)/bracketleftbig ˜ν1,2,3 p/bracketrightbig tr,lbin os- cillatory spectra: /bracketleftbig ν1,2,3 p/bracketrightbig tr,lb=c/bracketleftbig ˜ν1,2,3 p/bracketrightbig tr,lb(3.7) where (c) is light velocity. For water the most probable freq uencies of photons, corresponding to ( a⇔b)trtransitions of primary translational effectons are determined by maxima with the wave numbers: /tildewideν(1) p= 60cm−1;/tildewideν(2) p=/tildewideν(3) p= 190cm−1. The band ˜ ν(1) p= ˜ν(2) p= ˜ν(3) p= 700 cm−1corresponds to the ( a⇔b)lb transitions of primary librational effectons. The degenera teness of frequencies characterizes the isotropy of the given mobility type for mo lecules. The distribution (3.5) coincides with the Plank formula, fo r the case when frequency of a quantum oscillator is equal to the frequency o f photon and: νp=npνp (3.8) where ¯ np= [exp( hνp/kT−1)]−1is the mean number of photons with the fre- quency νp.. The transition a→bmeans that ¯npincreases by one νb=νa+νp=nνp+νp=νp(n+ 1) (3.9) The derivation of the formula (3.5) is based upon the assumpt ion that ( a⇔ b)1,2,3transitions are analogous to the beats in a system of two weak ly inter- acting quantum oscillators. In such a case the frequency ( ν1,2,3 p) of photons is equal to the difference between the frequencies of waves B forming a primary effecton s in (b) and ( a) states as a frequency of quantum beats [15]: 18/bracketleftBig ν1,2,3 p=νb 1,2,3−νa 1,2,3= ∆ν1,2,3 B/bracketrightBig tr,lb(3.10) where ∆ ν1,2,3 Bis the most probable difference between frequencies of waves B in the marked directions (1,2,3). The ratio of concentrations for waves B in aandbstates ( na B/nb B) at such consideration is equal to the ratio of wave B periods Ta,bor the inverse ratio of wave B frequencies in these states: (Ta/Tb)1,2,3= (νb/νa)1,2,3. At the same time, the ratio of concentrations is determined w ith the Boltzmann distribution. So, the formula is true: /parenleftbiggna B nb B/parenrightbigg 1,2,3=/parenleftbiggνb νa/parenrightbigg 1,2,3= exp/parenleftBigg hν1,2,3 B kT/parenrightBigg = exp/parenleftBigg hν1,2,3 p kT/parenrightBigg (3.11) Substituting the eq.(3.10) into (3.11) we derive the eq.(3. 5), allowing to find (νa 1,2,3)tr,lband(νb 1,2,3)tr,lbfrom the data of oscillation spectroscopy at every temperature. The energies of the corresponding three waves B(Ea 1,2,3andEb 1,2,3) and that of the primary effectons as 3D standing waves with energies ( Ea efandEb ef) in aandbstates are equal to: /bracketleftbig Ea 1,2,3=hνa 1,2,3/bracketrightbig tr,lb;/bracketleftbig Ea ef=h(νa 1+νa 2+νa 3/bracketrightbig tr,lb(3.12) /bracketleftbigEb 1,2,3=hνb 1,2,3/bracketrightbig tr,lb;/bracketleftbig Eb ef=h(νb 1+νb 2+νb 3/bracketrightbig tr,lb(3.13) In our model energies of quasiparticles in each state are thu s determined only by the three selected coherent modes in directions (1,2,3). All remnant degrees of freedom: (3 n−3), where nis the number of molecules forming effectons or deformons, are degenerated due to their coherence. The mean packing numbers for ¯ aand¯bstates are thereby expressed with the formula (1.27), and the mean energies ( ¯Ea 1,2,3=h¯νa 1,2,3and¯Eb 1,2,3=h¯νb 1,2,3) - with Bose-Einstein distribution (1.21;1.28), coincident with the Plank formula at chemical potential µ= 0. Finally, the averaged Hamiltonians of ( a,¯ a) and ( b,¯b) states of the system containing primary and secondary effectons (translational and librational) have such a form: /bracketleftbig¯Ha 1,2,3=Ea 1,2,3+Ea 1,2,3=hνa 1,2,3+h¯νa 1,2,3/bracketrightbig tr,lb(3.14) 19/bracketleftBig ¯Hb 1,2,3=Eb 1,2,3+Eb 1,2,3=hνb 1,2,3+h¯νb 1,2,3/bracketrightBig tr,lb(3.15) where /bracketleftBigg ¯νa 1,2,3=νa 1,2,3/bracketleftbigexp(hνa 1,2,3)/kT−1/bracketrightbig=¯va ph ¯λ1,2,3 a/bracketrightBigg tr,lb(3.16) /bracketleftBigg ¯νb 1,2,3=νb 1,2,3/bracketleftbig exp(hνb 1,2,3)/kT−1/bracketrightbig=¯vb ph ¯λ1,2,3 b/bracketrightBigg tr,lb(3.17) ¯νa 1,2,3and ¯νb 1,2,3are the mean frequency values of each of three types of co- herent waves B forming effectons in ( ¯ a) and ( ¯b) states; ¯ va phand ¯vb phare the corresponding phase velocities. The resulting Hamiltonian for photons, which form the primary deformons andphonons forming secondary deformons , are determined with the term-wise subtraction of the formula (3.14) from the formula (3.15): |∆¯H1,2,3|tr,lb=h|νb 1,2,3−νa 1,2,3|tr,lb+h|¯νb 1,2,3−¯νa 1,2,3|tr,lb= =h(ν1,2,3 p)tr,lb+h(ν1,2,3 ph)tr,lb (3.18) where the frequencies of six IR photons, propagating in dire ctions ( ±1,±2,±3) and composing in the interceptions primary deformons are eq ual to: (ν1,2,3 p)tr,lb=|νb 1,2,3−νa 1,2,3|tr,lb= (c/λ1,2,3 p·n)tr,lb (3.19) where: c and n are the light velocity and refraction index of m atter; λ1,2,3 phare the wavelengths of photons in directions (1,2,3); and (ν1,2,3 ph)tr,lb=|νb 1,2,3−νa 1,2,3|tr,lb= (vs/λ1,2,3 ph)tr,lb (3.20) are the frequencies of six phonons (translational and libra tional) in the directions (±1,±2,±3), forming secondary acoustic deformons; vsis the sound speed; ¯λ1,2,3 phare the wavelengths of phonons in three selected directions . The corresponding energies of photons and phonons are: E1,2,3 p=hν1,2,3 p;E1,2,3 ph=h¯ν1,2,3 ph(3.21) 20The formulae for the wave B lengths of primary and secondary e ffectons are derived from (3.5) and (3.16): λ1,2,3 a)tr,lb=λ1,2,3 b=va p/νa 1,2,3= = (va p/ν1,2,3 p)·/bracketleftbigexp(hν1,2,3 p)/kT−1/bracketrightbig tr,lb(3.22) ¯λ1,2,3 a)tr,lb=¯λ1,2,3 b= ¯va ph/¯νa 1,2,3= = (¯va ph/¯ν1,2,3 ph)·/bracketleftBig exp(h¯ν1,2,3 ph)/kT−1/bracketrightBig tr,lb(3.23) The wavelengths of photons and phonons forming the primary andsecondary deformons can be determined as follows (λ1,2,3 p)tr,lb= (c/nν1,2,3 p)tr,lb= 1/(/tildewideν)1,2,3 tr,lb where: (˜ ν)1,2,3 tr,lbare wave numbers of corresponding bands in the oscillatory spectra of condensed matter. (¯λ1,2,3 ph)tr,lb= (¯vs/¯ν1,2,3 ph)tr,lb For calculations according to the formulae (2.59) and (2.60 ) it is necessary to find a way to calculate the resulting phase velocities of wave s B forming primary and secondary effectons ( va phand ¯va ph). C. Phase velocities of standing de Broglie waves, forming ne w types of quasiparticles In crystals three phonons with different phase velocities ca n propagate in thedirection set by the longitudinal wave normal. In a general c ase, two quasi- transversal waves ”fast” ( vf ⊥) and ”slow” ( vs ⊥) and one quasi-longitudinal ( v/bardbl) wave propagate [7]. The propagation of transversal acoustic waves is known to be accompa- nied by smaller deformations of the lattice than that of longitudinal waves, when they are caused by external impulses. The thermal phonons, sponta- neously originating and annihilating under conditions of h eat equilibrium may be accompanied by even smaller perturbations of the structu re and could be considered as transversal phonons. Therefore, we assume, that in the absence of external impuls es in solid state: vf ⊥≈vs ⊥=v1,2,3 phand the resulting thermal phonons velocity is determined as: vres s= (v(1) ⊥v(2) ⊥v(3) ⊥)1/3=vph (3.24) In liquids the resulting sound speed has an isotropic value: 21vliq s=vph. According to our model, the resulting velocity of elastic wa ves in condensed me- dia is related to the phase velocities of primary and seconda ry effectons in both (acoustic and optic) states and that of deformons (translat ional and librational) in the following way: /bracketleftbigvs=fava ph+fbvb ph+fdvd ph/bracketrightbig tr,lb(3.25) /bracketleftbig¯vs=¯fa¯va ph+¯fb¯vb ph+¯fd¯vd ph/bracketrightbig tr,lb(3.26) where: va ph, vb ph,¯va ph,¯vb phare phase velocities of the most probable and mean effectons in the ”acoustic” and ”optic” states; and vd ph=vd ph=vs are phase velocities of primary and secondary acoustic defo rmons, equal to phonons velocity. Nevertheless, ( a→b)tr,lbor (b→a)tr,lbtransitions of primary effectons are mainly related with absorption or emission of photons, the r ate of such process (relaxation time) is limited by the rate of changing the mode of oscillations in (a) and ( b) state, i.e. by sound velocity ( vs=vph). The phonons [absorp- tion/radiation] during these transitions could accompani ed the like processes in composition of macrodeformons; fa=Pa Pa+Pb+Pd;fb=Pb Pa+Pb+Pd;fd=Pd Pa+Pb+Pd(3.27) and fa=¯Pa ¯Pa+¯Pb+¯Pd;fb=¯Pb ¯Pa+¯Pb+¯Pd;fd=¯Pd ¯Pa+¯Pb+¯Pd(3.28) are the probabilities of corresponding states of the primar y (f) and secondary quasiparticles; Pa, Pb, Pdand¯Pa,¯Pb,¯Pd- relative probabilities of excitation (thermoaccessibilities) of the primary and secondary effec tons and deformons (see eqs. 4.10, 4.11, 4.18, 4.19, 4.25 and 4.26). Using eq. (3.4c) it is possible to express the phase velociti es inband¯bstates of effectons ( vb phand ¯vb ph) via ( va phand ¯va ph) in the following way: /bracketleftBigg vb ph vbgr/bracketrightBigg tr,lb=/bracketleftbiggEb tot 2Tb k/bracketrightbigg tr,lb=/bracketleftbigghνb res m(vbgr)2/bracketrightbigg tr,lb(3.29) 22From this equation, we obtain for the most probable phase velocity in (b) state: (vb ph)tr,lib=/bracketleftbig λres phνres b/bracketrightbig tr,lib=/bracketleftbigg (va ph)νb res νares/bracketrightbigg tr,lb(3.30) We keep in mind that according to our model vb gr=va grand ¯vb gr= ¯va gr, i.e. the group velocities of both states are equal. Likewise for the mean phase velocity in ¯b-state of effectons we have: (¯vb ph)tr,lb=/bracketleftbigg/parenleftbig¯va ph/parenrightbig¯νres b ¯νresa/bracketrightbigg tr,lb(3.31) where in (3.30): /bracketleftbigg νb res= (νb 1νb 2νb 3)1/3 νa res= (νa 1νa 2νa 3)1/3/bracketrightbigg tr,lb(3.32) are the resulting frequencies of the most probable (primary ) effectons in band astates. They can be calculated using the eqs. (3.5 and 3.6); f requencies; and in (3.31): /bracketleftBig νb res= (νb 1νb 2νb 3)1/3/bracketrightBig tr,lb(3.33) /bracketleftBig νa res= (νa 1νa 2νa 3)1/3/bracketrightBig tr,lb(3.34) are the resulting frequencies of the mean effectons in ¯band¯ astates. They can be estimated according to eqs. (3.17 and 3.16). Using eqs. (3.25 and 3.30), we find the formulas for the resulting phase velocities of the primary translational and librational effectons in ( a) state: /parenleftbig va ph/parenrightbig tr,lb= vs(1−fd) fa 1 +Pb Pa/parenleftBig νbres νares/parenrightBig  tr,lb(3.35) Similarly, for the resulting phase velocity of secondary eff ectons in (a) state we get from (3.26) and (3.31): /parenleftbig ¯va ph/parenrightbig tr,lb= vs(1−¯fd) ¯fa 1 +¯Pb¯Pa/parenleftBig ¯νbres ¯νares/parenrightBig  tr,lb(3.36) 23As will be shown below, it is necessary to know va phand ¯va phto determine theconcentration of the primary and secondary effectons. When the values of resulting phase velocities in aand¯ astates of effectons are known, then from eqs. (3.30) and (3.31) it is easy to express resulting phase v elocities in band¯b states of translational and librational effectons. D. Concentrations of quasiparticles, introduced in Hierar chic model of condensed matter It has been shown by Rayleigh that the concentration of the st anding waves of any type with wave lengths within the range: λtoλ+dλis equal to: nλdλ=4πdλ λ4(3.37) or, expressing wave lengths via their frequencies and phase velocities λ=vph/ν we obtain: nνdν= 4πν2dν v2 ph(3.38) For calculation the concentration of standing waves within the frequency range from zero to the definite characteristic frequency, for exam ple, to the most probable ( νa) or mean (¯ νa) frequency of wave B, then eq. (3.38) should be integrated: na=4π v3 phνa/integraldisplay 0ν2dν=4 3π/parenleftbiggνa vph/parenrightbigg3 =4 3π1 λ3a(3.39) Jeans has shown that each standing wave formed by photons or p honons can be polarized twice. Taking into account this fact the concen trations of standing photons and standing phonons in the all three directions (1, 2,3) are equal to: n1,2,3 p=8 3π/parenleftBigνp 1,2,3 c1,2,3/n/parenrightBig3 ¯n1,2,3 ph=8 3π/parenleftbigg ¯νph 1,2,3 v1,2,3 ph/parenrightbigg3(3.40) where: c and n are the light speed in vacuum and refraction ind ex of matter; vph=vs- velocity of thermal phonons, equal to sound velocity. The standing waves B of atoms and molecules have only one line ar polariza- tion in directions (1,2,3). Therefore, their concentratio ns are described by an equation of type (3.39). According to our model (see Introduction), superposition o f each of three differently oriented (1,2,3) standing waves B forms quasi-p articles which we have 24termed effectons . They are divided into the most probable (primary) (with zer o resulting impulse) and mean (secondary) effectons. Quasipa rticles, formed by 3D superposition of standing photons and phonons, originat ing in the course of (a⇔b) and (¯ a⇔¯b) transitions of the primary and secondary effectons, respectively, are termed primary and secondary deformons (Table 1). Effectons and deformons are the result of thermal translations (tr) and libra- tions (lb) of molecules in directions (1,2,3). These quasiparticles a re generally approximated by a parallelepiped with symmetry axes (1,2,3 ). As far three coherent standing waves of corresponding nature take part in the construction of each effecton , it means that the concentration of such quasi- particles must be three times lower than the concentration o f standing waves expressed by eq. (3.39). The coherence of molecules in the vo lume of the ef- fectons and deformons due to partial Bose-condensation is t he most important feature of our model, which leads to degeneration of waves B o f these molecules. Finally, we obtain: Theconcentration of primary effectons, primary transitons and convertons: /parenleftbignef/parenrightbig tr,lb=4 9π/parenleftBigg νa res va ph/parenrightBigg3 tr,lb=nt=nc (3.41) where νa res= (νa 1νa 2νa 3)1/3 tr,lb(3.42) is the resulting frequency of a-state of the primary effecton; νa 1, νa 2, νa 3are the most probable frequencies of waves B in a-state in directions (1,2,3), which are calculated according to formula (3 .5);va ph- the resulting phase velocity of effectons in a-state, which corresponds to eq. (3.35 ). Theconcentration of secondary (mean) effectons and secondary t ransitons is expressed in the same way as eq. (3.41): (¯nef)tr,lb=4 9π/parenleftBigg ¯νa res ¯va ph/parenrightBigg3 tr,lb=nt (3.43) where phase velocity ¯va phcorresponds to eq. (3.36); νa res= (¯νa 1¯νa 2¯νa 3)1/3(3.44) - the resulting frequency of mean waves B in ¯ a-state. The mean values ¯ νa 1,2,3 are found by the formula (3.16). Maximum concentrations of the most probable and mean effectons ( nmax ef) and (¯nmax ef), as well as corresponding concentrations of transitons ( nmax t) and 25(¯nmax t) follow from the requirement that it should not be higher tha n the con- centration of atoms. If a molecule or elementary cell consists of [ q]atoms , which have their own degrees of freedom and corresponding impulses, then nmax ef=nmax t=nmax ef=nmax t=qN0 V0 The concentration of the electromagnetic primary deformon sfrom eq. (3.40): /parenleftbignd/parenrightbig tr,lb=8 9π/parenleftbiggνres d c/n/parenrightbigg3 tr,lb(3.46) where ( c) and ( n) are light speed and refraction index of matter; /parenleftbig νres d/parenrightbig tr,lb=/parenleftBig ν(1) pν(2) pν(3) p/parenrightBig1/3 tr,lb(3.47) - the resulting frequency of primary deformons, where /parenleftbigν1,2,3 p/parenrightbig tr,lb=c/parenleftbig˜ν1,2,3 p/parenrightbig tr,lb(3.48) are the most probable frequencies of photons with double pol arization, related to translations and librations; c - the speed of light; ˜ νp- the wave numbers, which may be found from oscillatory spectra of matter. Theconcentration of acoustic secondary deformons derived from eq. (3.40) is: /parenleftbig¯nd/parenrightbig tr,lb=8 9π/parenleftbigg¯νres d vs/parenrightbigg3 tr,lb(3.49) where v sis the sound velocity; and /parenleftbig¯νres d/parenrightbig tr,lb=/parenleftBig ¯ν(1) ph¯ν(2) ph¯ν(3) ph/parenrightBig1/3 tr,lb(3.50) is the resulting frequency of secondary deformons (translational and librational); in this formula: /parenleftBig ¯ν1,2,3 ph/parenrightBig tr,lb=/vextendsingle/vextendsingle¯νa−¯νb/vextendsingle/vextendsingle1,2,3 tr,lb(3.51) are the frequencies of secondary phonons in directions (1,2 ,3), calculated from (3.16) and (3.17). 26Since the primary and secondary deformons are the results of transitions (a⇔band ¯a⇔¯b)tr,lbof the primary and secondary effectons, respectively, then the maximum concentration of effectons, transitons and deformons must coincide: nmax d=nmax d=nmax ef=nmax t=nmax ef=nmax t=qN0 V0(3.52) IV. HIERARCHIC THERMODYNAMICS A. The internal energy of matter as a hierarchical system of c ol- lective excitations The quantum theory of crystal heat capacity leads to the foll owing equation for the density of thermal internal energy [7]: ǫ=1 Vi/summationtextEiexp(−Ei/kT) i/summationtextexp(−Ei/kT)(4.1) where V - the crystal volume; Ei- the energy of the i-stationary state. According to our Hierarchic theory, the internal energy of m atter is deter- mined by the concentration ( ni) of each type of quasiparticles, probabilities of excitation of each of their states ( Pi) and the energies of corresponding states (Ei). The condensed matter is considered as an ”ideal gas” of 3D s tanding waves of different types (quasiparticles and collective exc itations). However, the dynamic equilibrium between types of quasiparticles is very sensitive to the external and internal perturbations. The total partition function - the sum of the relative probab ilities of exci- tation of all states of quasiparticles (the resulting thermoaccessibility ) is equal to: Z=/summationdisplay tr,lb  /parenleftBig Pa ef+Pb ef+Pd/parenrightBig + +/parenleftBig ¯Pa ef+¯Pb ef+¯Pd/parenrightBig + +/bracketleftbig/parenleftbig PA M+PB M/parenrightbig +PM D/bracketrightbig   tr,lb+ + (Pac+Pbc+PcMd) +/parenleftbig PA S+PB S+Ps D∗/parenrightbig (4.2) Here we take into account that the probabilities of excitati on of primary and secondary transitons and deformons are the same ( Pd=Pt;¯Pd=¯Pt) and related to the same processes: (a⇔b)tr,lb and (¯ a⇔¯b)tr,lbtransitions. 27The analogous situation is with probabilities of a, b and cM convertons and corresponding acoustic deformons excitations: Pac, PbcandPcMd=PcMt. So it is a reason for taking them into account in the partition fu nction only ones. The final formula for the total internal energy of ( Utot) of one mole of matter leading from mesoscopic model, considering the system of 3D standing waves as an ideal gas is: Utot=V01 Z/summationdisplay tr,lb/braceleftbigg/bracketleftbigg nef/parenleftbigPa efEa ef+Pb efEb ef+PtEt/parenrightbig +ndPdEd/bracketrightbigg + +/bracketleftbig ¯nef/parenleftbig¯Pa ef¯Ea ef+¯Pb ef¯Eb ef+¯Pt¯Et/parenrightbig + ¯nd¯Pd¯Ed/bracketrightbig + +/bracketleftbig nM/parenleftbigPA MEA M+PB MEB M/parenrightbig +nDPD MED M/bracketrightbig tr,lb+ +V01 Z/bracketleftbig ncon/parenleftbigPacEac+PbcEbc+PcMtEcMt/parenrightbig + +(ncdaPacEac+ncdbPbcEbc+ncMdPcMdEcMd)/bracketrightbig + +V01 Zns/bracketleftbig/parenleftbig PA∗ SEA∗ S+PB∗ SEB∗ S/parenrightbig +nD∗PD∗ SED∗ S/bracketrightbig (4.3) where all types the effecton’s contributions in total intern al energy correspond to: Uef=V01 Z/summationdisplay tr,lb/bracketleftbignef/parenleftbigPa efEa ef+Pb efEb ef/parenrightbig + +¯nef/parenleftbig¯Pa ef¯Ea ef+¯Pb ef¯Eb ef/parenrightbig +nM/parenleftbigPA MEA M+PB MEB M/parenrightbig /bracketrightbig tr,lb+ +V01 Zns/parenleftbig PA∗ sEA∗ s+PB∗ sPB∗ s/parenrightbig (4.4) all types of deformons contribution in U totis: Ud=V01 Z/summationtext tr.lb/parenleftbig ndPdEd+ ¯nd¯Pd¯Ed+nMPD MED M/parenrightbig tr,lb+ +V01 ZnsPD∗ SED∗ S(4.5) and contribution, related to [ lb/tr] convertons: 28Ucon=V01 Z/bracketleftbig ncon/parenleftbig PacEac+PbcEbc+PcMtEcMt/parenrightbig + +(ncdaPacEac+ncdbPbcEbc+ncMdPcMtEcMd)/bracketrightbig Contributions of all types of transitons ( Ut) also can be easily calculated. The intramolecular configurational dynamics of molecules i s automatically taken into account in our approach as it has an influence on the intermolecular dynamics, dimensions, and on concentration of quasipartic les as well as on the energy of excitation of their states. These dynamics affects the positions of the absorption bands in oscillatory spectra and values of sound velocity, that we use for calculation of internal energy. The remnant small contribution of intramolecular dynamics to Utotis related to oscillation energy corresponding to fundamental molecu lar modes ( νi p). It may be estimated using Plank distribution: Uin=N0i/summationdisplay 1h¯νi p=N0i/summationdisplay 1hνi p/bracketleftbig exp/parenleftbighνi p/kT/parenrightbig −1/bracketrightbig−1 where ( i) is the number of internal degrees of freedom. i= 3q−6 for nonlinear molecules; i= 3q−5 for linear molecules qis the number of atoms forming a molecule. It has been shown by our computer simulations for the case of w ater and ice thatUin≪Utot. It should be general condition for any condensed matter. Let us consider now the meaning of the variables in formulae ( 4.2 - 4.5), necessary for the internal energy calculations: V0is the molar volume; nef,¯nefare the concentrations of primary (eq. 3.41) and secondary ( eq. 3.42) effectons; Ea ef, Eb efare the energies of the primary effectons in aandb states: /bracketleftbigEa ef= 3hνa ef/bracketrightbig tr,lb(4.6) /bracketleftbigEb ef= 3hνb ef/bracketrightbig tr,lb, (4.7) where /bracketleftbigνa ef=1 3/parenleftbigνa 1+νa 2+νa 3/parenrightbig/bracketrightbig tr,lb(4.8) 29/bracketleftbigνb ef=1 3/parenleftbig νb 1+νb 2+νb 3/parenrightbig/bracketrightbig tr,lb(4.9) are the characteristic frequencies of the primary effectons in the ( a) and ( b) - states; νa 1,2,3, νb 1,2,3are determined according to formulas (3.5 and 3.6); Pa ef, Pb ef- the relative probabilities of excitation (thermoaccessi bilities) of effectons in ( a) and ( b) states [2-4] introduced as:  Pa ef= exp −/vextendsingle/vextendsingle/vextendsingleEa ef−E0/vextendsingle/vextendsingle/vextendsingle kT = exp −3h/vextendsingle/vextendsingle/vextendsingleνa ef−ν0/vextendsingle/vextendsingle/vextendsingle kT   tr,lb(4.10)  Pb ef= exp −/vextendsingle/vextendsingle/vextendsingleEa ef−E0/vextendsingle/vextendsingle/vextendsingle kT = exp −3h/vextendsingle/vextendsingle/vextendsingleνb ef−ν0/vextendsingle/vextendsingle/vextendsingle kT   tr,lb(4.11) where E0= 3kT= 3hν0 (4.12) is the equilibrium energy of all types of quasiparticles det ermined by the tem- perature of matter (T): ν0=kT h(4.13) is the equilibrium frequency. ¯Ea ef,¯Eb efare the characteristic energies of secondary effectons in b aand states: /bracketleftBig Ea ef= 3h¯νa ef/bracketrightBig tr,lb(4.14) /bracketleftBig Eb ef= 3h¯νb ef/bracketrightBig tr,lb, (4.15) where /bracketleftbig¯νa ef=1 3/parenleftbig¯νa 1+ ¯νa 2+ ¯νa 3/parenrightbig/bracketrightbig tr,lb(4.16) /bracketleftbig¯νb ef=1 3/parenleftbig ¯νb 1+ ¯νb 2+ ¯νb 3/parenrightbig/bracketrightbig tr,lb(4.17) 30are the characteristic frequencies of mean effectons in aand¯bstates; ¯ νa 1,2,3,¯νb 1,2,3 determined according to formulae (3.16 and 3.17). ¯Pa ef,¯Pb efare the relative probabilities of excitation (thermoacces sibilities) of mean effectons in ¯ aand¯bstates (K¨ aiv¨ ar¨ ainen, 1989a) introduced as:  ¯Pa ef= exp −/vextendsingle/vextendsingle/vextendsingleEa ef−E0/vextendsingle/vextendsingle/vextendsingle kT = exp −3h/vextendsingle/vextendsingle/vextendsingle¯νa ef−ν0/vextendsingle/vextendsingle/vextendsingle kT   tr,lb(4.18)  ¯Pb ef= exp −/vextendsingle/vextendsingle/vextendsingleEb ef−E0/vextendsingle/vextendsingle/vextendsingle kT = exp −3h/vextendsingle/vextendsingle/vextendsingle¯νb ef−ν0/vextendsingle/vextendsingle/vextendsingle kT   tr,lb(4.19) Parameters of deformons (primary and secondary) [tr and lb] : nd,¯ndare the concentrations of primary (eq. 3.46) and secondary ( eq. 3.49) deformons; Ed,Edare the characteristic energies of the primary andsecondary defor- mons,equal to energies of primary and secondary transitons: /bracketleftbig Ed= 3hνres d=Et/bracketrightbig tr,lb(4.20) /bracketleftbig Ed= 3h¯νres d=Et/bracketrightbig tr,lb(4.20) where: characteristic frequencies of the primary and secon dary deformons are equal to: /bracketleftBig νres d=1 3/parenleftBig ν(1) p+ν(2) p+ν(3) p/parenrightBig /bracketrightBig tr,lb(4.22) /bracketleftBig ¯νres d=1 3/parenleftBig ¯ν(1) ph+ ¯ν(2) ph+ ¯ν(3) ph/parenrightBig /bracketrightBig tr,lb(4.23) The frequencies of the primary photons are calculated from t he experimental data of oscillatory spectra using (3.48). The frequencies of secondary phonons are calculated as: /parenleftBig ν1,2,3 ph/parenrightBig tr,lb=|νa−νb|1,2,3 tr,lb(4.24) where ν1,2,3 aandν1,2,3 bare founded in accordance with (3.16) and (3.17). Pdand ¯Pdare the relative probabilities of excitation of primary and secondary deformons in medium, surrounding effectons, intr oduced as the prob- abilities of intermediate transition states: 31(a⇔b)tr,lband (¯ a⇔¯b)tr,lb: /parenleftbigPd=Pa ef·Pb ef/parenrightbig tr,lb(4.25) /parenleftbig¯Pd=¯Pa ef·¯Pb ef/parenrightbig tr,lb(4.26) Parameters of transitons: (nt)tr,lband (¯nt)tr,lbare concentrations of primary and secondary transitons, equal to concentration of primary (3.41) and secondary (3.4 3) effectons: (nt=nef)tr,lb; (nt=nef)tr,lb (4.27) (Ptand¯Pt)tr,lbare the relative probabilities of excitation of primary and sec- ondary transitons, equal to that of primary and secondary de formons: (Pt=Pd)tr,lb; (¯Pt=¯Pd)tr,lb (EtandEt)tr,lbare the energies of primary and secondary transitons: /bracketleftBig Et=Ed=h(ν(1) p+ν(2) p+ν(3) p)/bracketrightBig tr,lb(4.28) /braceleftBig Et=Ed= 3h/bracketleftbig|¯νa ef−ν0|+|¯νb ef−ν0|/bracketrightbig1,2,3/bracerightBig tr,lb(4.29) Primary and secondary deformons in contrast to transitons, represent the quasielas- tic mechanism of the effectons interaction via medium. Parameters of macroeffectons [tr and lb]: (nM=nd)tr,lbare the concentrations of macroeffectons equal to that of primary deformons (3.46); (EA MandEB M)tr,lbare the energies of A and B states of macroeffectons; ( νA M andνB M)tr,lbare corresponding frequencies, defined as: /bracketleftbig EA M= 3hνA M=−kTlnPA M/bracketrightbig tr,lb(4.29a) 32/bracketleftbig EB M= 3hνB M=−kTlnPB M/bracketrightbig tr,lb(4.29b) where /bracketleftbig PA M=Pa·Pa/bracketrightbig tr,lb(4.29c) and /bracketleftbig PB M=Pb·Pb/bracketrightbig tr,lb(4.29d) are the relative probabilities of excitation of A and B state s of macroeffectons. Parameters of macrodeformons [tr and lb]: (nD M)tr,lbis the concentration of macrodeformons equal to that of macr oeffec- tons (macrotransitons) corresponding to concentration of corresponding primary deformons: see eq.(3.46); (PD M)tr,lb= (PA M·PB M)tr,lb (4.29e) are the probabilities of macrodeformons excitation; (EM D)tr,lb=−kTln(PD M)tr,lb= 3h(νD M)tr,lb (4.29f) are the energies of macrodeformons; Parameters of convertons and related excitations The frequency and energy of a-convertons and b- convertons: νac=|(νa ef)lb−(νa ef)tr|;Eac= 3hνac νbc=|(νb ef)lb−(νb ef)tr|;Ebc= 3hνbc (4.30) where: characteristic frequencies ( νa ef)lband (νa ef)trcorrespond to (4.8). where characteristic frequencies ( νb ef)lband (νb ef)trcorrespond to (4.9). Probabilities of (a) and (b) convertons, equal to that of cor responding acous- tic c-deformons excitations: /parenleftbiggPac= (Pa ef)tr·(Pa ef)lb Pbc= (Pb ef)tr·(Pb ef)lb/parenrightbigg (4.30a) 33Probability and energy of c - Macrotransitons (Macroconvertons) excitation [simultaneous excitation of (a) and (b) con- vertons ],equal to that of c- Macrodeformons is: PcMd=Pac·Pbc;EcMt=EcMd=−kT·lnPcMd (4.30b) The characteristic frequency of cM-transitons and cM-defo rmons is: νcMt=νcMd=EcMd/3h The concentrations of (a), (b)-convertons ( ncon) andc-Macrotransitons ( ncMd) are equal to that of primary effectons ( nef). The concentrations of acoustic deformons, excited by conve rtons.. The concentrations of ca-deformons andcb-deformons, representing 3D stand- ing phonons, excited by a-convertons and by b-convertons correspondingly are: /parenleftbign/parenrightbig cad,cbd=8 9π/parenleftbiggνac,bc vs/parenrightbigg3 (4.30c) where [ vs] is the sound velocity and νac= (νa ef)lb−(νa ef)tr, ν bc= (νb ef)lb−(νb ef)tr (4.30d) are characteristic frequencies of a- and b-convertons, equal to the difference between characteristic frequencies of primary librationa l and translational ef- fectons (see eqs.4.8 and 4.9) in aandbstates correspondingly. The concentration of cM-deformons , excited by cM-transitons (or Macro- convertons) is equal to: ncMd=8 9π/parenleftbiggνcMd vs/parenrightbigg3 (4.30e) where: νcMdis characteristic frequency of c-Macrodeformons, equal to that of c-Macrotransitons (Macroconvertons) . The maximum concentration of all convertons-related excit ations is also lim- ited by concentration of molecules Parameters of supereffectons: 34(nS=nd)lbis the concentration of supereffectons, equal to that of prim ary librational deformons (3.46); PA∗ S;PB∗ Sare the relative probabilities of excitation of A∗andB∗: PA∗ S= (PA M)tr·(PA M)lbPB∗ S= (PB M)tr·(PB M)lb (4.30f) andEA∗ S;EB∗ Sare the energies of A and B states of supereffectons from (3.27) and (3.28); EA∗ S=−kT·lnPA∗ S EB∗ S=−kT·lnPB∗ S Parameters of superdeformons: nD∗is the concentration of superdeformons, equal to that of sup ereffectons; PD∗ S= (PD M)tr·(PD M)lb (4.30g) is the relative probability of superdeformons; ED∗ Sis the energy of superdeformons, defined as: ED∗ S=−kT·lnPD∗ S (4.30h) Substituting the parameters of quasiparticles, calculate d in this way into eqs. (4.2 and 4.3), we obtain the total internal energy of one mole of matter in solid or liquid phase. For water and ice the theoretical re sults coincide with experimental one fairly well (see Fig. 2). It is important that our equations are the same for solid and l iq- uid states. The difference in experimental parameters, such as molar vol ume, sound velocity, refraction index, positions of translatio nal and librational bands determines the difference of internal energy and of more than 100 another pa- rameters of any state of condensed matter, which can be calcu lated using eq. (4.3). It is important to stress that our concept is general for soli ds and liquids, for crystals, glasses and amorphous matter. 4.2. The contributions of kinetic and potential energy to the total internal energy The total internal energy of matter ( Utot) is equal to the sum of total kinetic (Ttot) and total potential ( Vtot) energy: Utot=Ttot+Vtot 35The kinetic energy of wave B(TB) of one molecule may be expressed using its total energy ( EB), mass of molecule (m), and its phase velocity as wave B (vph): TB=mv2 gr 2=E2 B 2mv2 ph(4.31) The total mass ( Mi) of 3D standing waves B forming effectons, transitons and deformons of different types are proportional to number of mo lecules in the volume of corresponding quasiparticle ( Vi= 1/ni): Mi=1/ni V0/N0m (4.32) the limiting condition for minimum mass of quasiparticle is : Mmin i=m (4.33) Consequently the kinetic energy of each coherent effectons i s equal to /bracketleftBigg Ti kin=E2 i 2Miv2 ph/bracketrightBigg (4.34) where: E iis a total energy of given quasiparticle. The kinetic energy of coherent primary and secondary deform ons and tran- sitons we express analogously to eq. (4.34), but instead of t he phase velocity of waves B we use the light speed and resulting sound velocity vres(eq.3.24), respectively: /bracketleftbigg Ti kin=E2 i 2Mic2/bracketrightbigg dand/bracketleftbigg Ti kin=E2 i 2Mi(vress)2/bracketrightbigg d(4.35) The kinetic energies of [ tr/lb] convertons: /bracketleftbigg Ti kin=(Ei/3)2 2Mi(vress)2/bracketrightbigg =/bracketleftbigg Ti kin=E2 i 6Mi(vress)2/bracketrightbigg con(4.35a) According to our model, the kinetic energies of the effectons inaandband also in the ¯ aand¯bstates are equal. Using (4.34) and (4.35) we obtain from eq.( 4.3) the total thermal kinetic energy for 1 mole of matter: Ttot=V01 Z/summationdisplay tr,lb   nef/summationtext(Ea)2 1,2,3 2Mef(va ph)2∗/parenleftbig Pa ef+Pb ef/parenrightbig + ¯nef/summationtext/parenleftbig¯Ea/parenrightbig2 1,2,3 2Mef(va ph)2∗/parenleftbig¯Pa ef+¯Pb ef/parenrightbig + 36+ nt/summationtext(Et)2 1,2,3 2Mt(vress)2Pd+ ¯nt/summationtext/parenleftbig¯Et/parenrightbig2 1,2,3 2¯Mt(vress)2¯Pd + nd/summationtext(Ed)2 1,2,3 2Mdc2Pd+ ¯nd/summationtext/parenleftbig¯Ed/parenrightbig2 1,2,3 2Md(vress)2¯Pd + +/bracketleftBigg nM/parenleftbigEA M/parenrightbig2 6MM(vA ph)2∗/parenleftbigPA M+PB M/parenrightbig +nD/parenleftbigED/parenrightbig2 6MD(vress)2PM D/bracketrightBigg/bracerightBigg tr,lb+ +V0ncon Z/bracketleftBigg/parenleftbigEac/parenrightbig2 6Mc(vress)2Pac+/parenleftbigEbc/parenrightbig2 6Mc(vress)2Pbc+/parenleftbigEcMd/parenrightbig2 6Mc(vress)2PcMd/bracketrightBigg + V01 Z/bracketleftBigg ncda/parenleftbigEac/parenrightbig2 6Mc(vress)2Pac+ncdb/parenleftbigEbc/parenrightbig2 6Mc(vress)2Pbc+ncMd/parenleftbigEcMd/parenrightbig2 6Mc(vress)2PcMd/bracketrightBigg + +V01 Z/bracketleftBigg nS/parenleftbig EA∗ S/parenrightbig2 6MS/parenleftbig vA∗ ph/parenrightbig2∗/parenleftBig PA∗ S+PB∗ S/parenrightBig +nS(ED∗)2 6MS(vress)2PD∗ S/bracketrightBigg (4.36) where the effective phase velocity of A-state of macroeffecto ns is introduced as: /bracketleftBigg 1 vA ph=1 va ph+1 ¯va ph/bracketrightBigg tr,lb→/bracketleftBigg vA ph=va ph·¯va ph va ph+ ¯va ph/bracketrightBigg tr,lb(4.37) and the effective phase velocity of supereffecton in A∗-state: vA∗ ph=(vA ph)tr·(vA ph)lb (vA ph)tr+ (vA ph)lb(4.38) Total potential energy is defined by the difference between to tal internal (eq. 4.3) and total kinetic energy (eq. 4.36): Vtot=Utot−Ttot(4.39) Consequently, we can separately calculate the kinetic and p otential energy con- tributions to the total thermal internal energy of matter, u sing four experimental parameters, obtained at the same temperature and pressure: 1)density or molar volume; 2)sound velocity; 3)refraction index and 4)positions of translational and librational bands in oscillatory spectrum of condensed matter. 37It is important to stress that the same equations are valid fo r liquids and solids. The contributions of all individual types of quasiparticle s in thermodynamics as well as a lot of characteristics of these quasiparticles a lso may be calculated, using hierarchic theory. 4.3. Some useful parameters of condensed matter: The total Structural Factor can be calculated as a ratio of the kinetic to the total energy of matter: SF=Ttot/Utot(4.40) The structural factors, related to contributions of transl ations (SFtr) and to librations (SFlb) could be calculated separately as: SFtr =Ttr/Utotand SFlb =Tlb/Utot(4.41) The dynamic properties of quasiparticles, introduced in me so- scopic theory The frequency of c- Macrotransitons or Macroconvertons exc itation, repre- senting [dissociation/association] of primary libration al effectons - ”flickering clusters ”as a result of interconversions between primary [lb] and [tr ] effectons is: FcM=1 τMc·PMc/Z (4.42) where: PMc=Pac·Pbcis a probability of macroconverton excitation; Zis a total partition function (see eq.4.2); the life-time of macroconverton is: τMc= (τac·τbc)1/2(4.43) The cycle-period of (ac) and (bc) convertons are determined by the sum of life-times of intermediate states of primary translationa l and librational effec- tons: τac= (τa)tr+ (τa)lb; τbc= (τb)tr+ (τb)lb;(4.44) The life-times of primary and secondary effectons (lb and tr) ina- and b- states are the reciprocal values of corresponding state fre quencies: 38[τa= 1/νa;τa= 1/νa]tr,lb; (4.45) [τb= 1/νb;τb= 1/νb]tr,lb (4.45a) [(νa) and ( νb)]tr,lbcorrespond to eqs. 4.8 and 4.9; [(νa) and ( νb)]tr,lbcould be calculated using eqs.4.16; 4.17. The frequency of (ac) and (bc) convertons excitation [lb/tr ]: Fac=1 τac·Pac/Z (4.46) Fbc=1 τbc·Pbc/Z (4.47) where: PacandPbcare probabilities of corresponding convertons excitation s (see eq.4.29a). The frequency of Supereffectons and Superdeformons (bigges t fluc- tuations) excitation is: FSD=1 (τA∗+τB∗+τD∗)·PD∗ S/Z (4.48) Itis dependent on cycle-period of Supereffectons: τSD=τA∗+τB∗+τD∗ and probability of Superdeformon activation ( PD∗ S),like the limiting stage of this cycle. The averaged life-times of Supereffectons in A∗andB∗state are dependent on similar states of translational and librational macroeff ectons : τA∗= [(τA)tr·(τA)lb] = [(τaτa)tr·(τaτa)lb]1/2(4.49) and that in B state: τB∗= [(τB)tr·(τB)lb] = [(τbτb)tr·(τbτb)lb]1/2(4.50) The life-time of Superdeformons excitation is determined by frequency of beats between A∗and B∗states of Supereffectons as: τD∗= 1/|(1/τA∗)−(1/τB∗)| (4.51) The frequency of translational and librational macroeffect onsA⇋ Bcycle excitations could be defined in a similar way: 39/bracketleftbigg FM=1 (τA+τB+τD)·PD M/Z/bracketrightbigg tr,lb(4.52) where: (τA)tr,lb= [(τa·τa)tr,lb]1/2(4.53) and (τB)tr,lb= [(τb·τb)tr,lb]1/2(4.54) (τD)tr,lb= 1/|(1/τA)−(1/τB)|tr,lb(4.55) The frequency of primary translational effectons (a⇋b)trtransi- tions could be expressed like: Ftr=1/Z (τa+τb+τt)tr·(Pd)tr (4.56) where: ( Pd)tris a probability of primary translational deformons excita tion; [τa;τb]trare the life-times of (a) and (b) states of primary translational ef- fectons (eq.4.45). The frequency of primary librational effectons as ( a⇋b)lbcycles excitations is : Flb=1/Z (τa+τb+τt)lb·(Pd)lb (4.57) where: ( Pd)lbis a probability of primary librational deformons excitati on;τaand τbare the life-times of (a) and (b) states of primary libration al effectons defined as (4.45). The life-time of primary transitons (tr and lb) as a result of quantum beats between (a) and (b) states of primary effectons could be intro duced as: [τt=|1/τa−1/τb|−1]tr,lb (4.58) The fraction of molecules (Fr) in each type of independent ex cita- tion (quasiparticle): Fr(i) =P(i)/Z (4.59) 40where: P(i) is thermoaccessibility (relative probability) of given e xcitation andZis total partition function (4.2). The concentration of molecules in each type of independent e xci- tation (quasiparticles): Nm(i) =Fr(i)·(NA/V0) = [P(i)/Z]·(NA/V0) (4.60) where: NAandV0are the Avogadro number and molar volume of matter. The concentration of each type of independent excitations (quasiparticles): N(i) =Fr(i)·n(i) = [P(i)/Z]·n(i) (4.61) where: n(i) is a concentration of given type (i) of quasipart icles;Fr(i) is a fraction of corresponding type of quasiparticles. The average distance between centers of i-type of randomly d is- tributed quasiparticles: d(i) = 1/[N(i)]1/3= 1/[(P(i)/Z)·n(i)]1/3(4.62) The ratio of average distance between centers of quasiparti cles to their linear dimension [l= 1/n(i)1/3]: rat(i) = 1/[(P(i)/Z)]1/3(4.63) The number of molecules in the edge of primary translational and primary librational effectons: κtr=/parenleftbig Vtr ef/vm/parenrightbig1/3= (1/ntr ef)/(V0/NA) (4.63a) κlb=/parenleftbig Vlb ef/vm/parenrightbig1/3= (1/nlb ef)/(V0/NA) (4.63b) where: (1 /ntr,lb ef) is the volume of primary translational or librational effec - tons; ( V0/NA) is the volume, occupied by one molecule in condensed matter . A lot of other parameters, characterizing different propert ies of condensed matter are also possible to calculate, using Hierarchic mes oscopic theory and ourspecial software elaborated as will be shown in the next chapters. V. QUANTITATIVE VERIFICATION OF HIERARCHIC THEORY FOR ICE AND WATER 41All the calculations, based on Hierarchic mesoscopic conce pt, were performed on the IBM-compatible personal computers. The special software: ”Com- prehensive analyzer of matter properties” was elaborated. The pro- gram allows to evaluate more than hundred parameters of any c on- densed matter if the following data are available in the temp erature interval of interest: 1. Positions of translational and librational bands in osci llatory spectra; 2. Sound velocity; 3. Molar volume; 4. Refraction index . The basic experimental parameters for ice: The wave numbers (˜ νtr), corresponding to positions of translational and librational bands in oscillatory IR spectra were taken from book of Eisenberg and Kauzmann [16]. Wave numbers for ice at 0oCare: /parenleftBig ˜ν(1) ph/parenrightBig tr= 60cm−1; /parenleftBig ˜ν(2) ph/parenrightBig tr= 160 cm−1; /parenleftBig ˜ν(3) ph/parenrightBig tr= 229 cm−1 Accordingly to our model, the IR photons with corresponding frequencies are irradiated and absorbed a result of ( a⇔b) primary translational deformons in ice. Temperature shifts of these bands positions are close t o zero: ∂/parenleftBig ˜ν1,2,3 ph/parenrightBig tr/∂T≈0 Wave numbers of librational IR bands, corresponding to absorption of photons, related to ( a⇔b)1,2,3 lbtransitions of primary librational effectons of ice are: /parenleftBig ˜ν(1) ph/parenrightBig lb=/parenleftBig ˜ν(2) ph/parenrightBig lb=/parenleftBig ˜ν(3) ph/parenrightBig lb≈795cm−1. The equality of wave numbers for three directions (1,2,3) indicate the spatial isotropy of the librations of H2Omolecules. In this case deformons and effectons have a cube geometry. In general case they have a shape of para llelepiped (like quasiparticles of translational type) with each of three ribs, corresponding to most probable de Broglie wave length in selected direction. The temperature shift of the position of the librational ban d maximum for ice is: ∂/parenleftBig ˜ν1,2,3 ph/parenrightBig lb/∂T≈ −0.2cm−1/C0 42The resulting thermal phonons velocity in ice, responsible for secondary acoustic deformons, is taken as equal to the transverse sound velocit y [27]: vres s= 1.99·105cm/s This velocity and molar ice volume ( V0) are almost independent on temperature [16]: V0= 19.6cm3/M≃const The basic experimental parameters for Water The wave numbers of translational bands in IR spectrum, corr esponding to quantum transitions of primary translational effectons bet weenacoustic (a) and optical (b) states with absorption or emission of photons, forming electromag - netic 3D translational deformons at00Care [16]: /parenleftBig ˜ν(1) ph/parenrightBig tr= 60cm−1;/parenleftBig ˜ν(2) ph/parenrightBig tr≈/parenleftBig ˜ν(3) ph/parenrightBig tr≈199cm−1 with temperature shifts: ∂/parenleftBig ˜ν(1) ph/parenrightBig tr/∂T= 0; ∂/parenleftBig ˜ν(2,3) ph/parenrightBig tr/∂T=−0.2cm−1/C0 The primary librational deformons of water at 00Care characterized by follow- ing degenerated wave numbers of librational bands in it IR sp ectrum: /parenleftBig ˜ν(1) ph/parenrightBig lb≈/parenleftBig ˜ν(2) ph/parenrightBig lb≈/parenleftBig ˜ν(3) ph/parenrightBig lb= 700 cm−1 with temperature shift: ∂/parenleftBig ˜ν1,2,3 ph/parenrightBig lb/∂T=−0.7cm−1/C0 Wave numbers are related to the frequencies ( ν) of corresponding transitions via light velocity as: ν=c˜ν The dependence of sound velocity (vs)in water on temperature within the temperature range 0 −1000Cis expressed by the polynomial [17]: vs= 1402 .385 + 5 .03522 t−58.3087·10−3t2+ + 345 .3·10−6t3− 43−1645.13·10−9t4+ 3.9625·10−9t5(m/s). The temperature dependence of molar volume ( V0) ofwater within the same temperature range can be calculated using the polynomial [1 8, 19]: V0= 18000 /[(999,83952+ 16 .945176 t− −7.98704 ·10−3t2− −4.6170461 ·10−5t3+ 1.0556302 ·10−7t4− −2.8054253 ·10−10t5)/ /(1 + 1 .687985 ·10−2t)] (cm3/M) Therefraction index for ice was taken as an independent of temperature ( nice= 1.35) and that for water as a variable, depending on temperatur e in accordance with experimental data, presented by Frontas’ev and Schrei ber [20]. The refraction index for water at 200C is approximately: nH2O≃1.33 The temperature dependences of different parameters for ice and water, com- puted using the formulas of our mesoscopic theory, are prese nted in Figs.(1-4). It is only a small part of available information. In principl e, it is possible to cal- culate about 200 different parameters for liquid and solid st ate of any condensed matter [3]. A. Discussion of theoretical temperature dependences. It will be shown below that our hierarchic theory makes it pos sible to cal- culate unprecedently big amount of parameters for liquids a nd solids. Those of them that could be measured experimentally are in excelle nt correspondence with theory. 44Fig. 1 .(a, b, c).Temperature dependences of the resulting ther- moaccessibility ( Z) (eq.4.2) and contributions related to primary and secondary effectons and deformons for ice (a,b) and water (c). The resulting thermoaccessibility minimum (Fig. 1a) for ic e (Z) corresponds to the temperature of -1700C. The interval from -198 to -1730C is known indeed as anomalies one due to the fact that the heat equilibrium of i ce establishes very slowly in the above range [21]. This fact can be explained by t he less probable ice structure (minimum value of partition function Z ) near −1700C. For the other hand, experimental anomaly, related with maximum heat capacity ( Cp), also is observed near the same temperature. It can be expla ined, if we present heat capacity as: Cp=∂ ∂T(1 ZU∗) =−1 Z2∂Z ∂TU∗+1 Z∂U∗ ∂T One can see, that heat capacity is maximal, when ( ∂Z/∂T ) = 0 and Z is min- imal. It is a condition of Z(T) extremum, just leading from ou r theory at −170oC(Fig.3a). 45In liquid water the temperature dependences of Z and its comp onents are linear. The thermoaccessibility of mean secondary effecton s in water decreases, while that of primary effectons increases with temperature, just like in ice (Fig. 1 b,c). On lowering down the temperature the total internal energy o f ice (Fig. 2a) and its components decreases nonlinearly with temperature coming closer to absolute zero. The same parameters for water are decreasing almost linearly within the interval (100 −0)0C(Fig. 2b). In computer calculations, the values of Cp(t) can be determined by differen- tiating Utotnumerically at any of temperature interval. Fig. 2 . (a,b). Temperature dependences of the total internal energy ( Utot) and different contributions for ice (a) and water (b) (eqs. 4.3 - 4.5). Following contributions to Utotare presented: (Uef+¯Uef) is the contribution of primary and secondary effec- tons; (Ud+¯Ud) is the contribution of primary and secondary defor- mons; (Uef+Ud) is the contribution of primary effectons and deformons; (¯Uef+¯Ud) is the contribution of secondary effectons and defor- mons. The contributions of macro- and supereffectons to the total i n- ternal energy and corresponding macro- and superdeformons , as well as all types of convertons, are much smaller than those of pri mary and secondary effectons and deformons. 46It follows from Fig. 2a that the mean value of heat capacity fo r ice in the interval from -75 to 0oCis equal to: ¯Cice p=∆Utot ∆T≈39J/M·K= 9.3 cal/M·K For water within the whole range ∆ T= 1000C, the change in the internal energy is: ∆ U= 17−9.7 = 7.3kJ/M (Fig.2b). This corresponds to mean value of heat capacity of water: Cwater p =∆Utot ∆T= 73J/M·K= 17.5cal/M ·K These results of our theory agree well with the experimental mean values Cp= 18 Cal /M·Kfor water and Cp= 9cal/M ·Kfor ice. It can be seen in Fig. 3 a, bthat the total kinetic energy of water ( Tkin) is approximately 30 times less than the potential energy ( Vp) at the same tempera- tures. In the case of ice, they differ even more: ( Tkin/V)<1/100. The resulting Tkinof water increases almost twice over the range (0 −1000C) : from 313 to 585J/M. However, the change of the total internal energy ( Utot=Tkin+Vp) is determined mainly by the change in potential energy Vp(t) of ice and water. It is reasonable to analyze the above ratio between total kin etic and potential energies in terms of the Viral theorem (Clausius, 1870; see Prokhorov, 1988). This theorem for a system of particles relates the averaged k inetic ¯Tk(/vector v) =/summationtext imiv2 i/2 and potential ¯V(r) energies in the form: 2¯Tk(/vector v) =/summationdisplay imiv2 i=/summationdisplay i/vector ri∂V/∂/vector r i This equation is valid for both quantum-mechanical and clas sical systems. If the potential energy V(r) is a homogeneous n-order function like: V(r)∼rn(5.1) then average kinetic and average potential energies are rel ated as: Tk(/vector v) =nV(r) 2(5.2) For example, for a harmonic oscillator: n = 2 and ¯Tk=¯V. For Coulomb interaction: n=−1 and ¯T=−¯V /2. For water our calculation of TkandVgives: nw∼1/15 and for ice: nice∼ 1/50. It follows from (5.1) that in water and ice the dependence of potential energy on distance (r) is very weak: 47Vw(r)∼r(1/15);Vice∼r(1/50)(5.3) This result can be considered as indication of very distant i nter- action due to the expressed cooperative properties of water as an as- sociative hierarchic liquid. It points that the role of dist ant Van der Waals interactions, stabilizing primary effectons, repres enting molec- ular Bose condensate, is increasing with dimensions of thes e coherent clusters, i.e. with temperature decreasing. It is a strong e vidence that water and ice can not be considered as a classical system s. Fig. 3 . (a,b). Temperature dependences of the kinetic ( Tkin) and potential ( Vp) energy for the ice (a) and water (b). Note that Utot=Tkin+Vp. Forclassical equilibrium systems containing N-particles, the virial the- orem shows that average kinetic and potential energies rela ted to each degree of freedom are the same and are equal to: ¯Tk=1 2kT=¯V n = 2 (5.4) It means that in classical approximation the particles of co ndensed matter are considered as the harmonic oscillators. It leads from ou r theory and com- puter simulations that for real matter this approximation i s not valid in general case . 5.2. Explanation of Drost-Hansen temperature anomalies 48Our Hierarchic theory is the first one enable to predict and gi ve a clear expla- nation to deviations of temperature dependencies of some ph ysical parameters of water from monotonic ones. It clarify also the interrelation between these deviations (transitions) and corresponding temperature anomalies in properties of bios ystems, such as dy- namic equilibrium of [assembly-disassembly] of microtubu les and actin filaments, large-scale dynamics of proteins and the enzymes activity, etc. Fig. 4 .(a) : The temperature dependencies of the number of H2Omolecules in the volume of primary librational effecton ( nlb M)ef,left axis) and the number of H2Oper length of this effecton edge ( κ, right axis); (b): the temperature dependence of the water pr i- mary librational effecton (approximated by cube) edge lengt h [llib ef= κ(V0/N0)1/3]. The number of H2Omolecules within the primary libration effectons of water, which could be approximated by a cube, decreases fr omnM= 280 at 00tonM≃3 at 1000(Fig. 4a). It should be noted that at physiological temperatures (35 −400) such quasiparticles contain nearly 40 water molecules. This number is close to that of water molecules that can be enc losed in the open interdomain protein cavities judging from X-ray data. The flickering of these clusters, i.e. their ( dissociation ⇋association ) due to [ lb⇔tr] conversions in accordance with our model is directly related to the large -scale dynamics of proteins. It is very important that the linear dimensions of such water clusters (11 ˚A) at physiological temperature are close to common ones for pr otein domains (Fig. 4b). 49Such spatial correlations indicate that the properties of w ater ex- erted a strong influence on the biological evolution of macro molecules, namely, their dimensions and allosteric properties due to c ooperative- ness of intersubunit water clusters. We assume here that integer and half-integer values of numbe r of water molecules per effecton’s edge [ κ] (Fig. 4a) reflect the conditions of increased and decreased stabilities of water structure correspondingly . It is apparently related to the stability of primary librational effectons as coopera tive and coherent water clusters. Nonmonotonic behavior of water properties with temperatur e is widely known and well confirmed experimental fact (Drost-Hansen, 1976, 1 992; Clegg and Drost-Hansen, 1991; Etzler, 1991; Roberts and Wang, 1993; R oberts and Wang, 1993; Roberts, et al., 1993, 1994; Wang et al., 1994). We can explain this interesting and important for biologica l functions phe- nomenon because of competition between two factors: quantum and structural ones in stability of primary librational effecto ns.The quan- tum factor such as wave B length , determining the value of the effecton edge: /bracketleftBig lef=κ(V0/N0)1/3˜λB/bracketrightBig lb(5.5) decreases monotonously with temperature increasing. The structural factor is a discrete parameter depending on the water molecules effecti ve length: lH2O= (V0/N0)1/3and their number [ κ] in the effecton’s edge. In accordance with our model, the shape of primary libration al effectons in liquids and of primary translational effectons i n solids could be approximated by parallelepiped in general case or b y cube, when corresponding thermal movements of molecules (lb and/ or tr) and are isotropic. We suggest that when ( lef) corresponds to integer number of H2O, i.e. [κ= (lef/lH2O) = 2,3,4,5,6...]lb(5.6) thecompetition between quantum and structural factors is minimum and pri- mary librational effectons are most stable. On the other hand , when ( lef/lH2O)lb is half-integer, the librational effectons are less stable ( thecompetition is maxi- mum). In the latter case ( a⇔b)lbequilibrium of the effectons must be shifted rightward - to less stable state of these coherent water clus ters. Consequently, the probability of dissociation of librational effectons to a number of much smaller translational effecton, i.e. probability of [lb/tr ] convertons increases and concentration of primary librational effectons decreas es.Experimentally the nonmonotonic change of this probability with temperatu re could be registered by dielectric permittivity, refraction inde x measure- ments and by that of average density of water. The refraction index change should lead to corresponding variations of surface t ension, va- por pressure, viscosity, self-diffusion in accordance to ou r hierarchic theory (Kaivarainen, 1995). 50In accordance to our model the density of liquid water in comp o- sition of librational effectons is lower than the average in t he bulk . In the former case all hydrogen bonds of molecules are satura ted like in ideal ice in contrast to latter one. We can see from Fig.4a that the number of water molecules in primary lb effecton edge (κ) is integer near the following temperatures: 60(κ= 6); 170(κ= 5); 320(κ= 4); 490(κ= 3); 770(κ= 2) (5.7) These temperatures coincide very well with the maximums of relaxation time in pure water and with dielectric response anomalies (Roberts , et al., 1993; 1994; Wang, et al., 1994). The special temperatures predicted by o ur theory are close also to chemical kinetic (Aksnes, Asaad, 1989; Aksnes, Libn au, 1991), refrac- tometry (Frontas’ev, Schreiber 1966) and IR (Prochorov, 19 91) data. Small discrepancy may result from the high sensitivity of water to any kind of pertur- bation, guest-effects and additional polarization of water molecules, induced by high frequency visible photons. Even such low concentratio ns of inorganic ions ester and NaOH as used by Aksnes and Libnau (1991) may change w ater proper- ties. The increase of H2Opolarizability under the effect of light also may lead to enhancement of water clusters stability and to correspondi ng high-temperature shift of nonmonotonic changes of water properties. The semi-integer numbers of [ κ] for pure water correspond to temperatures: 00(κ= 6.5); 120(κ= 5.5); 240(κ= 4.5); 400(κ= 3.5); (5.7a) 620(κ= 2.5); 990(κ= 1.5) The conditions (5.7a) characterize the less stable water st ructure than con- ditions (5.7). The first order phase transitions - freezing a t 00and boiling at 1000of water almost exactly correspond to κ= 6.5 and κ= 1.5. This fact can be important for understanding the mechanism of first order p hase transitions. The temperature anomalies of colloid water-containing sys tems, discovered by Drost-Hansen (1976) and studied by Etzler and coauthors (1987; 1991) occurred near 14-160; 29-320; 44-460and 59-620C. At these tempera- tures the extrema of viscosity, disjoining pressure and mol ar excess entropy of water between quartz plates even with a separation 300-500 ˚A has been ob- served. These temperatures are close to predicted by our theory for bulk water anomalies, corresponding to integer values of [ κ](see 5.7). Some deviations can be a result of interfacial water perturb ations, induced by colloid particles and plates. It is a first theory w hich looks to be able to predict and explain the existence of Drost -Hansen temperatures. The dimensions, concentration and stability of water clust ers (primary li- brational effectons) in the volume of vicinal water should be bigger than that in bulk water due to their less mobility and to longer waves B l ength. Very interesting ideas, concerning the role of water cluste rs in biosystems were developed in works of John Watterson (Watterson, 1988a ,b). 51It was revealed in our laboratory (Kaivarainen, 1985; Kaiva rainen et al., 1993) that nonmonotonic changes of water near Drost Hansen temperatures are accompanied by in-phase change of different protein large-s cale dynamics, re- lated to their functioning. The further investigations of l ike phenomena are very important for understanding the molecular mechanisms of th ermoadaptation of living organisms. 5.3. Physiological temperature and the least action princi ple The Fig.5 a, bshows the resulting contributions to the total kinetic ener gy of water of two main subsystems: effectons and deformons. The minimum of deformons contribution at 430is close to the physiological temperatures for warm-blooded animals. Fig. 5. Temperature dependences of two resulting contributions - effectons ( Tef kin) and deformons ( Td kin) of all types to the total kinetic energy of water. The minima at temperature dependences of different contribu tions to the total kinetic energy of water at Fig.5 correspond to the best imple mentation of the least action principle in the form of Mopertui-Lagrange. In such a form, this principle is valid for the conservative h olonom systems, where limitations exist for the displacements of the particles of this sys- tem, rather than the magnitudes of their velocities. It states that among all the kinematically possible displacements of a system fr om one configuration to another, without changing total system energy, such a dis placements are most 52probable for which the action (W) is least: ∆ W= 0. Here ∆ is the symbol of total variation in coordinates, velocities and time. The action is a fundamental physical parameter which has the dimension of the product of energy and time characterizing the dynamics o f a system. According to Hamilton, the action: S=t/integraldisplay t0Ldt is expressed through the Lagrange function: L=Tkin−V, (5.4) where T kinand V are the kinetic and potential energies of a system or a su b- system. According to Lagrange, the action (W) can be expressed as: W=t/integraldisplay t02Tkindt We can assume that at the same integration limit the minimum v alue of the action ∆W ≃0 corresponds to the minimum value of T kin.. Then it can be said that at temperature about 430the subsystems of deformons is most stable (see Fig. 6). This means that the equilibrium between the acousti c and optic states of primary and secondary effectons should be most stable at th is temperature. 5.4. Mechanism of phase transitions in terms of the hierarch ic theory The abrupt increase of the total internal energy (U) as a resu lt of ice melting (Fig. 6a), equal to 6 .27kJ/M , calculated from our theory is close to the experi- mental data (6 kJ/M ) (Eisenberg, 1969). The resulting thermoaccessibility (Z) during ice →water transition decreases abruptly, while potential and k inetic energies increase (Fig. 6b). 53Fig. 6. The total internal energy ( U=Tkin+Vp) change during ice-water phase transition and change of the resulting ther moacces- sibility (Z) - (a); changes in kinetic ( Tkin) and potential ( Vp) energies (b) as a result of the same transition. It is important that at the melting point H2Omolecules number in a primary translational effecton (ntr M)efdecreases from 1 to ≃0.4 (Fig. 7a). It means that the volume of this quasiparticle type gets smaller than the v olume occupied by H2Omolecule. According to our model, under such conditions the individua l water molecules get the independent translation mobility. The number of water molecules forming a primary libration effecton decreases abruptly from about 3000 to 280, as a result of melting. The number of H2Oin the secondary librational effecton decreases correspondingly from ∼1.25 to 0.5 (Fig. 7b). Fig. 8 a, bcontains more detailed information on changes in primary li bra- tional effecton parameters in the course of ice melting. The theoretical dependences obtained allow us to give a clea r interpretation of the first order phase transitions. The condition of meltin g atT=Tcris realized in the course of heating when the number of molecule s in the volume of primary translational effectons nMdecreases: nM≥1(T≤Tcr)Tc→nM≤1(T≥Tcr) 54Fig. 7. Changes of the number of H2Omolecules forming pri- mary ( ntr M)efand secondary (¯ ntr M)eftranslational effectons during ice-water phase transition (a). Changes in the number of H2O molecules forming primary ( nlb M)efand secondary (¯ nlb M)eflibrational effectons (b) as a result of phase transitions. The process of boiling, i.e. [liquid →gas] transition, as seen from Fig. 7a, is also determined by condition (6.6), but at this case it is r ealized for primary librational effectons. In other words this means that [gas →liquid] transition is related to orig- ination (condensation) of the primary librational effectons which contain more than one molecule of substance. In a liquid as compared to gas, the quantity of rotational deg rees of freedom is decreased due to librational coherent effecton s forma- tion, but the number of translational degrees of freedom rem ains the same. The latter, in turn also decreases during [liquid →solid] phase transition, when the wave B length of molecules correspondi ng to their translations begins to exceed the mean distances betw een the centers of molecules (Fig. 7a). This process is accompanied by partial Bose-condensation of translational waves B and by the forma tion of coherent primary translational effectons, including more than one mo lecule. The size of librational effectons grows up strongly during this [ water →ice] transition. 55Fig. 8. Changes of the number of H2Omolecules forming a primary librational effecton ( nlb M)ef, the number of H2Omolecules (κ) in the edge of this effecton (a) and the length of the effecton edge: llb ef=κ(V0/N0)1/3(b) during the ice-water phase transition. In contrast to first order phase transitions the second order phase transitions are not related to the abrupt change of primary effectons volu me,but only to their shape and symmetry changes. Such phenomena may be the result of a gradual [temperature/pressure] - dependent decrease in t he difference between the dynamics and the energy of aiandbistates of one of three standing waves B, forming primary effectons: /bracketleftbig hνp=h/parenleftbig νb−νa/parenrightbig/bracketrightbigi tr,lbTc→0 at/bracketleftbig λTc b=λTca/bracketrightbigi tr,lb>/parenleftbigV0/N0/parenrightbig1/3   As a result of the second order phase transition a new type of p rimary effectons with the new geometry appears. It means the emergency of new v alues of ener- gies of ( a)1,2,3and (b)1,2,3-states and new constants of ( a⇔b)1,2,3equilibrium. Degree of polymerization of primary effectons, participati ng in the polyeffec- tons or orchestrated superclusters formation can also chan ge in the course of the second order phase transitions. The second order phase transitions of ice could be induced by pressure at certain temperatures. The second order phase transition usually is accompanied by nonmonotonic changes of sound velocity and the low-frequency shift of tra nslational and libra- tional bands in oscillatory spectra (so-called ”soft mode” ). According to our theory these changes should be followed by jump of heat capac ity, compress- ibility and coefficient of thermal expansion. It points that t he parameters of elementary cells, depending on the sizes and geometry of primary effectons have to change also. 565.5. The energy of quasiparticle discrete states. Activation energy of dynamics in water Over the entire temperature range for water and ice the energ ies of ”acoustic” a-states of primary effectons (translational and librationa l ones) are lower than the energies of ”optic” b-states (Fig.9). The energy of an ideal effecton (3RT) has the intermediate values. Fig. 9. Temperature dependences for the energy of primary effectons in ”acoustic” ( a) and ”optical” ( b) states and that for the energy of a harmonic 3D oscillator (the ideal thermal effecto n:E0= 3RT) for water and ice calculated according to the formulae (4.6 , 4.7 and 4.12): a) for primary translational effectons of wate r ina andbstates; b) for primary librational effectons of water in aandb states; c) for primary translational effectons of ice in aandbstates; d) for primary librational effectons of ice in aandbstates. According to the eq.(4.10 and 4.11) the thermoaccessibility of (a) and ( b) states is determined by the absolute value of the difference: 57|Ea ef−3kT|tr,lb;|Eb ef−3kT|tr,lb. where E0= 3kT= 3hν0is energy of an ideal effecton. The ( a⇔b) transitions (quantum beats) can be considered as autoosci l- lations of quasiparticles around the thermal equilibrium s tate ( E0), which is quantum-mechanically prohibited. In terms of synergetic t he primary effectons are the medium active elements. (b→a) transitions are related to origination of photons, and ele ctromagnetic deformons, while the reverse ones ( a→b) correspond to absorption of them, i.e. annihilation of deformons. The nonequilibrium conditions in the subsystems of effecton s and deformons can be induced by the competition between discrete quantum a nd continuous heat energy distributions of different quasiparticles. Som etimes these nonequi- librium conditions could lead to macroscopic long-period o scillations in con- densed matter. The temperature dependences of the excitation (or fluctuati on) energies for translational and librational macroeffectons in A(a,¯a) and B(b,¯b) states: (ǫA M)tr,lb; (ǫB M)tr,lband that for macrodeformons ( ǫM D)tr,lband superdeformons (ǫs D∗), for water (a,b) and ice (c,d) can be calculated according t o formulas (5.8, 5.9 and 5 .10). E0= 3RTis the energy of ideal quasiparticle, corresponding to thermal equilibrium energy. The knowledge of the excitation energies of macrodeformons is very impor- tant for calculation the viscosity and coefficient of self-di ffusion (see sections 6.6 and 6.8). The A and B states of macro- and supereffectons represent the s ignificant de- viations from thermal equilibrium. The transitions betwee n these states termed: macro- and superdeformons represent the strong fluctuation s of polarizabilities and, consequently, of refraction index and dielectric perm eability. The excitation energies of A and B states of macroeffectons ar e determined as: (ǫA M)tr,lb=−RTln(Pa efPa ef)tr,lb=−RTln(PA M)tr,lb (5.8) (ǫB M)tr,lb=−RTln(Pb efPb ef)tr,lb=−RTln(PB M)tr,lb (5.9) where Pa efand¯Pa efare the thermoaccessibilities of the ( a)−eq.(4.10) and (¯ a)− eq.(4.18) - states of the primary and secondary effectons, correspo ndingly; Pb ef and¯Pb efare the thermoaccessibilities of ( b)−eq.(4.11) and ( ¯b)−eq.(4.19) states. The activation energy for superdeformons is: 58ǫs D∗=−RTln(Ps D) =−RT[ln(PM D)tr+ ln(PM D)lb] = (5.10) = (ǫM D)tr+ (ǫM D)l The value ( ǫM D)tr≈11.7kJ/M ≈2.8 kcal/M characterizes the activation energy fortranslational self-diffusion of water molecules , and ( ǫM D)lb≈31kJ/M ≈7.4 kcal/M - the activation energy for librational self-diffusi on of H2O. The latter valueis close to the energy of the hydrogen bond in water (Eisenberg, 1969). On the other hand,the biggest fluctuations-superdeformons are responsible for the process of cavitational fluctuations in liquids and the emergency of defects in solids. They determine vapor pre ssure and sublimation, as it leads from our theory. Fig. 10. Temperature dependences of the oscillation frequen- cies in ( a) and ( b) state of primary effectons - translational and librational for water (a) and ice (b), calculated from (Fig. 9). The relative distribution of frequencies on Fig.10 is the sa me as of energies on Fig. 9. The values of these frequencies reflect the minimal life-times of corresponding states. The real life-time is dependent also on probability of ”jump” from this state to another one and on probability of st ates excitation. 5.6. The life-time of quasiparticles and frequencies of the ir excitations The set of formula, describing the dynamic properties of qua siparticles, in- troduced in mesoscopic theory was presented earlier. For the case of ( a⇔b)1,2,3transitions of primary and secondary effectons (tr and lb), their life-times in (a) and (b) states are the reciprocal val ue of 59corresponding frequencies: [ τa= 1/νaandτb= 1/νb]1,2,3 tr,lb. These parameters and the resulting ones could be calculated from eqs.(2.27; 2 .28) for primary effectons and (2.54; 2.55) for secondary ones. The results of calculations, using eq.(4.56 and 4.57) for fr equency of excita- tions of primary tr and lb effectons are plotted on Fig. 11a,b. The frequencies of Macroconvertons and Superdeformons wer e calculated using eqs.(4.42 and 4.48). Fig. 11. (a) - Frequency of primary [tr] effectons excitations, calculated from eq.(4.56); (b) - Frequency of primary [lb] effectons excitations, calcu lated from eq.(4.57); (c) - Frequency of [ lb/tr] Macroconvertons (flickering clusters) excitations, calculated from eq.(4.42); (d) - Frequency of Superdeformons excitations, calculated from eq.(4.48). At the temperature interval (0-100)0Cthe frequencies of translational and librational macrodeformons (tr and lb) are in the interval o f (1.3−2.8)·109s−1and(0.2−13)·106s−1 correspondingly. The frequencies of (ac) and (bc) converto ns could be defined also using our software and formulae, presented at the end of Section IV. 60The frequency of primary translational effectons [ a⇔b] transitions at 200C, calculated from eq.(4.56) is ν∼7·1010(1/s) It corresponds to electromagnetic wave length in water with refraction index (n= 1.33) of: λ= (cn)/ν∼6mm For the other hand, there are a lot of evidence, that irradiat ion of very different biological systems with such coherent electromagnetic fiel d exert great influences on their properties (Grundler and Keilman, 1983). Between the dynamics/function of proteins, membranes, etc . and dynamics of their aqueous environment the strong interrela tion exists. The frequency of macroconvertons, representing big densit y fluctuation in the volume of primary librational effecton at 370C is about 107(1/s) (Fig 11c), the frequency of librational macrodeformons at the same tem perature is about 106s−1,i.e. coincides with frequency of large-scale protein cavities pulsa- tions between open and closed to water states (see Fig.11). This confirm our hypothesis that the clusterphilic interaction is respo nsible for stabilization of the proteins cavities open state and that transition from the open state to the closed one is induced by coherent water cluster dissociatio n. The frequency of Superdeformons excitation (Fig.11d) is mu ch lower: νs∼(104−105)s−1 Superdeformons are responsible for cavitational fluctuati ons in liquids and orig- ination of defects in solids. Dissociation of oligomeric pr oteins, like hemoglobin or disassembly of actin and microtubules could be also relat ed with such big fluctuations. Superdeformons could stimulate also the reve rsible dissociation of water molecules, which determines the pH value. The parameters, characterizing an average spatial distrib ution of primary lb and tr effectons in the bulk water are presented on the next Fig .12. 61Fig. 12. Theoretical temperature dependencies of: (a) - the space between centers of primary [lb] effectons (cal cu- lated in accordance to eq.4.62); (b) - the ratio of space between primary [lb] effectons to thei r length (calculated, using eq.4.63); (c) - the space between centers of primary [tr] effectons (in a c- cordance to eq.4.62); (d) - the ratio of space between primary [tr] effectons to thei r length (eq.4.63). One can see from the Fig.12 that the dimensions of primary tra nslational effectons are much smaller and concentration much higher tha n that of primary librational effectons. We have to keep in mind that these are t he averaged spatial distributions of collective excitations. The form ation of polyeffectons - coherent clusters of lb (in liquids) and tr (in solids) prima ry effectons, interacting side-by-side due to Josephson effect is possible also. 62Fig. 13. (aandb):Temperature dependences for the con- centrations of primary effectons (translational and librat ional) in (a) and ( b) states: ( Na ef)tr,lb,(Nb ef)tr,lbfor water ( aandb):the similar dependencies for ice ( candd). Concentrations of quasiparticles were calculated from eqs .:(Na ef)tr,lb= (nefPa ef/Z)tr,lb; (Nb ef)tr,lb= (nefPb ef/Z)tr,lb;. These dependences can be considered as the quasiparticles d is- tribution functions. To get the information, presented in this article about meso scopic/cluster structure of solids and liquids, using conventional method s, i.e. by means of x-ray or neutron scattering methods is very complicated tas k. However, even in this case the final information about properties of collec tive excitations will not be so comprehensive. The results of computer simulations, confirms the correctne ss of our general model of condensed matter, as a hierarchic syste m of 3D standing waves of different nature. It is evident that the app lication of Hierarchic theory can be useful for elucidation and quant itative 63analysis of very different physical phenomena in liquids and solids. For example, for monitoring the processes in new materials t echnol- ogy, self-organization in colloid and biological systems ( Babloyantz, 1986; Chernikov, 1990; Kaivarainen, 1996). Our theory based idea of new optoacoustic device: Comprehen sive Analyzer of Matter Properties (CAMP) [see web site: www.kar elia.ru/˜alexk] may provide a huge amount of data (more than 300 parameters) o f any condensed system under study. REFERENCES: [1].W. Drost-Hansen, J.Singleton, in: Fundamentals of Medical Cell Biology, v.3A, Chemisrty of the living cell, JAI Press I nc.,1992, pp.157-180. [2].A.K¨ aiv¨ ar¨ ainen J.Mol.Liq .41,53 (1989). [3].A. 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Grundler, F.Keilmann, Phys. Rev. Lett. 51,1214, (1983). [38].Babloyantz A. Molecules, Dynamics and Life. An intro- duction to self-organization of matter. John Wiley & Sons, I nc. New York, (1986), pp.320. Haken H. Synergetics, computers a nd cognition. Springer, Berlin, 1990. [39]. F.R. Chernikov, Biofizika (USSR ),35,711 (1990) . [40]. A. Kaivarainen. In: Abstracts of conference: Toward a Science of Consciousness. Tucson, USA, p74, (1996). 65This figure "Fig1.png" is available in "png" format from: http://arxiv.org/ps/physics/0003044v1This figure "Fig2.png" is available in "png" format from: http://arxiv.org/ps/physics/0003044v1This figure "Fig3.png" is available in "png" format from: http://arxiv.org/ps/physics/0003044v1This figure "Fig4.png" is available in "png" format from: http://arxiv.org/ps/physics/0003044v1This figure "Fig5.png" is available in "png" format from: http://arxiv.org/ps/physics/0003044v1This figure "Fig6.png" is available in "png" format from: http://arxiv.org/ps/physics/0003044v1This figure "Fig7.png" is available in "png" format from: http://arxiv.org/ps/physics/0003044v1This figure "Fig8.png" is available in "png" format from: http://arxiv.org/ps/physics/0003044v1This figure "Fig9.png" is available in "png" format from: http://arxiv.org/ps/physics/0003044v1This figure "Fig10.png" is available in "png" format from: http://arxiv.org/ps/physics/0003044v1This figure "Fig11.png" is available in "png" format from: http://arxiv.org/ps/physics/0003044v1This figure "Fig12.png" is available in "png" format from: http://arxiv.org/ps/physics/0003044v1This figure "Fig13.png" is available in "png" format from: http://arxiv.org/ps/physics/0003044v1This figure "Table.png" is available in "png" format from: http://arxiv.org/ps/physics/0003044v1

arXiv:physics/0003045v1 [physics.gen-ph] 21 Mar 2000HIERARCHIC MODEL OFCONSCIOUSNESS: FROM MOLECULAR BOSECONDENSATION TO SYNAPTIC REORGANIZATION Alex Kaivarainen University of Turku, JBL, FIN-20520 Turku, Finland H2o@karelia.ru URL: http://www.karelia.ru/˜alexk Hierarchic Model of Consciousness, proposed in this work, i s based on Hierarchic Theory of Matter and Field, developed by the au thor ( Kaivarainen, 1995; 1998). In accordance to our Hierarchic Model of Consciousness (HMC ), each specific kind of neuron ensembles excitation - correspo nds to hierarchical system of three-dimensional (3D) standing wa ves of fol- lowing interrelated kinds: - thermal de Broglie waves (waves B), produced by unharmonic translations and librations of molecules and providing the possibility of high-temperature molecular Bose condensation; - electromagnetic (IR) waves; - acoustic waves; - vibro-gravitational waves, excited by coherent translat ional and librational oscillations of molecules, accompanied by alt ernating ac- celerations. Corresponding hierarchic system of 24 excitations, descri bing vir- tually all the properties of any condensed matter, can be gen erated by quantum transitions of the coherent water clusters, loca lized in cytoplasm and in the microtubules of neurons bodies. Most im por- tant primary collective excitations in form of coherent clu sters are resulted from high-temperature molecular Bose-condensat ion of wa- ter (see Appendix). We assume in our model the existence of feedback reaction be- tween the following hierarchy of phenomena, representing i nterrela- tion of quantum microscopic, mesoscopic and nonlinear macr oscopic events: a) hologram-like superposition of four types of 3D waves, li sted above; b) correlated MTs orientations, responsible for volume and ge- ometry of cells body in the tuned ensembles of neurons; c) quantity and distribution of synaptic contacts on the sur face of the nerve cells bodies, dependent on geometry of cells. 1It is assumed in our model that [gel →sol] transitions in cytoplasm of neuron’s body, produced by collective disassembly of big number of actin filaments and microtubules, as a consequence of nerve e xcitation and neuron’s body depolarization, - is accompanied by its vo lume and shape ”pulsation”. The twisting of centrioles in cells to or ientation, corresponding to maximum energy of the MTs distant interact ion by means 3D electromagnetic and vibro- gravitational waves - is an important stage of the excited neurons dynamic adaptation t o each other. Such adaptation becomes possible only in low viscosi ty - [sol] state of cytoplasm. As a result of cell’s volume/shape pulsation, induced by rev ersible osmotic equilibrium change, the distribution of synaptic c ontacts on the surface of cells and/or ionic channels activity - change . These changes provide the long-term and short-term memory corres pond- ingly. The synaptic redistribution is determined by MTs spa tial re- orientations and changes of their length. The processing of huge number of such ”informational acts as MTs - dependent synapt ic re- arragements”, accompanied by emergency of new states of neu ron ensembles and hologram-like fields, is responsible for cons ciousness and braining. The mechanism proposed needs the existence of feedback reac tion between following stages of HMC: a) simultaneous depolarization of big enough number of neur ons, forming ensemble, accompanied by opening the potential-de pendent channels and increasing the concentration of Ca2+in cytoplasm of neurons body; b) collective disassembly of actin filaments and MTs, accomp anied by [gel →sol] transition of big group of depolarized neurons due to their destruction by Ca2+−activatedproteins like gelsolin and villin; c) strong decreasing of cytoplasm viscosity and disjoining of the (+) ends of MTs from membranes, making possible the spatial r edis- tribution of MTs orientations, corresponding to maximum re sonance exchange interaction of MTs by means of IR photons and vibro- gravitational waves in simultaneously excited group of neu rons; d) volume/shape pulsation of neuron’s body and dendrites, i nduc- ing reorganization of ionic channels activity and synaptic contacts in the excited neuron ensembles. These volume/shape pulsatio ns occur due to reversible decrease of the intracell water activity a nd corre- sponding swallow of cell as a result of passive osmotic diffus ion of water from the external space. The probability of actin filaments and MTs disassembly and [g el→sol] transition can be enhanced not only by the Ca2+concentration in- creasing, but also by cavitational fluctuations, stimulate d by superde- formons and macroconvertons in accordance to our Hierarchi c the- ory of matter. Our model predicts that if the neurons or other cells, 2containing actin and MTs, will be treated by modulated acous- tic or electromagnetic field with resonance frequency of cyt oplasmic water cavitational fluctuations ( νres≥1·104s−1),it will activate si- multaneous disassembly of big number of actin filaments and M Ts, responsible for maintaining the specific cell volume and geo metry. As a result, it activates the neuron’s body volume/shape pul sation, depending on applied modulation. Such external stimulation of supercatastrophe or [gel →sol] transi- tions has two important consequences: -The first one is generation of strong high-frequency nerve i mpulse, propagating via axons and exciting huge number of other nerv e cells, i.e. distant nerve signal transmission in living organism; -The second one is stimulation the leaning process as far lon g- term memory in accordance to HMC, is related to synaptic cont acts reorganization, accompanied the neuron volume/shape puls ation. I. INTRODUCTION In accordance to our Hierarchic concept (Kaivarainen, 1992 , 1995) general for liquids and solids, any condensed matter, inclu ding the biosystems, represents cooperative [opto-acoustic-grav itating] system. Summary of this theory is presented in the Appendix. A usual hologram is a record of the object on the light-sensit ive material in the form of interference image, resulting from m ixing of the field, radiated by object with a coherent pilot wave. We assume here that interrelated subsystems of the effectons , transitons, acoustic, electromagnetic deformons and corr esponding vibro-gravitational excitations in biosystems, can be con sidered as superposition of three types of holograms: acoustic, optic and gravi- tational ones. An acoustic hologram (secondary acoustic de formons), as well as an optical and gravitational ones (formed by 3D ele ctromag- netic and gravitational standing waves), represent the int erference image, determined by cell’s components and different water f ractions properties. Under certain conditions an acoustic pilot signal can be substituted for an electric one (Prokhorov, 1988). This can occur during the transmission o f a nerve impulse along axons and neurons. The frequency of electromagnetic field related to change of i onic flux in excitable tissues usually does not exceed 1000 Hz (Kneppo an d Titomir, 1989). The electrical recording of human brain activity demonstra te a coherent (40 to 70 Hz) firing among widely distributed and distant brain ne urons (Singer, 1993). Such synchronization in a big population of groups of cells points to pos- sibility of not only regular axon-mediated, but also of phys ical fields-mediated distant or even quantum nonlocal interaction between them. 3We put forward a hypothesis here that a ”informational state ”, which reflects a kind of brain excitation, is related to certa in hi- erarchic hologram-like system of coherent electromagneti c,acoustic and vibro-gravitational 3D standing waves. Such a system ca n be produced by coherent unharmonic oscillations of water mole cules in composition of primary effectons and their quantum transiti ons. In microtubules the degree of Bose condensation, which determ ined the fraction of primary librational effectons and orchestratio n of all dy- namic process in water should be enhanced due to stabilizati on of water by walls of MTs. The coherent large-scale fluctuations of α and β tubulins, stimu- lated by water macroconvertons with frequency of about 107Hzalso could be a source of vibro-gravitational and electromagnet ic waves. Reorganization of actin and microtubule systems after coll ective disassembly (supercatastrophe), accompanied by [gel →sol] transition, induced by nerve excitation, is interrelated with cells vol ume and shape changes. The reorientation of centrioles of cells to s tate, corre- sponding to most effective resonance exchange between MTs by 3D IR photons and vibro-gravitational waves is one of the stage of the ”elementary act of consciousness”. Such ”tuning” of MTs bec omes possible because of strong declining of cytoplasm viscosit y, resulted from [gel-sol] transition. As a consequence of such reorganization of microfilaments an d mi- crotubules system in the nerve cell body and dendrites, the n umber of the active ionic channels and distribution of active synapt ic contacts on the surface of cells - will change. The processing of these ”in- formational acts” in the head brain is responsible for consc iousness, short and long-term memory and cognition. Such a mechanism means the existence of feedback reaction be - tween two stages of consciousness act: a) quantum exchange process in microtubule (MT) system of si - multaneously excited distant nerve cells, leading to spati al reorienta- tion of MTs and their tuning; b)reorganization of synaptic contacts in corresponding ne uron en- sembles, depending on the new ”tuned” spatial distribution of MTs. The vibro-gravitational 3D standing waves contain informa tion about all kind of dynamic processes in condensed matter (effe ctons and deformons), i.e. their informational volume is much mor e than that of IR- optic and acoustic deformons (see ”New articles” in http://www.karelia.ru/˜alexk). In such a way our model agrees with idea of Karl Pribram (Lan- guages of the Brain, CA, 1977) of holographic principles of m emory and braining and with its later development (Hameroff, 1987, 1994; Lechleiter et al., 1991; Jibu et all.., 1994). 4Nowadays to derive an acoustic hologram in the range of ultra sonic fre- quencies, the nematic liquid crystals have already been use d. The long axes of molecules in nematic liquid crystals are parallel to each ot her, like in biomem- branes. Biological membranes are formed by lipids representing two -chain amphiphylic molecules. They form double layers in the aqueous environme nt, where the nonpolar tails are turned to each other, and the polar ones - t o the aqueous environment (Cantor and Schimmel, 1980). Brain represents a complex liquid crystal system. The grey s ubstance con- sists mainly of cerebrosides, phosphatides, and glyceride s forming lyotropic liq- uid crystals. Myelin shell, which surrounds the conducting nerve fibers - a xons, has the properties of high-ordered nematic crystals (Chistyakov, 1966). Some types of holograms, e.g. echo-holograms, have a proper ties of not only spatial, but also temporal memory. Three-dimens ional holograms have a big informational capacity and an associat ive char- acter of memory. The pilotless writing of a hologram is possible, when the rad iation of every object’s point can be considered as a pilot one relat ive to its all other points. In this case, the three-dimensional hologram, can be restor ed by means of only the part of the object’s points, which is able to ”recollect” it as a whole. In contrast to usual holograms, reflecting the three-dimens ional geometric properties of objects in photo materials, the inf ormation in the membranes and cytoskeleton of nerve cells is encoded i n the form of the effectons and deformons. Hence, it is immediately related to the dynamic, mechanic and geometrical properties of memb ranes, cell’s filaments and even [DNA-protein] complexes. The code way of keeping the information in the form of the effec tons and de- formons as 3D standing waves (de Broglie waves, electromagn etic, acoustic and vibro-gravitational) in the intra-microtubule coherent w ater looks very effective and could be used in future quantum computer technology. It is an important point of our Hierarchic model of conscious ness that two subsystems: [microtubules + internal water] and [a ctin fila- ments+cytoplasmic water] are strongly interdependent and their in- teraction can be modulated by postsynaptic potential (PSP) changes, activating Ca2+−channels. Stability of MT, actin filaments and probability of their dis assem- bly is dependent also on the concentration of Ca2+and water activity changing in a course of membranes depolarization. Before depolarization the concentration of Ca2+outside of cell is about 10−3Mand inside about 10−7M.Such strong gradient provide fast increasing of these ions concentration in cell till 10−5M,enough to activate protein gelsolin. It is shown that [gel-sol] dynamic equilibrium (cycles) can be rapid and cor- 5relate with release of neurotransmitter vesicles from pres ynaptic axon terminals (Miyamoto, 1995; Muallem et al., 1995). Hameroff proposed that in his and Penrose ”orchestrated objective re- duction [orch. OR]” model (Hameroff and Penrose, 1996a,b; Hameroff, 1996, 1998) quantum computation/superposition phase occu rs in MTs during the gel phase of sol-gel cycles of frequency about 40 Hz. He as sumed that solid-like gel surrounding the MTs can serve as a shield of th ermal noise and provide a condition for MTs dynamics coherency, necessary f or OR. It is shown by Wachstock (1994), that gel state, depending on actin cros s linkers, can serve as a shock absorber indeed. In our model, in contrast to that of Hameroff’s, the key phenom ena - spatial adaptation of MTs of distant cells due to resonance electrom agnetic and vibro- gravitational interaction as an intermediate stage of syna ptic reorganization occur in low viscous sol state. However, our HMC as it will be shown below, does not contradic t the idea of Orch. OR model. Instead of coherent redistribution of mass o f MT’s proteins, which after Hameroff’ assumption should lead to new space-ti me geometry, our model propose the coherent change of water mass, involved in high-T molec- ular Bose condensate inside MTs in form of coherent water clu sters - primary librational effectons (see Appendix). HMC also have some common features with model of Quantum Brai n Dy- namics (QBD ), proposed by L.Riccardi and H.Umezawa in 1967 a nd devel- oped by C.I.Stuart, Y.Takahashi, H.Umezava (1978, 1979), M .Jibu and K.Yasue (1992, 1993). In addition to traditional electrical and che mical factors in the nerve tissue function, this group introduced two new types o fquantum excita- tions (ingredients), responsible for the overall control o f electrical and chemical signal transfer: corticons and exchange bosons (dipolar phonons). Corticons has a definite spatial localization and can be described by Pa uli spin matri- ces. The exchange bosons , like phonons are delocalized and follow Bose- Einstein statistics. ”By absorbing and emitting bosons coh erently, corticons manifest global collective dynamics, providing systemati zed brain functioning” (after M.Jibu and K.Yasue,1993). In other paper (1992) thes e authors gave more concrete definitions: ”Corticons are nothing but quanta of the molecular vibrational field of biomolecules (quanta of electric polarization, confined in protein filaments). Ex- change bosons are nothing but quanta of the vibrational field of water molec ules. . .”. One can find some analogy between spatially localized ”corti cons” and our effectons as well as between ”exchange bosons” and deform ons. Jibu, Yasue, S.Hagan and others (1994) discussed a possible role of quantum optical coherence in cytoskeleton microtubules: implicat ions for brain function. They considered MTs as wave guides for coherent superradiat ion - collective nonlinear phenomena. They assumed also that coherent photo ns, penetrating in MTs, lead to phenomenon called ”self-induced transparen cy”. II. PROPERTIES OF ACTIN FILAMENTS, MICROTUBULES 6AND INTERNAL WATER There are six main forms of actin existing. Most general F-ac tin is a polymer, constructed from globular protein G-actin with molecular m ass 41800. Each G-actin subunit is stabilized by one ion Ca2+and is in noncovalent complex with one ATP molecule. Polymerization of G-actin is accompa nied by splitting of the last phosphate group. The velocity of F-actin polymer ization is enhanced strongly by hydrolysis of ATP. However, polymeriz ation itself do not needs energy. Simple increasing of salt concentration (dec reasing of water activity), approximately till to physiological one - induc e polymerization and strong increasing of viscosity. The actin filaments are composed from two chains of G-actin wi th diameter of 40 ˚A and forming double helix. The actin filaments are the polar s tructure with different properties of two ends. Let us consider the properties of microtubules (MT) as one of the most important component of cytoskeleton, responsible for spatial organization and dynamic behavior of the cells. The [assembly ⇔disassembly] equilibrium of microtubules composed of α andβtubulins is strongly dependent on internal and external wat er activity aH2O(see Section 13.7 of book: Kaivarainen 1995; 1997), concent ration of Ca2+ and on the electric field gradient change due to MTs piezoelec tric properties. Theαandβtubulins are globular proteins with equal molecular mass (MM= 55.000), usually forming αβdimers with linear dimension 8nm. Poly- merization of microtubules can be stimulated by NaCl, Mg2+and GTP (1:1 tubulin monomer) (Alberts et al., 1983). The presence of hea vy water (deu- terium oxide) also stimulates polymerization of MTs. In contrast to that the presence of ions of Ca2+even in micromolar con- centrations, action of colhicine and lowering the temperat ure till 40C induce disassembly of MT. Due to multigene composition, αandβtubulins have a number of isoforms. For example, two-dimensional gel electrophoresis reveale d 17 varieties of βtubu- lin in mammalian brain (Lee et al., 1986). Tubulin structure may also be altered by enzymatic modification: addition or removal of amino acid s, glycosylation, etc. Microtubules are hollow cylinders, filled with water. Their internal diam eter about din=140˚A and external diameter dext=280˚A (Fig. 1). These data, in- cluding the dimensions of αβdimers were obtained from x-ray crystallography (Amos and Klug, 1974). However we must keep in mind that under the condi- tions of crystallization the multiglobular proteins and th eir assemblies tends to more compact structure than in solutions due to lower water a ctivity. This means that in natural conditions the above dimensions c ould be a bit bigger. The length of microtubules (MT) can vary in the interval: lt= (1−20)·105˚A (2.1) 7The spacing between the tubulin monomers in MT is about 40 ˚A and that between αβdimers: 80 ˚A are the same in longitudinal and transversal directions of MT. Microtubules sometimes can be as long as axons of nerve cells , i.e. tenth of centimeters long. Microtubules (MT) in axons are usually parallel and are arranged in bundles. Microtubules associated proteins (MA P) form a ”bridges”, linking MT and are responsible for their interaction and coo perative system for- mation. Brain contains a big amount of microtubules. Their most probable length is about 105˚A. The viscosity of ordered water in microtubules seems to be to o high for transport of ions or metabolites at normal conditions. All 24 types of quasiparticles, introduced in our hierarchi c theory of matter (Table 1 in book: Kaivarainen 1995; 1997), also can be pertin ent for ordered wa- ter in the microtubules (MT). However, the dynamic equilibrium between populations of different quasiparticles of water in MT must b e shifted towards primary librational effectons, comparing to bulk wa ter due to clusterphilic interactions (see section 13.3 of book: Kaiv arainen 1995 and ”New articles” in: http://www.karelia.ru/˜alexk). Th e dimen- sions of internal primary librational effectons have to be bi gger than in bulk water as a consequence of stabilization effect of MT wa lls on the thermal mobility of water molecules, increasing thei r most probable de Broglie wave length. Strong interrelation must exist between properties of inte rnal water in MT and structure and dynamics of their walls, depending on [ α−β] tubulins in- teraction. Especially important can be a quantum transitio ns like convertons [tr⇔lb] and a big fluctuations of internal water, like superdeformons, local- ized in the volume of primary IR librational deformon. The convertons in are accompanied by [dissociation/association] of primary librational effectons , i. e. flickering of coherent water clusters and can induce the change of angle between αandβsubunits in tubulin dimers. The biggest cavitational fluctuations - (superdeformons) c an in- duce total cooperative disassembly of MT .Superdeformons excitation in MT internal water could be an explanation of experimentally revealed dynamic instability (catastrophes) as a stochastic switching of MT growth to shrinkage (Mitchison and Kirschner, 1984; Horio and Hotani, 1986; Odd e at al., 1994). 8Fig. 1. Construction of microtubule from αandβtubulins, globular proteins with molecular mass 55 kD, existing in for m of dimers ( αβ). Eachαβdimer is a dipole with negative charges, shifted towards αsubunit (De Brabander, 1982). Consequently, microtubules, as an oriented elon- gated structure of dipoles system, have the piezoelectric p roperties (Athestaedt, 1974; Mascarennas, 1974). Intra-microtubular clusterphilic interactions (see Section 13.5) stimulate the growth of tubules from αβtubulin dimers. The structural physical-chemical asymmetry of αβdimers in composition of microtubules determines their dif - ferent rates of growth from the opposite ends ([+ ] and [- ]). The equilibrium of ”closed” (A) and ”open”(B) states of nonp olar cavities between αandβtubulins in ( αβ) dimers can be shifted to the (B) one under the change of external electric field in a course of membrane d epolarization. It can be a consequence of piezoelectric properties of MTs and w ill stimulate the formation of coherent water clusters in the open cavities of (αβ) dimers. The open cavities can serve as a centers of water cluster formati on and molecular Bose condensation. The coherent properties of water, in the hollow core of microtubules should be enhanced as a result of such process. The water in the microtubules, is orchestrated in the volume s of primary electromagnetic deformons (tr or lb). Water can exi st al- 9ternatively in the form of translational or in the form of lib rational effectons. Conversion from one type of collective excitatio n (tr) to another one (lb) can occur simultaneously in many parallel m icro- tubules with similar coherent properties of intra-MT water . It is a result of resonance-exchange process, mediated by electro magnetic, acoustic and vibro-gravitational deformons. The parallel orientation of MT in different cells, optimal fo r maxi- mum [MT-MT] resonance interaction could be achieved due to t wist- ing of centrioles, changing spatial orientation of MT. Howe ver, it looks that the normal orientation of MT as respect to each oth er corresponds to the most stable condition, i.e. minimum of po tential energy of interaction (see Albreht-Buehner, 1990). It is im portant to stress here that the orientation of two centrioles as a sourc e of MT bundles in each cell are always normal to each other. The stro nger is the nerve excitation, the bigger is population of coheren tly firing cells, tending to similar orientation of their internal MT. We suppose that the critical number of reorganized synaptic con- tacts is necessary for conversion of short-term memory to th e long- term one. The linear dimensions of the edge ( llb ef) of coherent water clusters - primary librational effectons in pure water at physiological temper ature (360C) - is about 11˚A and 45 ˚A in the ice at 00C. We assume that in the rigid internal core of MT, the linear dim ension (edge length) of librational effecton, approximated by cube is bet ween 11 ˚A and 45 ˚A i.e. about llb ef∼23˚A. It will be shown below, that this assumption fits the spatial a nd symmetry properties of MT very well. The most probable group velocity of water molecules in compo sition of pri- marylbeffectons is: vlb gr∼h/(mH2O·llb ef) (2.2) The librational mobility of internal water molecules in MT, which determines (vlb gr) should be about 2 times less than in bulk water at 370C,if we assume llb ef∼23˚A (see article 1.1 in ”New articles” http://www.karelia.ru ). The length of a orchestrated group of primary lbeffectons in the direction of microtubule main axis can be determined by the length of ed ge of primary librational IR deformons (see attachment), i.e. about 10 mi crons. Results of our computer simulations for pure bulk water shows, that the distance between centers of primary [lb] effectons, approxi mated by cube exceed their linear dimension to about 3.5 times (Fig 2b). For our ca se it means that the average distance between the effectons centers is about: d=llb ef·3.5 = 23 ·3.5∼80˚A (2.3) 10It gives a possibility for equidistant (80 ˚A) localization of the primary lbeffectons in clefts between αandβtubulins of each ( αβ) dimer in the internal core of MT. Such a regular spatial symmetry of the internal flickering clusters distribution in MT (Fig 2) is an important factor for realiza tion of the [opto- acoustic-conformational] signal propagation of configura tional waves along the MT, accompanied by their bending is related to alternating [ closing ⇋opening] clefts between αandβtubulins. This large-scale protein dynamics is correlated with dissociation/association of water clusters in clefts between ( αβ) dimers of MT (Fig.2) due to [ lb/tr] convertons excitation and phonons exchange. The size of trprimary effectons in MT is significantly smaller, than that oflbones and the microviscosity of water in regions, occupied by translational effectons - lower. The average angle between αandβtubulins change and the cavity’s [open ⇔closed] states equilibrium shifts to the closed one as a resu lt of conversion of lbeffectons to trones (dissociation of coherent water cluster). The dynamic equilibrium between trandlbtypes of the intra MT water effectons must to be very sensitive to α−βtubulins interactions, dependent on nerve excitation, in accordance to our model. Fig. 2. Theoretical temperature dependencies of: (a) - the space between centers of primary [lb] effectons (cal culated in ac- cordance to eq.4.62); (b) - the ratio of space between primary [lb] effectons to thei r length (calcu- lated, using eq.4.63); 11(c) - the space between centers of primary [tr] effectons (in a ccordance to eq.4.62); (d) - the ratio of space between primary [tr] effectons to thei r length (eq.4.63). Two statements of Hierarchic model of consciousness are imp or- tant: 1.The ability of intra-MT primary water effectons (tr and lb) fo r superradiation of six coherent IR photons from each of the eff ectons side, approximated by parallelepiped: two identical - ”longitudinal” IR photons, penetrating alo ng the core of microtubule, forming the longitudinal standing wav es inside it, and two pairs of identical - ”transverse” IR photons, also respon- sible for the distant, nonlocal interaction between microt ubules. In accordance to superradiation mechanism the intensity of lo ngitudinal radiation of MTs is much bigger than that of transverse one; 2. The parameters of the intra MT water radiation (frequency , coherency/amplitude, intensity) is regulated by the inter action of in- ternal water with MT walls, dependent on the [open ⇔closed] equi- librium of cavity between αandβtubulins, changing in the process of neuron depolarization. We have to stress here that our idea of IR superradiation , pro- duced by water in MT’s or in other condensed matter - is an in- herent property of our primary effectons, resulted from meso scopic molecular Bose condensation (Kaivarainen 1992, 1995). Thi s idea is independent on the model of MT’s as wave guide of superradiat ion for longitudinal photons, proposed by M. Jibu, S. Hagan, K. Y asue, et al., (1994). The difference of our approach from the latter one is that we assume in MT the existence of ”transverse” radiati on of IR photons as well as ”longitudinal” ones. In our model the de n- sity of electromagnetic energy is low enough and not destroy ing the protein’s of MT’s. Another advantage of our model - is the pos sibil- ity of electromagnetic interaction between MT’s by the exch ange of coherent transverse IR photons. III. THE SYSTEM OF 3D STANDING WAVES, PRODUCED BY MICROTUBULES The most probable length of microtubules ( l) satisfy the condi- tions of standing electromagnetic wave, corresponding to l ibrational longitudinal IR photons, radiated by primary effectons: llb=κλlb p 2=κ 2n¯νlbp(3.1) 12The similar dimension for translational IR photons is: ltr=κλtr p 2=κ 2n¯νtrp(3.2) hereκis the integer number; λlb pandλtr pare the librational and translational IR photons wave length: λlb p= (n˜νlb p)−1≃105˚A = 10 µ λtr p= (n˜νtr p)−1≃3.5·105˚A = 35 µ where: n≃1.33 is a refraction index of water; ˜ νlb p≃(700−750)cm−1is wave number of librational photons and ˜ νtr p≃200cm−1is the wave number of translational photons. When condition (3.1) of standing waves is violated, i.e. [κ]is non- integer, then the probability of IR radiation of librationa l photons increases strongly. Deviation of [ κ] from integer values due to change of microtubules length as a result of electrostriction, induced by electric field or as a result of Ca2+,induced disassembly in the process of nerve excitation, - should be a ccompanied by corresponding oscillations of microtubules radiation. Role of electromagnetic, acoustic and gravitational 3D sta nding waves in the adaptive reaction of nerve cells to excitation In the normal cells, microtubules grow from the cell center t o the cell periphery. In the animals cell center the pair of cen trioles is placed. In the center of plant cells the centrioles are absen t, only high electron density region is registered. Two centrioles in ce lls of animals are always oriented at the right angle with respect to each ot her. Centrioles represent a construction of 9 triplets of microt ubules (Fig. 3), i.e. two centriole are a source of: (2·27 = 54) microtubules. The centriole length is about 3000 ˚A and its diameter is 1000 ˚A. These dimensions mean that all 27 microtubules of each centr ioles can be or- chestrated in the volume ( vd) of one translational or librational electromagnetic deformon: [vd=9 4πλ3 p]tr,lb where: ( λp)lb∼105˚A and ( λp)tr∼3.5·105˚A 13Two centrioles with normal orientation as respect to each ot her and a lot of microtubules, growing from them, contain the internal orch estrated system of librational water effectons. It represent a quantum system w ith correlated ( a⇋ b)1,2,3 lbtransitions of the effectons. The resonance superradiation or absorption of anumber of librational photons (3q) in the process of above transiti ons, is dependent on the number of primary lbeffectons ( q) in the internal hollow core of a microtubule: q∼[L∗]/L/bardbl (3.3) where: [ L∗] is the length of microtubule; Llb /bardbl≃3.5·llb ef∼80˚A is the approximate space between internal water pri- mary librational effectons; llb ef∼23˚A is the edge length of the primary lbeffecton in MT at 360C. The value of qin (3.3) - determines the intensity (amplitude) of coherent longitudinal librational IR photons radiation from microt ubule with length L∗, for the case, when condition of standing waves (3.1) is viola ted. The frequency of this radiation is represented by eq.(3.5). Fig. 3 .(a) : The scheme of centriole construction from nine triplets of microtubules. The length and diameter of cylinder are 300 0˚A and 1000˚A, correspondingly. Each of triplets contain one complete m i- crotubule and two noncomplete MT; (b): the scheme of cross-s ection 14of cilia with number of MT doublets and MT-associated protei ns (MAP): [2 ·9 + 2] = 20. One of MT of periphery doublets is com- plete and another is noncomplete (subfibrilles A and B). It is important that the probabilities of pair of longitudin al and two pairs of transversal photons, emission as a result of superradiance by primary librational effectons are equal, being the consequence of the same collec tive (b→a)lb transition. These probabilities can be ”tuned” by the elect ric component of electromagnetic signals, accompanied axon polarization a nd nerve cell excitation due to piezoelectric properties of MT. Coherent longitudinal emission of IR photons from the ends of each pairof microtubules of two perpendicular centrioles of the same cell and from ends of onemicrotubule of other cell can form a 3 Dsuperposition of standing photons (primary deformons) as a result of 3 photons pairs intercept ion. This becomes possible, when the condition of standing waves for longitud inallbphotons in MT is violated. The system of such longitudinal electromagnetic deformons , as well as those formed by transversal photons, have a properti es of pilotless 3D hologram. Such an electromagnetic hologram ca n be responsible for the following physico-chemical phenomena : -Nonmonotonic distribution of intra-cell water viscosity and dif- fusion processes in cytoplasm due to corresponding nonmono tonic spatial distribution of macrodeformons (sections 11.6; 11 .8 of book: Kaivarainen 1995 and ”New articles” in http://www.karelia .ru/˜alexk); -Regulation of spatial distribution of water activity (aH2O)in cy- toplasm as a result of corresponding distribution of inorga nic ions (especially bivalent such as Ca2+)in the field of standing electromag- netic waves. Concentration of ions in the nodes of standing w aves should be higher than that between them. Water activity (aH2O) should vary in the opposite manner than ions concentration. The spatial variation of ( aH2O)means the equilibrium of [assem- bly⇔disassembly] modulation and regulation the length of actin and MTs filaments. As a consequence, the volume and shape of cell compartments will be modulated also. The activity of nu merous oligomeric allosteric enzymes could be regulated also in su ch a way. The system of coherent electromagnetic 3D standing waves (p ri- mary deformons) is interrelated with that of acoustic and gr avita- tional 3D waves, in accordance with our model. Microtubules may regulate very different processes in cells and in cells ensembles in space and time. The 3D holograms, creatin g by MTs, may represent the internal and external ”morphogenic fi elds” and be responsible for differentiation of cells. The following properties of microtubules can affect the prop erties of morphogenetic field: 15a) total number of microtubules in the cell; b) spatial distribution of microtubules in the volume of cyt oplasm; c) distribution of microtubules by their length. The constant of ( a⇔b) equilibrium of primary librational effectons (Ka⇔b)lb= exp[ −(Ea−Eb)/kT]lb (3.4) and that of ( A∗⇔B∗) equilibrium of supereffectons are dependent on the structure and dynamics of αβtubulin pairs forming MT walls. This equilibrium is interrelated, in turn, with lbphotons frequency ( νlb)1,2,3: [νlb=c(˜ν)lb= (Vb−Va)lb/h]1,2,3(3.5) which is determined by the difference of potential and total e nergies between (b) and (a) states of primary effectons in the hollow core of mi crotubules: [Vb−Va=Eb−Ea]1,2,3 lb(3.6) (˜ν)1,2,3 lbis the librational band wave number. The refraction index ( n) and dielectric constant of the internal water in MT depends on [ a⇔b] equilibrium of the effectons because the polarizability of water and their interaction in (a) state are higher, than tha t in (b) state (see Articles at: www.karelia.ru/˜alexk). IV. ROLE OF ACTIN FILAMENTS AND MICROTUBULES IN NEURON’S BODY VOLUME/SHAPE ADAPTATION TO NERVE EXCITATION The normal nerve cell contains few dendrites, increasing th e sur- face of cell’s body. It is enable to form synaptic contacts fo r reception the information from thousands of other cells. Each neurone has one axon for transmitting the ”resulting” signal in form of the e lectric im- pulses from the ends of axons of cells-transmitters to neuro n-receptor. The synaptic contacts, representing narrow gaps (about hun dreds of angstrom wide) could be subdivided on two kinds: the direct- electric and more univer- sal -chemical ones. In chemical synapsis the signal from the end of axon - is transmitted by neuromediator, i.e. acetylholine. The neuromediator molecules are stored in synaptic bubbles nearpresynaptic membrane. The releasing of me- diators is stimulated by ions of Ca2+.After diffusion throw the synaptic gap mediator form a specific complexes with receptors of post syn aptic membranes on the surface of neurons body or its dendrites. Often the rec eptors are the ionic channels like ( Na+, K+)- ATP pump. Complex - formation of differ- ent channels with mediators opens them for one kind of ions an d close for the 16other. Two kind of mediators interacting with channels: sma ll molecules like acetilholine, monoamines, aminoacids and big ones like set of neuropeptides are existing.. The quite different mechanism of synaptic transmission, rel ated to stimu- lation of production of secondary mediator is existing also. For example, activation of adenilatcyclase by first mediator increases t he concentration of intra-cell cyclic adenozin-mono-phosphate (cAMP). In tur n, cAMP can activate enzymatic phosphorylation of ionic channels, changing the electric properties of cell. This secondary mediator can participate in a lot of reg ulative processes, including the genes expression. In the normal state of dynamic equilibrium the ionic concent ration gradi- ent producing by ionic pumps activity is compensated by the e lectric tension gradient. The electrochemical gradient is equal to zero at this state. The equilibrium concentration of Na+and Cl+in space out of cell is bigger than in cell, the gradient of K+concentration has an opposite sign. The external concentration of very important for regulative processes Ca2+(about 10−3M) is much higher than in cytosol (about 10−7M).Such a big gradient provide fast and strong increasing of Ca2+internal concentration after activation of corre- sponding channels. At the ”rest” condition of equilibrium the resulting concen tration of internal anions of neurons is bigger than that of external ones, providing the difference of potentials equal to 50-100mV. As far the thickness of membrane is only about 5nm or 50 ˚A it means that the gradient of electric tension is about: 100.000V/sm i.e. it is extremely high. Depolarization of membrane usually is related to penetrati on ofNa+ions into the cell. This process of depolarization could be inhibited by selected diffusion ofCl−into the cell .Such diffusion can produce even hyperpolarization of mem- brane. The potential of action and nerve impulse can be excited in ne uron - receptor only if the effect of depolarization exceeds certain shresho ld. In accordance to our Hierarchic model of concsiousness (HMC ) three most important consequences of neuron’s body polariz ation can occur: -reorganization of MTs system and ionic channels activity, a ccom- panied by short-term memory emergency; -reorganization of synaptic contacts on the surface of neur on and its dendrites, leading to long-term memory; -generation of the nerve impulse, transferring the signal to another nerve cells. Propagation of nerve signal in axons may be related to intracellular water activity ( aH2O) decreasing due to polarization of membrane. As a result of 17feedback reaction the variation of aH2Oinduce the [ opening/closing ] of the ionic channels, thereby stimulating signal propagation. We suppose that the change of clusterphilic interaction of o rdered water between inter-lipid tails in nonpolar central region s of biomem- branes, could be responsible for lateral signal transmissi on in mem- branes (K¨ aiv¨ ar¨ ainen, 1985, 1995). Such mechanism can pr ovide dis- tant cooperative interaction between different receptors a nd channels on cell surface. As far the αβpairs of tubulins have the properties of ”electrets” (Debra - bander, 1982), the piezoelectric properties of core of microtubules can be predicted (Athenstaedt, 1974; Mascarenhas,1974). It means that structure and dynamics of microtubules can be r egu- lated by electric component of electromagnetic field, which accompa- nied the nerve excitation. In turn, dynamics of microtubule s hollow core affects the properties of internal ordered water. For example, shift of the [open ⇔closed] states equilibrium of cavity between αandβtubulins to the less stable open one in a course of excitation should lead to: [I]. Increasing the dimensions and life-time of coherent cl usters, represented by primary lbeffectons in the open states of inter α-βtubulins nonpolar cavities; [II]. Destabilization of MT, increasing the probability of its disassembly; [III]. Stimulation the distant interaction between MT of di fferent neurons as a result of increased frequency and amplitude/coherency of IR librational photons; [IV] The rightward shift of ( A∗⇔B∗) equilibrium could be stimulated by the elevation of lb IR- photons density due to enhancement of superradi- ance effect and IR photons pumping. This shift increases the p robability of cavitational fluctuation of the intra MT water and reversible disassembly of microtubules. The lower stability of MTs in the nerve cell body as respect to its bundles in axon or in cilia is a result of fact that microtu bules in bundles are interconnected by ”handle”-like proteins (den eins) and other microtubule associated proteins (MAP). Twisting of the centrioles of distant interacting cells and bending of MTs can occur after [gel →sol] transition. This tuning is necessary for enhancement o f the number of MTs with the parallel orientation, most effective f or their exchange interaction by means of 3D coherent IR photons and vibro-gra vitational waves. Reorganization of actin filaments and MTs system should be ac companied by corresponding changes of neuron’s body and its dendrites shape and activity of certain ionic channels and synaptic contacts redistribu tion; This stage is responsible for long-term memory emergency. At [sol]-state Ca2+- dependent K+channels turns to the open state and internal concentration of potassium decreases. The latter oppose the depolar- ization and decrease the response of neuron to external stim uli. Decay of neu- ron’s response is termed ”adaptation”. This response adaptation is accompanied 18byMTs-adaptation , i.e. their reassembly in conditions, when concentration o f Ca2+tends to minimum. The reverse [sol →gel] transition stabilize the new equilibrium state of given group of cells. The described hierarchic sequence of stages: from molecula r Bose condensation to synaptic reorganization, responsible for described cy- cles of nerve cells, reflects the idea of our model of consciou sness. The equilibrium constants of the intra-microtubule water convertons (K[tr/lb])and supereffectons (KA∗⇔B∗),which determines the probability of cav- itational fluctuations - superdeformons may also be depende nt on following ex- ternal as respect to MT factors: - the dynamics of MT (+) ends, fixed on cell’s membranes, depen ding on concentration of Ca2+, water activity ( aH2O) and cells swelling; - the frequency and intensity of the resonant IR-radiation o f MTs, external as respect to selected cell ; -resonant opto-acousto-gravitational fields, produced by water of MT s of other cells and large-scale (LS) and small scale (SS) dynami cs of αβtubulin pairs. The LS-dynamics of tubulin dimers represent the change of ”b ending” angle between αandβtubulins of about 210(Melki et al., 1989), corresponding to fluctuation of the inter-tubulins cavity between closed (A) and open (B) states. Such bending is a result of macroconvertons (flickering clus ters) excitations with frequency: (106−107)Hz(see Fig.48c). The[assembly ⇋disassembly ]dynamic equilibrium of the actin fil- aments and MT subsystems in cells in terms of colloid chemist ry rep- resents [coagulation ⇋peptization] or [gel ⇋sol] equilibrium .The increas- ing of cell’s volume, accompanied the MTs orchestrated disa ssembly is a result of cell ”swelling” due to osmotic diffusion of water from the e xtracell medium. The decreasing of water activity in cell, inducing such osmo se, is a consequence of increasing of ”bound” or ”hydration” water fraction afte r microfilaments and MT disassembly to huge number of subunits. The nerve cell body and dendrites swelling will induce the co llec- tive nonspecific opening of big number of ionic channels and s trong resulting postsynaptic potential (PSP) emergency. The big ger is re- sulting PSP the higher is frequency of the nerve impulses, ge nerated by this cell and penetrating via axon to other neurons (Coomb s, et al., 1957). The new assembly of MT-system in nerve cell’s body - stimulat es, in accordance to our model, the reorganization of synaptic c ontacts on the cell surface. It is accompanied by pumping out the extr a water from cell and restoring the rest - properties of ionic c hannels. However, the ”sensitivity” of certain ionic channels might be changed as a result of MTs and synaptic systems perturbation. This co uld be responsible for short-term memory. The cooperative disassembly/assembly of MTs, induced by ca vita- 19tional fluctuations (superdeformons) can be accompanied by coherent ”biophotons” emission/absorption in the ultraviolet (UV) and visible range due to water molecules [dissociation ⇋recombination ]reaction in a course of intra-MT water cavitational fluctuations. The se high- frequency coherent photons exchange, like the IR photons an d high- frequency nerve impulses, propagating via axons - may be res ponsi- ble for synchronized firing of distant neuron ensembles in he ad brain (Singer, 1993). The firing is a complex nonlinear process. It s char- acteristic time of about 1/50 of second (20ms) is much longer than pure quantum phenomena in MT like photons radiation and Bose [condensation ⇋evaporation], corresponding in our model to [lb/tr] convertons excitation. One of the important idea of HMC is that collective interacti ons of distant neurons in head brain could be realized not only by means of co nventional nerve impulse via axons. It happens more effectively by combinatio n of simultaneous cells bodies depolarization with resonant quantum exchang e between their MTs with similar orientation. 4.1 The entropy-driven information processing It leads from HMC is that changes of systems of electromagnet ic, acoustic and gravitational 3D standing waves and correspon ding holo- grams in the ensemble of nerve cells, produced by the orchest rated internal water of MTs in course of braining, enhance the quan tum exchange between neurons. This process induces redistribution of probabilities of di fferent wa- ter excitations in huge number of microtubules. It means cor respond- ing change of informational entropy <I>in accordance with known relations (Kaivarainen 1995; 1997): < I > =/summationdisplay iPilg(1/Pi) =−/summationdisplay iPilg(Pi) (4.1) where: Piis a probability of the ( i) state with energy ( Ei), defined as: Pi=exp(−Ei kT)/summationtext iexp(−Ei kT)(4.2) For the total system the relation between entropy (S) and inf ormation (I) is: S(e.u.) =k·lnW= (k·ln2)I= 2.3·10−24I(bit) (4.3) where: statistical weight of macrosystem: 20W=N! N1!N2!. . .N q!(4.4) where total number of internal water molecules in macrosyst em of interacting MT is: N=N1+N2+. . .+Nq; [q] is number of non degenerated states of 24 quasiparticles of intra MT water. Thereduced information of condensed matter [see Chapter 14 , eq.14.27b in book: Kaivarainen 1995; 1997] gives the quantitative cha racteristic not only about quantity (I) but also about the quality of the information : <Iq>=−[N0/V0]·/summationdisplay iPilg2(Pi)/ni (4.5) where N 0and V 0are the Avogadro number and molar volume; n iis a con- centration of excitation of (i)-type. The distant energy exchange between MT, accompanied by the change of Pifor different excitations can be considered as an infor- mational exchange between nerve cells. It is related to chan ge of fractions of water excitations in MT system with volume (vi=1/ni). The factors, affecting the equilibrium constant of two-stat e exci- tations of water in MT The dimensions of water librational effectons in given micro tubule and their life- time increases with probability of open states of nonpolar cleft between αandβtubulins. The equilibrium constants between ”acoustic” (a) and ”opti c” (b) states of primary effectons ( Ka⇔b)tr,lb, between trandlbprimary effectons ( Ktr⇔lb), sec- ondary effectons ( Ka⇔b)tr,lband that of supereffectons ( KA∗⇔B∗) are presented below: (Ka⇔b)tr,lb= exp[ −h(νa−νb)/kT]tr,lb= exp[ −hνp/kT]tr,lb (4.6) Ktr⇔lb= (Ka⇔b)tr·(Ka⇔b)lb= exp[ −h(νtr p+νlb p)/kT]lb (4.7) (Ka⇔b)tr,lb= exp[ −h(νa−νb)/kT]tr,lb= exp[ −hνph/kT]tr,lb (4.8) KA∗⇔B∗= exp[ −h(νA∗ −νB∗)/kT] = (KA⇔B)tr·(KA⇔B)lb== [Ka⇔b·Ka⇔b]tr·[Ka⇔b·Ka⇔b]lb (4.9) 21The primary effectons equilibrium constants ( Ka⇔b)tr,lbare related to difference between the total and potential energies of (b) and (a) state s and resulting frequency of coherent IR radiation ( νp)tr,lbof water primary tr and lib effectons. We have at least five defined parameters, involved in the quant um processes of nerve activity in our model: 1. Intensity of IR superradiance and vibro-gravitational w aves, radiated by MT system; 2. Frequency of coherent IR radiation, produced by water of t he microtubule system, responsible for distant cooperation b etween mi- crotubules; 3. The life-time of primary lbeffecton (τlb)responsible for [order ⇔disorder] equilibrium of water in MT, interrelated directl y with [B⇔A]equilibrium of nonpolar cavities between αandβtubulins; 4. Frequency (νA∗→B∗)of a big fluctuations- superdeformons, re- sponsible for MT reversible disassembly (catastrophe), co llective bend- ing and reorganization of microtubule system, leading to a n ew [vol- ume / shape] state of the nerve cell body; 5. Frequency and intensity of high-frequency ”biophotons” , re- sulted from recombination of water molecules after their di ssociation in a course of intra-MT water cavitational fluctuation (supe rdefor- mons). Thus, braining and consciousness in accordance with HMC is a process of synaptic contacts redistribution as a result of d irect axons- mediated interaction and distant quantum exchange between MT of different cells by means of coherent IR photons, and vibro-gr avitational waves (VGW). This exchange can be accompanied by the oscillations of cons tants of [ tr⇔ lb],(a⇔b) and ( A∗⇔B∗) equilibrium, as a result of periodic redistribu- tion of energy between subsystems of the effectons and deform ons (Kaivarainen 1995; ”New articles” in: http://www.karelia.ru/˜alexk). In the case of braining, however such autooscillations could be stimulated by perio dic excitation, which accompanied nerve impulse propagation along the axon. Autowaves originate as a result of interaction of given cell microtubu les with the microtubules of the surrounding cells, mediated by electromagnetic and, possibly, gravitational primary deformons. In this ca se, autowaves repre- sent spatially distributed oscillations of [ A∗⇔B∗] equilibrium constant in the active medium of nerve cells ensembles, accompanied by thei r reorganization. The active medium is defined as an two-level system which can r elax to the former energy distribution after excitation. As a result of competition, one of the sources of the autowave s with high- est frequency in form of rotating curl (reverberator) rises up and becomes the leading and dominating one. It is known that autowaves with h ighest frequency suppress other sources of autowaves in the active medium. Reverberat ors originate as a result of the autowave breach at non-homogene ities of the active medium and they are able to multiplication. 22Dissipative structures , introduced by Prigogin, can be considered as a private case of the autowaves and termed ”freezing out” autowaves (A ndronov et al., 1981). Autowaves in the neuron chains, related to oscillation of [A∗⇔B∗]and MTs [assembly ⇋disassembly ] equilibrium in a big groups of nerve cells, include two possi ble phenomena on mesoscopic and macroscopic scale: 1. Polarization/depolarization of neurons; 2. Collective activation/relaxation of synaptic connecti ons, accompanied by [assembly/disassembly] of MTs system. As was mentioned above, both of these processes can be accomp anied by the change of volume and shape of nerve cell body. Competition between autowaves in neuron ensembles is the crucial phenomena in ”selection of final result” as an elementary act of consciousness in the process of recalling and braining in accordance with o ur model. The subsystems of primary electromagnetic, acoustic and gr avitational de- formons with properties of pilotless hologram, produced by microtubules, are responsible for distant phenomena, necessary for autowave emergency. Resonance energy exchange between systems of microtubules ofdifferent ”normal” cells, leading to the change of KA∗⇋B∗andaH2Ocan be responsible also for differentiation and morphogenesis of cells. Frequency and amplitude of supereffectons equilibrium cons tant (KA∗⇔B∗)oscillations can serve as an additional informational pa- rameter, related to autowaves excitation in brain. Our model consider fluctuations and dissipation, stimulati ng [gel ⇌ sol] transitions and synaptic system reorganization, as a n ecessary phenomena for brain ”working”. However this CHAOS is organi zed by quantum phenomena ,like Bose-condensation of water in compo- sition of primary lbeffectons in MT, their superradiance and self- induced bistability. The higher is quantum order and cohere nce, the less is the number of mistakes in brain working. At the same ti me,the possibility of mistakes due to competition between discree t quantum and continuous thermal properties - make the process of brai ning NON-DETERMINISTIC and means its ability for creativity. Th e main difference between computer and brain looks be in the fac t that in the brain, in contrast to computer, the input and output th e in- formation is not always adequate to each other. The INTUITION from such point of view means the ability to choose one right solution (rigorously inadequate) from hug e number of wrong, but adequate to the available at the moment informa tion. It looks that associative memory, helping such choose, is th e most probable background for INTUITION. Von Neumann model of cellular automata was used by Hameroff et al. as a background for description of information proces sing. The 23essential features of cellular automata after Hameroff et al . (1992, 1994) are the following: 1. At a given time, each ”cell” is in one of a number of finite sta tes (usually two); 2. The ”cells” are organized according to a fixed geometry; 3. Each ”cell” communicates with other cells in its neighbor hood only. The size and shape are the same for all ”cells”; 4. There is a universal clock. The each ”cell” may change to a n ew state at each tick of the clock, depending on its present state and t he states of its neighbors. The time step of cellular automata clock in Hameroff’s model i s related with tubulins, composing MT, conformational changes with coher ent phonons fre- quency (1010−1011)Hz, proposed by Fr¨ olich (1968, 1975). Fr¨ olich assumed thatacousto-conformational transitions are coupled to charge redistributions, that accompanied the dipole oscillations or electron movem ents in nonpolar re- gions of protein with a low dielectric constant. Hameroff ass umed that such conformational transitions induce the acoustic waves, pro pagating across mi- crotubule diameter, providing ”clocking frequency” for ce llular automata. In our approach the ”clocking frequency” can be related to that of [A∗⇔B∗] transitions of supereffectons, having the order of 104Hz. The role of ”cell” in cellular automata could be played by definite par ts of microtubules, changing as a result of superdeformons excitation, between nodes of the stand- ing wave length of librational IR photons ( ∼105˚A). The number of hierarchic cellular automata can be very high. The notion of gliders, as of patterns, moving through the medium of cellu- lar automata unchanged, i.e. without dissipation is import ant for interaction between number of ”cells”. In our model the PRIMARY (electromagnetic, gravitational) de- formons for one side and SECONDARY acoustic deformons - coul d be termed the long distance and short distance gliders corresp ondingly. V. POSSIBLE MECHANISM OF WAVE FUNCTION COLLAPSING A lot of superimposed possible quantum states of any quantum system are always ”collapsed” or ”reduced” to single state ( or looks to be so) as a result of measurement, i.e. interaction with de tector. In accordance to ”Copenhagen interpretation”, the collaps ing of such system to one of possible states is unpredictable and pu rely random. Roger Penrose supposed (1989) that such a collapse is due to q uan- tum gravity, because the latter influences the quantum realm acting on space-time. After certain gravity threshold the system’ s wave function collapsed ”under its own weight”. Penrose (1989, 1994) considered the possible role of quantu m superposition in synaptic plasticity. He characterized the situation of l earning and memory by 24synaptic plasticity in which neuronal connections are rapi dly formed, activated or deactivated: ”Thus not just one of the possible alternati ve arrangements is tried out, but vast numbers, all superposed in complex linea r superposition”. The collapse of many cytoskeleton configuration to single on e is a nonlocal process, required for consciousness. Herbert (1993) estimated the mass threshold of wave function collapse roughly as 106daltons. Penrose and Hameroff (1995) calculated this thresh old as ∆Mcol∼1019D (5.1) Non-computable self-collapse of a quantum coherent wave fu nction within the brain may fulfill the role of non-deterministic free will aft er Penrose and Hameroff (1995). Our model, including the increasing of the total mass of wate r, in- volved in mesoscopic Bose-condensate (primary librationa l effectons), as a crucial stage of perception elementary act does not cont radict the above idea of nonlocal mechanism of collapsing of configu rational space of cytoskeleton. However, we explain the selection of certain configurationa l space of huge number of the excited neurons ensemble as a result of s pa- tial tuning of MTs orientations and corresponding redistri bution of synaptic contacts due to distant (not nonlocal) electromag netic and vibro-gravitational resonant interactions between MTs. The mass of water in one microtubule with most probable lengt h∼105˚A and diameter 140 ˚A is about mH2O∼108D In accordance with our calculations for bulk water , the fraction of molecules in composition of primary treffectons is about 23% and that in composition of primary lbeffectons is about ten times less (Fig.4). In MTs due to cluste rphilic interaction, the [lb] fraction, representing molecular Bo se condensate can be bigger. We assume, that in MTs at least 10% of the total water mass can b ead- ditionally converted to primary librational effectons as a result of neu ron de- polarization before MT disassembly. This corresponds to in creasing of mass of these quasiparticles in one MT as: ∆mH2O≃106D accompanied by decreasing of water mass, involved in other t ypes of excitations in MT. Based on known experimental data that each nerve cell contai ns about 50 MT , we assume that the maximum increasing of mass of primary l b effectons in one cell could be: ∆MH2O∼50·∆mH2O= 5·107D (5.2) 25If the true value of mass threshold, responsible for wave fun ction collapse, ∆Mcolis known, then the number ( Ncol) of neurons in assemblies, required for this process is Ncol∼(∆Mcol/∆MH2O) (5.3) Fig.4. Calculated ratio of water fractions involved in primary [lb] effectons to that, involved in primary [tr] effectons for the bulk water. The total number of nerve cells in human brain is about: Ntot∼1011. The critical fraction of cells population, participating in elementary act of perception can be calculated as: fc= (Ncol/Ntot) The [gel →sol] transition of simultaneously excited neurons body, followed by ”tuning” of their microtubules orientations, i s another explanation of coherent ”collapsing” of neurons group, lea ding to ”choosing” of one state from huge number of possible, after o ur HMC. Our model agree with general idea of Marshall (1989) that Bos e- conden- sation could be responsible for ”unity of conscious experie nce”. However, our 26model explains how this idea can work in detail and what kind o f Bose conden- sation is necessary. VI. Distant [cell-cell] interaction by means of high-frequency biophotons in our model We propose that all the intra-cell processes, accompanied b y drastic reorga- nization of the actin filaments and microtubules systems, i. e. collective disas- sembly (supercatastrophe), leading to [gel-sol] transiti on. This process can be stimulated by intra-MT water superdeformons, which are acc ompanied by high- frequency (UV, visible) biophotons radiation. It is be a res ult of water molecules recombination after their dissociation to the protons and h ydroxyl groups, ac- companied by superdeformons and cavitational fluctuations . It is possible that dynamic reorganizations of chromatin fib rils with diameter 30nm (elongated complexes of DNA with proteins-histones ), also can influence on water and other molecules recombination probability and biophotons emer- gency. Large- scale (LS) dynamics of the chromatin fibrils (CF) has a few hierarchic levels. The first one is due to relative fluctuations of nucleosomes, linked to eac h other with flexible regions of ”free” DNA. The width of caviti es between of nucle- osomes is about 11 ˚A. It is close to that of proteins domains and linear dimensio n of water lbeffectons at physiological temperature.(see http://www.k arelia.ru/˜alexk [New articles]). The second level of large scale (LS) dynamics of CF can be related to reversible cooperative dissociation of chromatin domai ns (composed from few nucleosomes) and histone H1, leading to unfolding of these d omains. The third level of LS dynamics of CF is assumed to occur in a course of relative fluctuations of big (400nm) loops of chromatin fib ril. Each level of LS dynamics of CF is correlated with water den- sity fluctuations like macroconverton and superdeformons ( see Ap- pendix). Biophotons radiation, related to MTs and chromatin large-s cale dynamics, their [assembly ⇋disassembly] could happen in a course of cells division or po i- soning. We suggest that this radiation could be responsible for ”mitogenic rays”, discovered by Gurvich in 1920. Later such effect was co nfirmed on divi- sion of synchronized yeast cultures. The ”Degradation radiation” was observed on strongly damaged or dying cells, regardless of the cause o f death. The”cytoxic effect” , revealed by Kaznacheev and others (1976) involves in- teraction between two cell cultures separated by quartz or r egular glass. In such a system a poisoned, dying culture shown to be able to communi cate via quartz glass only with normal cell culture initiating its patholog ical changes and even death. These experiments point to ultraviolet range of biophotons , ra- diated by dying culture, producing cytoxic effect in distant healthy 27cells. The ultra-weak radiation of coherent biophoton by different living organ- isms was profoundly studied by group of F. Popp (1992). Our model explains the emission of visible and UV-biophoton s as a result of water molecules recombination, accompanied the super- catastrophe of microtubules systems of cells. Thedissociation ⇋recombination of [anion–cation] bridges of chro- matin, induced by LS-dynamics like [DNA +H1hystons ]complexes reversible disassembly, also could be accompanied by bioph otons ra- diation. In course of cells division both kinds of these interrelated processes make contribution in coherent biophotons emission. The distant action of UV-biophotons on other cells could be a con- sequence of increasing of water molecule dissociation probability to proton (H+) and hydroxyl group ( HO−) due to photo-activation: hνp+H2O→[H2O]∗⇋HO−+H+(6.1) Shifting the equilibrium of this reaction to the right, lead s to perturbation of water structure in microtubules as a result of strong solvat ation of protons ( H+). In turn, this perturbation destabilize the primary librational effectons, increase the probabilities of superdeformons excitation and disassembly of microtubules and chromatin. The shift of equilibrium [assembly ⇋disassembly] of group of the actin fil- aments, microtubules and chromatin fibrils to the right in one conditions will stimulate cell division (when cell is ready for it), in o ther conditions - its morphogenic reorganization (differentiation) and in anoth er ones - death of cell. CONCLUSIONS OF HIERARCHIC MODEL OF CONSCIOUSNESS We can resume now, that in accordance with our HMC, the se- quence of following interrelated stages is necessary for el ementary act of perception and memory: 1. The change of the electric component of cell’s electromag netic field as a result of neuron depolarization; 2. Shift of A⇋Bequilibrium between the closed (A) and open to water (B) states of cleft, formed by αandβtubulins in microtubules (MT) to the right due to the piezoelectric effect; 3. Increasing the life-time and dimensions of coherent ”flic kering” water clus- ters, representing the 3D superposition of de Broglie stand ing waves of H20 molecules with properties of Bose-condensate ( effectons ) in hollow core of MT. This process is stimulated by the open nonpolar clefts of tub ulin dimers in MT with regular 80 ˚A spacing; 4. Increasing the superradiance of coherent IR photons indu ced by synchro- nization of quantum transitions of the effectons between acoustic andopticlike states; 285. Opening the potential dependent Ca2+channels and increasing the con- centration of these ions in cytoplasm; 6. Activation of Ca2+- dependent protein gelsolin, which induce fast dis- assembly of actin filaments and [gel-sol] transition, decre asing strongly the vis- cosity of cytoplasm and water activity; 7. Spatial ”tuning” of quasi-parallel MTs of distant simult aneously excited neurons due to distant electromagnetic and vibro-gravitat ional interaction be- tween them and centrioles twisting; 8. The coherent volume/shape pulsation of big group of depol arized cells as a consequence of (actin filaments+MTs) system disassembly and [gel →sol] transition. It happens as a result of [MF+MT] system reversi ble disassembly to huge number of subunits and increasing of water fraction in h ydration shell of actin and tubulin subunits due to increasing of their surfac e. This should de- crease the water activity in cytoplasm and increase the pass ive osmotic diffusion of water from the external volume to the cell. This stage should be accompanied by four effects: (a): Increasing the volume of the nerve cell body; (b): Disrupting the (+) ends of MTs with cytoplasmic membran es, making MTs possible to bend in cell and to collective spatial tuning of huge number of MTs in the ensembles of even distant excited ne urons; (c) Origination of new MTs system switch on/off the ionic chan nels and change the number and properties of synaptic contacts; (d): Decreasing the concentration of Ca2+to the limit one when its ability to disassembly of actin filaments and MT is stoppe d and [gel⇋.sol] equilibrium shifts to the left again, stabilizing a ne w MTs and synaptic configuration. This cyclic consequence of quantum mechanical, physico-ch emical and nonlinear events can be considered as elementary act of m emory and consciousness realization. This act can be as long as 500 ms, i.e. half of second. The elementary act of consciousness include a stage of coher ent electric firing in brain (Singer, 1993) of distant neurons groups with period of about 1/40 sec. Probability of superdeformons and cavitational fluctuatio ns in- creases after [gel →sol] transition. This process is accompanied by high-frequency (UV and visible) ”biophotons” radiation du e to re- combination of part of water molecules, dissociated as a res ult of cavitational fluctuation. The dimension of IR superdeformon edge is determined by the l ength of librational IR standing photon - about 10 microns. It is impo rtant that this dimension corresponds to the average microtubule length in cells confirming in such a way our idea. Another evidence in proof is that is that t he resonance wave number of excitation of superdeformons, leading from o ur model is equal to 1200 (1 /cm). 29The experiments of G.Albrecht-Buehler (1991) revealed tha t just around this frequency the response of surface extensions of 3T3 cel ls to weak IR irra- diation is maximum. Our model predicts that IR irradiation o f microtubules system in vitro with this frequency will dramatically increase the probabi lity of microtubules catastrophes. Except superradiance , two other cooperative optic effects could be in- volved in supercatastrophe realization: self-induced bistability and the pike regime of IR photons radiation (Bates, 1978; Andreev et al.,1988). Self-induced bistability is light-induced phase transition. It could be related to nonlinear shift of [ a⇔b] equilibrium of primary librational effectons of intra MT water to the right as a result of saturation of IR (lb)-photons absorption. As far the molecular polarizability and dipole moments in (a ) and (b) states of the primary effectons - differs, such shifts of [ a⇔b] equilibrium should be accompanied by periodic jumps of dielectric permeability and stability of coherent water clusters. These shifts could be responsible for the pike regime of librational IR photons absorption and radiation. As far the stability of b-states of lb effectons is less than th at of a-states, the characteristic frequency of pike regime can be correlated w ith frequency of MTs- supercatastrophe activation. This effect can orchestrate t he [gel-sol] transitions of neuronal groups in head brain. 30Fig.5. The schematic presentation of the local, acousto-conforma tional and distant - electromagnetic interactions between microt ubules (MT1 and MT2), connected by MAP. MAP– microtubules associated proteins stabilize the overa ll structure of MTs. They prevent the disassembly of MTs in bundles of axons a ndciliain a course of their coherent bending. In neuron’s body the concentration of MAP and their role in stabilization of MTs is much lower than in ci lia. The local acousto-conformational signals between MT are realized via MTs - associated proteins (MAP), induced by transitions of t he cleft, formed byαandβtubulins, between closed (A) and open (B) states. The orches trated dynamics of individual MT as quantum conductor is a result of phonons ( hνph) exchange between ( αβ) clefts due to [ lb/tr] conversions, corresponding to water clusters, ”flickering”, in-phase to [ B⇋A] pulsations of clefts. The distant electromagnetic and vibro-gravitational inte ractions be- tween different MT are the consequence of IR photons and coher ent gravitational waves exchange. The corresponding two types of waves are exc ited as a result of orchestrated ( a⇔b) transitions of water primary librational effectons, local ized in the open B- states of ( αβ) clefts. 31When the neighboring ( αβ) clefts has the alternative open and closed states like on Fig 5, the general spatial structure remains straigh t. However, when [A⇔B] equilibrium of all the clefts from one side of MT are shifted to the left and that from the opposite side are shifted to the right, it leads to bend- ing of MT. Coherent bending of MTs could be responsible for [v olume/shape] vibrations of the nerve cells and the cilia bending. Experimental data, confirming the HMC There are some experimental data, which support the role of m icrotubules in the information processing. Good correlation was found b etween the learn- ing, memory peak and intensity of tubulin biosyntesis in the baby chick brain (Mileusnic et al.1980). When baby rats begin their visual le arning phase after they first open eyes, neurons in the visual cortex start to pro duce vast quantities of tubulin (Cronley- Dillon et. al., 1974). Sensory stimula tion of goldfish leads to structural changes in cytoskeleton of their brain neuron s (Moshkov et al., 1992). There is evidence for interrelation between cytoskeleton p roperties and nerve membrane excitability and synaptic transmission (Matsumo to and Sakai, 1979; Hirokawa, 1991). It has been shown, that microtubules can tr ansmit electro- magnetic signals between membranes (Vassilev et al., 1985) . Desmond and Levy (1988) found out the learning-associated c hange in den- dritic spine shape due to reorganization of actin and microt ubules containing, cytoskeleton system. After ”learning” the number of recept ors increases and cytoskeleton becomes more dense. Other data suggest possibility that cytoskeleton regulate s the genome and that signaling along microtubules occurs as cascades of pho sphorylation/dephosphorylation linked to calcium ion flux (Puck, 1987; Haag et al, 1994). The frequency of superdeformons excitations in bulk water ( see Fig.48 d in book: Kaivarainen 1995) at physiological temperature (370C) in bulk water is around: νS= 3·104s−1 The frequency of such cavitational fluctuations of water in M T, stimulating in accordance to our model cooperative disassembly of MT, co uld differ a bit from the above value for bulk water. The difference is due to st abilizing influence of cavities between α and β tubulins on coherent water clusters (primary lb effectons). Our model predicts that if the neurons or other cells, contai ning MTs, will be treated by acoustic or electromagnetic field with resonance frequency of intra- MT water( νres˜νMT S≥1·104s−1),it can induce simultaneous disassembly of number of MTs, responsible for maintaining the specific ce ll volume and geometry. As a result, it activates the neuron’s body volume /shape pulsation. Such external stimulation of supercatastrophe has two impo rtant consequences: 32-The first one isgeneration of strong high-frequency nerve impulse, prop- agating via axons and exciting huge number of other nerve cel ls, i.e. distant nerve signal transmission in living organism; -The second one is stimulation the leaning process as far long-term mem- ory in accordance to HMC, is related to synaptic contacts reo rganization, ac- companied the neuron volume/shape pulsation. The first of these two consequences of HMC is in accordance with phenomena of ”ultrasound hearing”, discovered by P. Flanagan and used in his invented de- vice:”Neurophone” (Flanagan, 1996). It consists of a (30-50) kHz amplitude modulated by ordinary acoustic waves ultrasonic oscillato rthat generated 3.000 volts peak across two plastic insulated electrodes that were placed in contact with skin. It was shown that the skin under the electr odes was caused to vibrate by the energy field. Even some totally ne rve- deaf people could hear with Neurophone. The ordinary audio frequencies are in the range of 20 Hz to 20. 000 Hz. These ordinary audio frequencies are percepted by cochlea orinner ear through the air or through the bones. Lehardt et al. (1989) supposed that ultrasonic vibrations a re perceptive by another channel: tiny gland in the inner ear known as the Saccule. It looks that Saccule may have a dual functions of detection gravity and auditory signals. Cohlea could be a result of Saccule evolution in mammals. Lenhardt and colleagues constructed the an amplitude modul ated by audio- frequencies ultrasonic transmitter that operated at frequencies: (28-90) kHz. The output signal from their device was attached to the deaf p eople heads by means of piezo-electric ceramic vibrator. All people ”hear d” the modulated signal with clarity. The second mentioned above consequence of HMC also is confirmed by means of digital Neurophone version: Thinkman Model 50, develope d by Flanagan. It was demonstrated that if the educational tapes were playe d throw device, the information is very rapidly incorporated into long-ter m memory (Flanagan, 1996). Another very interesting optical phenomena , confirming HMC, was revealed by Flanagan also. It is a stimulation of light emission (visi ble and may be UV biophotons, affecting the photo film) by the human skin and any living things as well, induced by Neurophone’s radio-frequency (RF): 4 ·104Hz(Begich, 1996). It is in accordance with consequence of HMC that external fiel ds, inducing cavitational fluctuations and MTs supercatastrophe, shoul d lead to enhance- ment of ”biophotons” radiation in visible and UV regions, em itted as a result of recombination reaction: HO−+H+hνp⇋H2O (6.2) We can predict one more important consequence of HMC. Exci- tation of water superdeformons in MTs, leading to their coll ective disassembly (supercatastrophe), cell’s volume/shape pul sation and 33generation of high-frequency nerve impulse - could be stimu lated not only by US acoustic and RF electromagnetic signals, but as we ll by coherent laser emitted IR photons with frequency, correspo nding to excitation energy of superdeformons. We can calculate the photons frequency equal to νS p=c·/tildewideνS p=/parenleftbig 3·1010/parenrightbig ·1200 = 3 .6·1013c−1(6.3) and wave length: λS p=c//parenleftbig nH2O·νS p/parenrightbig ≃6.3·10−4cm= 6.3µ (6.4) where: /tildewideνS p= 1200 cm−1is wave number, corresponding to energy of su- perdeformons excitation; nH2O≃1.33 is refraction index of water. The idea of new device: Audio/Video Signals Skin Transmitter is proposed by us. In this device the laser beam with corresponding to (6.3) frequency and ultraweak intensity w ill be mod- ulated by acoustic and/or video signals. Then, the modulate d output optic signals will be transmitted to the nerve nodes of skin o r to chacras, using wave-guides. The nerve impulses, stimulate d by mod- ulated laser beam, can propagate via complex axon-synapse s ystem to brain centers, responsible for perception and processin g of audio and video information. The long-term memorizing process al so can be stimulated effectively by Skin Transmitter. The telepathic abilities of people could be enhanced strong ly due to increasing the coherency of quantum neurodynamics of the nerve nodes in chacras and brain. Another principle of neuromodulator can be based on ability of Aharonov-Bohm effect influence the biocells. The applicator s, con- taining solenoids, producing Aharonov-Bohm effect with fre quency of acoustic signals or brain’s [ α or β ]rhythms, applied to scull and chacras, can be of help for deaf people and that with nerve sys tem diseases. The direct and feedback reaction between brain centers, res pon- sible for audio and video information processing and certai n nerve nodes on skin is predictable. The coherent electromagnetic radiation of these nodes, including the acupuncture one can be respons ible for so-called aura. One of the important consequence of our Hierarchic model of consciousness is related to radiation of ultraviolet and vi sible pho- tons (”biophotons”) as a result of water molecules recombin ation af- ter their dissociation. Dissociation can be stimulated by c avitational fluctuation of water in the volume of superdeformons, induci ng re- versible disassembly of MTs. The frequency and intensity of this 34electromagnetic component of biofield, in turn, can affect th e kinetic energy of the electrons, emitted by skin in the process of Kir lian effect measurement. It is predictable that the above mention ed stim- ulation of psi-activity by resonant external radiation, sh ould influence on colors and character of Kirlian picture even from distant untreated points of skin. There are another resonant frequencies also, calculated fr om our theory, enable to stimulate big fluctuations of water in MTs a nd their disassembly. Verification of these important consequences of our model an d elaboration of Audio/Video Signals Skin - Transmitter is th e intrigu- ing task of future. The practical realization of Audio/Video Signals Skin Tran smitter will be a good additional evidence in proof of HMC and useful f or lot of people with corresponding diseases. The ways for experimental verification of HMC in vitro It is possible to suggest some experimental ways of verificat ion of our HMC using model systems. The important point of HMC is stabiliza tion of highly ordered water clusters (primary librational effectons) in t he hollow core of mi- crotubules. One can predict that in this case the IR libratio nal and translational bands of water in the oscillatory spectra of model system, co ntaining sufficiently high concentration of MT, must differ from IR spectra of bulk w ater as follows: - the shape of lbband in the former case must contain 2 components: the first one, big and broad, like in bulk water and the second one s mall and sharp, due to increasing coherent fraction of libeffectons and superradiance; - sound velocity in the system of microtubules must be bigger , than that in bulk pure water due to bigger fraction of ordered ice-like water ; - all the above mentioned parameters must be dependent on the external electromagnetic field, due to piezoelectric properties of M T; -the irradiation of MTs system in vitro by ultrasonic or electromagnetic fields with frequency of superdeformons excitation of the in ternal water of MTs at physiological temperatures (25 −400C) : νs= (2−4)·104Hz have to lead to increasing the probability of disassembly of MTs, induced by cavitational fluctuations. The corresponding effect of de creasing turbidity of MT-containing system could be registered by light scatteri ng method. Another consequence of superdeformons stimulation by exte rnal fields could be the increasing of intensity of radiation in visible and UV region due to emis- sion of corresponding ”biophotons” as a result of recombina tion reaction of water molecules: HO−+H+hν⇋H2O 35Cavitational fluctuations of water, representing in accord ance to our theory superdeformons excitations, are responsible for dissocia tion of water molecules, i.e. elevation of protons and hydroxyls concentration. The coherent transitions of ( αβ) dimers, composing MTs, between ”closed” (A) and ”open” (B) conformers with frequency ( νmc∼107s−1) are determined by frequency of water macroconvertons (flickering clusters ) excitation, local- ized in cavity between αandβtubulins. If the charges of (A) and (B) con- formers differ from each other, then the coherent (A ⇋B) transitions generate the vibro-gravitational and electromagnetic field with the same radio-frequency. The latter component of biofield could be detected by corresp onding radio waves receiver. The amplitude of corresponding vibro-gravitational waves (VGW) is not dependent on difference in electric charge, but on mass and accelerations of αandβtubulins in course of (A ⇋B) large-scale fluc- tuations. The VGW generated by relative LS oscillations of nucleosome s in composition of chromatin fibrils and electromagnetic waves generated by second level of the fibrils LS dynamics, described above - a lso could be responsible for distant [cell-cell] interaction. We can conclude that our Hierarchic theory of condensed matt er and its application to water and biosystems - provide reliab le models of informational exchange between different cells and corre lation of their activity. Hierarchic model of consciousness is based on proposed quantum exchange mechanism of interactions between neuron s, based on very special properties of microtubules, [gel-sol] tran sitions and interrelation between spatial distribution of MTs in neuro ns body and synaptic contacts on their surface. *************************************************** ****************************** APPENDIX: SUMMARY OF NEW HIERARCHIC THEORY OF CONDENSED MATTER by: Alex Kaivarainen The basically new hierarchic quantitative theory, general for solids and liquids, is developed. It is assumed, that unharmonic oscillations of particles in any con- densed matter lead to emergency of three-dimensional (3D) s uper- position of standing de Broglie waves of molecules, electro magnetic and acoustic waves. Consequently, any condensed matter cou ld be considered as a gas of 3D standing waves of corresponding nat ure. 36Our approach modify, unify and develops strongly the Einste in’s and Debye’s models. Collective excitations, like 3D standing de Broglie waves o f molecules, representing at certain conditions the mesoscopic molecul ar Bose con- densate, were analyzed, as a background of hierarchic model of con- densed matter. The most probable de Broglie wave (wave B) length is deter- mined by the ratio of Plank constant to the most probable impu lse of molecules, or by ratio of its most probable phase velocity to fre- quency. The waves B could be related to molecular translatio ns (tr) and librations (lb). As far the quantum dynamics of condensed matter did not follo w in general case the classical Maxwell-Boltzmann distribution, the re al most probable de Broglie wave length can exceed the classical thermal de Brog lie wave length and the distance between centers of molecules many times. This makes possible the atomic and molecular Bose condensation in solids and liq uids at tempera- tures, below boiling point. It is one of the most important re sults of our theory, confirmed by computer simulations on examples of water and ic e. Four strongly interrelated new types of quasiparticles (collective excita- tions) were introduced in our hierarchic model: 1.Effectons (tr and lb) , existing in ”acoustic” (a) and ”optic” (b) states represent the coherent clusters in general case ; 2.Convertons , corresponding to interconversions between trandlbtypes of the effectons (flickering clusters); 3.Transitons , presenting the intermediate [ a⇋b] transition states of the tr andlbeffectons; 4.Deformons , as the 3D superposition of IR electromagnetic or acoustic waves, activated by transitons andconvertons. Primary effectons (tr and lb) are formed by 3D superposition of the most probable standing de Broglie waves of the oscillating ions, atoms or molecules. The volume of effectons (tr and lb) could contain f rom less than one, to tens and even thousands of molecules. The first condition m eans validity ofclassical approximation in description of the subsystems of the effect ons. The second one points to quantum properties of coherent clusters due to molecular Bose condensation . The liquids are semiclassical systems, as far their (tr) pri mary effectons con- tain less than one molecule and (lb) primary effectons - more t han one molecule. The solids are quantum systems totally because both kind of t he effectons (tr and lb) are molecular Bose condensates. These consequences of our theory are confirmed by computer calculations. The 1st order gas-liquid transition is accompanied by strong decreasing of rotational degrees o f freedom due to emergency of primary (lb) effectons and liquid-solid transi tion - by decreasing of translational degrees of freedom due to Bose-condensati on of primary (tr) effectons. In general case the effecton can be approximated by parallele piped 37with ribs corresponding to de Broglie waves length in three s elected directions (1, 2, 3), related to symmetry of molecular dynam ics. In the case of isotropic molecular motion the effectons shape co uld be approximated by cube. The edge’s length of primary effectons (tr and lb) can be consi d- ered as the parameter of order. The in-phase oscillations of molecules in the effectons corr espond to the effecton’s (a)- acoustic state and the counterphase oscillations correspond to their (b)- optic state. States (a) and (b) of the effectons differ in potential energy only, however, their kinetic energies, impulses and spatial dimensions - are the same. The b-state the effectons has a common features with Fr¨ olich’s polar mode. The(a→b)or(b→a)transition states of the primary effectons (tr and lb), defined as primary transitons, are accompanied b y a change in molecule polarizability and dipole moment withou t density fluctuations. At this case they lead to absorption or radiati on of IR photons, respectively. Superposition (interception) of three internal standing I R pho- tons of different directions (1,2,3) - forms primary electro magnetic deformons (tr and lb). On the other hand, the [lb/tr] convertons andsecondary transitons are accompanied by the density fluctuations, leading to absorption or radiation of phonons with corresponding frequencies . Superposition - interception of standing phonons of three directions (1,2,3), formssecondary acoustic deformons (tr and lb). Correlated collective excitations of primary and secondary effectons and deformons (tr and lb) ,localized in the volume of primary trandlb electromag- netic deformons ,lead to origination of macroeffectons, macrotransitons andmacrodeformons (tr and lb respectively) . Correlated simultaneous excitations of tr and lb macroeffec tonsin the vol- ume of superimposed trandlbelectromagnetic deformons lead to origination ofsupereffectons. In turn, the coherent excitation of both: tr andlb macrodeformons and macroconvertons in the same volume means origination of superdeformons. Superdeformons are the biggest (cavitational) fluctuation s, leading to microbub- bles in liquids and to local defects in solids. Total number of quasiparticles of condensed matter equal to 24 = 4! reflects the all of possible combinations of the four basi c ones [1-4], introduced above. This set of collective excitation s in the form of ”gas” of 3D standing waves of three types: de Broglie, acou stic and electromagnetic - is shown to be able to explain virtuall y all the properties of any condensed matter. The important positive feature of our hierarchic model of ma tter is that it does not need to use the semi-empiric intermolecular potent ials for calculations, which are unavoidable in existing theories of many body syst ems. The potential 38energy of intermolecular interaction is involved indirect ly in dimensions and stability of quasiparticles, introduced in our model. The main formulae of theory are the same for liquids and solid s and include following experimental parameters, which take into ac- count their different properties: [1]- Positions of (tr) and (lb) bands in oscillatory spectra; [2]- Sound velocity; [3]- Density; [4]- Refraction index (extrapolated to the infinitive wave leng th of photon ). The knowledge of these four basic parameters at the same temp erature and pressure makes it possible using our computer program, t o evaluate more than 150 important characteristics of any condensed matter . Among them are such as: total internal energy, kinetic and potential energ ies, heat-capacity and thermal conductivity, surface tension, vapor pressure, vi scosity, coefficient of self-diffusion, osmotic pressure, solvent activity, etc. M ost of calculated param- eters are hidden, i.e. inaccessible to direct experimental measurement. The new interpretation and evaluation of Brillouin light sc attering and M¨ ossbauer effect parameters also are done on the basis of hie rarchic model. Mesoscopic scenarios of turbulence, superconductivity an d superfluidity are elaborated. Some original aspects of water in organization and large-sc ale dynamics of biosystems: proteins, DNA, microtubules, membranes and regulative role of water in cytoplasm, cancer emergency, quantum neurodyna mics, etc. are analyzed in the framework of Hierarchic theory. Computerized verification of our Hierarchic concept of matt er on examples of water and ice has been performed, using specia l computer program: Comprehensive Analyzer of Matter Proper ties (CAMP, copyright, 1997, Kaivarainen). The new opto-acoust ical de- vice (CAMP), based on this program, with possibilities much wider, than that of IR, Raman and Brillouin spectrometers, has been pro- posed (see URL: http://www.karelia.ru/˜alexk). It is a first theory enable to predict all known experimental t em- perature anomalies for water and ice. The conformity betwee n theory and experiment is very good even without fit parameters. 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arXiv:physics/0003046v1 [physics.chem-ph] 21 Mar 2000Ortho-Para Conversion in CH 3F. Self–Consistent Theoretical Model∗ Pavel L. Chapovsky† Institute of Automation and Electrometry, The Russian Acad emy of Sciences, 630090 Novosibirsk, Russia; and Laboratoire de Physique de s Lasers, Universit´ e des Sciences et Technologies de Lille, F-59655 Villeneuve d’Ascq Cedex, Fr ance (February 2, 2008) Abstract A complete theoretical model of the nuclear spin conversion in13CH3F in- duced by intramolecular ortho-para state mixing is propose d. The model contains parameters determined from the level-crossing sp ectra of the13CH3F spin conversion. This set of parameters includes the ortho- para decoherence rate, the magnitude of the hyperfine spin-spin interaction b etween the molec- ular nuclei and the energy gap between the mixed ortho and par a states. These parameters are found to be in a good agreement with thei r theoretical estimates. Typeset using REVT EX ∗Presented at the VI International Symposium on Magnetic Fie ld and Spin Effects in Chemistry and Related Phenomena, Emmetten, Switzerland, August 21-2 6, 1999. †E-mail: chapovsky@iae.nsk.su 1I. INTRODUCTION The study of nuclear spin isomers of molecules was started by the discovery of the ortho and para hydrogen in the late 1920’s [1]. It became clea r already in that time that many other symmetrical molecules should have nuclear spin i somers too. Nevertheless, their investigation has been postponed by almost 60 years. T he reason for this delay was severe difficulties in the enrichment of spin isomers. The sit uation is improving now (see the review in Ref. [2]) but yet we are at the very early stage of this research: in addition to the well-known spin isomers of H 2only a few molecules have been investigated so far. Among them, the CH 3F nuclear spin isomers occupy a special place being the most s tudied and understood. The conversion of CH 3F nuclear spin isomers has been explained [3,4] in the framew ork ofquantum relaxation [5]. which is based on the intramolecular ortho-para state m ixing and on the interruption of this mixing by collisions. This me chanism of spin conversion has a few striking features. The nuclear spin states of CH 3F appeared to be extremely stable surviving 109−1010collisions. Each of the collision changes the energy of the m olecule by 10 −100 cm−1and shuffles the molecular rotational state substantially. N evertheless, the model predicts that the spin conversion is governed by ti ny intramolecular interactions having the energy ∼10−6cm−1. Under these circumstances, the validity of the proposed the oretical model should be checked with great care. This is especially important becau se the CH 3F case gives us the first evidence of the new mechanism behind the nuclear spin co nversion in molecules. Hy- drogen spin conversion, which is the only other comprehensi vely studied case, is due to the completely different process based on direct collisional tr ansitions between ortho and para states of H 2. Presently there is substantial amount of the experimental d ata on CH 3F isomer conver- sion (see [2] and references therein). Theory and experimen t on the CH 3F isomer conversion were compared in a number of papers but these comparisons wer e never aimed to determine a complete set of parameters necessary for a quantitative description of the process. The purpose of the present paper is to construct such self-consi stent theoretical model of the CH3F isomer conversion. 2II. QUANTUM RELAXATION The CH 3F molecule is a symmetric top having the C 3vsymmetry. The total spin of the three hydrogen nuclei in the molecule can be equal to I= 3/2 (ortho isomers), or I= 1/2 (para isomers). The values of the molecular angular momentu m projection on the molecular symmetry axis ( K) are specific for these spin isomers. Only Kdivisible by 3 are allowed for the ortho isomers. All other Kare allowed for the para isomers. Consequently, the rotational states of CH 3F form two subspaces which are shown in Fig. 1 for the particul ar case of the13CH3F molecules. Let us briefly recall the physical picture of the CH 3F spin conversion by quantum re- laxation. Suppose that a test molecule was placed initially in the ortho subspace of the molecular states (Fig. 1). Due to collisions in the bulk the t est molecule will undergo fast rotational relaxation inside the ortho subspace. This running up and down along the ortho ladder proceeds until the molecule reaches the ortho state mwhich is mixed with the para statenby the intramolecular perturbation ˆV. Then, during the free flight just after this collision, the perturbation ˆVmixes the para state nwith the ortho state m. Consequently, the next collision is able to move the molecule to other para s tates and thus to localize it inside the para subspace. Such mechanism of spin isomer conv ersion was proposed in the theoretical paper [6]. The quantum relaxation of spin isomers can be quantitativel y described in the framework of the kinetic equation for density matrix [3]. Let us consid er first a free molecule which is not subjected to an external field. One needs to split the mole cular Hamiltonian into two parts ˆH=ˆH0+ ¯hˆV , (1) where the main part of the Hamiltonian, ˆH0, has pure ortho and para states as the eigen- states; the perturbation ˆVmixes the ortho and para states. In the first order perturbati on theory the nuclear spin conversion rate, γ, is given by γ=/summationdisplay α′∈p,α∈o2Γα′α|Vα′α|2 Γ2 α′α+ω2 α′α(Wp(α′) +Wo(α)), (2) 3where Γ α′αis the decay rate of the off-diagonal density matrix element ρα′α(α′∈para;α∈ ortho); ¯hωα′αis the energy gap between the states α′andα;Wp(α′) and Wo(α) are the Boltzmann factors of the corresponding states. The paramet ers Γ α′α,Vα′α, and ωα′αare crucial for the quantitative theoretical description of th e13CH3F spin isomer conversion. All previous comparisons between the experiment on the CH 3F spin conversion and the theory were performed using “total” rates of conversion which summarize all contributions to the rate from many ortho-para level pairs. The “total” rat e is just a single number and obviously cannot provide unambiguous determination of all parameters which are present in the expression (2). One may combine the experimental data on “total” rates with theoretical calculations of some parameters but it is not easy. In this ca se one has to perform extensive calculations of the intramolecular ortho-para state mixin g. Even more difficult is to calculate the decoherence rates Γ α′α. Consequently, development of the self-consistent model o f the nuclear spin conversion in which all parameters are unambig uously determined should be based on a different approach. III. LEVEL–CROSSING RESONANCES Theoretical model of spin conversion predicts strong depen dence of the conversion rate, γ, on the level spacing ωα′α(see Eq. (2)). This can be used to single out the contribution to the conversion from each level pair which should substant ially simplify the quantitative comparison between theory and experiment. It was proposed i n [7] and performed in [8] to use the Stark effect for crossing the ortho and para states of C H3F. These crossings result in sharp increase of the conversion rate γgiving the conversion spectra if electric field is varied. The experimental data [8] are presented in Fig. 2. It is evide nt that such spectrum contains much more information than the ”total” conversion rate whic h is just a single number. Comparison of the conversion spectrum in Fig. 2 with the theo ry needs a modification of the model in order to incorporate the Stark effect. Homogeneo us electric field lifts partially the degeneracy of the α-states of CH 3F (see Appendix). The new states, µ-basis, can be found in a standard way [9]: |µ >≡ |β, ξ > |σF>|σC>;ξ= 0,1. (3) 4Because electric field in the experiment is relatively small , it is sufficient to consider only diagonal matrix elements of the Stark perturbation over ang ular momentum Jwhen calcu- lating the µ-states. Energy of the µ-states are given by the expression E(µ) =Efree(J, K) + (−1)ξK|M| J(J+ 1)|dE|, (4) where Efree(J, K) is the energy of free molecule; dis the molecular permanent electric dipole moment; Eis the electric field strength. The new states are still degen erate with respect to the spin projections σ,σF, and σC, and to the sign of M. An account of the Stark effect in the spin conversion model is straightforward. Eq. (2) sho uld be rewritten in the µ-basis with the level energies determined by the Eq. (4). IV. FITTING OF THE EXPERIMENTAL DATA Nuclear spin conversion in13CH3F at zero electric field is governed almost completely by mixing of only two level pairs ( J′=11,K′=1)–( J=9,K=3) and (21,1)–(20,3) [4,10]. The spectrum presented in Fig. 2 is produced by crossings of the M-sublevels of the para (11,1) and ortho (9,3) states. This pair of states is mixed by the spi n-spin interaction between the molecular nuclei [4]. There is no contribution to the mix ing of this level pair from the spin-rotation interaction because of the selection rule fo r spin-rotation interaction |∆J| ≤1 [11]. This is fortunate because the spin-spin interaction c an be calculated rather accurately. Contrary to that, the spin-rotation interaction in CH 3F is known only approximately. For more details on spin-rotation contribution to the CH 3F spin conversion see Refs. [11–15] The second pair of ortho-para states (21,1)–(20,3), which i s also important for the spin conversion in13CH3F at zero electric field, is mixed by both the spin-spin and spi n-rotation interactions. The magnitude of the latter is presently unkn own. Nevertheless, it does not complicate the fitting procedure because in the vicinity of the (11,1)–(9,3) resonances presented in Fig. 2, the (21,1)–(20,3) pair gives very small and almost constant contribution. Let us find out now an analytical expression for modelling the experimental data. We start by analyzing the contribution to the conversion rate p roduced by the level pair (11,1)– (9,3) which will be denoted as γa(E). This contribution can be obtained using the results of Refs. [4,7]: 5γa(E) =/summationdisplay M′∈p;M∈o2Γ|VM′M|2 Γ2+ω2 M′M(E)(Wp(µ′) +Wo(µ)); |VM′M|2= (2J+ 1)(2 J′+ 1) J′J 2 −K′K K′−K 2 J′J 2 −M′M M′−M 2 T2. (5) HereVM′M≡< µ′|V|µ >are the matrix elements of the perturbation ˆVin which only M-indexes were shown explicitly; (:::) stands for the 3j-sym bol;Tis the magnitude of the spin-spin interaction. Note, that the selection rules for t he ortho-para state mixing by the spin-spin interaction result from Eq. (5): |∆K|;|∆J|;|∆M| ≤2. In the fitting procedure T will be considered as an adjustable parameter. In Eq. (5) we h ave assumed all Γ M′Mbeing equal: Γ M′M≡Γ. This property of Γ is the consequence of the spherical symm etry of the media. The decoherence decay rate Γ is an another unknown par ameter which needs to be determined. The spacing between the M′andMstates in an electric filed follows directly from the Eq. (4) ωM′M(E) =ω0+/parenleftiggK′|M′| J′(J′+ 1)−K|M| J(J+ 1)/parenrightigg |dE|, (6) where ω0is the gap between the states ( J′,K′) and ( J,K) at zero electric field. We have considered in Eq. (6) only pairs of states which have ξ′=ξ. They are the only pairs which contribute to the spectrum in the electric field range of Fig. 2. The level spacing ω0will be considered as an adjustable parameter in the fitting. The d ipole moment of13CH3F in the ground state, which is necessary for the calculation o fωM′M(E), was determined very accurately from the laser Stark spectroscopy of13CH3F and was found equal d= 1.8579 ±0.0006 D [16]. At zero electric field the level pair (21,1)–(20,3) contribu tes nearly 30% to the total conversion rate [10]. At electric fields, where γa(E) has peaks, this contribution is on the order of 10−2in comparison with γa(E). The first crossing of the pair (21,1)-(20,3) occurs at ≃4000 V/cm thus having its peaks far away from the electric fiel d range of Fig. 2. In the electric field range of Fig. 2 (1–1200 V/cm) the contribution from the pair (21,1)-(20,3) is changing by 10% only. Consequently, in the fitting procedure the (21,1)-(20,3) contribution is assumed to be constant. This quantity will be denoted as γb. 6To summarize, the function which will be used to model the exp erimental data is γ(E) =γa(E) +γb. (7) This function contains adjustable parameters T, Γ,ω0, and γb. The result of the least-square fit is shown in Fig. 2 by solid li ne. The error of the individual experimental points in Fig. 2 was estimated as 7% . The values of the parameters are given in the Table 1, where one standard deviation of stat istical error is indicated. Electric field in the Stark cell was determined in experiment [8] by measuring the voltage applied to the electrodes and assuming the distance between them equal to 4.18 mm, which is the spacer thickness. It was found out after the experimen t [8] was performed that the thickness of the glue used to attach the Stark electrodes was not negligible. The updated spacing between the electrodes in the Stark cell is l= 4.22±0.02 mm. Such correction of the spacing gives 1% systematic decrease of the experimenta l electric field values given in [8]. This shift is taken into account in Fig. 2. V. THEORETICAL ESTIMATION OF THE PARAMETERS Let us compare the parameters obtained in the previous secti on with their theoretical estimates. We start from the analysis of the level spacing ω0. The best sets of the ground state molecular parameters of13CH3F are given in Ref. [17]. The spacing between the levels (11,1) and (9,3) is presented in the Table 1 where the set havi ng most accurate molecular parameter A0was used. The theoretical value appears to be close to the exp erimental one obtained from the spin conversion spectra. The difference be tween them is ω0(exp)−ω0(theor) = 1.0±0.3 MHz , (8) which is less than 1% in comparison with ω0itself. Next, we calculate the parameter Twhich characterizes the spin-spin mixing of the level pair (11,1)–(9,3) in13CH3F. The spin-spin interaction between the two magnetic dipol esm1 andm2separated by the distance rhas the form [9]: ¯hˆV12=P12ˆI(1)ˆI(2)• •T(12), T(12) ij=δij−3ninj;P12=m1m2/r3I(1)I(2), (9) 7where ˆI(1)andˆI(2)are the spin operators of the particles 1 and 2, respectively ;nis the unit vector directed along r;iandjare the Cartesian indexes. For the spin-spin mixing of the ortho and para states in13CH3F one has to take into ac- count the interaction between the three hydrogen nuclei ( ˆVHH), between the three hydrogen and fluorine nuclei ( ˆVHF), and between the three hydrogen and carbon nuclei ( ˆHHC). Thus the total spin-spin interaction responsible for the mixing in13CH3F is ˆVSS=ˆVHH+ˆVHF+ˆVHC. (10) The complete expressions for all components of ˆVSScan be written by using Eq. (9) for the spin-spin interaction between two particles. For example, forˆVHFone has ˆVHF=PHF/summationdisplay nˆI(n)ˆIF• •TnF;n= 1,2,3. (11) HerePHFis the scaling factor analogous to P12in Eq. (9); nrefers to the hydrogen nuclei in the molecule. Tcan be calculated in a way similar to that used previously [4] . It gives |T |2= 3|PHHT(12) 2,2|2+ 2|PHFT1F 2,2|2+ 2|PHCT1C 2,2|2. (12) HereT1q 2,2is the spherical component of the second rank tensor T1qcalculated in the molecular system of coordinates. The superscripts 1 qindicate the interacting particles: 1 refers to the hydrogen nucleus H(1)andqrefers to the nucleus of H(2), or F, or C. The calculation of Tneeds the knowledge of the molecular structure. We used the g round state structure of13CH3F determined in [18,19]: rCF= 1.390(1) ˚A,rCH= 1.098(1) ˚A, and β(F−C−H) = 108 .7o(2). The numbers in parentheses represent the error bars in u nits of the last digit. By using these parameters one can obtain the v alue of Twhich is given in the Table 1. The difference between the experimental and theo retical values of Tis equal to Texp− Ttheor=−5.1±0.5 kHz . (13) 8VI. DISCUSSION Small difference between the experimental and theoretical v alues of ω0unambigiously confirms that the mixed ortho-para level pair (9,3)–(11,1) w as determined correctly. From the spectroscopical data [17] one can conclude that there ar e no other ortho-para level pairs which can mimic the level spacing ω0= 130 .99 MHz. It is also true if one takes into account even all ortho-para level pairs ignoring the restrictions i mposed by the selection rules for the ortho-para state mixing. The difference between experimental and theoretical values of the level spacing at zero electric field, ω0, is only 1 .0±0.3 MHz. The main error in theoretical value of ω0is caused by the error in the molecular parameter A0. It gives nearly half of the error indicated in the Table 1. On the other hand, the JandKdependences of the molecular electric dipole moment are too small [16] to affect our determination of the th eoretical value of ω0. It is possible that the experimental value of ω0is affected by the pressure shift, which magnitude we presently do not know. Further investigations can precis e the frequency gap between the states (9,3) and (11,1). The difference between the experimental and theoretical val ues of Tis rather small (≃7%) but well outside the statistical error. This difference m ay originate from our method of calculating T⊔/angb∇ack⌉⊔l⌉f⊔⌉≀∇in which we used the molecular structure (bond lengths and an gles) averaged over ground state molecular vibration. More corre ct procedure would be to av- erage an exact expression for Tover molecular vibration. This requires rather extensive calculations. There are a few contributions to the systematic error of valu e ofT⌉§√. The response time of the setup used to measure the concentration of ortho m olecules ( ≃1 sec) was not taken into account in the processing of the experimental dat a. This gives ≃2% systematic decrease in the value of T⌉§√. Another few percent of the systematic error may appear due to the procedure employed in [8] to find out the conversion rat e inside the Stark cell. This procedure relies on the ratio of the Stark cell volume to the v olume outside the electric field. Taking these circumstances into account we can estima te that up to ≃10% difference between the experimental and theoretical values of Tcan be explained by the systematic errors. Despite this difference, it is rather safe to conclud e that our analysis has proven 9that the levels (9,3) and (11,1) are indeed mixed by the spin- spin interaction between the molecular nuclei. It is impressive that the level-crossing spectrum in the13CH3F isomer conversion has allowed to measure the hyperfine spin-spin co upling with the statistical error of 0.5 kHz only. Comparison between the measured spectrum and the model supp orts our choice for the ΓM′Mbeing independent on MandM′. Independence of this parameter on Mis the direct consequence of the spatial isotropy of the media. The indepe ndence on M′−Mis more intricate. This will be discussed in more detail elsewhere. The value of Γ obtained from the fitting procedure, Γ = (1 .9±0.1)·108s−1/Torr, appeared to be close to the level population decay rate 1 .0·108s−1/Torr measured in Ref. [20] for the state ( J=5,K=3) of13CH3F. The factor 2 difference is not surprising. Γ refers to the decay rate of the off-diagonal density matrix element ρµ′µbetween the states (11,1) and (9,3) which should be different from the population decay rat e. In addition, the rotational quantum numbers in these two cases are different too. Column designated as γ(0) in the Table 1 gives the rates at zero electric filed. The “theoretical value” is the magnitude of γ(0) given by the solid line in Fig. 2. The theoretical value coincides well with the experimental one from Ref. [21 ]. Finally we would like to mention that our analysis of the spin conversion spectrum ha s allowed to disentangle for the first time the contributions to the conversion rate which ari se from the mixing of the two level pairs: (9,3)–(11,1) and (20,3)–(21,1). VII. CONCLUSIONS We have performed the first quantitative comparison of the le vel-crossing spectrum of the nuclear spin conversion in13CH3F with the theoretical model. This approach has allowed to single out the contribution to the spin conversion caused by the mixing of one particular pair of the ortho-para rotational states of the molecule and confirmed unambiguously that the mechanism of the intramolecular state mixing is the spin -spin interaction between the molecular nuclei. All important parameters of the theoretical model which des cribe the nuclear spin con- version in13CH3F due to the spin-spin mixing of the ortho-para level pair (9, 3)–(11,1) are 10determined quantitatively. These parameters are the decoh erence rate, Γ, the spin-spin mixing strength, T, the level spacing, ω0, and the contributions to the conversion rate from the two level pairs separately (9,3)-(11,1) and (20,3)-(21 ,1). While the decoherence rate Γ is difficult to estimate on the basis of independent informati on, the experimental values for the spin-spin mixing, T, and the level spacing, ω0, are found to be close to their theoretical values. These results prove that the nuclear spin conversio n in the13CH3F molecules is indeed governed by the quantum relaxation. ACKNOWLEDGMENTS This work was made possible by financial support from the the R ussian Foundation for Basic Research (RFBR), grant No. 98–03–33124a, and the R´ eg ion Nord Pas de Calais, France. VIII. APPENDIX The CH 3F quantum states in the ground electronic and vibration stat e can be classified as follows [22–24]. CH 3F is a rigid symmetric top but it is more transparant to take mo lecular inversion into account and classify the states in D 3hsymmetry group. First, one has to introduce an additional (molecular) system of coordinates which has the orientation defined by the numbered hydrogen nuclei and z–axis directed along the molecular symmetry axis. Next, one introduces the states |β >≡ |J, K, M > |I, σ, K > ;K≥0, (14) which are invariant under cyclic permutation of the three hy drogen nuclei: P123|β >=|β >. In Eq. (14), |J, K, M > are the familiar rotational states of symmetric top, which a re characterized by the angular momentum ( J), its projection ( K) on the z-axis of the molecular system of coordinates and the projection ( M) on the laboratory quantization axis Z.Iandσ are the total spin of the three hydrogen nuclei and its projec tion on the Z-axis, respectively. The explicit expression for the spin states |I, σ, K > is given in [22]. 11Permutation of any two hydrogen nuclei in CH 3F inverts z-axis of the molecular system of coordinates. Consequently, the action of such operation ( P23, for instance) on the molecular states reads: P23|β >=|β >, where β≡ {J,−K, M, I, σ }. Note that the complete set of the molecular states comprises both βandβsets. Using the states |β >and|β >one can construct the states which have the proper symmetry with respect to the permutation of any two hydrogen nuclei: |β, κ > =1√ 2[1 + (−1)κP23]|β >;κ= 0,1. (15) The action of the permutation of two hydrogen nuclei on the st ate|β, κ > is defined by the rule:P23|β, κ > = (−1)κ|β, κ > and by similar relations for the permutations of the other two pairs of hydrogen nuclei. In the next step, one has to take into account the symmetric ( |s= 1>) and antisymmet- ric (|s= 0>) inversion states. The action of the permutation of the two h ydrogen nuclei on these states, for example P23, reads P23|s= 0>=−|s= 0>;P23|s= 1>=|s= 1> . (16) Evidently, the cyclic permutation of the three hydrogen nuc lei of the molecule does not change the inversion states. The total spin-rotation states of CH 3F should be antisymmetric under permutation of any two hydrogen nuclei, because protons are fermions. Cons equently, the only allowed states of CH 3F are |β, κ=s >|s >. Finally, the description of the CH 3F states should be completed by adding the spin states of fluorine and carbon (13C) nuclei, both having spin equal 1/2: |α >=|β, κ=s >|s >|σF>|σC>, (17) where σFandσCare the Z-projections of the F and13C nuclei’ spins, respectively. In the following, we will denote the states (17) of a free molecu le asα-basis. For the rigid symmetric tops, as CH 3F is, the states |α >are degenerate over the quantum numbers s, M,σ,σF,σC. 12REFERENCES [1] A. Farkas, Orthohydrogen, Parahydrogen and Heavy Hydrogen (Cambridge University Press, London, 1935), p. 215. [2] P. L. Chapovsky and L. J. F. Hermans, Annu. Rev. Phys. Chem .50, 315 (1999). [3] P. L. Chapovsky, Zh. Eksp. Teor. Fiz. 97, 1585 (1990), [Sov. Phys. JETP. 70, 895 (1990)]. [4] P. L. Chapovsky, Phys. Rev. A 43, 3624 (1991). [5] P. L. Chapovsky, Physica A (Amsterdam) 233, 441 (1996). [6] R. F. Curl, Jr., J. V. V. Kasper, and K. S. Pitzer, J. Chem. P hys.46, 3220 (1967). [7] B. Nagels, M. Schuurman, P. L. Chapovsky, and L. J. F. Herm ans, Chem. Phys. Lett. 242, 48 (1995). [8] B. Nagels, N. Calas, D. A. Roozemond, L. J. F. Hermans, and P. L. Chapovsky, Phys. Rev. Lett. 77, 4732 (1996). [9] L. D. Landau and E. M. Lifshitz, Quantum Mechanics, 3rd ed. (Pergamon Press, Oxford, 1977), p. 673. [10] P. L. Chapovsky, D. Papouˇ sek, and J. Demaison, Chem. Ph ys. Lett. 209, 305 (1993). [11] K. I. Gus’kov, Zh. Eksp. Teor. Fiz. 107, 704 (1995), [JETP. 80, 400-414 (1995)]. [12] P. L. Chapovsky, in 12th Symposium and School on High-Resolution Molecular Spe c- troscopy , edited by L. N. Sinitsa, Y. N. Ponomarev, and V. I. Perevalov (SPIE, Wash- ington, 1997), Vol. 3090, pp. 2–12. [13] K. Bahloul, M. Irac-Astaud, E. Ilisca, and P. L. Chapovs ky, J. Phys. B: At. Mol. Opt. Phys.31, 73 (1998). [14] E. Ilisca and K. Bahloul, Phys. Rev. A 57, 4296 (1998). [15] K. I. Gis’kov, J. Phys. B: At. Mol. Opt. Phys. 32, 2963 (1999). 13[16] S. M. Freund, G. Duxbury, M. Romheld, J. T. Tiedje, and T. Oka, J. Mol. Spectr. 52, 38 (1974). [17] D. Papouˇ sek, J. Demaison, G. Wlodarczak, P. Pracna, S. Klee, and M. Winnewisser, J. Mol. Spectr. 164, 351 (1994). [18] T. Egawa, S. Yamamoto, M. Nakata, and K. Kuchitsu, J. Mol . Structure 156, 213 (1987). [19] T. Egawa, private communication, 1998. [20] H. Jetter, E. F. Pearson, C. L. Norris, J. C. McGurk, and W . H. Flyger, J. Chem. Phys. 59, 1796 (1973). [21] B. Nagels, M. Schuurman, P. L. Chapovsky, and L. J. F. Her mans, Phys. Rev. A 54, 2050 (1996). [22] C. H. Townes and A. L. Shawlow, Microwave Spectroscopy (McGraw-Hill Publ. Comp., New York, 1955), p. 698. [23] P. R. Bunker, Molecular symmetry and spectroscopy (Academic Press, New York, San Francisco, London, 1979). [24] J. Cos´ eou, F. Herlemont, M. Khelkhal, J. Legrand, and P . L. Chapovsky, The Eur. Phys. J. D (1999), accepted for publication. 14Table 1. Experimental and theoretical parameters of the nuc lear spin conversion in 13CH3F by quantum relaxation. ω0/2π T Γ γ(0) γb (MHz) (kHz) (108s−1/Torr) (10−3s−1/Torr) (10−3s−1/Torr) Experiment 132.06 ±0.27 64.1±0.5 1.9±0.1 12.2±0.6(4)4.6±0.7 Theory 130.99±0.15(1)69.2±0.2(2)1.0(3)12.04±0.5(5)– Difference 1.0±0.3 -5.1±0.5 – 0.15±0.8 – (1)Calculated using the molecular parameters from Ref. [17], ( Table 1, column 2). (2)Calculated using the molecular structure determined in Ref . [18,19]. (3)The level population decay rate from Ref. [20]. (4)Experimental value from Ref. [21] (5)Zero-field value predicted by the theoretical curve in Fig. 2 . 15FIGURES 050100150200 para orthon mVmnLevel energy (cm -1) FIG. 1. Ortho and para states of13CH3F in the ground vibrational state. The level energies were calculated using the molecular parameters from [17]. T he (11,1)–(9,3) pair of states is shown to be mixed by the intramolecular perturbation. The bent lin es indicate the rotational relaxation induced by gas collisions. These collisions do not provide d irect ortho-para transitions. 16/G01 /G02/G01/G01 /G03/G01/G01 /G04/G01/G01 /G05/G01/G01 /G06/G01/G01/G01 /G06/G02/G01/G01/G01/G07/G01/G01/G01/G07/G01/G02/G01/G07/G01/G03/G01/G07/G01/G04 /G08/G09/G0A/G0B/G07 /G08/G0C/G0D/G0E/G0F/G10/G11Conversion rate (1/s) Electric field (V/cm) FIG. 2. Experimental [8] and theoretical ortho–para conversion sp ectrum in13CH3F. 17

arXiv:physics/0003048v1 [physics.med-ph] 21 Mar 2000On the uniqueness of the surface sources of evoked potential s Alejandro Cabo∗, Carlos Handy and Daniel Bessis. Center of Theoretical Studies of Physical Systems Clark Atlanta University ,Atlanta U.S.A. Abstract. The uniqueness of a surface density of sources localized ins ide a spatial region Rand producing a given electric potential distribution in its bo undary B0is revisited. The situation in which Ris filled with various metallic subregions, each one having a definite constant value for the electric conductivity is considered . It is argued that the knowledge of the potential in all B0fully determines the surface density of sources over a wide c lass of surfaces supporting them. The class can be defined as a union o f an arbitrary but finite number of open or closed surfaces. The only restriction upon them is that no one of the closed surfaces contains inside it another (nesting) of the closed or open surfaces. * On leave in absence from the: Group of Theoretical Physics, Instituto de Cibern´ etica, Matem´ atica y F ´isica, Calle E, No. 309 Vedado, La Habana, Cuba. email: cabo@cidet .icmf.inf.cu 1I. INTRODUCTION. The uniqueness problem for the sources of the evoked potenti al in the brain is a relevant research question due to its role in the development of cereb ral electric tomography [1], [2] , [3], [4] . Since long time ago, it is known that the genera l inverse problem of the determination of volumetric sources from the measurement o f the potential at a surface is not solvable in general [5], [6]. However, under additional assumptions about the nature of the sources, solutions can be obtained [7], [8], [9]. The s upplementary assumptions can be classified in two groups: the physically grounded ones, wh ich are fixed by the nature of the physical problem and the ones which are imposed by invo king their mathematical property of determining a solution, but having in another ha nd, a weak physical foundation. The resumed situation implies that the determination of phy sical conditions implying the uniqueness of the sources for the evoked potentials remains being an important subject of study. Results in this direction could avoid the imposition of artificial conditions altering the real information on the sources to be measured. The question to be considered in this work is the uniqueness o f the sources for evoked potentials under the assumption that these sources are loca lized over surfaces. This issue was also treated in Ref. [1] by including also some specially defined volumetric sources. The concrete aim here is to present a derivation of the results en unciated in [1] for the case of open surfaces and to generalize it for a wider set of surfaces including closed ones. We consider that the results enunciated in Ref. [1] are valid and useful ones. Even more, we think that a relevant merit of that paper is to call for the a ttention to the possibility for the uniqueness for classes of surface density of sources . Specifically, in our view, the conclusion stated there about the uniqueness of the sources of evoked potentials as restricted to sources distributed in open surfaces is effectively valid . In the present work, the central aim is to extend the result for a wider set of surfaces includi ng closed ones by also furnishing an alternative way to derive the uniqueness result. The uniq ueness problem for the special class of volumetric sources discussed in [1] is not consider ed here in any way. 2The physical system under consideration is conformed by var ious volumetric regions, each of them having a constant value of the conductivity, sep arated by surface boundaries at which the continuity equations for the electric current i s obeyed. It should pointed out that the special volumetric sources examined in Ref. [1] are not addressed here. The precise definition of the generators under examination is the follow ing. The sources are assumed to be defined by continuous and smooth surface densities lyin g over a arbitrary but finite number of smooth open or closed surfaces. The unique constra int to be imposed on these surfaces is that there is no nesting among them. That is, ther e is no closed surface at which interior another open or closed of the surfaces resides. Thi s class of supports expands the one considered in Ref. [1] and in our view is sufficiently general t o create the expectative for the practical applications of the results. It should be stresse d that the boundaries between the interior metallic regions are not restricted by the ”non-ne sting” condition. That is, the fact that the skull and the few boundaries between cerebral tissu es can be visualized as nearly closed surface does not pose any limitation on the conclusio n. The ”non-nesting” condition should be valid only for the surfaces in which the sources can be expected to reside. For example, if by any mean we are sure that the sources stay at the cortex surface, then the uniqueness result apply whenever the portion of the cortex i mplied does not contains any closed surface. The paper is organized as follows. An auxiliary property is d erived in the form of a theorem in the Section II. In Section III the proof of uniquen ess for the kind of sources defined above is presented. II. GREEN THEOREM AND FIELD VANISHING CONDITIONS Let us consider the potential φgenerated by a source distribution concentrated in the ”non-nested” set of open or closed surfaces defined in last Se ction, which at the same time are contained within a compact and simply connected spatial reg ionR.The set R,as explained before, is formed by various connected subregions Ri, i= 0,1, ...neach of them filled with 3a metal having a constant conductivity σi. Also, let Bijthe possibly but non necessarily existing, boundary between the subregions RiandRjandB0the boundary of R.For the sake of a physical picture, we can interpret B0as the surface of the skull, Ras the interior of the head and the subregions Rias the ones containing the various tissues within the brain. It is defined that the exterior space of the head corres ponds to R0. In addition, let Si, i= 1, ...m the surfaces pertaining to the arbitrary but finite set Sof non-nested open or closed surfaces in which the sources are assumed to be loca lized. The above mentioned definitions are illustrated in Fig.1. Then, the Poisson equation satisfied by the potential φin the interior region of Rcan be written as ∇2φ(− →x) =g(− →x) σ(− →x), (1) g(− →x) =−− →∇.− →J(− →x), (2) where− →Jare the impressed currents (for example, generated by the ne uron firings within the brain) and the space dependent conductivity is defined by σ(− →x) =σifor− →x∈Ri. (3) It should be noticed that the conductivities are different fr om zero only for the internal regions to R.The vacuum outside is assumed to have zero conductivity and t he field satis- fying the Laplace equation. In addition outside the support of the sources where g= 0 the Laplace equation is also satisfied. The usual boundary conditions within the static approximat ion, associated to the con- tinuity of the electric current at the boundaries, take the f orm σi∂φ ∂ni|x∈Bij=σj∂φ ∂nj|x∈Bij, (4) where ∂nisymbolizes the directional derivative along a line normal t oBijbut taken in the limit of x−> B ijfrom the side of the region Ri. A main property is employed in this work in obtaining the clai med result. In the form of a theorem for a more precise statement it is expressed as 4Theorem. Letφis a solution of the Laplace equation within an open and conne cted spatial region R∗. Assume that ϕhave a vanishing electric field over an open section of certai n smooth surface S∗which is contained in an open subset QofR∗. Let the points of the boundaries between QandR∗have a minimal but finite distance among them. Then, the poten tialφ is a constant over any open set contained in R∗. As a first stage in the derivation of this property, let us writ e the Green Theorem as applied to the interior of the open region Qdefined in the Theorem 1 in which a field ϕsatisfies the Laplace equation. Then, the Green Theorem expr essesϕevaluated at a particular interior point− →xin terms of itself and its derivatives at the boundary BQas follows. ϕ(− →x) =/integraldisplay BQd− → s′. 1/vextendsingle/vextendsingle/vextendsingle/vextendsingle− →x−− → x′/vextendsingle/vextendsingle/vextendsingle/vextendsingle− →∇x′ϕ/parenleftbigg− → x′/parenrightbigg −− →∇x′ 1/vextendsingle/vextendsingle/vextendsingle/vextendsingle− →x−− → x′/vextendsingle/vextendsingle/vextendsingle/vextendsingle ϕ/parenleftbigg− → x′/parenrightbigg  (5) where the integral is running over the boundary surface BQwhich is described by the coordi- nates− → x′.This relation expresses the potential as a sum of surface int egrals of the continuous and bounded values of ϕand its derivatives. Those quantities are in addition analy tical in all the components of− →x ,if the point have a finite minimal distance to the points in BQ. These properties follow because Q⊂R∗and then, ϕsatisfies the Laplace equation in any open set in which Qand its boundary is included. But, due to the finite distance c ondition among the point− →xand the points of BQ, the expression (5) for ϕshould be an analytical function of all the coordinates of− →x .Figure 2 depicts the main elements in the formulation of the Green Theorem. Further, let us consider that S∗is siting inside the region Q.Then, as this surface is an equipotential and also the electric field over it vanishes, i t follows that no line of force can have a common point with it. This is so because the divergence of the electric field vanishes, then it is clear that the existence of nonvanishing value of t he electric field at another point of the line of force will then contradicts the assumed vanish ing of the divergence. Therefore, the lines of forces in any sufficiently small open neighborhoo d containing a section of S∗ 5should tend to be parallel to this surface on approaching it, or on another hand, the electric field should vanish. Next, it can be shown that in such neighbo rhoods the lines of forces can not tend to be parallel. Let us suppose that lines of forces exist and tend to be tangen t to the surface S∗and consider the integral form of the irrotational property of t he electric field as /contintegraldisplay C− →E . d− →l=/integraldisplay C1− →E . d− →l+/integraldisplay C2− →E . d− →l=/integraldisplay C1− →E . d− →l= 0 (6) where the closed curve Cis constructed as follows: the piece C1coincides with a line of force, the piece C2is fixed to rest within the surface S∗and the other two pieces necessary to close the curve are selected as being normal to the assumed existin g family of lines of forces. The definitions are illustrated in Fig. 3. By construction, the e lectric field is colinear with the tangent vector to C1and let us assume that we select the segment of curve C1for to have a sufficiently short but finite length in order that the cosine as sociated to the scalar product will have a definite sign in all C1. This is always possible because the field determined by (5) should be continuous. Then Eq. (6) implies that the electric field vanish along all C1as a consequence of the integrand having a definite sign and then s hould vanish identically. Since this property is valid for any curve pertaining to a sufficient ly small open interval containing any particular open section of S∗,it follows that in certain open set containing S∗there will be are no lines of forces, or what is the same, the electric fiel d vanish. To finish the proof of the theorem, it follows to show that if ϕand the electric field vanish within a certain open neighborhood N,included in an arbitrary open set Opertaining to the region R∗in which the Laplace equation is obeyed, then ϕand the electric field vanish in all O. Consider first that Qis an open set such that O⊂Qand also suppose that the smallest distance form a point in Oto the boundary BQofQhas the finite value δ. Then, the Green Theorem (5) as applied to the region Qexpresses that the minimal radius of convergence of ϕconsidered as analytical function of any of the coordinates is equal or greater than δ. Imagine now a curve Cstarting in an interior point PofNand ending at any point P1of O.Assume that Cis formed by straight lines pieces (See Fig. 4). It is then pos sible to define 6ϕas a function of the length of arc sofCas measured form the point P. It should be also valid that in any open segment of C,not including the intersection point of the straight lines, the potential ϕis an analytical function of s.Furthermore, let consider Cas partitioned in a finite number of segments of length σ < δ. Suppose also, that the intersection points of the straight lines are the borders of some of the segments. It can be noticed that ϕvanishes in any segment of Cstarting within Nbecause it vanishes in Nexactly. Thus, if ϕand the electric field are not vanishing along all C,there should be a point over the curve in which the both quantities do not vanish for an open region sat isfying s > s o,and vanish exactly for another open interval obeying s < s o. However, in this case, all the derivatives ofϕof the electric field over svanish at so.This property in addition with the fact that the Taylor series around soshould have a finite radius of convergence r > δ, as it assumed in the Theorem 1, leads to the fact that ϕand the electric field should vanish also for s > s o. Henceforth, the conclusion of the Theorem 1 follows: the pot ential ϕand its corresponding electric field vanish at any interior point of R∗. III. UNIQUENESS OF THE NON-NESTING SURFACE SOURCES Let us argue now the uniqueness of the sources which are define d over a set of non nested surfaces Sproducing specific values of the evoked potential φat the boundary B0of the region R.For this purpose it will be assumed that two different source d istributions produce the same evoked potential over B0.The electrostatic fields in all space associated to those sources should be different as functions defined in all space. They will be called φ1andφ2. As usual in the treatment of uniqueness problems in the linea r Laplace equation, consider the new solution defined by the difference φ=φ1−φ2.Clearly ϕcorresponds to sources given by the difference of the ones associated to φ1andφ2.It is also evident that φhas vanishing values at B0.Then, since the sources are localized at the interior of Randφsatisfies the Laplace equation with zero boundary condition at B0and at the infinity, it follows that the field vanishes in all R0,that is, in the free space outside the head. Therefore, it fol lows 7that the potential and the electric field vanish in all B0when approaching this boundary from the free space ( R0).The continuity of the potential, the boundary conditions (3 ) and the irrotational character of the electric field allows to co nclude that φand the electric field also vanish at any point of B0but now when approaching it from any interior subregion Rihaving a boundary Bi0with the free space. Moreover, if the boundary surface of any of these regions which are in contact with the boundary of Ris assumed to be smooth, then it follows from Theorem 1 that the potential φand its the electric field vanish in all the open subsets of Riwhich points are connected through its boundary Bi0with free space by curves non-touching the surfaces of S. It is clear that this result hold for all the open subsets of theseRiin which Laplace equation is satisfied excluding those which are also residing inside one of the closed surfaces Siin the set S. It is useful for the following reasoning to remark that if we h ave any boundary Bij between to regions RiandRj,and the potential φand the electric field vanish in certain open (in the sense of the surface) and smooth regions of it, th en Theorem 1 implies that the potential and the electric field also vanish in all the open su bsets of RiandRjwhich are outside any of the closed surfaces in S.Since the sources stay at the surfaces in Sthe field φ in some open region of Rincluded inside certain of the closed surfaces Siwill not necessarily satisfy the Laplace equation in any interior point of Rand Theorem 1 is not applicable. Let us consider in what follows a point Pincluded in a definite open vicinity of a subregion Ri.Suppose also that Pis outside any of the closed surfaces in S. Imagine a curve Cwhich joinPwith the free space and does not touch any of the surfaces in S. It is clear that, if appropriately defined, Cshould intersect a finite number of boundaries Bijincluding always a certain one Bj0with free space. Let us also assume that Cis adjusted in a way that in each boundary it crosses, the intersection point is contained in a smooth and open vicinity (in the sense of the surface) of the boundary (See Fig. 1 and 5). Th en, it also follows that the curve Ccan be included in open set OChaving no intersection with the non-nested surfaces inS.This is so because the region excluding the interior of the cl osed surfaces in Sis also connected if the Siare disjoint . But, from Theorem 1 it follows that φand the electric field 8must vanish in all OC.This should be the outcome because the successive applicati on of the Theorem 1 to the boundaries intersected by the curve Cpermits to recursively imply the vanishing of φand the electric field in each of the intersections of OCwith the subregions Ri through which Cpasses. The first step in the recursion can be selected as the i ntersection ofCwithBj0at a point which by assumption is contained in an open neighbo rhood of the boundary Bj0. As the electric field and φvanish at free space, the fields in the first of the considered intersection of Ocshould vanish. This fact permits to define another open and smooth neighborhood of the next boundary intersected by Cin which the field vanish and so on up to the arrival to the intersection with the boundary o f the region Ricontaining the ending of Cat the original point P. Therefore, the electric field and the potential should vanish at an arbitrary point PofRwith only two restrictions: 1) Pto be contained in an open neighborhood of some Riand 2) Pto reside outside any of the surfaces in S.Thus, it is concluded that the difference solution φand its corresponding electric field, in all the space outside the region containing the sources vanish. Hencefor th, it implies that the difference between the two source distributions also should be zero ove r any of the open surface in the setS.This is necessary because the flux going out from any small pie ce of the considered surface is zero, which means that the assumed continuous den sity of surface sources exactly vanish. This completes the proof of the conclusion of Ref. [1 ] in connection with sources supported by open surfaces. It only rests to show that the sou rces are also null over the closed Si. Before continuing with the proof, it is illustrative to exem plify from a physical point of view how the presence of nested surfaces among the Sidestroys the uniqueness. For this aim let us let us consider that a closed surface Sihas another open or closed of the surface Sj properly contained inside it. That means that an open set con taining Sjis contained inside Si.Imagine also that Siis interpreted as the surface of a metal shell connected to th e ground; that is, to a zero potential and that the surface Sjis the support of an arbitrary density of sources. As it is known from electrostatics theory, the char ge density of a metal connected to the ground is always capable to create a surface density of ch arge at Sisuch that it exactly 9cancels the electric field and the potential at the outside of Si,in spite of the high degree of arbitrariness of the charge densities at the interior. Th at is, for nested surfaces in S, it is not possible to conclude the uniqueness, because at the in terior of a nesting surface, and distributed over the nested ones, arbitrary source distrib utions can exist which determine exactly the same evoked potential at the outside boundary B0. Let us finally show that if no nesting exists the uniqueness al so follows. Consider any of the closed surfaces, let say Si.As argued before φand the electric field vanish at any exterior point of Sipertaining to certain open set containing Si.Then, the field created by the difference between the sources associated to the two diffe rent solutions assumed to exist should be different from zero only at the interior region. Tha t zone, in the most general situation can be filled by a finite number of metallic bodies wi th different but constant con- ductivities. The necessary vanishing of the interior field f ollows from the exact conservation of the lines of forces for the ohmic electric current as expre ssed in integral form by /integraldisplay d− →s . σ(− →x)− →E(− →x) = 0. (7) Let us consider a surface Tdefined by the all the lines of forces of the current vector passing through an arbitrarily small circumference cwhich sits on a plane being orthogonal to a particular line of force passing through its center. Let the center be a point at the surface Si. Because, the above defined construction, all the flux of the c urrent passing trough the piece of surface of Si(which we will refer as p) intersected by Tis exactly equal to the flux through any intersection of Twith another surface determining in conjunction withpa closed region. By selecting a sufficiently small radius for t he circumference cit can be noticed that the sign of the electric field component along the unit tangent vector to the central line of forces should be fixed. This is so because on th e other hand there will be an accumulation of charge in some closed surface. Now, let us co nsider the fact that the electric field is irrotational and examine a line of force of the curren t density which must start at the surface Si.It should end also at Si, because in another hand the current density will not be divergence less. After using the irrotational condition for the electric field in the form 10/contintegraldisplay C− →E . d− →l=/integraldisplay C1− →E . d− →l+/integraldisplay C2− →E . d− →l=/integraldisplay C1− →E . d− →l= 0 (8) in which C1is the line of force starting and ending at SiandC2is a curve joining the mentioned points at Sibut with all its points lying outside Siwhere φ=φ1-φ2and the electric field vanish. Let us notice that the electric field an d the current have always the same direction and sense as vectors, because the electric co nductivity is a positive scalar. In addition, as it is argued above, the current can not revers e the sign of its component along the tangent vector of line of forces. Therefore, it fol lows that also the electric field can‘t revert the sign of its component along a line of force. T hus, the integrand of the line integral over the C1curve should have a definite sign at all the points, hence impl ying that φand the electric field should vanish exactly in all C1.Resuming, it follows that the electric field vanish also at the interior of any of the closed surfaces Si.Therefore, the conclusion arises that the difference solution φ=φ1-φ2= 0 in all the space, thus showing that the evoked potential at B0uniquely fixes the sources when they have their support in a se t of non nesting surfaces S. Acknowledgments We would like to thank the helpful discussions with Drs. Augu sto Gonz´ alez , Jorge Riera and Pedro Vald´ es. One of the authors ( A.C.) also would like a cknowledge the support for the development of this work given by the Christopher Reynolds F oundation (New York,U.S.A.) and the Center of Theoretical Studies of Physical Systems of the Clark Atlanta University (Atlanta, U.S.A). The support of the Associateship Program me of the Abdus Salam Inter- national Centre for Theoretical Physics (Trieste Italy) is also greatly acknowledged. 11Figure Captions Fig.1 . An illustration of a simply connected region Rconstituted in this case by only two simply connected subregions R1andR2having a boundary B12.The boundary with free space is denoted by B0.The set of non-nesting surfaces Shave four elements Si,i= 1, ..4. two of them open and other two closed ones. A piece wise straig ht curve Cjoining any interior point PofRand a point Oin the free space is also shown. Fig.2. Picture representing the region Qin which a field ϕsatisfies the Laplace equation and its value at the point− →xis given by the Green integral (5). Fig.3. The contour employed in the line integral in Eq. (6). Fig.4. Picture of the region Riand the open neighborhood Nin which the field ϕvanish exactly . A piece wise straight line curve Cjoining a point P∈Nand certain point P1in Riis also shown. Fig.5. Scheme of the curve Cand the open region OCcontaining it. 12REFERENCES [1] A. Amir, IEEE Trans. Biomed. Eng. 41 (1994)1. [2] J.J. Riera, Physical Basis of Electrical Cerebral Tomography (in Spani sh) PhD. Dissertation, Cuban Center for Neuro Sciences, Havana , 1999. [3] P. Nunez, Electric Fields of the Brain, Oxford University Press, New York 1981. [4] C. J. Aine, Critical Review in Neurobiology 9 (1995) 229. [5] H. Hemholtz, Ann. Phys. Chem.(ser 3) 29 (1853)211 and 353 . [6] J. Hadamard, Lecture on the Cauchy Problem in linear partial differential equations, Yale University Press, New Haven, 1923. [7] M. Scherg and D. Von Cramon, Neurophysiology 62 (1985)44 . [8] G. Wahba, Spline Models for Observational Data, SIAM, Phyladelfia, 1990. [9] J.J. Riera, E. Aubert,, P. Vald´ es, R. Casanova and O. Lin s, Advances in Biomagnetism Research 96, Springer Verlag, New York, 1996. 13

arXiv:physics/0003049v1 [physics.atom-ph] 22 Mar 2000Ultrastable CO2Laser Trapping of Lithium Fermions K. M. O’Hara, S. R. Granade, M. E. Gehm, T. A. Savard, S. Bali, C. Freed†, and J. E. Thomas Physics Department, Duke University, Durham, North Caroli na 27708-0305 (March 8, 1999) We demonstrate an ultrastable CO 2laser trap that provides tight confinement of neutral atoms w ith negligible optical scattering and minimal laser noise indu ced heating. Using this method, fermionic 6Li atoms are stored in a 0.4 mK deep well with a 1/e trap lifetim e of 300 seconds, consistent with a background pressure of 10−11Torr. To our knowledge, this is the longest storage time ever achieved with an all-optical trap, comparable to the best reported ma gnetic traps. PACS numbers: 32.80.Pj Copyright 1999 by the American Physical Society Off-resonance optical traps have been explored for many years as an attractive means of tightly confin- ing neutral atoms [1]. Far off resonance optical traps (FORTs) employ large detunings from resonance to achieve low optical heating rates and high density, as well as to enable trapping of multiple atomic spin states in nearly identical potentials [2–6]. For CO 2laser traps [7], the extremely large detuning from resonance and the very low optical frequency lead to optical scattering rates that are measured in photons per atom per hour. Hence, op- tical heating is negligible. Such traps are potentially im- portant for development of new standards and sensors based on spectroscopic methods, for precision measure- ments such as determination of electric dipole moments in atoms [8], and for fundamental studies of cold, weakly interacting atomic or molecular vapors. However, all-optical atom traps have suffered from unexplained heating mechanisms that limit the mini- mum attainable temperatures and the maximum storage times in an ultrahigh vacuum [4,9,10]. Recently, we have shown that to achieve long storage times in all-optical traps that are not limited by optical heating rates, heat- ing arising from laser intensity noise and beam point- ing noise must be stringently controlled [11,12]. Prop- erly designed CO 2lasers are powerful and extremely sta- ble in both frequency and intensity [13,14], resulting in laser-noise-induced heating times that are measured in hours. Hence, in an ultrahigh vacuum (UHV) environ- ment, where loss and heating arising from background gas collisions are minimized [15,16], extremely long stor- age times should be obtainable using ultrastable CO 2 laser traps. In this Letter, we report storage of6Li fermions in an ultrastable CO 2laser trap. Trap 1/e lifetimes of 300 sec- onds are obtained, consistent with a background pressure of 10−11Torr. This constitutes the first experimental proof of principle that extremely long storage times can be achieved in all-optical traps. Since arbitrary hyperfine states can be trapped, this system will enable exploration of s-wave scattering in a weakly interacting fermi gas. The well-depth for a focused CO 2laser trap is deter- mined by the induced dipole potential U=−αg¯E2/2, where αgis, to a good approximation, the ground statestatic polarizability [7], and¯E2is the time average of the square of the laser field. In terms of the maximum laser intensity Ifor the gaussian CO 2laser beam, the ground state well-depth U0in Hz is U0 h(Hz) = −2π hcαgI. (1) In our experiments, a laser power of P=40 W typically is obtained in the trap region. A lens is used to focus the trap beam to field a 1/e radius of af= 50µm, yielding an intensity of I= 2P/(πa2 f)≃1.0 MW /cm2. For the I-P(20) line with λCO2≃10.6µm, the Rayleigh length isz0=πa2 f/λCO2= 0.74 mm. Using the Li ground state polarizability of αg= 24.3×10−24cm3[17] yields a well depth of U0/h=−8 MHz, which is approximately 0.4 mK. For this tight trap, the6Li radial oscillation frequency is 4.7 kHz and the axial frequency is 0.22 kHz. For6Li in a CO 2laser trap, both the excited and the ground states are attracted to the well. The excited state static polarizability is αp= 18.9×10−24cm3[17], only 20% less than that of the ground state. With a ground state well depth of 8 MHz, the frequency of the first res- onance transition in the trap is shifted by only 1.6 MHz at the center of the trap and thus does not significantly alter the operation of the magneto-optical trap (MOT) from which the trap is loaded. The optical scattering rate Rsin the CO 2laser trap arises from Larmor scattering [7] and can be written as Rs=σSI/(¯hck), where the Larmor scattering cross sec- tionσSis σS=8π 3α2 gk4. (2) Here, k= 2π/λCO2. Using αg= 24.3×10−24cm3yields σS= 5.9×10−30cm2. At 1.0 MW/cm2, the scattering rate for lithium is then 2 .9×10−4/sec, corresponding to a scattering time of ≃3400 sec for one photon per atom. As a result, the recoil heating rate is negligible. Heating can arise from laser intensity noise and beam pointing fluctuations [11,12]. For simplicity, we estimate the noise-induced heating rates for our trap using a har- monic oscillator approximation which is valid for atoms 1near the bottom of the well. This provides only a rough estimate of the expected heating rates in the gaussian well, since the trap oscillation frequency decreases as the energy approaches the top of the well. A detailed discus- sion of noise-induced heating in gaussian potential wells will be given in a future publication. In the harmonic os- cillator approximation, intensity noise causes parametri c heating and an exponential increase in the average en- ergy for each direction of oscillation, /angbracketleft˙E/angbracketright= Γ/angbracketleftE/angbracketright, where the rate constant in sec−1is Γ =π2ν2SI(2ν). (3) Here νis a trap oscillation frequency and SI(2ν) is the power spectrum of the fractional intensity noise in fraction2/Hz. For our CO 2laser, SI(9.4 kHz) ≤1.0× 10−13/Hz, where it is comparable to the detector noise. This is three orders of magnitude lower than that mea- sured for an argon ion laser [11]. The corresponding heat- ing time for radial oscillation in our trap at ν= 4.7 kHz is Γ−1≥4.6×104sec. For the axial oscillation, ν= 220 Hz,SI(440Hz)≃1.1×10−11/Hz and Γ−1≃2×105sec. Fluctuations in the position of the trapping laser beam cause a constant heating rate /angbracketleft˙E/angbracketright=˙Q, where ˙Q= 4π4Mν4Sx(ν). (4) HereMis the atom mass and Sxis the position noise power spectrum in cm2/Hz at the trap focus. For6Li, one obtains ˙Q(nK/s) = 2 .8×10−4ν4(Hz)Sx(µm2/Hz). Position noise only couples directly to the radial motion where ν≃4.7 kHz. For our laser, Sx(4.7 kHz) ≤3.4× 10−10µm2/Hz, where the upper bound is determined by the noise floor for our detection method. This yields ˙Q≤ 46 nK/s. Hence, we expect the trap lifetime to be limited by the background pressure of our UHV system. The expected number of trapped atoms NTcan be estimated as follows. We take the trapping potential to be approximately gaussian in three dimensions: U(/vector x) =−U0exp(−x2/a2−y2/b2−z2/z2 o), (5) where a=b=af/√ 2 is the intensity 1/e radius and zois the Rayleigh length. Here, the lorentzian depen- dence of the trap beam intensity on the axial position z is approximated by a gaussian dependence on z. We assume that after a sufficient loading time, atoms in the CO 2laser trap will come into thermal and dif- fusive equilibrium with the MOT atoms that serve as a reservoir [18]. The density of states in the gaussian trap and the occupation number then determine the number of trapped atoms, which takes the form NT=n VFORT F[U0/(kBT)]. (6) Here the volume of the CO 2laser trap is defined as VFORT =a2zoπ3/2. Hence, n VFORT is the total number of atoms contained in the volume of the FORT at the MOT density n.F(q) determines the number of trapped atoms com- pared to the total number contained in the FORT volume at the MOT density. It is a function only of the ratio of the well depth to the MOT temperature, q≡U0/(kBT): F(q) =q3/2 2/integraldisplay1 0dxx2g(x) exp[ q(1−x)]. (7) Hereg(x) is the ratio of the density of states for a gaus- sian well to that of a three dimensional harmonic well: g(x) =β3/2(1−x)1/2 x216 π/integraldisplay1 0du u2/radicalbig exp[β(1−u2)]−1, (8) where β≡ −ln(1−x). The variable x= (E+U0)/U0 is the energy of the atom relative to the bottom of the well in units of the well depth, where −U0≤E≤0, and g(0) = 1. For our MOT, the typical temperature is 1 mK, n≃1011/cm3, and n VFORT = 5×105atoms. Using the well depth of U0= 0.4 mK in Eq. 6 shows that NTis of the order of 6 ×104atoms. Much higher numbers are obtainable for a deeper well at lower temperature. The experiments employ a custom-built, stable CO 2 laser. High-voltage power supplies, rated at 10−6frac- tional stability at full voltage, proper electrode design, and negligible plasma noise enable highly stable cur- rent. Heavy mechanical construction, along with ther- mally and acoustically shielded invar rods, reduces vi- bration. The laser produces 56 W in an excellent TEM 00 mode. The CO 2laser beam is expanded using a ZnSe telescope. It is focused through a double-sealed, differentially-pumped, 5 cm diameter ZnSe window into a UHV system. The vacuum is maintained at ≃10−11 Torr by a titanium sublimation pump. The trap is at the focus of a 19 cm focal length ZnSe lens. The trap is continuously loaded from a6Li MOT employing a standard σ±configuration [19] with three orthogonal pairs of counterpropagating, oppositely- polarized 671 nm laser beams, each 2.5 cm in diameter and 8 mW. Power is supplied by a Coherent 699 dye laser that generates 700 mW. The MOT magnetic field gradient is 15 G/cm (7.5 G/cm) along the radial (ax- ial) directions of the trap. The MOT is loaded from a multicoil Zeeman slower system [20] that employs a dif- ferentially pumped recirculating oven source [21]. Using a calibrated photomultiplier, the MOT is estimated to trap approximately 108 6Li atoms. The MOT volume is found to be ≃1 mm3. This yields a density of 1011/cm3, consistent with that obtained for lithium in other ex- periments [22,23]. Using time-of-flight methods, we find typical MOT temperatures of 1 mK. We initially align the CO 2laser trap with the MOT by using split-image detection of the fluorescence at 671 nm to position the focusing ZnSe lens in the axial direction. The focal point for the trapping beam is positioned in the center of the MOT, taking into account the differ- ence in the index of refraction of the optics at 671 nm 2and 10.6 µm. A 671 nm laser beam is aligned on top of the CO 2laser beam to align the transverse position of the focal point in the MOT. Since the Rayleigh length is short and the focus is tight, this method does not reli- ably locate the actual focus of the CO 2beam. Hence, a spectroscopic diagnostic based on the light shift induced by the CO 2laser is employed for final alignment of the trapping beam. While the near equality of the Li excited and ground state polarizabilities is ideal for continuous loading fro m the MOT, it makes locating the CO 2laser focus in the MOT by light shift methods quite difficult. To circum- vent this problem, a dye laser at 610 nm is used to ex- cite the 2p-3d transition for diagnostics. At the 10.6 µm CO2laser wavelength, we estimate that the 3d state has a scalar polarizablity of approximately 700 ×10−24cm3[24], nearly 30 times that of the 2s or 2p state. In the focus of the CO 2laser, the corresponding light shift is ≃ −300 MHz. Chopping the CO 2laser beam at 2 kHz and using lock-in detection of fluorescence at 610 nm yields a two- peaked light shift spectrum. This two-peaked structure arises because the lock-in yields the difference between signals with the CO 2laser blocked and unblocked. At the ideal focusing lens position, the amplitude and the frequency separation of these peaks are maximized. Op- tical alignment remains unchanged for months after this procedure. Measurement of the trapped atom number versus time is accomplished by monitoring the fluorescence at 671 nm induced by a pulsed, retroreflected probe/repumper beam (1 mW, 2 mm diameter). The probe is double- blinded by acousto-optic (A/O) modulators to minimize trap loss arising from probe light leakage. The loading sequence is as follows: First, the CO 2laser trap is contin- uously loaded from the MOT for 10 seconds. This pro- vides adequate time for the MOT to load from the Zee- man slower. Then the MOT repumping beam is turned off, so that atoms in the upper F= 3/2 hyperfine state are optically pumped into the lower F= 1/2, M=±1/2 states. After 25 µsec, the optical MOT beams are turned off using A/O modulators, and a mechanical shutter in front of the dye laser is closed within 1 ms to eliminate all MOT light at 671 nm. The MOT gradient magnets are turned off within 0.2 ms. After a predetermined time interval between 0 and 600 sec, the probe beam is pulsed to yield a fluorescence signal proportional to the number of trapped atoms. The detection system is calibrated and the solid angle is estimated to determine the atom number. Typical trapped atom numbers measured in our initial experiments are ≃2.3×104. This corresponds to the predictions of Eq. 6 for a well depth of 0.25 mK. Since we expect the potential of the MOT gradient magnet to lower the effective well depth from 0.4 mK by ≃0.15 mK during loading, the measured trap number is consistent with our predictions. Fig. 1 shows the decay of the trapped atom number on a time scale of 0-600 seconds. Each data point is the mean obtained from four separate measurement se-quences through the complete decay curve. The error bars are the standard deviation from the mean. Atoms in the F= 1/2 state exhibit a single exponential decay with a time constant of 297 sec, clearly demonstrating the potential of this system for measurements on a long time scale. We have observed that an initial 10-15% decrease in the signal can occur during the first second. This may arise from inelastic collisions between atoms in the F= 1/2 state with atoms that are not optically pumped out of the upper F= 3/2 state. During optical pumping, fluorescence from the F=3/2 state decays in ≃5µsec to a≃5% level which persists for a few milliseconds, consistent with a residual F= 3/2 population. The lifetime of atoms in the F= 1/2 state can be limited by processes that cause heating or direct loss. If we attribute the trap lifetime entirely to residual heat- ing, the heating rate from all sources would be at most 400µK/300sec ≃1µK/sec, which is quite small. How- ever, if the loss were due to heating, one would expect a multimodal decay curve, analogous to that predicted in Ref. [12]. Instead, we observe a single exponential de- cay as expected for direct loss mechanisms, such as colli- sions with background gas atoms or optical pumping by background light at 671 nm (into the unstable F= 3/2 state). If we assume that the lifetime is background gas limited and that Li is the dominant constituent, the mea- sured lifetime of 297 sec is consistent with a pressure of ≃10−11Torr. The long lifetime of the F= 1/2 state is expected, based on the prediction of a negligible s-wave elastic scat- tering length ( <<1 Bohr) at zero magnetic field [25]. Hence, spontaneous evaporation should not occur. We have made a preliminary measurement of trap loss aris- ing from inelastic collisions when the F= 3/2 state is occupied. This is accomplished by omitting the optical pumping step in the loading sequence described above. The trap is found to decay with a 1/e time <1 sec when 2.3×104atoms are loaded (density ≃109/cm3). A de- tailed study of elastic and inelastic collisions at low mag- netic field is in progress. In conclusion, we have demonstrated a 300 sec 1/e life- time for lithium fermions in an ultrastable CO 2laser trap with a well depth of 0.4 mK. By using an improved as- pherical lens system, an increase in trap depth to 1 mK is achievable. Further, Eq. 6 shows that, if the MOT tem- perature is reduced to 0.25 mK, more than 106atoms can be trapped in a 1 mK deep well. Since the ground and excited state trapping potentials are nearly identi- cal, exploration of optical cooling schemes may be par- ticularly fruitful in this system. Currently, we are ex- ploring6Li as a fundamental example of a cold, weakly- interacting fermi gas. By trapping multiple hyperfine states, it will be possible to study both elastic and inelas- tic collisions between fermions. The combination of long storage times and tight confinement obtainable with the CO2laser trap, as well as the anomalously large scatter- ing lengths for6Li [26,27], make this system an excellent 3candidate for evaporative cooling and potential observa- tion of a Bardeen-Cooper-Schrieffer transition. Further, this system is well suited for exploring novel wave op- tics of atoms and molecules, such as coherent changes of statistics by transitions between free fermionic atoms and bosonic molecules, analogous to free to bound transitions for bosonic atoms [28]. We thank Dr. R. Hulet for stimulating conversations regarding this work. We are indebted to Dr. C. Primmer- man and Dr. R. Heinrichs of MIT Lincoln Laboratory for the loan of two stable high voltage power supplies and to Dr. K. Evenson of NIST, Boulder for suggestions regard- ing the laser design. This research has been supported by the Army Research Office and the National Science Foundation. †Permanent Address, Department of Electrical Engi- neering and Computer Science, MIT, Cambridge, MA 02139. [1] See, D. J. Wineland and W. M. Itano, Phys. Rev. A 20, 1521 (1979) and references therein; J. P. Gordon and A. Ashkin, Phys. Rev. A21, 1606 (1980). [2] J. D. Miller, R. A. Cline, and D. J. Heinzen, Phys. Rev. A47, R4567 (1993). [3] S. Rolston et al., in Proceedings of the 1992 Shanghai International Symposium on Quantum Optics, ed. Yuzhu Wang, Yiqiu Wang and Z. Wang [Proc. SPIE 1726, 205 (1992)]. [4] H. J. Lee et al., At. Phys. 14, 258 (1995). [5] C. S. Adams et al., Phys. Rev. Lett. 74, 3577 (1995). [6] D. M. Stamper-Kurn et al., Phys. Rev. Lett. 80, 2027 (1998). [7] T. Takekoshi and R. J. Knize, Opt. Lett. 21, 77 (1996). [8] M. V. Romalis, and E. N. Fortson, to appear in Phys. Rev. A 59(1999); M. Bijlsma, B. J. Verhaar, and D. J. Heinzen, Phys. Rev. A 49, R4285 (1994); N. Davidson et al., Phys. Rev. Lett. 74, 1311 (1995). [9] H. J. Lee, C. S. Adams, M. Kasevich, and S. Chu, Phys. Rev. Lett. 76, 2658 (1996). [10] C. S. Adams and E. Riis, Prog. Quant. Elec. 21, 1-79 (1997). [11] T. A. Savard, K. M. O’Hara, and J. E. Thomas, Phys. Rev. A 56, R1095 (1997). [12] M. E. Gehm et al., Phys. Rev. A 58, 3914 (1998). [13] C. Freed, Ultrastable CO2lasers (Lincoln Laboratory, Bedford, MA, 1990) Vol. 3, 479-500. [14] J. E. Thomas et al., Rev. Sci. Instrum. 51, 240 (1980). [15] C. R. Monroe et al., Phys. Rev. Lett. 70, 414 (1993). [16] The significance of quantum diffractive background gas collisions for heating and loss in shallow traps is discusse d by S. Bali, K. M. O’Hara, M. E. Gehm, S. R. Granade, and J. E. Thomas, Phys. Rev. A, to be published. [17] L. Windholz et al., Phys. Rev. A 46, 5812 (1992). [18] A model of the spatial loading dynamics of a MOT-loaded gaussian laser beam trap will be discussed else- where. [19] E. Raab et al., Phys. Rev. Lett. 59, 2631 (1987). [20] T. A. Savard et al., Quantum Electronics and Laser Sci- ence Conference , Vol. 12, 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 107-108. [21] The design of the recirculating oven source was suggest ed to us by R. Hulet, Rice University. [22] J. Kawanaka et al., Phys. Rev. A 48, R883 (1993); Z. Lin et al., Jap. Jour. Appl. Phys. 30, L1324 (1991). [23] B. P. Anderson, T. L. Gustavson, and M. A. Kasevich, Phys. Rev. A 53, R3727 (1996). [24]Atomic Transition Probabilities , Vol.I, eds. W. L. Wiese, M. W. Smith, and B. M. Glennon, National Standard Reference Data Series-NBS 4 (Washington, DC, 1966), pp.16-18. [25] M. Houbiers, H. T. C. Stoof, W. I. McAlexander, and R. G. Hulet, Phys. Rev. A 57, R1497 (1998). [26] H. T. C. Stoof, M. Houbiers, C. A. Sackett, and R. G. Hulet, Phys. Rev. Lett. 76, 10 (1996). [27] M. Houbiers et al., Phys. Rev. A 56, 4864 (1998). [28] D. J. Heinzen, P. D. Drummond, and K. V. Kherunstyan, Bull. Am. Phys. Soc., 44, 1007 (1999); J. Javanainen and M. Mackie, Bull. Am. Phys. Soc., 44, 1007 (1999). FIG. 1. Trapped number of atoms versus time for an ultra- stable CO 2laser trap. The solid line is a single exponential fit,N(t) =Aexp(−t/τ), and gives τ= 297 sec. We believe that a small fraction of atoms are lost at short times ≤1 sec, (see insert, 0-10 sec) from collisions with atoms that remai n in the F= 3/2 state after optical pumping. Hence, the first two points at 0.1 and 0.3 sec are neglected in the fit. The trap lifetime for the remaining F= 1/2 atoms is 297 seconds, to our knowledge the longest ever obtained with an all-optical trap. 4

1RF-induced evaporative cooling and BEC in a high magnetic field P. BOUYER1, V. BOYER1, S.G. MURDOCH1, G. DELANNOY1, Y. LE COQ1, A. ASPECT1 AND M. LÉCRIVAIN2 1Groupe d'Optique Atomique Laboratoire Charles Fabry de l'Institut d'Optique UMRA 8501 du CNRS Orsay, France. 2L.E.Si.R. URA 1375 du CNRS - ENS Cachan, France 1. INTRODUCTION Bose-Einstein condensates1-3 are very promising for atom optics4-6, where they are expected to play a role as important as lasers in photon optics, since they are coherent sources of atoms with a very large luminosity. In view of applications, it is crucial to develop apparatuses that produce BEC faster - the average production rate of a condensate is 0.01 Hz - and with more versatile designs, by reducing, for example, the power consumption of the electromagnets . For this purpose, we have developed a magnetic trap for atoms based on an iron core electromagnet, in order to avoid the large currents, electric powers, and high pressure water cooling, required in schemes using simple coils. The latest developments allow us to achieve a very high confinement that will permit to achieve much higher production rates. In this chapter, we will first present the design of the iron core electromagnet and how to solve the specific experimental problems raised by this technique. After presenting the experimental set-up, we will address the interruption of runaway evaporative cooling when the Zeeman effect is not negligible compared to the hyperfine structure. We will then present two ways to circumvent this problem: use of multiple RF frequencies and2 RF-induced evaporative cooling and BEC in a high magnetic field sympathetic cooling . Another method, hyperfine evaporation, was used in Ref. 2. In conclusion, we will present some applications of these high magnetic fields. 2. IRON-CORE ELECTROMAGNET TRAP FOR ATOMS Our iron-core electromagnet is shown Fig.2. It follows the scheme of Tollett et al.7. Instead of using permanent magnets, we use pure iron pole pieces excited by coils, which allows us to vary the trap configuration. The use of ferromagnetic materials was reported in Ref.8 .The role of the pole pieces is to guide the magnetic field created by the excitation coils far away from the center of the trap towards the tips of the poles. To understand this effect, let us consider the magnetic circuit represented Fig. 1. Figure 1. a) System of two anti-Helmholtz coils of excitation NI separated by e. b) Equivalent magnetic circuit. l: length of the ensemble poles + yoke. e: gap between the tips. The two tips are separated by a gap e of a few centimeters. The ferromagnetic structure has a total length l and a section S. The whole structure is excited by a coil of 2 N loops driven by a current I, leading to an excitation 2 NI. From the Ampere's theorem9, we can introduce the reluctance Riron: 0 0iron imm mm r rl dlR ≅ =∫Γ(1) inside the iron core and Rgap: 0 0gap gm me dlR ≅ =∫Γ(2)P. Bouyer et al. 3 in the gap between the tips. A simple relation between the excitation 2 NI and the magnetic flux BS can be written: gap iron2 R RNIBS+= . (3) In our case, the gap e and the size of the ferromagnets l are comparable. Since mr is very important (mr > 10 4) for ferromagnetic materials, only the gap contribution is importanti. A more complete calculation shows that the field created in the gap is similar to that created with two coils of excitation NI placed close to the tips as represented10 Fig. 1. Thus, guiding of the magnetic field created by arbitrary large coils far away from the rather small trapping volume is achieved. All this demonstration is only valid if a yoke links a north pole to a south pole. If not, no guiding occurs and the field in the gap is significantly reduced. ZY X Figure 2. Diagram showing the position of the pole pieces of the electromagnet. The tip-to-tip spacing is 3 cm and the section of the cell is 1 × 2 cm2. We will focus now on our Ioffe-type trap for Rubidium 87 (Fig. 2), which consists of a superposition of a linear quadrupole field and dipole field. The linear quadrupole field of gradient G is produced by two pairs of coils in an anti-Helmholtz configuration, along the x- and y-axis, and the i The case of very small gap where Riron > Rgap was studied in Ref.[10]. In this case, the ferromagnetic materials amplify the magnetic field in the gap.4 RF-induced evaporative cooling and BEC in a high magnetic field dipole field of curvature C is produced by two coils along the z-axis in a Helmholtz configuration8. The magnitude of the total magnetic field can be approximated by: ()2 2 2 02 0 2 2Cz y xC BGB ++   − +=B (4) leading to an anisotropic axial harmonic potential for trapping states, in the linear Zeeman effect regime. The use of ferromagnetic materials raises several problems: Geometry. As mentioned previously, a ferromagnetic yoke has to link a north pole to a south pole. A bad coupling between two poles can result in reduced performances of the electromagnet. Our solution shown Fig. 2 optimizes the optical access around the vacuum cell while keeping the required coupling efficient. In addition, the geometry of the magnetic field relies on the shape of the pole pieces rather than the geometry of the exciting coils. This results in the fact that the bias field B0 cannot be easily decreased without also canceling the dipole curvatureii C. This implies a high bias field, for which the Zeeman shift is no longer linear in the magnetic field, due to a contamination between hyperfine levels. In fact, a more complicated design of the poles along the z-axis allows for canceling the bias field while keeping an important dipole curvature. This new design will be discussed in the last section of this chapter. Hysteresis. Hysteresis prevents from returning to zero magnetic field after having switched ON and OFF the electromagnet. A remanent field of a few Gauss remains, as shown Fig. 3. This remanent magnetization needs to be cancelled in order, for example, to release the atoms and make a velocity (temperature) time-of-flight (TOF) measurement. Extra coils around the magnetic poles (see Fig. 4), carrying a DC current will shift the hysteresis cycle so that it crosses zero again. The current is adjusted to provide the coercive excitation which cancel exactly the remanent magnetic field when the large coils are switched off. This compensation is valid as long as we remain on the same excitation cycle. This stability is achieved thanks to a computer control of the experiment. ii In the systems using coils, an additional compensation coils is used the reduce the bias field. In our case, this additional external excitation would couple into the ferromagnetic structure, decreasing both the bias field and the curvature of the field.P. Bouyer et al. 5 Figure 3. Hysteresis cycle of a ferromagnetic structure. The insert shows the typical remanent field value before compensation. Dynamic properties. The use of big coils (lots of loops) results in a big inductance ( ≅ 100 mH), leading to a switching time τ = L/R ≅ 100 ms, too long to allow a good transfer of atoms into the magnetic trap. By assisting the switching with a capacitor, we are able to reduce τ to less than a millisecond. Eddy currents are expected to seriously slow down the switching, and indeed a field decay constant of more than 10 ms was found in our first electromagnet8. The use of laminated material (stacked 100 mm/1 mm thick layers of ferromagnetic materials isolated by epoxy) solves this problem and allows to switch ON or OFF the field within 100 ms. quadrip ole cores Dipole cores Master c oils Coerc ive coils Figure 4. Photo of the poles, the excitation coils and the compensation coils.6 RF-induced evaporative cooling and BEC in a high magnetic field 3. EXPERIMENTAL SETUP The experimental setup is shown on Fig. 5. The electromagnet is placed around a glass cell of inner section of 1 × 2 cm2, pumped with two ion pumps and a titanium sublimation pump. The background pressure is of the order of 10-11 mbar. The tip to tip spacing is 3 cm for the poles of the dipole, and 2 cm for the poles of both quadrupoles. The power consumption is 25 W per coil for a gradient of 900 Gauss/cm, and the maximum gradient at saturation is 1400 Gauss/cm. Figure 5. Experimental apparatus. Our source of atoms is a Zeeman slowed atomic beam of 87Rb. The beam is collimated with a transverse molasses and is decelerated in a partially reversed solenoid. It allows us to load a MOT with 1.5 × 109 atoms in 5 s. In order to increase the density, we then switch to a forced dark MOT by suppressing the repumper in the center and adding a depumper tuned to the F = 2 → F′ = 2 transition12,13. We obtain 8×108 atoms at a density of 1.5 × 1011 cm-3. After additional molasses cooling, we optically pump the atoms into either the F = 2 or the F = 1 state. We then switch on the electromagnet in a configuration adapted to the phase space density of the atomic cloud. The bias field B0 is fixed to ∼ 140 Gauss for F = 2, or to ∼ 207 Gauss for F = 1. The corresponding oscillation frequency is Ω/2π = 21 Hz for F = 2 and Ω/2π = 18 Hz for F = 1. We end up with N = 4 × 108 trapped atoms at a temperature of 120 mK, with a peak density of 5 × 1010 cm-3. All this information is obtained by conventional absorption imaging on a CCD camera.P. Bouyer et al. 7 4. INTERRUPTED EVAPORATIVE COOLING IN A HIGH MAGNETIC FIELD To achieve Bose-Einstein Condensation (BEC), we use RF-induced evaporative cooling of the 87Rb atoms confined in the magnetic trap. In the approximation of the magnetic moment of the atom adiabatically following the direction of the field during the atomic motion, the magnetic potential is a function of the modulus of the field and the projection hm of the total angular momentum on the field axis. Depending on the sign of m, the Zeeman sublevel will be confined towards (trapping state) or expelled from (non-trapping state) a local minimum of the field modulus. RF-induced evaporative cooling consists of coupling the trapping state to a non-trapping state with a radio-frequency field (RF knife), in order to remove the most energetic atoms from the trap. Efficient evaporative cooling14-16 relies on fast thermal relaxation, and thus on the ability of increasing the collision rate by adiabatic compression of the atomic cloud. The most widely used mean to increase the curvature of the trapping potential is to partially cancel the bias field B0 with two additional coils in Helmholtz configuration along the z-axis. As seen in Eq.(4), this increases the radial curvature without changing the axial curvature. Typical values of the compensated bias field in previous experiments are 1 to 10 Gauss. One can also radially compress the atomic cloud by increasing the gradient G without modifying the bias field. This is the approach for our trap. However, the quadratic Zeeman effect is not negligible anymore. Defining hfL hf0 2 ww wme = = hBB(5) as the ratio of the linear Zeeman effect ?L o the hyperfine splitting of the ground state ?hf, we may write the Zeeman effect for 87Rb to the second order in e as () () ( ) { }2 2 hf hf ,4 1 e w e w e F FF F mFm m E EF− + −=− h h (6) with () ( )F FE 1 12hf−+ =wh(7)8 RF-induced evaporative cooling and BEC in a high magnetic field the hyperfine structure. It gives a quadratic Zeeman effect of about 2 MHz at 100 Gauss (a typical bias field for our trap). An immediate consequence is that the ( F = 2, mF = 0) state is a trapping state. For a small magnetic field, as in most experiments, the second order in Eq.(6) is negligible. The RF coupling between adjacent Zeeman sublevels ( |Δm| = 1) results in an adiabatic multiphoton transition to a non-trapping state, leading to efficient RF induced evaporative cooling . In the case of a high bias field, the RF couplings are not resonant at the same location because of the quadratic Zeeman effect . Depending on the hyperfine level, this effect leads to different scenarios17, that we have experimentally identified18 thanks to our magnetic trap allowing strong confinement with a high bias field. -400-200200400 Energy (MHz) Position in the trap 0 rmF=2 mF=1 mF=0 mF=-1 mF=-2 Figure 6. Forced evaporative cooling , in the presence of the quadratic Zeeman shift. We have plotted the magnetic energy of the ( F = 2, mF = 0) sublevels as a function of a trap coordinate r (the magnetic field B(r) = B 0+b r2). For a given RF frequency, the various transitions do not happen at the same position. We have indicated with black arrows the resonant transitions and with white arrows what would happen in the absence of quadratic corrections. For atoms in the ( F = 2, mF = 2) state, forced evaporative cooling will be subject to unwanted effects as the RF knife gets close to the bottom of the potential well. Indeed, we can only cool down the sample to about 50 mK until the atoms cannot be transferred to a non-trapping state and cooling stops. In addition, a careful analysis of the evaporation shows that it can only be optimized to rather poor efficiency. In order to give an insight of the efficiency of the forced evaporation in such a situation, let us consider an atom initially in the mF = 2 trapping state, and following the path represented in Fig. 7 to connect to the mF = -1 non-trapping state.P. Bouyer et al. 9 Figure 7. One short paths connecting the trapping state (F = 2, mF = 2) to the non-trapping state ( F = 2, mF = −1). The atom crosses 5 times the RF knife without making a transition at point B and E. The atom, travelling from the center of the trap, reaches the RF knife at A, and makes a transition to the mF = +1 state at B with a transition probability P. From there, it continues to move away from the center. When it comes back towards the center of the trap, the atom passing on B must not make a transition in order to reach the RF knife on C. The probability to reach BC from OA is P(1 - P). Assuming the same probability P for all the RF transitions, the probability that the atoms follows the path shown in Fig. 7 and leaves the trap on EU is P3(1 - P)2. There are 4 analogous paths involving 5 crossings of the RF knife. Consequently, neglecting interference effects, the total probability associated to these 4 short evaporation paths is 4 P3 (1 - P)2. This probability has a maximum value of about 10% for a transition probability P of 53, and is associated to a precise value of the atomic velocity. When considering all possible velocities, the probability of leaving the trap on mF = -1 averages to less than 10%, much less than for the standard situation where the adiabatic passage has 100% efficiency for almost all velocities17,19. The experimental observation on Fig. 8 supports this simple analysis: when we increase the RF power, the efficiency of the evaporation reaches a maximum and then decreaseiii. In addition, all paths longer than the 5 crossings path as in Fig. 7 contribute to build up a macroscopic population in the intermediate levels mF = 1 and mF = 0, as soon as evaporation starts. This results in the presence of the atoms intermediate sublevels during the evaporation and the observation of a heating of 5 mK/s of the mF = 2 cloud, when we remove the RF knife at the end of the evaporation19. iii Of course, with sufficient RF power (P > 100 W), we would eventually reach a situation where all the various transitions merge, and a direct adiabatic transition to a non-trapping state with 100% efficiency would be obtained.10 RF-induced evaporative cooling and BEC in a high magnetic field efficiency (a.u.) -45 -30 -20 RF power from synth. (dBm)F=1 -45 -30 -20 RF power from synth. (dBm)F=2 efficiency (a.u.) Figure 8. Efficiency of the evaporation for F = 1 (left) and F = 2 (right) as observed via the optical thickness of the atomic cloud. A clear optimum can be observed in the F = 2 case (the relative height between F = 1 and F = 2 is arbitrary). Forced evaporative cooling for atoms trapped in ( F = 1, mF = −1) is not adversely affected by the quadratic Zeeman effect at a bias field B 0 of 207 Gauss, since the mF = 0 state is non -trapping because of the sign of the quadratic term. The RF power has to be large enough to ensure an adiabatic transfer to mF = 0 with an efficiency close to 1. For atoms in the F = 1 state, after adiabatic compression, the oscillation frequencies are Ω||/2π = 18 Hz along the dipole axis, and Ω⊥/2π = 55 Hz along both quadrupole axis. We could successfully cool down the sample, and we obtained a condensate of a few 106 atoms as shown Fig. 9. Figure 9. Bose Einstein condensation in F = 1, mF = -1P. Bouyer et al. 11 5. REACHING BEC IN F=2 IN HIGH MAGNETIC FIELD Several strategies can circumvent the adverse consequences of the quadratic Zeeman effect , and achieve efficient forced evaporative cooling of87Rb in F = 2. 5.1 Evaporation with 3 RF knives22 When evaporating the ( F = 2, mF = 2) state of 87Rb in a high bias field trap like ours, the RF couplings between the adjacent magnetic sublevels are not resonant at the location in the trap. Thus the transfer of atoms from trapping to non-trapping states is inefficient (or even non-existent). This problem can be overcome if we evaporate with three distinct RF frequencies chosen so that a direct transition to a non trapping state is always possible, as shown in Fig. 10. Position in the trap (a.u.)Energy (a.u.)K Figure 10. Implementation of 3 RF knives to evaporate in a high magnetic field. All possible transitions are represented. Frequency matching is only achieved at the RF knife K. However the requirement of three independent frequency sources is not very practicaliv. Rather a simpler solution involving the mixing of two frequencies will compensate the quadratic term of the Zeeman shift. As calculated on Eq.(6), the difference between successive RF transition is the same. Thus, compensation is achievable by mixing one independent frequency source (the carrier) with another RF frequency source of frequency dv to obtain a pair of sidebands of fixed detuning and iv Apart from the technological complexity, the mixing of 3 different frequencies can generate sidebands that will induce stray RF knives, reducing the evaporation efficiency.12 RF-induced evaporative cooling and BEC in a high magnetic field approximately the same power as the carrierv. This detuning by is chosen as to align the three knives perfectly at the end of the evaporation ramp. This approach will be limited to magnetic fields where the higher order Zeeman terms are not significantvi. Indeed, one has to compare the quadratic correction in Eq.(6) with the well-known Breit-Rabi formula: () () ( )1 1212 hf n ,−++ −+ −=− xxwm FF IF F mFm B gm E B EFh(8) with ( ) e wm m x 4 hfn≅ − = hBg g I Bs(9) where gS and gI are the electronic and nuclear g-factors, and mn the nuclear magneton. The RF frequencies between the sublevels calculated from Eq.(8) are shown in table 1. We list only the transitions required to transfer the atoms to the first non-trapping state ( F = 2, mF = −1). Table 1. Zeeman effect without approximations for different bias fields B |2,2〉 → |2,1〉 |2,1〉 → |2,0〉 |2,0〉 → |2,-1〉 56 (Gauss) 39.08-0.44 (MHz) 39.08 (MHz) 39.08+0.45 (MHz) 111 77.12-1.66 77.12 77.12-1.78 207 142.18-5.43 142.18 142.18+6.12 From table 1 we can immediately see that for a bias field of 207 Gauss this approach will not work : it is impossible to choose a sideband detuning dv for which either the (F = 2, mF = 2) → (F = 2, mF = 1) or the ( F = 2, mF = 0) → (F = 2, mF = −1) transition will not be detuned from resonance by at least 500 kHz. This is much larger than the available RF power broadening estimated to be of about 10 kHz. Indeed, experimentally when evaporating (F = 2, mF = 2) with the 3 RF knifes in this bias field we are unable to cool the atoms below 15 mK. However this is an order of magnitude lower than the lowest temperature we can obtain when evaporating ( F = 2, mF = 2) with only one RF knife (100 mK). In a bias field of 111 Gauss the situation is already better with a optimum detuning of the two sidebands from their v Additional sidebands of much lower power are also generated. Consequently, this forced us to reduce the RF power of the 3 knives to avoid unwanted evaporation effects. vi Of course, even then the evaporation efficiency at the start of the ramp will not be optimal as the frequency detuning of sidebands has been optimised for magnetic field at the end of the ramp.P. Bouyer et al. 13 respective resonance of 50 kHz. Here we can cool the ( F = 2, mF = 2) cloud down to 500 nK, and obtain a phase space density of 0.1. We believe that with just a little more RF power or better initial conditions for the evaporation the condensation of ( F = 2, mF = 2) should be possible with this technique for this bias field. When we again lower the bias field by a factor of two to 56 Gauss the effect of the nonlinear terms of the Zeeman shift higher than the quadratic correction becomes negligible compared to the RF power broadening. Here we were able to cool atoms below 100 nK and could attain BEC in ( F = 2, mF = 2) as desired. It should be noted that the effect of the quadratic correction to the Zeeman shift is significant here, since with one RF knife we are unable to cool the cloud below 10 mK. Figure 11 shows a graph of the measured number of atoms in a condensate of ( F = 2, mF = 2) as a function of the sideband detuning dv. The optimal detuning of the RF sidebands from the central carrier is measured to be 0.45 MHz in good agreement with the prediction of table 1. The width of the curve in Fig. 11 is in good agreement with the estimated Rabi frequency and with the residuals calculated with Eq.(8). From this, we can conclude that the average Rabi frequency of our RF knives is indeed of the order of 10 kHz. 30 20 10 0 Number of condensed atoms 0.50 0.48 0.46 0.44 0.42 0.40 du (MHz)x103 Figure 11 Bose Einstein condensation with 3 RF knives : number of atoms in the condensate versus dv. 5.2 Sympathetic Cooling Another possible path to condensation in the ( F = 2, mF = 2) state of 87Rb in a high magnetic field is to use sympathetic cooling20. In sympathetic cooling , one evaporatively cools one species of atom, a second species being cooled simply by thermal contact with the former. In our case, we evaporate ( F = 1, mF = −1) which we know may be efficiently cooled by the standard method, even in a high magnetic field, and use them to cool atoms in (F = 2, mF = 2). This cooling method is nearly lossless for the ( F = 2,14 RF-induced evaporative cooling and BEC in a high magnetic field mF = 2) atoms as they are evaporated in a potential twice as strong as the (F = 1, mF = −1) atoms. The efficiency of sympathetic cooling can be estimated with a simple model assuming that the two species are always at thermal equilibrium. The total energy of the system can be written TkN TkN E B B 2 13 3 + = . (10) The energy taken away by dN1 atoms evaporated at height h14-16 is ( )Tk dN dE B1 31− = h (11) and the energy of the atoms remaining trapped after evaporation of these dN1 atoms is ( )( ) ( )dTTkN dTTkdN N dEE B B− +− −=− 2 1 13 3 (12) since the number N2 of atoms in ( F = 2, mF = 2) is nearly constant during evaporation. If we assume that evaporation is performed at fixed height h, one can simply integrate Eq.(12) by replacing E and dE by their expression in Eqs. (10) and (11). This results in the equation () 32 2i 12f 1 if−     ++=h N NN N TT(13) relating the ratio between initial temperature Ti and final temperature Tf with the loss of atoms in ( F = 1, mF = −1). For example, if we choose h to be 5 (a typical value for experiments) and if we suppose that i 1 2N N<<one can immediately see that the minimum achievable temperature Tmin scales as the initial ratio i 1 2/NN . We can now estimate if the initial conditions are sufficient to achieve BEC. For that, we need to compare Tmin to the critical temperature () 3 f 1 02 3f 1 2 1.2021 NKMBG NFTB c=    ==mh (14a) ( ) 3 2 c 2 2 NK FT == (14b)P. Bouyer et al. 15 for each of the 2 species. We can easily see that the initial ratio i 1 2/NN can be chosen to either condense ( F = 1, mF = −1) before ( F = 2, mF = 2), or ( F = 2, mF = 2) before ( F = 1, mF = −1). If N2 is too large, no condensation is possible and if N2 is too small, only the ( F = 1, mF = −1) atoms can be condensed. This happens for a critical number of atoms cN2 ()()3 1 3 i2i 13 c 222−−     ≅hh TNKN . (15) In order to keep evaporative cooling efficient all the way towards BEC, one has to insure that the atoms remain in good thermal contact. Because of gravity the ( F = 1, mF = −1) cloud is centered below the ( F = 2, mF = 2) cloud as it is more weakly trapped. This displacement between the two clouds is given by 20 GBMgD Bm= . (16) For a fixed gradient G = 900 Gauss/cm this gives a variation with B0, D =0.16 mm/Gauss. The RMS width of a thermal cloud decreases with the square root of the cloud temperature during the cooling, so assuming the two clouds must be within one RMS width of each other for good thermal contact we can obtain an estimate for the minimum temperature to which ( F = 2, mF = 2) can be sympathetically cooled by atoms in ( F = 1, mF = −1), namely, 0 222 limit 24BG kgMT B Bm= (17) proportional to the bias field B0. For a gradient G = 900 Gauss/cm (Eq. (17)) may be evaluated to give a variation with B0 of Tlimit ≅ 1.4 nK/Gauss. The physical interpretation of this is simple; for our trap, the higher the value of the bias field, the weaker the confinement of the quadrupole, the larger the displacement between the two species and hence the higher the minimum possible temperature. Experimentally this simple theory was in good agreement with our observations. By a careful adjustment in the optical pumping cycle during the transfer to the magnetic trap we could start the evaporation with a small but controllable fraction of the atoms in the ( F = 2, mF = 2) state and the rest in the ( F = 1, mF = −1) state. For a bias field of 20716 RF-induced evaporative cooling and BEC in a high magnetic field Gauss we found we were able to cool the atoms in ( F = 2, mF = 2) down to a temperature of 400 nK, in rough agreement with the simple estimate of Eq. (17) of 290 nK. The phase space density for the ( F = 2, mF = 2) cloud at this point was 0.05. Further cooling the ( F = 1, mF = −1) cloud did not reduce the temperature of the atoms in ( F = 2, mF = 2). When we repeated this experiment for a bias field of 56 Gauss, we were able to condense ( F = 2, mF = 2) sympathetically in the presence of ( F = 2, mF = −1) for a sufficiently small initial number of atoms in ( F = 2, mF = 2). When the proportion of atoms in the ( F = 2, mF = 2) state is too large their rethermalization heats the cooling atoms in ( F = 1, mF = −1) too much for an efficient evaporation. Fig. 12 shows phase space density in each state as a function of the final frequency of the evaporation ramp, for three different initial numbers of atoms in the ( F = 2, mF = 2) state. 2.612|F=1,m=-1> |F=2,mF=2>F3 2 1 0n l3 39.60 39.55 39.50 stop (MHz) stop (MHz)3 2 1 0 39.60 39.55 39.50b) n l32.612 c)1.0 0.5 0.0T (µK) 39.60 39.55 39.50 stop (MHz) stop (MHz)3 2 1 0 39.60 39.55 39.50n l32.612 a) d) Figure 12. Sympathetic cooling : representation of the phase space density of both species as a function of the RF frequency. a) The number of atoms in ( F = 2, mF = 2) is small enough so as not to significantly affect the evaporation of the atoms in ( F = 1, mF = −1): in the total absence of atoms in ( F = 2, mF = 2) we could produce condensate in ( F = 1, mF = −1) with a similar transition temperature and number of atoms. In this case the cloud of atoms in ( F = 1, mF = −1) attains sufficient phase space density nl3 to condense before the atoms in ( F = 2, mF = 2). b) The number of atoms in ( F = 2, mF = 2) is two times higher and the evaporation of the (F = 1, mF = −1) cloud is significantly hampered by the sympathetic cooling of (F = 2, mF = 2) atoms. Here the atoms in ( F = 2, mF = 2) attain sufficient phase space density to condense before the ( F = 1, mF = −1), as even though there are less of them they are more tightly confined in the magnetic trap. c) The number of atoms in ( F = 2, mF = 2) is too large too be cooled sympathetically to the temperature required for condensation d) Diagram showing the 2 clouds remain in good thermal equilibrium during all the evaporation.P. Bouyer et al. 17 6. AN APPLICATION OF HIGH BIAS FIELD: COUPLING BETWEEN 2 POTENTIAL WELLS The quadratic Zeeman effect can be an asset rather than a nuisance once condensation is reached. For instance, one can make a selective transfer of part of the condensate from the ( F = 1, mF = −1) state to the ( F = 2, mF = 0) state by using a 6.8 GHz pulse. Thanks to the quadratic Zeeman effect , the (F = 2, mF = 0) state is a very shallow trapping state (for a bias field of 56 Gauss, the oscillation frequencies are Hz400≤Ω⊥ along the quadrupole and Hz100≤Ωz along the dipole), some features of a trapped Bose gas can eventually be observed more easily. We studied the weak coupling between those two states by turning on a weak 6.8 GHz RF knife. The two-coupled potential wells are represented on Fig. 13. Because of gravity, the centers of these two harmonic traps are displaced by typically of 300 mm. -0.5 0Energy (a.u.) Pos ition (mm )(F=2,mF=0)(F=1,mF=-1) D6.8 GHz Figure 13. Energy diagram of the ( F = 1, mF = −1) and ( F = 2, mF = 0) states We start with a condensate in the ( F = 1, mF = −1) state. We will restrict ourselves to the vertical dimension, where the two traps are offset. In the Thomas-Fermi approximation, this condensate is described by the wave function ()() UxUx ~12 − =mj (18) where m is the chemical potential, a the scattering length and U1(x) the trapping potential for the atoms in F = 1. We define MaU24~ hp= . (19)18 RF-induced evaporative cooling and BEC in a high magnetic field This state has a size of typically sx ≅ 10mm. The origin of coordinates is taken at the center of this trap. We will neglect the interactions in the ( F = 2, mF = 0) potential well. Consequently, the weak RF knives will couple the wave function j(x) to the eigenstates Ψn(x - D) of the F = 2 trapping potential. In good approximation, this potential is that of a harmonic oscillator of oscillation frequency 0 ⊥Ω offset down by D from the Bose- Einstein condensate. Thus, we can write () ()x H e nDxnx n nbpbb 241222 !21 −    =−Ψ (20) where Hn (x) is a Hermite Polynomial of order n and h0 ⊥Ω=mb (21) a scaling parameter. The size of these eigenstates is given by ( ) b21+=Δnxn. (22) The coupling efficiency is proportional to the overlap integral between j(x) and Ψn (x). Roughly, only the eigenstates n such as D-sx ≤ Δxn/2 ≤ D+sx will be efficiently coupled. Since the condensate is highly coherent, the resulting wavefunction will be the coherent sum of those coupled eigenstatesvii. This allows us to evaluate the atomic density created in the (F = 2, mF = 0) state () ()2 \x x nDxnx nΨ∑≅Π±≅Δ s(23) as shown in Fig. 14. One clearly sees beatnotes between the different atomic modes. On the contrary, in the case of a thermal cloud of F = 1 atoms with approximately the same size, the resulting density distribution in the F = 2 trap will be the sum of the single eigenstates density profiles, since the coupled eigenstates will incoherently add-up. vii Experimental studies21 showed that the trapped condensate is coherent over its full length.P. Bouyer et al. 19 () ()2 \x x nDxn x nΨ∑≅Π±≅Δ s(24) this results in the disappearance of the periodic structure. Figure 14. Simple picture of the effect of a weak coupling between the ( F = 1, mF = −1) state and the ( F = 2, mF = 0) state. Left, incoherent superposition of different eigenstates of the ( F = 2, mF = 0) trapping potential. Right, Atomic density profile Π(x) of the coherent sum of the coupled eigenstates. A more complete analysis can be done by computing a numerical solution of the coupled Gross-Pitaevskii equations. ( ) 2 12 22 1 11 122 1 2~ 2 tj jj j j j jRFU UM ddiΩ+ + ++∇−=h hh (25a) ( ) 1 22 22 1 22 222 2 2~ 2 tj jj j j j jRFU UM ddiΩ+ + + +∇−=h hh .(25b) For simplicity, we supposed that the scattering length is the same for any binary elastic collision. A comparison of the numerical calculation and of preliminary experimental results is shown Fig. 15. Figure 15. Effect of a weak coupling between the ( F = 1, mF = −1) state and the ( F = 2, mF = 0) state. Top, result of a numerical integration of the coupled Gross-Pitaevskii equation. The20 RF-induced evaporative cooling and BEC in a high magnetic field image takes into account corrections from the resolution of the imaging system and the camera. Bottom, experimental result of the weak coupling versus the time after the coupling was switched ON. One can clearly see the apparition of beatnotes between the atomic mode. With a cold but thermal cloud, no similar feature could be observed. 7. IMPROVED IRON CORE ELECTROMAGNET TRAPS A new design of the pole pieces allow for a compensated bias field B0 - on the order of 1 Gauss - while keeping a significant value for the dipole curvature C - on the order of 100 Gauss/cm2. This, combined with an improved quadrupole gradient to 2400 Gauss/cm allows for a very high compression ratio. Depending on the initial number of atoms, this would allow to reach BEC in a few seconds. The parameters of this new trap will also allow for studying new properties of BEC. Given a bias field of 80mG, this trap has a transverse field curvature of 7 ×107G/cm2, such that the ratio of the transverse to longitudinal field curvatures is 106: 1. This large asymmetry in the trapping potential will allow to form a 1D system. When the temperature of the system is low enough, particles are frozen into the quantum mechanical ground state of the transverse dimensions. However, since the ground state energy in the longitudinal direction is roughly 103 times smaller than that of the transverse direction (since ground state energy scales as the square root of the field curvature), excited longitudinal states can still be occupied. In this one-dimensional regime, the physics of collisions, thermalization, and quantum degeneracy follow laws which are qualitatively different from those of the typical three-dimensional system. ACKNOWLEDGMENTS This work is supported by CNRS, MENRT, Région Ile de France and the European Community. SM acknowledges support from Ministère des Affaires Étrangères. REFERENCES 1. Anderson, M.H., Ensher, J.R., Matthews, M.R., Wieman, C.E., Cornell, E.A., 1995, Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor, Science 269, 198.P. Bouyer et al. 21 2. Bradley, C.C., Sackett, C.A., Tollett, J.J., Hulet, R.G., 1995, Evidence of Bose-Einstein Condensation in an Atomic Gas with Attractive Interactions, Phys. Rev. Lett . 75, 1687; Bradley C.C, et al., 1997, Bose Einstein Condensation of Lithium: Observation of Limited Condensate Number, Phys. Rev. Lett. 78, 985. 3. Davis, K.B., Mewes, M.-O., Andrews, M.R, van Druten, N.J., Durfee, D.S., Kurn, D.M., and Ketterle, W., 1995, Bose-Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett. 75, 3969. 4. Bloch, I., Hänsch, T.W., and Esslinger, T., 1999, Atom Laser with CW Output Coupler, Phys. Rev. Lett. 82, 3008. 5. Bongs, K., Burger, S., Birkl, G., Sengstock, K., Ertmer, W., Rzazewski, K., Sanpera, A., and Levenstein, M., 1999, Coherent Evolution of Bouncing Bose-Einstein Condensates, Phys. Rev. Lett. 83, 3577. 6. Deng L., et al., 1999, 4-wave mixing with matter wave, Nature , 398, 218. 7. Tollett, J.J., Bradley, C.C., Sackett, C.A., and Hulet, R.G., 1995, Permanent magnet trap for cold atoms, Phys. Rev. A 51, R22. 8. Desruelle, B., Boyer, V., Bouyer, P., Birkl, G., Lécrivain, M., Alves, F., Westbrook, C.I., and Aspect, A., 1998, Trapping cold neutral atoms with an iron-core electromagnet, Eur. Phys. J . D 1, 255. 9. Jackson, J., 1962, Classical Electrodynamics, Wiley, New York. 10. Vuletic, V., Hänsch, T. W., and Zimmermann, C., 1996, Europhys. Lett. 36 (5) 349. 11. Desruelle, B., 1999, PhD Thesis. 12. Anderson, M.A., Petrich, W., Ensher, J.R, and Cornell, E.A., 1994, Reduction of light- assisted collisional loss rate from a low pressure vapor-cell trap, Phys. Rev. A 50, 83597. 13. Ketterle, W., Davis, K.B., Joffe, M.A., Martin, A., and Pritchard, D.E., 1993, Phys. Rev. Lett. 70, 2253. 14. Ketterle W., and Druten, N.J., 1996, Advances in Atomic, Molecular and Optics Physics 37 edited by Bederson B. and Walther H. 15. Luiten, O.J., Reynolds, M.W., and Walraven, J.T.M., 1996, Kinetic theory of evaporative cooling of a trapped gas, Phys. Rev. A 53, 381. 16. Walraven, J., 1996, Quantum dynamics of simple systems, Proceedings of the 44th Scottish University Summer School in Physics , Stirling. 17. Pakarinen, O.H., and Suominen, K.-A., 1999, Atomic dynamics in evaporative cooling of trapped alkali atoms in strong magnetic field e-print physics/9910043. 18. Desruelle, B., Boyer, V., Murdoch, S.G., Delannoy, G., Bouyer, P., and Aspect, A., 1999, Interrupted evaporative cooling of 87Rb atoms trapped in a high magnetic field, Phys. Rev. A 60, 81759. 19. Suominen, K.-A., Tiesinga, E., and Julienne, P., 1998, Phys. Rev. A 58, 3983. 20. Hall D.S., et al., 1998, Phys. Rev. Lett. 81, 1543. 21. Hagley, E.W., e t al., 1999, Measurement of the Coherence of a Bose-Einstein Condensate, Phys. Rev. Lett. 83, 3112. 22. Boyer V., Delannoy G., Le Coq Y., Bouyer, P., and Aspect, A., 1999, Multi Frequency Evaporative Cooling towards BEC in a High Magnetic Field, In prep .

arXiv:physics/0003051v1 [physics.flu-dyn] 22 Mar 2000SYMMETRY BREAKING of VIBRATING INTERFACES: A MECHANISM for MORPHOGENESIS N.Garc´ ıa and V.V.Osipov Laboratorio de F´ ısica de Sistemas Peque˜ nos y Nanotecnolo g´ ıa, Consejo Superior de Investigaciones Cient´ ıficas, c/Serra no 144, 28006 Madrid, Spain (February 2, 2008) We show that very small-amplitude oscillations of a highly s ymmetric, spheric or cylindrical, interface (thin membrane) between two fluids can result in in homogeneous instability and breaking of the interface symmetry: the frequency of the breathing vi bration selects the spatial symmetry. This mechanism may govern morphogenesis. 05.65.+b, 47.20.-k, 87.10.+e The nature of spontaneous symmetry breaking remains one of t he most enigmatic questions of modern science. This problem emerges in connection with the equilibrium phase tr ansitions, self-organization in nonequilibrium systems and many other areas in physics, chemistry and biology (see, e.g., [1]), as well as with cell fission and morphogenesis, i.e., the development and spatial differentiation of comple x body structures during their growth [2]. In 1952 Turing showed that the homogeneous state of some spec ific chemical reactions can lose stability with regard to a spontaneous increase of perturbations of certain form [ 3]. Since then the chemical basis is the prevalent idea of phenomenological theory of morphogenesis (see, e.g., [1,2 ]). Turing’s model is based on chemical or biological proces ses of local self-reproduction of some chemical agent (the acti vator) and far-ranging inhibition. As a consequence of such processes a very small increase of the activator concentrat ion in a local region results in a global redistribution of th e substance concentrations and formation of more complex str ucture [1,2]. However, the Turing’s chemical reactions are uncommon, unique and very complex processes. In this work we develop a new mechanism, without complexity, that breaks the symmetry by creating an instability in an oscillating interface, thin membrane, separating two different fluids. In other words, we show that if, for example, a spherical or cylindrical structure vibrates with a breath ing symmetric mode for a given set of the frequencies the symmetry breaks with respect to bimodal, trimodal, pentago nal, etc. modes, i.e., the vibration frequency selects the spatial symmetry of the interface. We consider a thin symmetric membrane, spherical or cylindr ical interface, with the radius R0which separates two fluids with densities ρ1andρ2(ρ1≃ρ2≃ρ) respectively. Owing to Archimed’s force the effective grav ity acceleration operating on the internal fluid is g=ge(1−ρ1/ρ2)<< g e.We propose R0is small enough, so the condition γk2 m/ρ=γm2/R2 0ρ >> g is valid. Here γis the surface tension and k=m/R 0is the typical wave vector of the increasing deformation of the symmetric interface, m= 1,2,3, .... This is the condition when we can neglect the gravity and consider only the effect due to the surface tensio n of the interface. Let us take, at first, for definiteness, a spherically symmetr ic interface Swhose radius, R, oscillates with a frequency ω:R=R0−dcosωt.From the incompressibility of the fluid it follows that its ra dial velocity is vr0=vR0(t)R2 0/r2, where vR0(t) =dR/dt =dωsinωt. (This means that some source, for example, a small pulsatin g ball has to be inside the interface.) The vortex-free motion of an ideal liquid (w e consider the effect of the viscosity below) is described by the Euler and the continuity equations: dv dt=∂v ∂t+ (v∇)v=−1 ρ∇p, (1) ∇2Φ =∇2 rΦ +∇2 ⊥Φ = 0 (2) where Φ is the velocity potential, v=∇Φ and∇2 ⊥is the part of Laplacian depending only on coordinates of the surface S. For the undistorted spherical surface, from the symmetry o f the problem, it follows that v⊥= 0,i.e.,∇2 ⊥Φ = 0 . Then, from Eq.2, we can write that ∇2 rΦ =∇r(vr0) =∂vr0/∂r+ 2vr0/r= 0,in accord with vr0=vR0(t)R2 0/r2.In the presence of a distortion, ς, of the spherical surface Sthe interface radial velocity is vr=vR0(t) +∂ς/∂t. Using this, we find from Eq.1 that near the interface dvr dt=F(t) +∂2ς/∂t2=−1 ρ∂p ∂r, (3) 1∂v⊥ ∂t=−1 ρ∇⊥p (4) where F(t) =dω2cosωtis the acceleration of the interface and we neglect the term ( v⊥∇)v⊥in Eq. 4 by virtue of smallness of ς[4]. Owing to smallness of ςwe can write the pressure near the surface as p=ρF(t)(r−R−ς) +γ∇2 ⊥ς+po(t) (5) Here we took into account that the pressure at the interface ( when r=R+ς) isp=γ(σ1+σ2) +po(t) where σ1and σ2are the principal curvatures of the interface [4]: ( σ1+σ2) =∇2 ⊥ςsince∇2 ⊥ς > R−1 0. Substituting Eq.5 into Eq.4 we obtain ∂v⊥ ∂t=F(t)∇⊥ς−γ ρ∇3 ⊥ςor∂ ∂t∇2 ⊥Φ =F(t)∇2 ⊥ς−γ ρ∇4 ⊥ς (6) We will seek solutions of the problem in the following form ζ=∞/summationdisplay m=0am(t)Smand Φ =∞/summationdisplay m=0cm(t)Ψm(r)Sm−vR0(t)R2 0/r (7) where Smis the complete orthogonal set of eigenfunctions depending only on the coordinates of the undisturbed surface Sand satisfying the following equation (∇2 ⊥+k2 m)Sm= 0 (8) forr=R0and the boundary conditions corresponding to the symmetry o f the problem. In the spherical case Sm= Cl,mP|m| l(cosθ)exp(imϕ) are the spherical functions of angles ϕandθandk2 m=l(l+1)R−2 0where m=l, l−1, ...,−l andl= 0,1,2, ... Substituting Φ from Eq.7 into Eq.2, using Eq.8 and the condi tion∇r(vr0) = 0 cited above, we obtain the equation for Ψ m(r) : (∇2 r−k2 m)Ψm(r) = 0 (9) with the boundary conditions ∇rΨm→0 when r→0 and Ψ m(r) =Aatr=R0where Ais some constant which does not reveal itself in the final results. Near the interfac e∇rΦ =vr=vR0(t) +∂ς/∂t and so from Eq.7 it follows thatcm(t) =dam/dt(∇rΨm)−1 r=R0.Substituting Φ from Eq.7 into Eq.2 and using Eq.9 and cm(t),we find that ∇2 ⊥Φ =−∞/summationdisplay m=0k2 mκ−1 mSmdam/dt (10) where κm= [∇rΨm/Ψm(r)]r=R0does not depend on the constant A. Then from Eq.10 and Eq.6, we obtain d2am/dt2+ [γk2 mκmρ−1−κmF(t)]am= 0. (11) Using T=ωt/2 we can rewrite Eq.11 as d2am/dT2+ (pm−2qmcosωt)am= 0, (12) where qm= 2κmdandpm= Ω2 mω−2where Ω2 m= 4k2 mκmγρ−1. (13) For the spherical interface kmR0>1 and κm≃km= [l(l+ 1)]1/2R−1 0and so Ω2 m= 4[l(l+ 1)]3/2R−3 0γρ−1and qm= 2d[l(l+ 1)]1/2R−1 0. These results can be extended easily to other cases. For exam ple, when the interface have a form of a cylinder with vibrating radius, then Sm= cos( klz)exp(imϕ) and in Eq.13 κm≃kmandk2 m=m2/R2 0+π2l2/h2 0where h0is height of the cylinder. This vibrating cylindrical body can sponta neously distort in the axis zor with respect the azimuthal perturbations. We emphasize that Eq.12 coincides with Eq.(2.12) of Ref. [5] to describe the Faraday’s instability [6] of the plane free surface of an ideal liquid under vertical periodic vibratio ns. These equations differ in the values of the parameters pm andqm. Moreover, in contrast to the Faraday’s instability when th e vibrations are reduced to trivial renormalization 2of the gravity, in this work we consider spherical or cylindr ical oscillating interfaces when the vertical direction, a xial gravity, is not distinguished from other directions. Benja min and Ursell [5] have constructed the stability diagram for Eq.12 with respect to the universal parameters pmandqmusing the analogy between Eq.12 and the Mathieu’s equation [7]. From this diagram it follows that the instabil ity is realized only in regions near the points pm=n2 where n= 1,2,3,4, .... In other words, the condition ω=ωn,m≃n−1Ωm= 2n−1km(κmγ/ρ)1/2(14) determines the resonant vibration frequencies when the sym metric interface spontaneously deforms with respect to the standing wave with the azimuthal number m. However, the greater is n, the narrower is the width E(n) g(qm) of the n-th region of the instability for given qm[5,7]. For the widest instability region, with n= 1,the value E(1) g(qm)≃2qm forqm<1. It means that the instability takes place when (1 −qm)< pm<(1 +qm),i.e., the symmetry breaking is realized for the vibration frequency lying within the spect ral range: Ωm(1−κmd)< ω < Ωm(1−κmd). (15) The threshold of the vibration amplitude dis limited by the fluid viscosity. For real fluid Eqs.11 and 12 i nclude the additional terms γmdam(t)/dtand Γ mdam(T)/dT, respectively, where γm= 2νk2 mC1mand Γ m= 4νk2 mC1m/ωare proportional to the kinematic viscosity νandCmis some constant of the order of unity [8]. The threshold vibr ation amplitude, d=dt,for the instability region with n= 1,can be estimated from the condition E(1) g(qm)>2Γm, i.e., qm= 2κmd >Γm.This condition follows practically from results of Refs. [8 ] and is obtained in [9]. Using Eq.15, this condition can be written as d > d t= 2νCmk2 m/κmω≃ν(ρ/γκm)≃ν(ρR0/γm). (16) For parameters of water dt≃4µm, i.e., the threshold vibration is a very small flutter of the i nterface. We propose that the results above may be used as a basis for a si mple, without complexity, mechanism to trigger the fanciful morphogenesis appearing in nature. The freque ncy of homogeneous interface vibrations self-selects the space symmetry. If the interface oscillates with a characte ristic frequency the germ symmetry will break when its radius R0amounts to the quantity satisfied by Eq.15 or Eq.14 for n >1. After the new symmetry appears the growth rate increases with surface curvature as is usual for many of Stefan-like problems [10]. The mathematical results reported here will be applied in a f orthcoming paper to explain the morphogesis of the acetabularia, equinoderms and cell fision. This work has been supported by the Spanish DGCIYT and by a NAT O fellowship Grant. [1] G. Nicolis and I. Prigogine, Self-Organization in nonequilibrium systems, Wiley, N.Y., London, 1977; M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65,851 (1993); B. S. Kerner and V. V. Osipov, Autosolitons: A new Approach to Problems of Self-Organization and Turbulence, Kluwer, Dordrecht, Boston, London, 1994; C. B. Muratov and V . V. Osipov, Phys. Rev.E 54, 4860 (1996) and Phys. Rev. E 53, 3101 (1996). [2] H. Mainhardt, Models of Biological Formation ( Academic, NY,1982 );H. Mainhardt, Patttern formation in Biology, Rep. Prog. Phys. 55,797 (1992); B. N. Belintsev, Usp. Fiz. Nauk 141, 55 (1983) [So v.Phys. Usp. 26,775 (1983)]; L. A. Segel, Modelling Dynamic Phenomena in Molecular and Cellular Biol ogy(Cambridge University Press, Cambridge, U.K., 1984); G. M. Malacinski and S. V. Bryant, Eds., Pattern Formation, A Primer in Developmental Biology (Macmillan, NY, 1984); J. D. Murray, Mathematical Biology (Springer-Verlag, Berlin,1989); B.S.Kerner and V.V.Osip ov,Structures in Different Models in Morphogenesis, inSelf-organization, Autowaves and Structures Far from Equi librium, edited by V.I.Krinsky (Springer-Verlag, Berlin), p. 265-319, 1984. [3] A. M. Turing, Phil. Trans. R. Soc. London B 237, 37 (1952). [4] L. D. Landay and E. M. Lifshits, Fluid Mechanics (Pergamon, 1987). [5] T. B. Benjamin and F. Ursell, Proc. R. Soc. London, Ser.A 225, 505 (1954). [6] M. Faraday, Phil. Trans. R. Soc. London 52, 319 (1831). [7] N.W.McLachlan, Theory and application of Mathieu functions (Oxford University Press, Oxford, 1947). [8] E. A. Cerda and E. L. Tirapequi, J. Fluid Mech. 368,195 (1998). W. J. Miles, Proc. R. Soc. Lond A 297, 459 (1967); S. T. Milner, J. Fluid Mech. 225,81 (1991); K. Kumar and K. M. S. Bajaj K, J. Fluid Mech. 278,123 (1994); K. Kumar, Proc. R. Soc. London, Ser.A 452, 1113 (1996); K. Kumar and L. S. Tuckerman, J. Fluid Mech. 279, 49 (1994); H. W. M¨ uller, H. Wittmer, C. Wagner, J. Albers, and K. Knorr, Phys . Rev. Lett. 78,2357 (1997).. 3[9] N. Garcia and V. V. Osipov, (unpublished). [10] J. S. Langer, Revs. Mod. Phys. 52,1(1980); J. Chadam and P. Ortoleva, Moving interfaces and their stability (Gordon and Breach, N.Y., 1982); V. V. Gafiichuk, I. L. Lubashevkii and V. V. Osipov, Dynamics of the formation of surface structures (Naukova Dumka, Kiev, 1990). 4

arXiv:physics/0003052v1 [physics.flu-dyn] 22 Mar 2000NATURE OF SONOLUMINESCENCE: NOBLE GAS RADIATION EXCITED BY HOT ELECTRONS IN ”COLD” WATER N.Garc´ ıa1, A.P.Levanyuk2and V.V.Osipov1,3 1Laboratorio de F´ ısica de Sistemas Peque˜ nos y Nanotecnolo g´ ıa, Consejo Superior de Investigaci´ ones Cient´ ıficas, c/Serr ano 144, 28006 Madrid, Spain 2Departamento de F´ ısica de la Materia Condensada, Universidad Aut´ onoma de Madrid, 28049 Madrid, Spain 3Department of Theoretical Physics, Russian Science Center ”ORION”, Plekhanova str. 2/46, 111123 Moscow, Russia (July 24, 2013) It was proposed before that single bubble sonoluminescence (SBSL) may be caused by strong electric fields occurring in water near the surface of collap sing gas bubbles because of the flexoelectric effect involving polarization resulting from a gradient of p ressure. Here we show that these fields can indeed provoke dynamic electric breakdown in a micron-size region near the bubble and consider the scenario of the SBSL. The scenario is: (i) at the last stage of incomplete collapse of the bubble, the gradient of pressure in water near the bubble surface has suc h a value and a sign that the electric field arising from the flexoelectric effect exceeds the threshold fi eld of the dynamic electrical breakdown of water and is directed to the bubble center; (ii) mobile ele ctrons are generated because of thermal ionization of water molecules near the bubble surface; (iii ) these electrons are accelerated in ”cold” water by the strong electric fields; (iv) these hot electrons transfer noble gas atoms dissolved in water to high-energy excited states and optical transitions betw een these states produce SBSL UV flashes in the trasparency window of water; (v) the breakdown can be r epeated several times and the power and duration of the UV flash are determined by the multiplicit y of the breakdowns. The SBSL spectrum is found to resemble a black-body spectrum where te mperature is given by the effective temperature of the hot electrons. The pulse energy and some o ther characteristics of the SBSL are found to be in agreement with the experimental data when real istic estimations are made. 78.60.Mq I. INTRODUCTION Sonoluminescence refers to the phenomenon of light emissio n during acoustic radiation of a liquid and is associated with cavitation bubbles present in the liquid. The most cont rollable and promising experimental data were obtained for single bubble sonoluminescence (SBSL): picosecond UV r adiation of a single bubble pulsating in the field of the sound wave [1,2]. Though the case of a single pulsating bubble is, of course, mu ch simpler than cavitation in general, it turns out that the SBSL is a very complex phenomenon which still remains not completely understood. There is a vast literature devoted to SBSL that was recently reviewed in Ref. [2]. Inter est in this phenomenon is stimulated, on the one hand, by the fact that in relatively simple and controllable exper iments extraordinary conditions (ultra-high pressures, temperatures, and ultra-short light flashes) are realized a t the final stage of the bubble collapse; on the other hand, it looks promising to use the SBSL to construct a source of ult raviolet ultra-short flashes that is much cheaper than lasers. The highly involved hydrodynamics of bubble collapse has be en addressed in many papers that are reviewed in Ref. [2]. One of the most important problems to explain here is the very existence of stable pulsation of bubbles. Recently a theory has been developed [3,4] which explains this regime in terms of the dissolved gas diffusion and chemical reactions in the gas within the bubble. It has been shown that an accumulation of noble gas in the bubble takes place during pulsation of the bubble and that in a stable situation the gas in the bubble consists, almost entirely, of noble gas and, of course, of water vapor. These theoretical conclu sions were supported by the experimental data of Ref. [5]. Another key problem in SBSL is the mechanism of the light emis sion. Many mechanisms have been proposed, criticized, and reviewed to explain UV flashes radiated by th e collapsing bubble [2]. The most popular of the suggested mechanisms are the adiabatic heating of the bubbl e gas [2,6,7], shock wave-Bremsstrahlung model [2,10–12], and, recently proposed proton-tunneling radiation as a res ult of a phase transition in water at ultra-high pressure [13]. A general feature of these mechanisms is that extraord inary conditions are needed which can only be realized, if at all, at a very small bubble radius when the density of the bubble gas is close to the water density and both the 1hydrodynamics of the bubble and the properties of the bubble content and the neighboring liquid are not known from experiment but inferred from numerical simulations. In this paper we show that the main characteristic features o f SBSL can be explained even without making any assumptions about the extraordinary conditions. We are far from saying that these extraordinary conditions are not present in experiments. What we mean is that there is anot her mechanism of the SBSL which occurs in water near the bubble surface but not in the bubble gas as has been as sumed in the most popular models of SBSL [2]. Extraordinary conditions are not necessary for the operati on of this mechanism even though it may well be the main cause of UV radiation. The mechanism under consideration is based on the idea put forward in Ref. [14] that SBSL occurs because of electric breakdown in strong electric fiel ds arising near the bubble surface as a result of flexoelectri c effect, that is the effect of polarization of water because of g radients of pressure [16]. Here we present a scenario and estimations that show that within the hypothesis the mai n features of SBSL can be explained using relatively moderate parameters, e.g., temperatures (5 ÷10)103K for the bubble gas [6], and the natural (expected) value of t he flexoelectric coefficient of water [14]. Let us mention that Ref. [14] has left many questions to answe r. The origin of the optical radiation of the bubble remaines unexplained. The problem is that in the visible reg ion pure water has very low levels of absorbtion and radiation due to interband optical transition [17]. Moreov er the breakdown scenario is far from being clear. The mechanism of the breakdown in water (see, e.g., [18]) involv es ”lucky electrons” whose acceleration in the electric fiel d leads to development of an avalanche. At ambient temperatur e the concentration of free electrons in water is quite negligible (the band gap is about 6.5 eV [17,18]) and it is imp ossible to find a ”lucky electron” in a small volume near the bubble surface, i.e. in the region of high electric field, during the short time of the existence of this field. The spectacular synchronization of the emission pulses [2,19] that in the case of breakdown appears, at least at first sight, to be hardly compatible with the fact that we are dealing with a probabilistic situation. In addition, the reference to the Penning effect to explain the role of the noble gases is n ot convincing because the ionization energy of the metastable state is more than the band gap of water as distinc t from the case of breakdown in gases where the Penning effect is pronounced. In this paper we consider in more detail possible processes a ssociated with the electric field arising from the flexoelectric effect near a bubble exhibiting SBSL. A common d ifficulty with theories of SBSL is that little is known about the parameters of gas in the bubble when the bubble radi us is near its minimum value. Consensus exists about one point: it is far from being a gas, because the lowest limit of the radius is governed by the van der Waals repulsion and the minimum volume of the bubble is close to the van der Waa ls hard core volume. The equations of state used for these conditions are not reliable at present. This is a ch allenging and fundamental problem but it is beyond the scope of this paper. What is now possible is to make order-of- magnitude estimations. That is why we shall first explain what values of relevant parameters are necessary fo r our scenario to be operative and then discuss whether our assumptions about these parameters are realistic. We in clude the value of the flexoelectric coefficient of water among these parameters as well. This coefficient has not been m easured, unfortunately, and we use its estimated ”natural” value but, from the other hand, the knowledge of it s precise value would hardly be of decisive importance because the bubble gas parameters are not known precisely. Let us describe shortly the proposed scenario. At certain sh ort time interval, τc∼1ns, when the bubble radius is near its minimum, the acceleration of the radius and, theref ore, the pressure gradient, assume gigantic values [2] and the sign of the pressure gradient is such as to create a strong (depolarizing) electric field directed to the centre of the bubble in a thin water layer, ∼1µm, near the bubble surface, because of the flexoelctric effect. What happens then is, in effect, screening of this depolarizing field by free electr ons. Indeed, during the same time interval the temperature of the gas sharply increases up to at least several thousand K , which owing to thermal excitation of the bubble gas and water in a thin layer near the bubble surface makes the concen tration of free electrons appreciable. The free electrons are accelerated by the strong ”flexoelectric” field up to ener gies sufficient to generate additional free electrons as a result of the electric breakdown of water. The hot electrons also collide with noble gas atoms dissolved in water and transfer them to higher energy excited states. Optical tran sitions from these states produce light radiation with a broad spectrum whose shape is determined mainly by the energ y distribution of the hot electrons. The latter has the form of a Maxwell distribution with an effective electron temperature which can be very high. This is due to the fact that the noble gas atoms have a huge number of excited sta tes with very high probabilities of optical transitions between them (see, e.g., [20,21]). Since the radiation occu rs in the region of very strong and inhomogeneous electric fields the observed radiation spectrum is featureless. The c haracteristic time of the electric breakdown is much less than the characteristic time for the last stage of the collap seτc, i.e. the polarization can continue to change after the first breakdown and the electric field can reach the breakdown threshold value more than once. As a result, several breakdowns can take place during the time interval τc. During each breakdown the noble gas atoms are excited. An important specific feature of the noble gas atoms is the exist ence of long-living (metastable) excited states with a life time of up to several milliseconds (see, e.g., [20,21]). The refore, once excited the noble gas atom can remain in the metastable state for the entire time interval τc∼1ns, and, possibly, for many periods of the acoustic wave (th e period 2is about 30 µs). That means that in the stationary state the longer is the p ulse the more is its intensity. The paper is organized as follows. In Sec.2 we discuss the flex oelectric effect in water in more detail than in [14] and consider the sign of the flexoelectric field, which is cruc ial for the proposed scenario, at different stages of the bubble collapse. The scenario of SBSL is presented in Sec.3 w here the values of relevant parameters necessary for realizing the scenario are estimated. In Sec.4 we discuss th e energy distribution functions of hot electrons and the breakdown conditions in strong flexoelectric fields as well a s the spectrum of SBSL. In Conclusions we summarize the results of our theory, discuss their relation to the experim ental data, speculate about some possible modifications of the scenario and point out some problems to solve. II. FLEXOELECTRIC EFFECT AND ELECTRIC FIELD NEAR THE BUBBLE We have already mentioned that the proposed mechanism of SBS L is based on the flexoelectric effect, namely the appearance of a polarization (electric field) due to gradien t of density (pressure) [14]. This effect is not widely known and it was discussed in [14] but shortly. Besides, the flexoel ectric coeficients are still not determined experimentally [15]. That is why we think it is worthwhile considering in mor e detail the appearance of electric field due to flexoelectric effect . The flexoelectric effect is a particular case of a more general phenomenon: appearance of electrical polarization, P,as a result of gradient of some scalar quantity, e.g., temper ature, concentration of a component, mass density, ρ. This effect should take place in any substance [16]. As usual, the material equation can be written in several forms. In particular, considering polarization as a function of di latation, u= ∆ρ/ρ, and the electric field, E, one has for an isotropic medium P=α∇u+ε0χE, (1) where χis the electric susceptibility and αis one of the flexoelectric coefficients. We shall be intereste d in the case of spherical symmetry and the absence of free charges. In this c aseD=ε0E+P= 0.Then from Eq.1 it follows that E=−α εε0∇u=−αβ εε0∇p≡f∇p, (2) sinceu=βp, where pis the (excess) pressure, βis the compressibility. To estimate the flexoelectric coefficients it is convenient to consider separately two main mechanisms of polarization: charge displacement and dipole ordering. We shall see that i n both cases the coefficient of proportionality between polarization and gradient of pressure, f, has the same characteristic value. The charge displacement polarization is realized, e.g., in ionic crystals. When a gradient of dilatation takes place there are less and more compressed regions in each unit cell. The ions of larger radius tend to displace to the less compressed region while ions of less radius displace into th e opposite direction. Since cations and anions usually have different radii their displacements produce polarizat ion. The coefficient αcan be estimated as follows [16]. The maximum (”atomic”) gradient of dilatation is equal to 1 /d,where dis the interatomic distance.This gradient has to produce ”atomic” polarization Pat∼e/d2, where eis the electron charge, when the electric field is absent (compensated by some charges), i.e. αion∼e/d. Then from Eq.2 it follows that f∼αionβ εε0∼eβ dε0(3) where we have taken into account that ε∼1 for ionic crystals. We shall argue that the same estimation is aplicable for mate rials with dipole ordering. The dipole ordering polarization in the absence of an external electric field but under a gradient of pressure arises because of geometric asymmetry of the molecular dipoles. For example, the geomet ric shape of water molecule is highly asymmetric: the negatively charged end is much more compact than the positiv e one: the negative charge is concentrated in the oxygen ion (O−−) while the positive charge is shared by two hydrogen ions (H+) located fairly far from each other. Under a pressure gradient a water molecule tries to orient itself i n such a way that the oxygen ion would be located in the region of higher density while the hydrogens would be loc ated in the region of lesser density. In other words, a gradient of pressure leads to orientation of the water molec ular dipoles. It is well known that polarization due to dipole ordering is much more effective than polarization due to charge displacements. This is reflected in the fact that dielectric constants of dipolar materials are, normally, m uch larger than of non-polar materials. For the same reason it is natural to estimate αdip∼εαion. Then from Eq.2 it follows that for water 3f∼αdipβ εε0∼eβ dε0∼10−7V m2 N, (4) where we have taken into accout that for water β≈5·10−10m2 N, d∼10−10m. For what follows the sign of the flexoelectric coefficients is o f importance. As it has been mentioned above the negative tip of the water molecule, the oxygen ion, tends to b e located in the region of higher pressure. This means that the polarization vector is directed opposite to the gra dient of pressure, i.e., the flexoelectric coefficient αis negative and the coefficient fis positive. Let us mention that we rather underestimated the flexoelectr ic coefficient of water than overestimated. Indeed, what we call αionis replaced, in fact, by αatdefining an ”atomic” flexoelectric coefficient that is of the sa me order of magnitude for all substances, i.e. it does not take into ac count specific features of the substance in question. It is natural to expect that a substance consisting of molecules w hose electric asymmetry (existence of a dipolar moment) is accompanied by a pronounced geometric asymmetry (as it is for water) will exhibit a stronger flexoelectric effect than that estimated above. However, while this coefficient is not measured (reliable calculations seem much more problematic than measurements) we shall assume the value gi ven by Eq.4. On the contrary, the conclusion reached about the sign of the coefficient f, which is crucial for the mechanism discussed in this paper, seems much more definite. Above we have neglected the conductivity of water. This may s eem questionable because the Debye radius of the electric field screening in water is comparable with the char acteristic size of the strong field region ( ∼1µm). However, this neglection is justified because, as a rule, the dielectr ic relaxation time, τD,is much greater than τc≃1ns, which is the maximal characteristic time of the polarization change in our case. Increasing the ionic conductivity of water by adding, for example, NaCl, one can, according to our estimat ions, decrease τDdown to 0.1ns. In such an electrolyte the mechanism of SBSL discussed in this paper might be less eff ective. III. THE SCENARIO: ESTIMATION OF THE MAIN PARAMETERS The dynamics of bubble cavitation have been studied in many p apers [2]. Here we will only discuss the short time interval when the bubble radius Ris close to its minimum value Rc(Fig.1a). Fig.1a reflects the most essential features of Fig.4 of Ref. [8] and Fig.12 of Ref. [2] where expe rimental results were presented. Within this interval the velocity of the bubble surface v=dR/dt reaches its maximum and reaches zero at the point R=Rc(Fig.1b) and the acceleration a=d2R/dt2reverses its sign and can achieve huge values (Fig.1c). Duri ng the negative acceleration period ( t∼t1, see Fig. 1c) the gradient of pressure ∇p=−ρais directed from the bubble center to its periphery, thus according to Eq.4 the flexoelectric depolarizing field has th e same direction (Fig.1d). That means that free electrons that could be generated in the gas or in water near the bubble s urface cannot be accelerated. During the positive acceleration period ( t1< t < t 2, see Fig. 1e) the situation is opposite: the arising flexoele ctric field is directed to the bubble center and the generated electrons can be acceler ated by the field and produce a breakdown (Fig. 1e). According to experimental data [8,9] for a bright SBSL the ch aracteristic value of the acceleration is about (1011÷ 1013) m/s2during an interval, τc=t2−t1∼1ns , so the pressure gradient ∇p=−ρacan reach (1014÷1016)N/m3. Note that the gradient of pressure decays with distance from the bubble surface . For purpose of estimations one can consider water as incompressible liquid where ∇p= (∇ps)R2/r2where ∇psis the gradient of pressure at the bubble surface. The same estimation for the gradient of pressure ca n be obtained if we take into account that the pressure is about (106÷108) N/m2at the final stage of the bubble collapse and the extension of t he high pressure region is about 1µm [2]. It follows from Eqs.4 and 2 that for ∇p=(1014÷1016)N/m3the value of the flexoelectric field, E,can reach (107÷ 109)V/m. According to Eq.4 the electric field decays just as the g radient of pressure, i.e., the flexoelectric field in water is given by the formula E≃EsR2/r2where Esis the field at the bubble surface, i.e. the radial extension o f the strong field region is about 1 µm. The field Ecan essentially exceed the threshold field of dynamic electr ic breakdown of water Ethwhich is about (1 ÷3)·108V/m [18]. It does not mean, however, that electric breakdown will occur: a ”lucky electron” which capable of provoking an avalanche is needed . A similar situation takes place with the laser breakdown [18]. At the same time the conduction electron concentratio n,n,in water at room temperature is practically zero (n∼10−90cm−3): water could be considered as a wide gap amorphous semicond uctor with Eg≃6.5eV [18,17,22–24]. The breakdown starts only the moment when the strong flexoele ctric field is directed to the bubble center (Fig. 1f) and conduction electrons appear because of the sharp increa se of temperature near the bubble surface . The latter takes place when the bubble radius R≃Rswhich is close to Rc≃(0.5÷1)µ. At this moment the flexoelectric field Efmay be much more greater than Ethso the coefficient of avalanche multiplication of an electron can be practically infinite and in this case just several electrons are enough to provoke the breakdown and to screen the field. 40 tv= R( b ) hνhν+++++ + + + +++++++++++++ + + +++++++++ ------------------ e ArAr+hν hν ArAr e hν hνArAre( f ) t = t+++++ + + + + ++++++++++++++ + + + ++++++++++ ------------------( e ) t t≥----- - - - - -------------- - - - ---------- +++ + ++++++++ + +++++( d ) t < t RR, µm RR 0246 ttt tt( a ) s c 1smin mint1 0t t tta R= ( c ) 1 2 S1 1 SS FIG. 1. Illustration of dynamics of the pulsating bubble and of formation of the flexoelectric field and the electric break down at the bubble collapse. Time dependence of the bubble radius (a), velocity (b) and acceleration (c) of the bubble surface . The flexoelectric polarization at different moments: (d) - at the beginning of the collapse ( t < t 1), (e) - not very close to the turning point, (f) - in the vicinity of the turning point ( R=Rc). Let us show that even relatively moderate temperatures near the bubble surface (which seem to be generally acceptable for the regimes without the shock waves) are quit e sufficient to provide a sufficient number of conduction electrons to ensure the breakdown. As in Ref. [6] we assume th at the gas has the temperature of 7000K. That means that the water layer with a thickness of the thermal penetrat ion length, δT, has a temperature of about 3000K. For δTone has: 5δT= (2κ ρct)1/2(5) where κ, the thermal conductivity of water is about 0.4 J/m s K; c, the specific heat, is about 4 ·103J/m3K;ρ, the density, is 103kg/m3. Bearing in mind that the characteristic time for the last st age of the collapse is τc∼1ns [2], one finds that the thermal penetration length is δT∼0.1µm. Following [17,18,24] we will consider water as an amorpho us semiconductor with a band gap Eg≃6.5eV and an effective density of states N∗≃1021forT≃3000K. The equilibrium concentration of conduction electrons in wate r atT≥3000K is n=N∗exp(−Eg/2kT)≥3·1015cm−3. Taking into account that the volume of the hot surface layer i s approximately V≃10−12÷10−13cm3we find that the equilibrium number of conduction electrons in the layer isN >300÷3000.It is important also to find the time needed for the equilibrium concentration of electrons to be established. The thermal ionization frequency (see, e.g., [25]) is νT= (N∗σvT)exp(−Eg/2kT), (6) where σis the cross-section of the free carrier recombination, vTis the thermal velocity of conduction electrons in water. Taking into account that a reasonable value of σis about 10−15cm−2andvT∼107cm/s we find from Eq.6 that for T >3000K the transient time to the equilibrium τT=ν−1 T≤10−8s. From that follows that during the characteristic time for the last stage of the bubble collaps eτc∼1ns the number of conduction electrons in the layer, N,exceeds 30 ÷300 electrons, which is quite enough to provoke breakdown. Note that a similar number of free electrons can be generated by thermal ionization of the gas. Indeed, the assumed gas temperature (7000K) is about two times larger and the ion ization energy both of the water vapor and Ar, Kr, Xe is about 12 ÷16eV, i.e. also about two times larger than Eg. Authors of many works (see [2] and the references therein) state that much higher temperatures can be reached because of shock waves forming in the bubble gas during the collapse. In principle, one can imagine a situation wher e high temperatures are not reached (and free electrons are not generated) before the shock waves are formed. In this case the shock wave, when it explodes, will start the electric breakdown. As a result of breakdown the depolarizing flexoelectric field becomes screened. The total transmitted charge in the process of the screening is Qt=PS= 4πR2 sε0Es (7) where we have taken into account that in our case D=P+ε0E= 0, i.e., P=−ε0E. Assuming Rs≃1µm and Es= (108÷109) V/m we find that the maximum total number of transmitted elec trons is Nt=Qt/e≃(105÷106). Note that for considered electric fields the time of the scree ning (breakdown), τs, is determined by the time it takes for the conduction electrons to cross the region of the stron g electric field whose size is l∼Rs. Thus τs=l/vd where vdis the electron drift velocity which in strong electric field s saturates at some value vd≃105m/s (see Sec.4) . Assuming l∼1µm we find τs∼10ps. Note that the value of τsis less than the observed SBSL pulse width [2,26,27]. The whole period of positive acceleration, τc, (see Sec.3) is about two orders of magnitude larger than τb.Therefore, after the breakdown is finished and the depolarization field i s screened the polarization continues to change, because of change in acceleration. The electric field arises once mor e and can exceed the breakdown threshold. As a result a new breakdown will take place. Such a situation can be repea ted several times. Effectively, it manifests itself in an increase of the pulse duration which may achieve a fractio n of a nanosecond. Therefore, within our scenario the greater is the pulse duration the greater is its energy. Anal ogous interrelation is observed in experiment. [2,26,27]. IV. ENERGY DISTRIBUTION OF HOT ELECTRONS IN THE ELECTRIC FIE LD AND SPECTRUM OF THE SONOLUMINESCENCE We have already mentioned that the electric fields under cons ideration are very strong, inhomogeneous and change over time. However, the characteristic space and time scale s of the field change are correspondingly l∼1µm and τc∼1ns and they are much more than the relaxation length λand relaxation time τεof the hot electron energy which are respectively about (10 ÷100)˚Aand 10−13s. [17,18,24]. Therefore one can consider the local electro n energy distribution function using well known results for homogen eous static electric fields [28–30]. The considered electric fields are superstrong, i.e. the fol lowing condition is valid: qEλ≫hωph≡εph, (8) where ǫphis the characteristic energy of local oscillations in water which is practically equal to the energy of optical phonons in ice, ǫph≃(80÷100)meV [17,18,24]. Using λ≃(10÷100)˚Aone sees that the condition 8 is satisfied for 6fields E >107V/m. The condition 8 means that in these fields an electron acq uires, on average, an energy qEλwhich is much more than ǫph. Since in the process of the acceleration an electron genera tes many phonons the electron energy distribution is nearly isotropic in the momentum spa ce. Specifically, in this case the energy distribution of hot conduction electrons is approximated with high accurac y, up to energies of electrical breakdown ǫ=Eg, by the Maxwell function [28–30]: f(ǫ)∼exp(−ǫ kTe) (9) with an effective electron temperature Tewhich is determined by the balance equations for energy and m omentum of the hot electrons dǫ dt=eEvd−ǫphvT λ= 0, (10) dp dt=eE−mevdvT λ= 0. (11) where vTis the effective thermal velocity of electrons. Since for the Maxwell distribution 3 2kTe=1 2mev2 T, (12) from Eqs.10, 11 and 12 it follows that vd=/radicalBig εph/me (13) and kTe= (eEλ)2/3εph. (14) From Eqs 12 and 13 as well from Eq. 8 one sees that vT/vdr=/radicalbig kTe/εph=qEλ/ε ph>>1.This is precisely the condition of validity for the thermalization of the hot elec trons and use of the Maxwell distribution. Conduction electrons in water form polarons which are conve ntionally called hydrated electrons [22,24]. However, in the strong electric field the polaron states decay and the c urrent carriers are the usual conduction electrons. For purpose of estimation we assume that their effective mass meis close to that of free electrons m0. Then from Eq.13 it follows that vd≃105m/s for ǫph≃100meV. It has already been mentioned that the electric field s under consideration areE≃(108÷109)V/m. It follows from Eq.14 that for such fields kTe≃(1÷10)eV, i.e., Te≃(104÷105)K for λ≃(10÷100)˚A. One sees that the electron temperatures can be several orde rs higher than the gas temperature in the bubble. Recall that for our scenario (see Sec.3) it suffici ent that the gas temperature be about 7000K. Now we will discuss the role of noble gases dissolved in water . An important feature of an noble gas atom is the existence of metastable states. The life time of the metasta ble states can reach several milliseconds when the noble gas atoms are impurities in solids (see, e.g., [20,21]). Sin ce the nearest order in water is essentially the same as in solids it would be natural to expect that the life-time of the metastable states for the noble gas atoms in water is not less than 1ns . Hot electrons not only generate new conductio n electrons (the breakdown avalanche) but also excite the noble gas atoms into metastable states. Below we will est imate from the experimental data that the characteristic value of kTe≃(2÷5)eV. However, a considerable number of electrons may have e nergies about 10eV and can excite the noble gas atoms. Note that the energies necessary for excitation to the metas table states, εm, for Xe, Kr, Ar, Ne, He are about 10eV and increase monotonically from Xe to He. In other words , it is much easier to excite Xe, Kr and Ar than Ne and He. This could be the reason for the increase in SBSL inten sity in the series He-Xe, which has been observed experimentally [2,31]. As we have mentioned above, the life-time of the metastable s tates of noble gas atoms in water is expected to be fairly long. Thus, once excited a noble gas atom can remain in a metastable state for the entire time interval of positive acceleration and multiple breakdowns, τc∼1ns. The major part of electrons has the energies ε < ε m,Eg;they collide with these metastable atoms and transfer them to hig her excited states. The radiation transitions between high-energy excited states of noble gas atoms govern the SBS L spectrum. Note that the life-time of the metastable states can exceed t he period of the acoustic wave. In this case there will be an effect of accumulation of the noble gas atoms in metastab le states resulting in the gradual build up of the SBSL power during several periods of the acoustic wave. 7A specific feature of a noble gas is an abundance of excited sta tes with energies higher than those of metastable states and with high probabilities of the radiation transit ions between them. That is why the optical spectra of the noble gas atoms contain many lines [20,21]. Because of the St ark effect in the strong electric field these lines are split. Besides, in the active region the strong electric fiel d changes at least several times and the observed radiation spectrum is a superposition of the spectra of the atoms in diff erent strong electric fields. In other words one can consider the density of the atomic excited states as a consta nt. The probability of a hot electron having an energy ε < E gis given by Eq.9 and at every collision it transfers the noble gas atom in a metastable state to a state with energy ε, the reference point of energy being the energy of the metastable state (for our estimations we consider only one m etastable state). The concentration of atoms excited during the time of a single electric breakdown, i.e. the scre ening time, τs,to energies within an interval dεreads dn∗∗ n=n∗ n(σexvTn)τsexp(−ε kTe)·dε kTe(15) where n∗ nis the number of the noble gas atoms in the metastable state, σexis the cross section of the impact excitation of an atom from the metastable state to a state with energy ε. Atoms excited to states with the energy εgenerally do not go to the ground state directly but through intermedia te excited states. It reasonable to assume that with a high probability they radiate phonons with energy hν≃ε. So the spectral density of the SBSL radiation energy per pulse for unit volume can be written as /tildewideP(hν)d(hν) =hνw rn∗ n(σexvTn)τsexp(−hν kTe)d(hν) kTe(16) where wris the probability of the spontaneous radiation transition . Taking into account that wr=4(2π)4ν3 3c3hD2where Dis modulus of matrix element of the dipole moment of the trans ition [32] we find /tildewideP(hν)d(hν) =4(2π)4D2 3c3n∗ n(σexvTn)τsν4exp(−hν kTe)d(hν) kTe. (17) Recall that Tein Eq. 17 is a function of the coordenates because Te∼E2(see Eq.14 ). Putting, as before, E=f∇p≃EsR2 s/r2and integrating approximately Eq.17 we find the spectral pow er of SBSL for a single breakdown is P=/integraldisplay /tildewideP(hν, r)dV=4(2π)4D2 3c3hn∗ n(σexvTNt)τsν3exp(−hν kTes) (18) where kTes= (eEsλ)2/3εphandNtis, practically, the total number of electrons participati ng in the breakdown. Note that the approximative Eq.18 is not sensitive to the form of d ecay of the field, it is important only that Edecays more steeply than r−1, i.e. it is not important that for ∇pwe use an expression for an incompressible liquid. The observed spectra are cut off in the shortwave region becau se of the absorption of water. This can be taken into account by multiplying Eq.18 by exp(- α(hυ)L) where Lis the size of the acoustic resonator ( L≃2.5cm [2]). Because of the Urbach absorption tails [17,23] the radiation maximu m is located considerably below than hν≃Eg≃6.5eV. It should be emphasized that the spectrum given by Eq.18 rese mbles the black-body spectrum but temperature is given here by the effective temperature Tesof hot electrons near the bubble surface. The value of Tesmay be much higher than the bubble gas temperature. Experimenta lly observed spectra can be fitted, in the wavelength interval (200 ÷700) nm, to the black-body ones with temperatures (2 ÷5)104K [2,33]. According to Eq.14 such electron temperatures are reached at electric fields Es≃(2÷10)108V/m for values of εph≃0.1eV and λ≃(10÷100)˚A characteristic for water [18]. As follows from Eqs.4 and ∇p=−ρasuch fields are realized at accelerations a≃ (1013÷1014) m/s2.Similar accelerations are reported in [2,9]. Integrating Eq.18 multiplied by exp(- α(hυ)L) we find an estimate of the radiation energy for a single break down Wr=/integraldisplay Pw−1 rexp(−α(hυ)L)d(hν)≃(σexvTNt)τsn∗ nhν≃vT vd(σexRsn∗ n)Nthν (19) where hνis the characteristic photon energy which is close to the ene rgy of the maximum of the observed spectrum, which is about (5 ÷6)eVin the case of strong electric field of our interest. Recall th at in Eq.19 τs≃Rs/vdis the screening (breakdown) time and Nt≃4πR2 sε0Es/eis the number of transmitted electrons in the screening proc ess (Sec. 3) and n∗ nis the concentration of noble gas atoms in a metastable state near the bubble surface. As we have mentioned before it was shown in [3,4] that because of dissolved gas diffusion and chemical reactions an accumulation of noble gas in the bubble takes place while t he bubble pulsates and in stationary conditions the 8gas in the bubble consists almost entirely of noble gas. Ther efore one can assume that the concentration of the noble gas atoms, nn,near the bubble surface is close to its saturation value. Not e that this value has to be estimated for atmospheric pressure and ambient temperature rather than f or the high pressures and temperatures existing at the last stage of the collapse over a very short time. This is why w e assume that nn≃(1018−1019) cm−3. Now we will argue that the value of n∗ ncan be only one order of magnitude less than nn. Indeed, for kTe≃(3÷5) eV obtained above by fitting the SBSL spectrum to the black-body spectrum , the number of electrons having an energy higher than 10eV, Nhmay be about 10−3÷10−2of the total electron number Nt≃106i.e.,Nhis about 103÷104. Such a ”superhot” electron excites a noble gas atom to the metastab le state over the time τex= (σexvTnn)−1≃10−12s for σex≃10−15cm2, vT≃106m/s and nn≃(1018÷1019) cm−3.Therefore a ”superhot” electron during its participation in screening excites τs/τexnoble gas atoms and the total number of the excited atoms is Nhτs/τex≃104÷105if one takes into account that τs≃10τex≃10−11s. The total number of the noble gas atoms in the region of the s trong field (∼1µm3) is about 106÷107, i.e. the percentage of the noble gas atoms excited to the met astable state during a single breakdown can be about 10%. Since the life time of the metastable state is much more than τcand during this time interval several breakdowns can take place (Sec.3 ), one may expect that the number of the noble gas atoms excited to the metastable state during the interval τcis only one order of magnitude less than the total number of these atoms i.e. n∗ n∼1018cm−3. If the life time of the metastable states exceeds the acoust ic period ( ∼30µs) an accumulation of noble gas atoms in the metastable states can also occur during several acoustic periods. Now we can estimate the total radiation energy. We have menti oned above that the maximum value of Nt≃106, Rs≃1µm,σex≃10−15cm2andvT>> v d. Taking into account that the positive acceleration period τc∼1ns and the breakdown time τsis about 0.01ns one can assume that the number of breakdowns i s not less than 10. Thus one finds from Eq.19 that the maximum photon number in the SBSL pul se can be more than 107. This value corresponds to the maximum photon number observed experimentally [2,33 ,34]. Note that the radiation energy is a small part of the total energy of the flexoelectric field (see Ref [35]). V. CONCLUSIONS Let us emphasize once more that within the mechanism of SBSL c onsidered the main processes occur in water near the bubble surface unlike the most widely discussed mechani sms of SBSL [2]. Within the framework of this mechanism many experimental data about the SBSL can be explained quite naturally. Of course, since the parameters of the collapsing bubble are not reliably known when the bubble rad ius is close to its minimum value we can present no more than order-of-magnitude estimations. Let us summarize the main results. (i). The minimum duration of the SBSL flash is determined by th e single breakdown (screening) time, τs≃10ps. A larger time is possible because of the multiplicity of brea kdowns. The maximum duration is limited by τc∼1ns. The longer is the duration the greater is the energy of the flas h. This is in agreement with experiments where it was found that the pulse duration changes from 30ps to 400ps when its energy increases [2,26,27]. (ii). According to our estimations the maximum energy of the flash corresponds to 107÷108photons with energy (5÷6)eV, which also agrees with the experiment [2,33,34]. (iii). Within our scenario the noble gas atoms play an import ant role. They do not reduce so much the breakdown threshold as they govern the radiation process. Abundance o f optical transitions in these atoms and inhomogeneous broadening because of the Stark effect explain the practical ly continuous character of the SBSL spectrum. (iv). In agreement with the experiment [2,33] the theoretic al spectrum of the SBSL resembles the black-body spectrum but the temperature is given here by the effective te mperature of the hot electrons which can be about several eV, what corresponds to the observed apparent radia tion temperature [2,33]. At the same time the gas temperature is not directly related to the radiation temper ature and can be considerably less than 1eV. (v). The increase of influence on SBSL intensity in the noble g as series He-Xe observed in the experiment [2] is connected with the decrease, in this series, of the energy of the lowest metastable state. (vi). The pulse width does not depend on the spectral range of the radiated photons. This also agrees with experiment [26]. (vii). The effect of synchronization of the light pulses obse rved experimentally [2,19] is explained. Note also that within our scenario of SBSL the influence of mag netic fields on SBSL is fairly weak. Their influence becomes appreciable when they are high enough to hamper the h eating of electrons [36]. This might be the reason for the decrease of the SBSL intensity in strong magnetic fiel ds which has been observed experimentally [37]. It seems that the considered mechanism of SBSL is especially effective for water because of a lucky coincidence of several conditions: 9(i) the strong geometric asymmetry of the water molecule, wh ich accompanies its electric asymmetry, provides the needed sign of the flexoelectric coefficient. This sign is s uch that the electric field has the ”correct direction” (accelerates electron) over the same (very short) period in which electrons are generated because of the sharp increase of the bubble gas temperature. The temporal coincidence of t hese two cicumstances is a possible reason of the effect of synchronization of the light pulses observed experiment ally [2,19]. (ii) water is a semiconductor with a relatively narrow band g ap (Eg= 6.5eV) and sufficienly wide conduction band, what is necessary for the heating of the electrons in strong e lectric field. (iii) high solubility of noble gases in water what makes poss ible high concentrations of noble gas in water near the bubble filled by the noble gas accumulated there in the proces s of the bubble pulsations. Fulfillment of these conditions gives the answer to the quest ion why water is the friendliest liquid for SBSL [2]. Some experimentally found characteristics of SBSL have not been explained in this paper, in particular the de- pendence of the energy of the pulse upon the partial pressure of noble gases and water temperature [2]. These dependences may be not necessarily a manifestation of the ra diation processes studied in this paper but they could be a consequence of a set of various factors. They may be deter mined by the hydrodynamics of the pulsating bubble, the solubility of noble gases in water and also by the tempera ture dependence of the flexoelectric coefficient of water which still remains, unfortunately, unmeasured. Although the main features of SBSL seem to be explainable wit hin our scenario, these explanations are qualitative rather than quantitative. We are still a long way from a quant itative theory at present. One of the main aims of the paper is to stimulate some experiments that could either sup port or discard the proposed mechanis of SBSL. (i) Measurements of flexoelectric coeficients of water. (ii) A detailed study of radiation spectra at electric break down of water and ice with different concentrations of noble gases. (iii) Study of influence of water conductivity on the SBSL int ensity. After performing these experiments it makes sense to develo p the theory further. In particular, it will be necessary to study in more detail the kinetics of the screening process and of the excitation of noble gas atoms while taking into account the time variation of the distribution functio n of the hot electrons. This problem should be considered together with that of the determination of the spatial distr ibution of excited noble gas atoms near the bubble surface. Diffusion and relaxation of the excited atoms should be consi dered together, of course, with the processes that lead to noble gas accumulation in the bubble [3,4]. In our estimations we assumed that flexoelectric coefficient o f water has its ”natural” value. This was sufficient for the theoretical estimations to be in agreement with the expe rimental data. In fact, because of the above-mentioned strong geometric asymmetry of the water molecule, the coeffic ient could be even larger. If this is the case some new phenomena will take place including influence of the flexoele ctric effect on hydrodynamics of the pulsating bubble and X-ray radiation induced by electrons of very high energy . We are grateful to V.Kholodnov for useful discussions. NG an d VVO thank EU ESPRIT, the Spanish CSIC and NATO for Linkage Grant Ref. OUTRLG 970308. Also, APL thanks N ATO for Linkage grant HTECHLG 971213. [1] D.F.Gaitan, Ph.D. Thesis, University of Mississippi (1 990); D.F.Gaitan, L.A.Crum, C.C.Church, and R.A.Roy, J.Acoust.Soc.Am. 91, 3166 (1992). [2] B.P.Barber, R.A.Hiller,R. L¨ ofstedt, S.J.Puttermane t, K.R.Weninger, Phys.Reports 281, 65 (1997), and references therein. [3] D.Lohse, M.Brenner, T.Dupont, S.Hilgenfeldt, and B.Jo hnston, Phys. Rev. Lett. 78, 1359 (1997). [4] D.Lohse and S.Hilgenfeldt, J.Chem. Phys. 107, 6986 (1997). [5] T.J.Matula and L.A.Crum, Phys. Rev. Lett. 80, 865 (1998). [6] R.L¨ ofstedt, B.P.Barber, and S.J.Putterman, Phys. Flu ids A5, 2911 (1993). [7] S.Hilgenfeldt, S.Grossmann, and D.Lohse, Nature 398, 402 (1999). [8] B.P.Barber and S.J.Putterman, Phys.Rev.Lett. 69, 3839 (1992). [9] K.R.Weninger, B.P.Barber, and S.J.Putterman, Phys. Re v. Lett. 78, 1799 (1997). [10] P.Jarman, J.Acoust.Soc.Am. 32, 1459 (1960). [11] R.L¨ ofstedt, B.P.Barber, and S.J.Putterman, J.Acous t.Soc.Am. 92, S2453 (1992). [12] C.C.Wu and P.H.Roberts, Phys.Rev.Lett. 70, 3424 (1993). [13] J.R.Willison, Phys. Rev. Lett. 81, 5430 (1998). [14] N.Garc´ ıa and A.P.Levanyuk, JETP Lett. 64, 907 (1996). [15] The term ”flexoelectric effect” is also used in the liquid crystals physics but it has a different meaning. 10[16] A.K.Tagantsev, Usp. Fiz. Nauk 152, 423 (1987) [Sov. Phys. Usp. 30, 588 (1987)]. [17] F.Williams, S.P.Varma, and S.Hillenius, J.Chem.Phys .64,1549 (1976). [18] C.A.Sacchi, J.Opt.Soc.Am.B 8, 337 (1991) and references therein. [19] B.P.Barber and S.J.Putterman, Nature 352,318 (1991); L.A.Crum, Phys.Today, September #8, 22 (1994). [20] R.P.Bauman, Absorption Spectroscopy (John Willey & Sons, Inc., NY, London, 1962); G.R.Harrison a nd R.C.Lord, Prac- tical Spectroscopy (Blackie & Son, London, 1949). [21] M.J.Beesley, Lasers and Their Applications ( Taylor and Francis LTD, London, 1972 ). [22] J.W.Boyle, J.A.Chromley, C.J.Hochanadel, and J.F.Ri ley, J.Phys.Chem. 73,2886 (1969). [23] D.Grand, A.Bernas, and E.Amoyal, Chemical Physics 44, 73 (1979). [24] P.Krebs, J.Phys.Chem. 88,3702 (1984). [25] K.Seeger, Semiconductor Physics( Springer, Wien, NY, 1973 ).. [26] B.Gompf, R.G¨ unther, G.Nick, R.Pecha, and W.Eisenmen ger, Phys. Rev. Lett. 79, 1405 (1997). [27] R.Hiller, S.J.Putterman, and K.R.Weninger, Phys. Rev . Lett. 80, 1090 (1998). [28] P.A.Wolf, Phys.Rev. 95,1415 (1954). [29] G.A.Baraff, Phys.Rev. 128, 2507 (1962). [30] L.V.Keldysh, Sov.Phys. JETP 21, 1135 (1965). [31] R.Hiller, K.Wenninger, S.J.Putterman, B.P.Barber, S cience 266, 248 (1994). [32] H.A.Bethe, Intermediate Quantum Mechanics (W.A.Benjamin, Inc., NY, Amsterdam, 1964). [33] R.Hiller, S.J.Putterman, and B.P.Barber, Phys. Rev. L ett.69, 1182 (1992). [34] B.P.Barber, C.C.Wu, R.Lofstedt, P.H.Roberts, S.J.Pu tterman, Phys. Rev. Lett. 72, 1380 (1994). [35] N.Garc´ ıa, A.P.Levanyuk, and V.V.Osipov, accepted to JETP Letters. [36] F.G.Bass, Yu.G.Gurevich, Hot Electrons and Strong Electomagnetic Waves in Semicondu ctors and Gas Discharge Plasma, Nauka, Moscow, 1975. [37] J.B.Young, T.Schmiedel, and Woowon Kang, Phys. Rev. Le tt.77, 4816 (1996). 11

arXiv:physics/0003053v1 [physics.acc-ph] 23 Mar 2000Radiation from a Superluminal Source Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princet on, NJ 08544 (Nov. 26, 1986) Abstract The sweep speed of an electron beam across the face of an oscil loscope can exceed the velocity of light, although of course the velocity of the ele ctrons does not. Associated with this possibility there should be a kind of ˇCerenkov radiation, as if the oscilloscope trace were due to a charge moving with superluminal velocity. 1 Introduction The possibility of radiation from superluminal sources was first considered by Heaviside in 1888 [1]. He considered this topic many times over the next 20 years, deriving most of the formalism of what is now called ˇCerenkov radiation. However, despite being an early proponent of the concept of a velocity-dependent electroma gnetic mass, Heaviside never acknowledged the limitation that massive particles must ha ve velocities less than that of light. Consequently many of his pioneering efforts (and thos e of his immediate followers, Des Coudres [2] and Sommerfeld [3]), were largely ignored, and t he realizable case of radiation from a charge with velocity greater than the speed of light in a dielectric medium was discovered independently in an experiment in 1934 [4]. In an insightful discussion of the theory of ˇCerenkov radiation, Tamm [5] revealed its close connection with what is now called transition radiati on,i.e., radiation emitted by a charge in uniform motion that crosses a boundary between met allic or dielectric media. The present paper was inspired by a work of Bolotovskii and Ginzb urg [6] on how aggregates of particles can act to produce motion that has superluminal as pects and that there should be corresponding ˇCerenkov-like radiation in the case of charged particles. T he classic example of aggregate superluminal motion is the velocity of the poin t of intersection of a pair of scissors whose tips approach one another at a velocity close to that of light. Here we consider the example of a “sweeping” electron beam in a high-speed analog oscilloscope such as the Tektronix 7104 [7]. In this device t he “writing speed”, the velocity of the beam spot across the faceplate of the oscilloscope, ca n exceed the speed of light. The transition radiation emitted by the beam electrons just before they disappear into the faceplate has the character of ˇCerenkov radiation from the superluminal beam spot, accord ing to the inverse of the argument of Tamm. 2 Model Calculation As a simple model suppose a line of charge moves in the −ydirection with velocity u≪c, where cis the speed of light, but has a slope such that the intercept w ith the xaxis moves with velocity v > c . See Figure 1a. If the region y <0 is occupied by, say, a metal the charges will emit transition radiation as they disappear in to the metal’s surface. Interference 1among the radiation from the various charges then leads to a s trong peak in the radiation pattern at angle cos θ=c/v, which is the ˇCerenkov effect of the superluminal source. Figure 1: a) A sloping line of charge moves in the −ydirection with veloc- ityvy=u≪csuch that its intercept with the xaxis moves with velocity vx=v > c . As the charge disappears into the conductor at y <0 it emits transition radiation. The radiation appears to emanate fro m a spot moving at superluminal velocity and is concentrated on a cone of ang le cos−1(c/v). b) The angular distribution of the radiation is discussed in a s pherical coordinates system about the xaxis. To calculate the radiation spectrum we use equation (14.70) from the textbook of Jackson [8]: dU dωdΩ=ω2 4π2c3/bracketleftbigg/integraldisplay dt d3r/hatwiden×j(r, t)eiω(t−(/hatwiden·r)/c)/bracketrightbigg2 , (1) where dUis the radiated energy in angular frequency interval dωemitting into solid angle dΩ,jis the source current density, and /hatwidenis a unit vector towards the observer. The line of charge has equation y=u vx−ut, z = 0, (2) 2so the current density is j=−/hatwideyNeδ(z)δ/parenleftbigg t−x v+y u/parenrightbigg , (3) where Nis the number of electrons per unit length intercepting the xaxis, and e <0 is the electron’s charge. We also consider the effect of the image current, jimage= +/hatwidey(−Ne)δ(z)δ/parenleftbigg t−x v−y u/parenrightbigg . (4) We will find that to a good approximation the image current jus t doubles the amplitude of the radiation. For u∼cthe image current would be related to the retarded fields of th e electron beam, but we avoid this complication when u≪c. Note that the true current exists only for y >0, while the image current applies only for y <0. We integrate using rectangular coordinates, with componen ts of the unit vector ngiven by nx= cos θ, n y= sinθcosφ, and nz= sinθsinφ, (5) as indicated in Fig. 1b. The current impinges only on a length Lalong the xaxis. The integrals are elementary and we find, noting ω/c= 2π/λ, dU dωdΩ=e2N2L2 π2cu2 c2cos2θ+ sin2θsin2φ (1−u2 c2sin2θcos2φ)2 sin/bracketleftBig πL λ(c v−cosθ)/bracketrightBig πL λ(c v−cosθ) 2 . (6) The factor of form sin2χ/χ2appears from the xintegration, and indicates that this leads to a single-slit interference pattern. We will only consider the case that u≪c, so from now on we approximate the factor 1−u2 c2sin2θcos2φby 1. Upon integration over the azimuthal angle φfrom−π/2 toπ/2 the factor cos2θ+ sin2θsin2φbecomesπ 2(1 + cos2θ). It is instructive to replace the radiated energy by the numbe r of radiated photons: dU= ¯hωdN ω. Thus dNω dcosθ=α 2πdω ωN2L2u2 c2(1 + cos2θ) sin/bracketleftBig πL λ(c v−cosθ)/bracketrightBig πL λ(c v−cosθ) 2 , (7) where α=e2/¯hc≈1/137. This result applies whether v < c orv > c . But for v < c , the argument χ=πL λ(c v−cosθ) can never become zero, and the diffraction pattern never achieves a principal maximum. The radiation pattern remain s a slightly skewed type of transition radiation. However, for v > c we can have χ= 0, and the radiation pattern has a large spike at angle θˇCsuch that cosθˇC=c v, which we identify with ˇCerenkov radiation. Of course the side lobes are still prese nt, but not very prominent. 33 Discussion The present analysis suggests that ˇCerenkov radiation is not really distinct from transition radiation, but is rather a special feature of the transition radiation pattern which emerges under certain circumstances. This viewpoint actually is re levant to ˇCerenkov radiation in any real device which has a finite path length for the radiatin g charge. The walls which define the path length are sources of transition radiation which is always present even when the ˇCerenkov condition is not satisfied. When the ˇCerenkov condition is satisfied, the so-called formation length for transition radiation becomes longer t han the device, and the ˇCerenkov radiation can be thought of as an interference effect. IfL/λ≫1, then the radiation pattern is very sharply peaked about th eˇCerenkov angle, and we may integrate over θnoting dχ=πL λdcosθand/integraldisplay∞ −∞dχsin2χ χ2=π (8) to find dNω∼α 2π(Nλ)2dω ωL λu2 c2/parenleftBigg 1 +c2 v2/parenrightBigg . (9) In this we have replaced cos2θbyc2/v2in the vicinity of the ˇCerenkov angle. We have also extended the limits of integration on χto [−∞,∞]. This is not a good approximation for v < c, in which case χ>0 always and dNωis much less than stated. For v=cthe radiation rate is still about one half of the above estimate. For comparison, the expression for the number of photons rad iated in the ordinary ˇCerenkov effect is dNω∼2παdω ωL λsin2θˇC. (10) The ordinary ˇCerenkov effect vanishes as θ2 ˇCnear the threshold, but the superluminal effect does not. This is related to the fact that at threshold ordina ryˇCerenkov radiation is emitted at small angles to the electron’s direction, while in the sup erluminal case the radiation is at right angles to the electron’s motion. In this respect the mo ving spot on an oscilloscope is not fully equivalent to a single charge as the source of the ˇCerenkov radiation. In the discussion thus far we have assumed that the electron b eam is well described by a uniform line of charge. In practice the beam is discrete, wit h fluctuations in the spacing and energy of the electrons. If these fluctuations are too large w e cannot expect the transition radiation from the various electrons to superimpose cohere ntly to produce the ˇCerenkov radiation. Roughly, there will be almost no coherence for wa velengths smaller than the actual spot size of the electron beam at the metal surface, Th us there will be a cutoff at high frequencies which serves to limit the total radiated energy to a finite amount, whereas the expression derived above is formally divergent. Similarly the effect will be quite weak unless the beam current is large enough that Nλ≫1. We close with a numerical example inspired by possible exper iment. A realistic spot size for the beam is 0.3 mm, so we must detect radiation at longer wa velengths. A convenient choice is λ= 3 mm, for which commercial microwave receivers exist. The b andwidth of a candidate receiver is dω/ω = 0.02 centered at 88 GHz. We take L= 3 cm, so L/λ= 10 4and the ˇCerenkov ‘cone’ will actually be about 5◦wide, which happens to match the angular resolution of the microwave receiver. Supposing the electr on beam energy to be 2.5 keV, we would have u2/c2= 0.01. The velocity of the moving spot is taken as v= 1.33c= 4×1010 cm/sec, so the observation angle is 41◦. If the electron beam current is 1 µA then the number of electrons deposited per cm along the metal surface is N∼150, and Nλ∼45. Inserting these parameters into the rate formula we expect a bout 7×10−3detected photons from a single sweep of the electron beam. This suppos es we can collect over all azimuth φwhich would require some suitable optics. The electron beam will actually be swept at about 1 GHz, so we can collect about 7 ×106photons per second. The corresponding signal power is 2 .6×10−25Watts/Hz, whose equivalent noise temperature is about 20 mK . This must be distinguished from the background of thermal ra diation, the main source of which is in the receiver itself, whose noise temperature is a bout 100◦K [9]. A lock-in amplifier could be used to extract the weak periodic signal; an integra tion time of a few minutes of the 1-GHz-repetition-rate signal would suffice assuming 100 % collection efficiency. Realization of such an experiment with a Tektronix 7104 osci lloscope would require a custom cathode ray tube that permits collection of microwav e radiation through a portion of the wall not coated with the usual metallic shielding laye r [10]. 4 Appendix: Bremsstrahlung Early reports of observation of transition radiation were c onsidered by sceptics to be due to bremsstrahlung instead. The distinction in principle is that transition radiation is due to acceleration of charges in a medium in response to the far fi eld of a uniformly moving charge, while bremsstrahlung is due to the acceleration of t he moving charge in the near field of atomic nuclei. In practice both effects exist and can be sep arated by careful experiment. Is bremsstrahlung stronger than transition radiation in th e example considered here? As shown below the answer is no, but even if it were we would the n expect a ˇCerenkov-like effect arising from the coherent bremsstrahlung of the elect ron beam as it hits the oscilloscope faceplate. The angular distribution of bremsstrahlung from a nonrelat ivistic electron will be sin2θ withθdefined with respect to the direction of motion. The range of a 2.5-kev electron in, say, copper is about 5 ×10−6cm [11] while the skin depth at 88 GHz is about 2 .5×10−5cm. Hence the copper is essentially transparent to the backward hemisphere of bremsstrahlung radiation, which will emerge into the same half space as the t ransition radiation. The amount of bremsstrahlung energy dUBemitted into energy interval dUis just Y dU where Yis the so-called bremsstrahlung yield factor. For 2.5-keV e lectrons in copper, Y= 3×10−4[11]. The number dNof bremsstrahlung photons of energy ¯ hωin a bandwidth dω/ω is then dN=dUB/¯hω=Y dω/ω . For the 2% bandwidth of our example, dN= 6×10−6per beam electron. For a 3-cm-long target region there will be 50 0 beam electrons per sweep of the oscilloscope, for a total of 3 ×10−4bremsstrahlung photons into a 2% bandwidth about 88 GHz. Half of these emerge from the faceplate as a backgroun d to 7×10−3transition- radiation photons per sweep. Altogether, the bremsstrahlu ng contribution would be about 1/50 of the transition-radiation signal in the proposed exp eriment. 55 References [1] O. Heaviside, Electrical Papers , The Electrician Press (London, 1892), reprinted by Chelsea Publishing Company (Bronx, New York, 1970), Vol. II , pp. 492-494, 496-499, 515-518; Electromagnetic Theory , The Electrician Press (London, 1893, 1899, 1912), reprinted by Chelsea Publishing Company (Bronx, New York, 1 971), Vol. II, pp. 533- 555, Vol. III, pp. 15-130, 373-380, 485-489. [2] Th. Des Coudres, “Zur Theorie des Kraftfeldes elektrish er Ladungen, die sich mit ¨Uberlichtgeschwindigkeit bewegen”, Arch. N´ eer. (Harlem )5652-664 (1900). [3] A. Sommerfeld, “Simplified Deduction of the Field and For ces of an Electron, Moving in Any Given Way”, K. Akad. Weten. Amsterdam 7, 346-367 (1905); “Zur Elektronentheo- rie III. ¨Uber Lichtgeschwindigkeits- und ¨Uberlichtgeschwindigkeits-Elektronen”, Nachr. Kgl. Ges. d. Wiss. G¨ ottingen 5, 201-235 (1905). [4] P.A. ˇCerenkov, C.R. Acad. Sci. U.S.S.R. 2, 451 (1934). [5] I. Tamm, “Radiation Emitted by Uniformly Moving Electro ns”, J. Phys. U.S.S.R. 1, 439-454 (1939). [6] B.M. Bolotovskii and V.L. Ginzburg, “The Vavilov- ˇCerenkov Effect and the Doppler Effect in the Motion of Sources with Superluminal Velocity in Vacuum”, Usp. Fiz. Nauk 106, 577-592 (1972); Sov. Phys. Uspekhi 15, 184-192 (1972). [7] H. Springer, “Breakthroughs Throughout Push Scope to 1 G HZ”, Electronic Design 2, 60-65 (Jan. 18, 1979). [8] J.D. Jackson, Classical Electrodynamics , 2nd ed. (Wiley, New York, 1975). [9] D.T. Wilkinson (private communication). [10] D. Stoneman, New Ventures Engineering, Tektronix, Inc . (private communication). [11] Extrapolated from the table on p. 240 of Studies in Penetration of Charged Particles in Matter , National Academy of Sciences – National Research Council, PB-212 907 (Washington, D.C., 1964). 6

1A new volatility term in the theory of options with transaction costs Alexander Morozovsky1 Bridge, 57/58 Floors, 2 World Trade Center, New York, NY 10048 E-mail: alex@nyc.bridge.com Phone: (212) 390-6126 Fax: (212) 390-6498 The introduction of transaction costs into the theory of option pricing could lead not only to the change of return for options, but also to the change of the volatility. On the base of assumption of the portfolio analysis, a newequation for option pricing with transaction costs is derived. A new solution for the option price is obtained for the time close to expiration date. Keywords: Option, Transaction costs, Market, Volatility. 1 The ideas expressed in this article are the author’s only and do not necessarily correspond to the views of Bridge.2Introduction. The problem of option pricing with transaction costs is one of the most interesting and important problems in the theory of option pricing [1,3]. The problem could be rephrased as a try to extend the results from Black-Scholes theory to the incomplete market. In the standard approach thepossible return for the option is decremented by the transaction costs (theresults derived by Leland and Wilmott [3] ). However, the introduction of transaction costs could also lead to the change of volatility. Because of this, new terms could appear in the option equation. On the base of this idea thenew equation for option pricing with transaction costs is derived in this article. The new solutions are obtained and the conditions of their applicability are discussed. Method. First of all, let us reproduce the derivation of Black - Scholes equation for call option on the base of risk-return argument (for simplicity’s sake, we will consider the European call option). In order to do this we will write achange in security value in the specific form. We will assume that dP in time3is determined by the sum of two factors: a factor that depends on the term proportional to dt and the factor proportional to dz: dP = P t*dt + P z*dz, (1) where t is time and z is Wiener process. Now we will use these notations in order to connect risk and return for different instruments. Proportionality of risk and return in these terms could be expressed in the form: ||zt PrP Pk−= (2) For option we could write P t as: 22 2 2 21 SPSSPStPPt∂∂+∂∂+∂∂= σ µ (3) and P z : SPS Pz∂∂=σ (4) Because the investor will make sure that relationship between risk and return for option and stock would be the same we could write that : ) (σµλλλr s s op−== (5) where opλis the market price of risk for option and sλis the market price of risk for stock. Combining (2) and (5) we could immediately obtain: σµr PrP P zt −=− || (6)4Substituting P t and | P z | by (3) and (4) and considering that SP ∂∂ > 0 for call options we immediately get: σµ σσ µr SPSrPSPSSPStP −= ∂∂−∂∂+∂∂+∂∂ 22 22 21 (7) From (7) we receive usual Black - Scholes equation: rPSPSSPrStP=∂∂+∂∂+∂∂ 222 21σ (8) Now, let's consider the same equation in the case of transaction costs. We will assume that we are using option for hedging and we are changing portfolio all the time. In this case: we need to find new changed factors fromequation (2) - new return and new volatility. In order to do this let's rewrite differential dP into a new form: ), || ( )21(22 22Ldt dzA dzSPS Ldt dtSPSSPStPdP + −∂∂+−∂∂+∂∂+∂∂= σ σ µ (9) or 1 0 dP Ldt dP dP +−= , (10) where dtSPSSPrStPdP )21(222 0∂∂+∂∂+∂∂= σ , (11) Ldt dzA dzSPS dP + −∂∂= ||1σ , (12)5dt SPSk L SPSk A12| |21|, |21 22 2 22 2 πσ σ ∂∂= ∂∂= , (13) and A, L – are terms connected with transactions cost. From (13) one can immediately obtain that on average: 0 |,|1>=< = dP dzA Ldt . (14) This means that the term dP 1 corresponds to volatility term in (2). When dP 0 - Ldt is proportional to the usual return for derivatives with transaction costs [3], dP 1 is constructed from different terms, one of which is the usual volatility term and the others appear because of transaction costs.Let's calculate an average value of dP 12 <dP 12> : dtdz AL dtL dz A dtSPS dP | | 2 ||2 2 2 2 22 22 2 1 −+ +∂∂>=< σ (15) because all terms, proportional to dz disappear (<dz> = 0). Let's remind also that: <A|dz|> = Ldt (16) However, because |dz|2 = dz2 -> dt, we could rewrite (15) as: dtL dtL dtA dtSPS dP2 2 2 2 22 2 1~2~)( −++∂∂>=< σ , (17) where dtLL=~.6This means that ><2 1dP is different from dtSPS2 22)(∂∂σ : dtL A dtSPS dP )~( )(2 2 2 22 2 1 −+∂∂>=< σ (18) The origin of a new term in volatility is simple. Even, if < A|dz| – Ldt > = 0, it doesn't mean that < (A|dz| – Ldt)2 > = 0. From (18) and (14), (17) we immediately obtain: dtSPSk dtSPS dP | |)21(21)(22 2 2 22 2 1∂∂− +∂∂>=<πσ σ (19) Because we could write dP 0 – Ldt = P t*dt and dP 12 = P z2*dt, we could apply (6) for new P t and P z: σµr PrPLP zt −=−− (20) Results Now, we could rewrite (20) as: σµ πσ σσ µr SPSkSPSrPL SPSSPStP −= − ∂∂+∂∂−− ∂∂+∂∂+∂∂ 2 2 22 2 222 22 )21()21() (21 (21) This a new equation for P (price of option with transaction costs). This equation confirms that the value of option price with transactions costs changed not only because transactions costs lead to changed return, but7because transactions costs could also influence volatility. This equation equally shows the importance of both. Let us find the solution of this equation for two special cases:1. Small transaction costs (k – small and we could consider )21(| |21 22 2 πσ − ∂∂= SPSk C as much smaller term than | |SPS B∂∂=σ : C << B (22) 2. Price of option near time of expiration. In small interval close to time of expiration it is possible to assume [for example in [3], for the case of Asian options] that dominant term is proportional to 22 SP ∂∂ in comparison with SP ∂∂. Because of this we will consider situation when B << C, and denominator in (21) is mostly determined by the term proportional to | |22 SP ∂∂. Small transactions costs. Substituting (22) into (21) we obtain: σµσ µr BCBrPL SPSSPStP −= +−− ∂∂+∂∂+∂∂ )211(21 2222 22 (23) or8 2 22 2 222 22|) |)21(21( ||21||) (21 SPSk SPSr SPSr rPLSPSSPStP ∂∂− ∂∂−+∂∂−=−−∂∂+∂∂+∂∂ πσ σµµ σµ (24) The equation similar to well known equation ( [3] ) from the theory of options with transactions costs could be obtained from (24) in case we will completely neglect the last term D. Than the equation (22) become: | |) (21 22 22 SPSr rPLSPSSPStP ∂∂−=−−∂∂+∂∂+∂∂µ σ µ (25) This equation is similar to the new equation suggested in the paper [2], but with the influence of transactions costs (term L). Also it is usual equationfrom the theory of option pricing with transactions costs. Time close to the expiration. If we completely neglect SP ∂∂ in the denominator of (21) we could get the following equation: )21(| |21 21 22 2 22 22 πσσµσµ − ∂∂−=−− ∂∂+∂∂+∂∂ SPSkrrPL SPSSPStP (26)9This equation is very similar to the usual Black-Scholes equation, and if we assume that 022 >∂∂ SP (what is correct for options without transactions costs), the equation (26) could be rewritten as: rPL SPkrSSPStP=− ∂∂−−− +∂∂+∂∂ 22 2 2))21( (21σπσµσ µ (27) The solution of this equation is Black-Scholes solution for the case of options with dividends, when we need to assume that one of parameters in this formula (rate of dividend) is negative: q = -(µ - r) (28) and volatility parameter σ~ is equal to: )2)21)( ( (~ 2 tkk rπδσπµσσ −−−−= (29) The solution would be Black - Scholes solution: P = SN(d 1) - Xe-(r-q)tN(d 2), (30) where volatility parameter used in this formula is σ~.10References. 1. Antonio Paras (1995) Non-Linear Diffusion Equations in Mathematical Finance: A Study of Transaction Costs and Uncertain Volatility, NewYork. 2. Ralf Korn, Paul Wilmott (1998) A General Framework for hedging and speculating with options, International Journal of Theoretical andApplied Finance, vol. 1, N4 , 507 –522. 3. Wilmott P. et al. (1993) Option Pricing: Mathematical Models and Computation, Oxford, Oxford Financial Press.

arXiv:physics/0003055v1 [physics.atom-ph] 23 Mar 2000LETTER TO THE EDITOR Enhanced dielectronic recombination of lithium-like Ti19+ions in external E×Bfields T Bartsch †, S Schippers †∝bardbl, M Beutelspacher ‡, S B¨ ohm †, M Grieser ‡, G Gwinner ‡, A A Saghiri ‡, G Saathoff ‡, R Schuch §, D Schwalm ‡, A Wolf ‡and A M¨ uller † †Institut f¨ ur Kernphysik, Universit¨ at Gießen, 35392 Gieß en, Germany ‡Max-Planck-Institut f¨ ur Kernphysik and Physikalisches I nstitut der Universit¨ at Heidelberg, 69117 Heidelberg, Germany §Department of Physics, Stockholm University, 10405 Stockh olm, Sweden Abstract. Dielectronic recombination(DR) of lithium-like Ti19+(1s22s) ions via 2 s→2p core excitations has been measured at the Heidelberg heavy i on storage ring TSR. We find that not only external electric fields (0 ≤Ey≤280 V/cm) but also crossed magnetic fields (30 mT ≤Bz≤80 mT) influence the DR via high-n2pjnℓ-Rydberg resonances. This result confirms our previous findi ng for isoelectronic Cl14+ions [Bartsch T et al,Phys. Rev. Lett. 82, 3779 (1999)] that experimentally established the sensitivity of DR to E×Bfields. In the present investigation the larger 2 p1/2−2p3/2fine structure splitting of Ti19+allowed us to study separately the influence of external fields via the tw o series of Rydberg DR resonances attached to the 2 s→2p1/2and 2 s→2p3/2excitations of the Li-like core, extracting initial slopes and saturation fiel ds of the enhancement. We find that for Ey/greaterorsimilar80 V/cm the field induced enhancement is about 1.8 times stronger for the 2 p3/2series than for the 2 p1/2series. PACS numbers: 34.80.Lx,31.50.+w,31.70.-f,34.80.My Dielectronic recombination (DR) is a fundamental electron -ion collision process well known to be important in astrophysical and fusion plasm as (Dubau and Volont´ e 1980). It proceeds in two steps e−+Aq+→[A(q−1)+]∗∗→A(q−1)++hν (1) where in the first step the initially free electron is capture d into a bound state nℓ of the ion with simultaneous excitation of a core electron. T his dielectronic capture (DC, time inverse of autoionization) can only occur if the en ergyEof the incident free electron in the electron-ion center-of-mass (c.m.) fr ame matches the resonance condition E=Eres=Ed−Eiwhere EiandEdare the total energies of all bound electrons in the initial and in the doubly excited state, res pectively. Employing the principle of detailed balance the DC cross section can be calculated from the autoionization rate Aa(d→i) for a transition from the doubly excited state dto the initial state i: σ(DC)(E) =S0gd 2gi1 EAa(d→i)Γd (E−Eres)2+ Γ2 d/4(2) /bardblemail: Stefan.E.Schippers@strz.uni-giessen.deLetter to the Editor 2 withS0= 7.88×10−31cm2eV2s, statistical weights gdandgiand Γ d=/planckover2pi1[/summationtext kAa(d→ k) +/summationtext f′Ar(d→f′)] denoting the total width of the doubly excited state d. The summation indices kandf′run over all states which from dcan be either reached by autoionization or by radiative transitions with rates Aa(d→k) and Ar(d→f′), respectively. In the second step of reaction (1) the new charge state is stab ilized by photon emission from the intermediate doubly excited state, there by transferring the ion into a final state fbelow the first ionization limit. This radiative stabilizat ion competes with autoionization which would transfer the ion back into i ts initial charge state with the net effect being resonant electron scattering. Acco rdingly, in order to obtain the cross section for DR one has to multiply the DC cross secti on from equation (2) by the branching ratio [/summationtext fAr(d→f)]/Γdfor radiative stabilization. Integrating the resulting expression over the c. m. energy and assuming Γ d≪Eresyields the DR resonance strength due to the intermediate state din the isolated resonance approximation (Shore 1969), i.e. ¯σd=S0gd 2gi2π EresAa(d→i)/summationtext fAr(d→f)/summationtext kAa(d→k) +/summationtext f′Ar(d→f′). (3) In case of ∆ n= 0 DR of Li-like ions, i.e. for 1 s22s→1s22pcore excitations, the dominant decay channels of the doubly excited intermediate state (for very high n) can be identified as 2 pjnℓ→2snℓradiative and 2 pjnℓ→2sEresℓ′autoionizing transitions with rates denoted as ArandAnℓ, respectively. Since the radiative transition only involves the excited core electron, its rate Arto a good approximation is independent of the quantum numbers nandℓof the excited Rydberg electron. Neglecting all other transitions such as 2 p3/2nℓ→2p1/2E′ℓ′or 2pjnℓ→2pjn′ℓ′, equation (3) simplifies to ¯σnℓ= (2j+ 1)(2 ℓ+ 1)S0π EnAnℓAr Anℓ+Ar≈(2j+ 1)(2 ℓ+ 1)S0π EnA<(4) using the Rydberg resonance energy Eres=En,gd= 2(2 j+ 1)(2 ℓ+ 1) and gi= 2; in the r.h.s approximation A<denotes the lesser of ArandAnℓ. Autoionizing rates decrease as ∝n−3and even more rapidly with ℓ, such that at a given nthe relation Anℓ> Arholds only for nℓ-Rydberg states with angular momentum ℓbelow a limit ℓc. Thus ( ℓc+ 1)2sublevels dominantly contribute to DR for a given core state jand consequently for the n-manifold 2 pjnℓof doubly excited states the resonance strength is given as ¯σn≈(2j+ 1)[ℓc(n) + 1]2S0π EnAr. (5) Within this ‘counting of states’ picture the effect of extern al electric fields on DR is readily explained. In external electric fields Stark mixing of high- ℓwith low- ℓlevels occurs. This yields autoionization rates which are lower fo r low-ℓand higher for high- ℓ states as compared to the field free situation. The net effect i s an increase of ℓc, i.e. an increase of the number of states participating in DR. Sinc e high- nRydberg states are more easily perturbed by external electric fields than lo w-nstates the electric field induced enhancement of DR is stronger for higher- n2pjnℓDR resonances. This effect of external electric fields on DR was recognized early by Burgess and Summers (1969) and Jacobs et al (1976). Electric field enhancement of DR was subsequently found in numerous theoretical calculatio ns (Hahn 1997). The first clear experimental verification of this effect has been given by M¨ uller et al (1986) who investigated DR in the presence of external fields (DRF) o f singly charged Mg+Letter to the Editor 3 ions under controlled conditions. Further DRF experiments with multiply charged C3+ions (Young et al 1994, Savin et al 1996) and Si11+ions (Bartsch et al 1997) also revealed drastic DR rate enhancements by electric field s. Especially the Si11+ experiment which employed merged electron and ion beams at a heavy ion storage ring equipped with an electron cooler produced results with unprecedented accuracy, enabling a detailed comparison with theory. Whereas the ove rall agreement between experiment and theory as for the magnitude of the effect was fa ir, discrepancies remained in the functional dependence of the rate enhanceme nt on the electric field strength (Bartsch et al1997). This finding stimulated theoretical investigations of the role of the additional magnetic field which in storage ring DR experiments is always present, since it guides and confines the electron beam withi n the electron cooler. In a model calculation Robicheaux and Pindzola (1997) found th at in a configuration of crossed EandBfields indeed the magnetic field through the mixing of mlevels influences the rate enhancement generated by the electric fie ld through the mixing ofℓlevels. More detailed calculations (Griffin et al 1998a, Robicheaux et al 1998) confirmed these results. It should be noted that in theoretic al calculations by Huber and Bottcher (1980) no influence of a pure magnetic field ( E= 0) on DR was found up to at least B= 5 T. Inspired by these predictions we previously performed stor age ring DRF experiments using Li-like Cl14+ions and crossed EandBfields (Bartsch et al 1999) where we clearly discovered a distinct effect of the magnetic field strength on the magnitude of the E-field enhanced DR rate. Shortly after that Klimenko and coworkers (1999) experimentally verified that for the m-mixing to occur the crossed EandBarrangement is essential. For the case of parallel BandEfields, where m remains a good quantum number, they did not observe any influe nce of the magnetic field on the measured recombination signal. The aim of the present investigation with Li-like Ti19+is to confirm the novel E×Bfield effect on DR for a heavier Li-like ion. Because of the stro ng scaling of the fine-structure splitting with the nuclear charge the T i18+(1s22p1/2nℓ) and Ti18+(1s22p3/2nℓ) Rydberg series of DR resonances are well separated in energ y. The corresponding series limits occur at 40.12 eV and 47.81 eV, r espectively (Hinnov et al 1989). This energy difference is large enough that, in contra st to the Cl14+experiment, here our experimental resolution permits to study the effect of external fields on both Ti18+(1s22pjnℓ) Rydberg series individually. The experiments were carried out at the heavy ion storage rin g TSR of the Max- Planck-Institut f¨ ur Kernphysik in Heidelberg. Here we onl y give a brief account of the experimental procedure for DRF measurements. Details will be given in a forthcoming publication by Schippers et al(2000, and references therein) on DRF measurements with lithium-like Ni25+ions. Beams of48Ti19+ions with intensities up to almost 80 µA were stored in the ring at energies of 4.6 MeV/u. The ion beams were cooled by int eraction with a velocity-matched cold beam of electrons which was confined b y a magnetic field B; the direction of Bdefines that of the electron beam. The electron beam diameter was 30 mm, while that of the cooled ion beam was of the order of 2 mm. First, as in the standard tuning procedure of the electron cooler, the elect ron beam was steered so that, along the straight interaction region of 1.5 m length, the ion beam travelled on the electron beam center line and the guiding field Bpointed exactly along the ion beam; this minimized the electric field in the frame of the ion s originating from space charge and motional ( v×B) fields. A reasonably ‘electric-field free’ measurement ofLetter to the Editor 4 the DR rate coefficient (with an estimated residual field of at m ost±10V/cm) could then be obtained at high energy resolution by switching the e nergy of the electrons in the cooler to different values. The energy range thus cover ed in the center-of- mass frame includes all Ti19+(1s22pjnℓ) ∆n= 0 DR resonances due to the 2 s→2pj core excitations. Recombined Ti18+ions were magnetically separated from the parent Ti19+beam and detected with an efficiency ≥95% downbeam from the cooler behind the first bending magnet. Controlled motional electric fields in the frame of the ions w ere then applied by superimposing in the interaction region a defined transvers e (horizontal) magnetic fieldBx≪Bzin addition to the unchanged longitudinal field Bzalong the ion beam direction ( z). This field was generated by the electron-beam steering coi ls along the complete straight section of the electron cooler and create d a motional electric field Ey=vBxin the frame of the ions at a beam velocity v; the total magnetic field strength ( B2 x+B2 z)1/2, however, remained almost unchanged. Progressively differ ent electric fields were produced by varying the transverse magn etic field strength Bx. At a given transverse field Bxthe electron beam (following the magnetic field lines) and the ion beam are misaligned by the small angle Bx/Bzso that the distance of the ion beam from the center of the electron beam varies alo ng the interaction region. This leads to unwanted effects due to the electron spa ce charge: (I) a variation of the (temperature average) relative velocity b etween electrons and ions along the interaction path, resulting in a degraded energy r esolution; (II) creation of an additional electric field Exwhich, in contrast to the imposed field Ey, varies along the interaction region. Low electron densities were c hosen in order to keep these effects small. With electron currents of 20 mA, measure ments were performed at an electron density of 6 .2×106cm−3. The cooler was operated at longitudinal field strengths Bz= 30, 41.8, 60.0 and 80.1 mT. Transverse fields of |Bx| ≤0.7mT (measured with an uncertainty of ±3%) were applied, corresponding to controlled motional electric fields |Ey|up to 280 V/cm; in all measurements the misalignment angle was kept below |Bx/Bz|/lessorsimilar0.02. The ratio Ex/Eyof the unwanted electric space charge field and the applied motional field is expected t o vary linearly along the interaction region with |Ex/Ey|remaining always below 0.07 for all measurements with different experimental parameters. Before each energy scan with an imposed electric field Ey, ions were injected into the ring, accumulated and then cooled for 1s. After that, the cathode potential of the cooler was offset from cooling by about 1 kV (corresponding to 55 eV in the center- of-mass frame) and then, the steering coils were set to produ ce a defined transverse magnetic field Bx. Next, the center-of-mass energy was ramped down from about 55 eV to 1 eV within 4 s thus completing a first mini-cycle. Afte r new ion injection and cooling (at Bx= 0), the next magnetic steering field Bx(i.e., next Ey) was automatically set and a new energy scan started. The mini-cy cles, covering one complete energy scan each, were repeated for a set of pre-cho sen magnetic steering fields. A grand cycle through typically 11 values of Ey=vBxthus took about 2 minutes and such cycles were repeated until a satisfying lev el of statistical uncertainty (below 3% per channel) had been reached. Sets of recombination rate measurements were made for differ ent longitudinal fields Bz. Using measured beam currents the spectra were calibrated, reaching an uncertainty of ±20% for absolute and ±5% for relative rate coefficients. The center- of-mass energies were determined (with uncertainties belo w±1%) from the average relative velocities of electrons and ions, accounting for t he angle between the electronLetter to the Editor 5 10 20 30 40 50 6005101520 n = 11 1213 n = 12 2p3/22p1/2|Ey| Recombination rate coefficient (10-10 cm3/s) Relative energy (eV) Figure 1. Absolute recombination rate coefficients measured for 4.6 Me v/u Ti19+ ions at applied motional electric fields |Ey|increasing nearly linearly from 0 to 265 V/cm; longitudinal magnetic field Bz=69 mT, electron density 6 ×106cm−3. Energetic positions of the 2 p1/2nland 2 p3/2nlresonances according to the Rydberg formula are indicated. and the ion beam due to the applied transverse field Bx. A typical set of measurements is presented in figure 1 and show s the two series of Rydberg resonances converging to the 2 p1/2and 2 p3/2core excitation limits. A significant enhancement of the rate coefficient with increasi ng electric field Eyis observed for high Rydberg states n/greaterorsimilar27, while for the lower-lying resonances the rate coefficient remains constant. The enhancement of the DR via high Rydberg states is quantifie d by extracting rate coefficients integrated over different energy regions of the measured spectra. The integrals I1/2(Ey, Bz) and I3/2(Ey, Bz) extend over the energy ranges 33.4–40.57 eV and 40.57–50 eV, respectively, and represent the high-Rydb erg contributions of the 2p1/2nℓand 2 p3/2nℓseries of Rydberg resonances with n≥27. For normalization purposes we also monitor the integral I0(integration range 4–24 eV) comprising DR contributions from lower n. It should be noted that the maximum quantum number of Rydberg resonances contributing to the measured recombina tion rate is limited by field ionization in the charge analyzing dipole magnet. Taking in to account also radiative decay of high Rydberg states on the way from the cooler to the d ipole magnet, we estimate the maximum quantum number to be nc= 115. The high-Rydberg contributions Ij(Ey, Bz) (j= 1/2,3/2) monotonically increase with|Ey|, while the lower- ncontribution I0(Ey, Bz) remains constant. In order to provide a quantity for the following discussion that is inde pendent of the normalization of the individual spectra, we consider ratios Ij/I0of the high- nto the low- nLetter to the Editor 6 1,01,52,02,5 1,01,52,02,5 1,01,52,02,5 0 50 100 150 200 250 3001,01,52,02,51,01,52,02,5 1,01,52,02,5 1,01,52,02,5 0 50 100 150 200 250 3001,01,52,02,52p1/2, Bz = 30.0 mT 2p1/2, Bz = 41.8 mT 2p1/2, Bz = 60.0 mTRate enhancement factor rj 2p1/2, Bz = 80.1 mT Electric field |Ey| (V/cm) 2p3/2, Bz = 30.0 mT 2p3/2, Bz = 41.8 mT 2p3/2, Bz = 60.0 mT 2p3/2, Bz = 80.1 mT Electric field |Ey| (V/cm) Figure 2. Measured field enhancement factors (cf. equation (6)) r1/2(left) and r3/2(right) as a function of the applied motional electric field s trength |Ey|for different longitudinal magnetic field strengths Bz= 30.0,41.8,60.0 and 80 .1 mT (from top to bottom). Triangles pointing up (down) mark data points which have been measured with positive (negative) Ey. The full lines have been fitted to the data points (cf. equation (7)). The dashed straight lines ar e tangents to the fit at Ey= 0.Letter to the Editor 7 contribution in a single DR spectrum. The electric-field enh ancement factor rj(Ey, Bz) =Cj(Bz)Ij(Ey, Bz) I0(Ey, Bz)(6) then directly measures the influence of the external electri c field on the DR rates via high Rydberg states. The constants Cj(Bz) have been chosen such that fits to the data points (see below) yield r(fit) j(0, Bz) = 1.0. The field enhancement factors r1/2(Ey, Bz) and r3/2(Ey, Bz) found for different Bzare shown in figure 2 as a function of |Ey|. The enhancement factors turn out to be independent of the sign of Ey, as expected. The formula r(fit) j(Ey, Bz) = 1 + sj(Bz)Ej(Bz){1−exp[−Ey/Ej(Bz)]} (7) which we have fitted to the measured enhancement factors, pro vides a useful parameterization of our data. The parameters which have bee n varied during the fits (at fixed values of Bz) are the saturation field Ej(Bz) and the initial slope sj(Bz); tangents to r(fit)atEy= 0, representing the initial slopes are also displayed in fig ure 2. We note that the Ey= 0 data points are slightly above the fitted lines. This is due to the fact that zero applied field Ey= 0 still implies a residual electric field /lessorsimilar10V/cm so that the measured dependence of rj(Ey, Bz) near Ey= 0 is washed out to some extent. At higher electrical field strengths the m easured data points drop below the (dashed) straight lines (cf. figure 2). This is an in dication that the electric field effect is subject to saturation which occurs at higher el ectric field strength where the mixing of ℓ-levels is complete. Presently, higher electric fields are n ot accessible in our DR experiment. The fit parameter Ej(Bz) indicates how fast the saturation regime will be reached. The values for the parameters sj(Bz) and Ej(Bz), which along with their uncertainties have been obtained from the fits, are displaye d in figures 3a and 3b, respectively, as a function of the magnetic field Bz. Both sjandEjexhibit a strong dependence on the strength of the magnetic guiding field. The slopes sjdecrease with increasing magnetic field both for the 2 p1/2and 2p3/2series of Rydberg resonances. This confirms our recent finding for Cl14+ions (Bartsch et al1999) where the existence of a sensitivity of DR to external magnetic fields in an E×Bfield configuration was experimentally demonstrated for the first time. The paramet ersEjincrease with increasing magnetic field strength, i.e. at higher Bzthe saturation regime will be reached at higher electric fields |Ey|. While the parameters Ejare not markedly different for the 2 p1/2and 2p3/2series, the slopes s3/2are steeper than the slopes s1/2(open and closed circles in figure 3a, respectively), i.e. the relative increase of the DR line str ength is stronger for the 2 p3/2 series of Rydberg resonances than for the 2 p1/2series. For a comparison to recent theoretical predictions (Griffin et al1998a, 1998b) we consider the ratio R(Ey, Bz) =I3/2(Ey, Bz)−I3/2(0, Bz) I1/2(Ey, Bz)−I1/2(0, Bz)(8) of absolute Eyinduced DR rate enhancements for the 2 p3/2and 2p1/2series of Rydberg resonances, which is practically independent of the integr ation ranges used for the determination of I1/2andI3/2as long as they cover nearly all DR resonances affected by the external fields. As a function of Eythe ratio Rrises up to |Ey|= 80 V/cm and then essentially stays constant at higher electric field s. Values ∝an}bracketle{tR∝an}bracketri}htaveraged over the interval 80 V/cm ≤ |Ey| ≤280 V/cm are plotted in figure 3c which shows that ∝an}bracketle{tR∝an}bracketri}ht= 1.77±0.06 independent of Bz.Letter to the Editor 8 30 40 50 60 70 80012(c)〈R〉 30 40 50 60 70 800246810(b) Ej (100 V/cm) Magnetic field Bz (mT)30 40 50 60 70 800123456789(a) sj (10-3 cm/v) Figure 3. Dependence of the fit parameters (cf. equation (7)) s1/2(closed circles), s3/2(open circles), E1/2(closed squares) and E3/2(open squares) on the longitudinal magnetic field strength Bz. The error bars were obtained from the fits. The lines are drawn to guide the eye. The diamonds in p anel (c) represent ratios R(cf. equation (8)) averaged over the electric field interval 80 V/cm ≤ |Ey| ≤280 V/cm as a function of Bz. The error bars correspond to one standard deviation. The dashed straight line represe nts the mean value 1.77±0.06. In view of the fact that for a given nthe manifold of 2 p3/2nℓresonances contains twice as many sublevels that can be mixed by external fields as the manifold of 2 p1/2nℓ resonances (8 n2vs. 4n2, cf. equation (4)) one expects a value of 2 for the ratio ∝an}bracketle{tR∝an}bracketri}ht. We here observe a ratio somewhat lower than 2 similar to the va lue∝an}bracketle{tR∝an}bracketri}ht ∼1.5 found in our experiments with lithium-like Ni25+ions (Schippers et al2000). In calculations for lithium-like Si11+ions (Griffin et al 1998a) and C3+ions (Griffin et al 1998b) ratios even less than 1 have been found. This has been attribu ted to the electrostatic quadrupole-quadrupole interaction between the 2 pand the nℓRydberg electron in the intermediate doubly excited state, which more effectively l ifts the degeneracy between the 2p3/2nℓthan between the 2 p1/2nℓlevels. Our experimental results suggest that this effect might be weaker than theoretically predicted. Our data emphasize the relevance of the effect of small magnet ic fields on DR via high Rydberg levels in conjunction with the well-known elec tric-field enhancement. This result bears important implications upon the charge st ate balance of ions in astrophysical and laboratory plasmas where both, electric and magnetic fields are ubiquitous. We gratefully acknowledge support by BMBF, Bonn, through co ntracts No. 06 GI 848 and No. 06 HD 854 and by the HCM Program of the European Co mmunity.Letter to the Editor 9 References Bartsch T, M¨ uller A, Spies W, Linkemann J, Danared H, DeWitt D R, Gao H , Zong W , Schuch R, Wolf A, Dunn G H, Pindzola M S and Griffin D C 1997 Phys. Rev. Lett. 792233 Bartsch T, Schippers S, M¨ uller A, Brandau C, Gwinner G, Sagh iri A A, Beutelspacher M, Grieser M, Schwalm D, Wolf A, Danared H and Dunn G H 1999 Phys. Rev. Lett. 823779 Burgess A and Summers H P 1969 Astrophys. J. 1571007 Dubau J and Volont´ e S 1980 Rep. Prog. Phys. 43199 Griffin D C, Robicheaux F and Pindzola M S 1998a Phys. Rev. A572798 Griffin D C, Mitnik D, Pindzola M S and Robicheaux F 1998b Phys. Rev. A584548 Hahn Y 1997 Rep. Prog. Phys. 60691 Hinnov E and the TFTR operating team, Denne B and the JET opera ting team 1989 Phys. Rev. A 404357 Huber W A and Bottcher C 1980 J. Phys. B: At. Mol. Phys. 13L399 Jacobs V L, Davies J and Kepple P C 1976 Phys. Rev. Lett. 371390 V. Klimenko, L. Ko, and T. F. Gallagher 1999 Phys. Rev. Lett. 833808 M¨ uller A, Beli´ c D S, DePaola B D, Djuri´ c N, Dunn G H, Mueller D W and Timmer C 1986 Phys. Rev. Lett. 56127 Robicheaux F and Pindzola M S 1997 Phys. Rev. Lett. 792237 Robicheaux F, Pindzola M S, and Griffin D C 1998 Phys. Rev. Lett. 801402 Savin D W, Gardner L D, Reisenfeld D B, Young A R, and Kohl J L 199 6Phys. Rev. A53280 Schippers S, Bartsch T, Brandau C, M¨ uller A, Gwinner G, Wiss ler G, Beutelspacher M, Grieser M, Wolf A and Schuch R 2000 to be published Shore B W 1969 Astrophys. J. 1581205 Young A R, Gardner L D, Savin D W, Lafyatis G P, Chutjian A, Blim an S and Kohl J L 1994 Phys. Rev.A49357

arXiv:physics/0003056v1 [physics.optics] 23 Mar 2000Axicon Gaussian Laser Beams Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princet on, NJ 08544 (March 14, 2000) 1 Problem Deduce an axicon solution for Gaussian laser beams, i.e., one with radial polarization of the electric field. 2 Solution If a laser beam is to have radial transverse polarization, th e transverse electric must vanish on the symmetry axis, which is charge free. However, we can ex pect a nonzero longitudinal electric field on the axis, considering that the rays of the be am that converge on its focus each has polarization transverse to the ray, and hence the projec tions of their electric fields onto the axis all have the same sign. This contrasts with the case o f linearly polarized Gaussian laser beams [2, 3, 4, 5] for which rays at 0◦and 180◦azimuth to the polarization direction have axial electric field components of opposite sign. The lo ngitudinal electric field of axicon laser beams may be able to transfer net energy to charged part icles that propagate along the optical axis, providing a form of laser acceleration [6, 7, 8]. Although two of the earliest papers on Gaussian laser beams [ 9, 10] discuss axicon modes (without using that term, and without deducing the simplest axicon mode), most subse- quent literature has emphasized linearly polarized Gaussi an beams. We demonstrate that a calculation that begins with the vector potential (sec. 2. 1) leads to both the lowest order linearly polarized and axicon modes. We include a discussio n of Gaussian laser pulses as well as continuous beams, and find condition (8) for the tempo ral pulse shape in sec. 2.2. The paraxial wave equation and its lowest-order, linearly p olarized solutions are reviewed in secs. 2.3-4. Readers familiar with the paraxial wave equati on for linearly polarized Gaussian beams may wish to skip directly to sec. 2.5 in which the axicon mode is displayed. In sec. 2.6 we find an expression for a guided axicon beam, i.e., one that requires a conductor along the optical axis. 2.1 Solution via the Vector Potential Many discussions of Gaussian laser beams emphasize a single electric field component such asEx=f(r,z)ei(kz−ωt)of a cylindrically symmetric beam of angular frequency ωand wave numberk=ω/cpropagating in vacuum along the zaxis. Of course, the electric field must satisfy the free-space Maxwell equation ∇ ·E= 0. IfEy= 0, we must then have nonzero Ez. That is, the desired electric field has more than one vector c omponent. To be able to deduce all components of the electric and magnet ic fields of a Gaussian laser beam from a single scalar wave function, we follow the s uggestion of Davis [11] and seek solutions for a vector potential Athat has only a single component. We work in the 1Lorentz gauge (and Gaussian units), so that the scalar poten tial Φ is related to the vector potential by ∇ ·A+1 c∂Φ ∂t= 0. (1) The vector potential can therefore have a nonzero divergenc e, which permits solutions having only a single component. Of course, the electric and magneti c fields can be deduced from the potentials via E=−∇Φ−1 c∂A ∂t, (2) and B=∇ ×A. (3) For this, the scalar potential must first be deduced from the v ector potential using the Lorentz condition (1). The vector potential satisfies the free-space wave equation , ∇2A=1 c2∂2A ∂t2. (4) We seek a solution in which the vector potential is described by a single component Ajthat propagates in the + zdirection with the form Aj(r,t) =ψ(r⊥,z)g(ϕ)eiϕ, (5) where the spatial envelope ψis azimuthally symmetric, r⊥=√x2+y2,gis the temporal pulse shape, and the phase ϕis given by ϕ=kz−ωt. (6) Inserting trial solution (5) into the wave equation (4) we fin d that ∇2ψ+ 2ik∂ψ ∂z/parenleftBigg 1−ig′ g/parenrightBigg = 0, (7) whereg′=dg/dϕ . 2.2 A Condition on the Temporal Pulse Shape g(ϕ) Sinceψis a function of rwhilegandg′are functions of the phase ϕ, eq. (7) cannot be satisfied in general. Often the discussion is restricted to t he case where g′= 0,i.e., to continuous waves. We learn that a pulsed laser beam must obey g′≪g. (8) It is noteworthy that a “Gaussian” laser beam cannot have a Ga ussian temporal pulse. That is, ifg= exp[−(ϕ/ϕ 0)2], theng′/g=−2ϕ/ϕ2 0, which does not satisfy condition (8) for |ϕ|large compared to the characteristic pulsewidth ϕ0=ω∆t,i.e., in the tails of the pulse. A more appropriate form for a pulsed beam is a hyperbolic seca nt (as arises in studies of solitons): g(ϕ) = sech/parenleftBiggϕ ϕ0/parenrightBigg . (9) Then,g′/g=−(1/ϕ0) tanh(ϕ/ϕ 0), which is less than one everywhere provided that ϕ0≫1. 22.3 The Paraxial Wave Equation In the remainder of this paper, we suppose that condition (8) is satisfied. Then, the differ- ential equation (7) for the spatial envelope function ψbecomes ∇2ψ+ 2ik∂ψ ∂z= 0. (10) The function ψcan and should be expressed in terms of three geometric param eters of a focused beam, the diffraction angle θ0, the waistw0, and the depth of focus (Rayleigh range) z0, which are related by θ0=w0 z0=2 kw0,andz0=kw2 0 2=2 kθ2 0. (11) We therefore work in the scaled coordinates ξ=x w0, υ =y w0, ρ2=r2 ⊥ w2 0=ξ2+υ2,andς=z z0, (12) Changing variables and noting relations (11), eq. (10) take s the form ∇2 ⊥ψ+ 4i∂ψ ∂ς+θ2 0∂2ψ ∂ς2= 0, (13) where ∇2 ⊥=∂2 ∂ξ2+∂2 ∂υ2=1 ρ∂ ∂ρ/parenleftBigg ρ∂ψ ∂ρ/parenrightBigg , (14) sinceψis independent of the azimuth φ. This form suggests the series expansion ψ=ψ0+θ2 0ψ2+θ4 0ψ4+... (15) in terms of the small parameter θ2 0.Inserting this into eq. (13) and collecting terms of order θ0 0andθ2 0, we find ∇2 ⊥ψ0+ 4i∂ψ0 ∂ς= 0, (16) and ∇2 ⊥ψ2+ 4i∂ψ2 ∂ς=−∂2ψ0 ∂ς2, (17) etc. Equation (16) is called the the paraxial wave equation, whos e solution is well-known to be ψ0=fe−fρ2, (18) where f=1 1 +iς=1−iς 1 +ς2=e−itan−1ς √ 1 +ς2. (19) 3The factor e−itan−1ςinfis the so-called Guoy phase shift [2], which changes from 0 to π/2 aszvaries from 0 to ∞, with the most rapid change near the z0 The solution to eq. (17) for ψ2has been given in [11], and that for ψ4has been discussed in [12]. With the lowest-order spatial function ψ0in hand, we are nearly ready to display the electric and magnetic fields of the corresponding Gaussian b eams. But first, we need the scalar potential Φ, which we suppose has the form Φ(r,t) = Φ(r)g(ϕ)eiϕ, (20) similar to that of the vector potential. Then, ∂Φ ∂t=−ωΦ/parenleftBigg 1−ig′ g/parenrightBigg ≈ −ωΦ, (21) assuming condition (8) to be satisfied. In that case, Φ =−i k∇ ·A, (22) according to the Lorentz condition (1). The electric field is then given by E=−∇Φ−1 c∂A ∂t≈ik/bracketleftbigg A+1 k2∇(∇ ·A)/bracketrightbigg , (23) in view of condition (8). Note that (1 /k)∂/∂x = (θ0/2)∂/∂ξ,etc., according to eqs. (11)-(12). 2.4 Linearly Polarized Gaussian Beams Taking the scalar wave function (18) to be the xcomponent of the vector potential, Ax=E0 ikψ0g(ϕ)eiϕ, A y=Az= 0, (24) the corresponding electric and magnetic fields are found fro m eqs. (3), (23) and (24) to be the familiar forms of a linearly polarized Gaussian beam, Ex=E0ψ0geiϕ+O(θ2 0)≈E0fe−fρ2geiϕ =E0e−ρ2/(1+ς2)g(ϕ)√ 1 +ς2ei[kz+ςρ2/(1+ς2)−ωt−tan−1ς], =E0e−r2 ⊥/w2(z)g(ϕ)/radicalBig 1 +z2/z2 0ei{kz[1+r2 ⊥/2(z2+z2 0)]−ωt−tan−1(z/z0)}, Ey= 0, (25) Ez=iθ0E0 2∂ψ0 ∂ξgeiϕ+O(θ3 0)≈ −iθ0fξE x, Bx= 0, By=Ex, (26) Bz=iθ0E0 2∂ψ0 ∂υgeiϕ=−iθ0fυE x, 4where w(z) =w0/radicalBig 1 +z2/z2 0 (27) is the characteristic transverse size of the beam at positio nz. Near the focus ( r⊥<∼w0,|z|< z0), the beam is a plane wave, Ex≈E0e−r2 ⊥/w2 0ei(kz−ωt−z/z0), E z≈θ0x w0E0e−r2 ⊥/w2 0ei(kz−ωt−2z/z0−π/2), (28) For largez, Ex≈E0e−θ2/θ2 0ei(kr−ωt−π/2) r, E z≈ −x rEx, (29) wherer=/radicalBig r2 ⊥+z2andθ≈r⊥/r, which describes a linearly polarized spherical wave of extentθ0about thezaxis. The fields ExandEz,i.e., the real parts of eqs. (29), are shown in Figs. 1 and 2. Figure 1: The electric field Ex(x,0,z) of a linearly polarized Gaussian beam with diffraction angle θ0= 0.45, according to eq. (27). The fields (25)-(26) satisfy ∇ ·E= 0 =∇ ·Bplus terms of order θ2 0. Clearly, a vector potential with only a ycomponent of form similar to eq. (24) leads to the lowest-order Gaussian beam with linear polarization in theydirection. 5Figure 2: The electric field Ez(x,0,z) of a linearly polarized Gaussian beam with diffraction angle θ0= 0.45, according to eq. (27). 2.5 The Lowest-Order Axicon Beam An advantage of our solution based on the vector potential is that we also can consider the case that only Azis nonzero and has the form (18), Ax=Ay= 0, A z=E0 kθ0fe−fρ2gei(kz−ωt). (30) Then, ∇ ·A=∂Az ∂z≈ikAz/bracketleftBigg 1−θ2 0 2f(1−fρ2)/bracketrightBigg , (31) using eqs. (11)-(12) and the fact that df/dς =−if2, which follows from eq. (19). Anticipating that the electric field has radial polarization, we work in cy lindrical coordinates, ( r⊥,φ,z), and find from eqs. (3), (23), (30) and (31) that E⊥=E0ρf2e−fρ2geiϕ+O(θ2 0), Eφ= 0, (32) Ez=iθ0E0f2(1−fρ2)e−fρ2geiϕ+O(θ3 0). 6The magnetic field is B⊥= 0, B φ=E⊥, B z= 0. (33) The fieldsExandEzare shown in Figs. 3 and 4. The dislocation seen in Fig. 4 for ρ≈ςis due to the factor 1 −fρ2that arises in the paraxial approximation, and would, I beli eve, be smoothed out on keeping higher-order terms in the expansion (15). Figure 3: The electric field Er(r⊥,0,z) of an axicon Gaussian beam with diffraction angle θ0= 0.45, according to eq. (32). The transverse electric field is radially polarized and vani shes on the axis. The longitu- dinal electric field is nonzero on the axis. Near the focus, Ez≈iθ0E0and the peak radial field isE0/√ 2e= 0.42E0. For large z,E⊥peaks atρ=ς/√ 2, corresponding to polar angle θ=θ0/√ 2. For angles near this, |E⊥| ≈ρ|f|2≈1/z, as expected in the far zone. In this region, the ratio of the longitudinal to transverse fields is Ez/E⊥≈ −iθ0fρ≈ −r⊥/z, as expected for a spherical wave front. The factor f2in the fields implies a Guoy phase shift of e−2itan−1ς, which is twice that of the lowest-order linearly polarized beams. It is noteworthy that the simplest axicon mode (32)-(33) is n ot a member of the set of Gaussian modes based on Laguerre polynomials in cylindrica l coordinates (see, for example, sec. 3.3b of [1]). 7Figure 4: The electric field Ez(r⊥,0,z) of an axicon Gaussian beam with diffraction angle θ0= 0.45, according to eq. (32). 2.6 Guided Axicon Beam We could also consider the vector potential Ar⊥∝ψ0geiϕ, A φ=Az= 0, (34) which leads to the electric and magnetic fields Er=E0fe−fρ2geiϕ, Eφ= 0, Ez=−iθ0fρE r, B r= 0, Bφ=Er, Bz= 0, (35) and the potential Ar⊥= 0, A φ∝ψ0geiϕ, A z= 0, (36) which leads to Er= 0, Eφ=E0fe−fρ2geiϕ, Ez= 0, B r=−Eφ, Bφ= 0, Bz=−iθ01−2fρ2 2ρEφ.(37) The case of eqs. (36)-(37) is unphysical due to the blowup of Bzasr⊥→0. The fields of eqs. (34)-(35) do not satisfy ∇ ·E= 0 atr⊥= 0, and so cannot correspond to a free-space wave. However, these fields could be supporte d by a wire, and respresent a 8TM axicon guided cylindrical wave with a focal point. This is in contrast to guided plane waves whose radial profile is independent of z[13, 14]. Guided axicon beams might find application when a focused beam is desired at a point where a s ystem of lenses and mirrors cannot conveniently deliver the optical axis, or in wire-gu ided atomic traps [15]. Figures 1 and 2 show the functional form of the guided axicon beam (35) , when coordinate xis reinterpreted as r⊥. 3 References [1] H. Kogelnik and T. Li, Laser Beams and Resonators , Appl. Opt. 5, 1550-1567 (1966). [2] A.E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986), chaps. 1 6-17. [3] P.W. Milonni and J.H. Eberly, Lasers (Wiley Interscience, New York, 1988), sec. 14.5. [4] A. Yariv, Quantum Electronics , 3rd ed. (Wiley, New York, 1989), chap. 6. [5] K.T. McDonald, Time Reversed Diffraction (Sept. 5, 1999). [6] J.A. Edighoffer and R.H. Pantell, Energy exchange between free electrons and light in vacuum , J. Appl. Phys. 50, 6120-6122 (1979). [7] E.J. Bochove, G.T. Moore and M.O. Scully, Acceleration of particles by an asymmetric Hermite-Gaussian laser beam , Phys. Rev. A 46, 6640-6653 (1992). [8] L.C. Steinhauer and W.D. Kimura, A new approach to laser particle acceleration in vacuum , J. Appl. Phys. 72, 3237-3245 (1992). [9] G. Goubau and F. Schwering, On the Guided Propagation of Electromagnetic Wave Beams , IRE Trans. Antennas and Propagation, AP-9 , 248-256 (1961). [10] G.D. Boyd and J.P. Gordon, Confocal Multimode Resonator for Millimeter Through Optical Wavelength Masers , Bell Sys. Tech. J. 40, 489-509 (1961). [11] L.W. Davis, Theory of electromagnetic beams , Phys. Rev. A 19, 1177-1179 (1979). [12] J.P. Barton and D.R. Alexander, Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam , J. Appl. Phys. 66, 2800-2802 (1989). [13] A. Sommerfeld, Electrodynamics (Academic Press, New York, 1952), secs. 22-23. [14] J.A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), secs. 9.16-17. [15] J. Denschlag, D. Cassettari and J. Schmiedmayer, Guiding neutral atoms with a wire , Phys. Rev. Lett. 82, 2014-2017 (1999). 9

arXiv:physics/0003057v1 [physics.optics] 23 Mar 2000Diffraction as a Consequence of Faraday’s Law Max. S. Zolotorev Center for Beam Physics, Lawrence Berkeley National Labora tory, Berkeley, CA 94720 Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princet on, NJ 08544 (Jan. 11, 1999) 1 Problem A linearly polarized plane electromagnetic wave of frequen cyωis normally incident on an opaque screen with a square aperture of edge a. Show that the wave has a longitudinal magnetic field once it ha s passed through the aperture by an application of Faraday’s Law to a loop paralle l to the screen, on the side away from the source. Deduce the ratio of longitudinal to tra nsverse magnetic field, which is a measure of the diffraction angle. 2 Solution Consider a linearly polarized wave with electric field Exei(kz−ωt)incident on a perfectly ab- sorbing screen in the plane z= 0 with a square aperture of edge acentered on the origin. We apply the integral form of Faraday’s Law to a semicircular loop with its straight edge bisecting the aperture and parallel to the transverse elect ric field Ex, as shown in the figure. The electric field is essentially zero close to the screen on t he side away from the source. Then, at time t= 0,/contintegraldisplay E·dl≈Exa/negationslash= 0. (1) If the loop were on the source side of the screen, the integral would vanish. Faraday’s Law tells us immediately that the time derivative of the magnetic flux through the loop is nonzero. Hence, there must be a nonzero longitudi nal component, Bz, to the magnetic field, once the wave has passed through the aperture . In Gaussian units, Bya=Exa≈/contintegraldisplay E·dl=−1 cd dt/integraldisplay B·dS≈ −1 cdBz dta2 2, (2) whereBzis a characteristic value of the longitudinal component of t he magnetic field over that half of the aperture enclosed by the loop. The longitudi nal magnetic field certainly has time dependence of the form e−iωt, sodBz/dt=−iωB z=−2πicB z/λ, and (2) leads to Bz By≈ −iλ πa. (3) By a similar argument for a loop that enclosed the other half o f the aperture, Bz/By≈iλ/πa in that region; Bz= 0 in the plane y= 0. 1Figure 1: A screen with a square aperture of edge ais illuminated by a linearly polarized electromagnetic wave. The imaginary lo op shown by the dashed curve lies close to the screen, on the side away from th e source, and so is partly in the shadow of the wave. We see that the wave is no longer a plane wave after passing thr ough the aperture, and we can say that it has been diffracted as a consequence of Farad ay’s Law. This argument emphasizes the fields near the aperture. A deta iled understanding of the fields far from the aperture requires more than just Faraday’ s Law. A simplified analysis is that that magnitude of the ratio (3) is a measure of the spread of angles of the magnetic field vector caused by the diffraction, and so in the far zone th e wave occupies a cone a characteristic angle λ/πa. 3 Comments Using the fourth Maxwell equation including the displaceme nt current, we can make an argument for diffraction of the electric field similar to that given above for the magnetic field. After the wave has passed through the aperture of size a, it is very much like a wave that has been brought to a focus of size a. Hence, we learn that near the focus ( x,y,z ) = (0,0,0) of a linearly polarized electromagnetic wave with E=Eˆxand propagating in the z direction, there are both longitudinal electric and magnet ic fields, and that EzandBzare antisymmetric about the planes x= 0 andy= 0, respectively. Also, eq. (3) indicates that near the focus the longitudinal and transverse fields are 90◦ 2out of phase. Yet, far from the focus, the transverse and long itudinal fields become in phase, resulting in spherical wavefronts that extend over a cone of characteristic angle λ/πa. For this to hold, the longitudinal and the transverse fields must experience phase shifts that differ by 90◦between the focal point and the far zone. It is only a slight leap from the present argument to conclude that the transverse fields undergo the extra phase shift. This was first deduced (or noti ced) by Guoy [1] in 1890 via the Huygens-Kirchhoff integral [2]. The latter tells us that the secondary wavelet ψat a large distance rfrom a small region of area Awhere the wave amplitude is ψ0e−iωtis ψ=kψ0A 2πiei(kr−ωt) r=kψ0A 2πei(kr−ωt−π/2) r. (4) The possibly mysterious factor of iin the denominator of the Huygens-Kirchhoff integral implies a 90◦phase shift between a focus and the far field of a beam of light. Here, we have seen that this phase shift can also be considered as a consequ ence of Faraday’s Law. 4 References [1] A.E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986), sec. 17. 4. [2] L. Landau and E.M. Lifshitz, The Classical Theory of Fields , 4th ed. (Pergamon Press, Oxford, 1975), sec. 59. 3

arXiv:physics/0003058v1 [physics.optics] 23 Mar 2000Time-Reversed Diffraction Max. S. Zolotorev Center for Beam Physics, Lawrence Berkeley National Labora tory, Berkeley, CA 94720 Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princet on, NJ 08544 (Sep. 5, 1999) 1 Problem In the usual formulation of the Kirchhoff diffraction integra l, a scalar field with harmonic time dependence at frequency ωis deduced at the interior of a charge-free volume from knowledge of the field (or its normal derivative) on the bound ing surface. In particular, the field is propagated forwards in time from the boundary to the d esired observation point. Construct a time-reversed version of the Kirchhoff integral in which the knowledge of the field on the boundary is propagated backwards in time into the interior of the volume. Consider the example of an optical focus at the origin for a sy stem with the zaxis as the optic axis. In the far field beyond the focus a Gaussian beam ha s cone angle θ0≡√ 2σθ, and thexcomponent of the electric field in a spherical coordinate sys tem is given approximately by Ex(r,θ,φ,t ) =E(r)ei(kr−ωt)e−θ2/θ2 0, (1) wherek=ω/candcis the speed of light. Deduce the field near the focus. Since the Kirchhoff diffraction formalism requires the volum e to be charge free, the time- reversed technique is not applicable to cases where the sour ce of the field is inside the volume. Nonetheless, the reader may find it instructive to at tempt to apply the time-reversed diffraction integral to the example of an oscillating dipole at the origin. 2 The Kirchhoff Integral via Green’s Theorem A standard formulation of Kirchhoff’s diffraction integral f or a scalar field ψ(x) with time dependence e−iωtis ψ(x)≈k 2πi/integraldisplay Seikr′ r′ψ(x′)dArea′, (2) where the spherical waves ei(kr′−ωt)/r′are outgoing, and r′is the magnitude of vector r′= x−x′. For a time-reversed formulation in which we retain the time d ependence as e−iωt, the spherical waves of interest are the incoming waves e−i(kr′+ωt)/r′. In brief, the desired time- reversed diffraction integral is obtained from eq. (2) on rep lacingiby−i: ψ(x)≈ik 2π/integraldisplay Se−ikr′ r′ψ(x′)dArea′. (3) For completeness, we review the derivation of eqs. (2)-(3) v ia Green’s theorem. See also, sec. 10.5 of ref. [1]. 1Green tells us that for any two well-behaved scalar fields φandψ, /integraldisplay V(φ∇2ψ−ψ∇2φ)dVol =/integraldisplay S(φ∇′ψ−ψ∇′φ)·dS′. (4) The surface element dS′is directly outward from surface S. We consider fields with harmonic time dependence at frequency ω, and assume the factor e−iωt. The wave function of interest, ψ, is assumed to have no sources within volume V, and so obeys the Helmholtz wave equation, ∇2ψ+k2ψ= 0. (5) We choose function φ(x) to correspond to waves associated with a point source at x′. That is, ∇2φ+k2φ=−δ3(x−x′). (6) The well-known solutions to this are the incoming and outgoi ng spherical waves, φ±(x,x′) =e±ikr′ r′, (7) where the + sign corresponds to the outgoing wave. We recall t hat ∇′r′=−r′ r′=−ˆno, (8) where ˆnopoints towards the observer at x. Then, ∇′φ±=∓ikˆno/parenleftbigg 1±1 ikr′/parenrightbigg φ. (9) Inserting eqs. (5)-(9) into eq. (4), we find ψ(x) =−1 4π/integraldisplay Se±ikr′ r′ˆn′·/bracketleftbigg ∇′ψ±ikˆno/parenleftbigg 1±1 ikr′/parenrightbigg ψ/bracketrightbigg dArea′, (10) where the overall minus sign holds with the convention that ˆn′is the inward normal to the surface. We only consider cases where the source of the wave ψis far from the boundary surface, so that on the boundary ψis well approximated as a spherical wave, ψ(x′)≈Aeikrs rs, (11) wherersis the magnitude of the vector rs=x′−xsfrom the effective source point xsto the point x′on the boundary surface. In this case, ∇′ψ=ikˆns/parenleftbigg 1±1 ikrs/parenrightbigg ψ, (12) where ˆns=rs/rs 2We also suppose that the observation point is far from the bou ndary surface, so that kr′≪1 as well as krs≪1. Hence, we neglect the terms in 1 /ikr′and 1/ikr sto find ψ(x) =−ik 4π/integraldisplay Se±ikr′ r′ˆn′·(ˆns±ˆno)ψ(x′)dArea′. (13) The usual formulation, eq. (2), of Kirchhoff’s law is obtaine d using outgoing waves (+ sign), and the paraxial approximation that ˆn′≈ˆno≈ˆns. The latter tacitly assumes that the effective source is outside volume V. Here, we are interested in the case where the effective source is inside the volume V, so that the paraxial approximation is ˆn′≈ˆno≈ −ˆns. When we use the incoming wave function to reconstruct ψ(x,t) from information on the boundary at time t′>t, we use the −sign in eq. (13) to find eq. (3). Note that in this derivation, we assumed that ψobeyed eq. (5) throughout volume V, and so the actual source of ψcannot be within V. Our time-reversed Kirchhoff integral (3) can only be applied when any source inside Vis virtual. This includes the interesting case of a focus of an optical system (secs. 4 and 5). However, we canno t expect eq. (3) to apply to the case of a physical source, such as an oscillating dipole, inside volume V(sec. 6). The laws of diffraction do not permit electromagnetic waves to conver ge into a volume smaller than a wavelength cubed, and so eq. (3) cannot be expected to descr ibe the near fields around a source smaller than this. 3 A Plane Wave The time-reversed Kirchhoff integral (3) for the xcomponent of the electric field is Ex(obs,now) =ik 2π/integraldisplaye−ikr′ r′Ex(r,θ,φ, future)dArea , (14) wherer′is the distance from the observation point robs= (x,y,z ) in rectangular coordinates to a point r=r(sinθcosφ,sinθsinφ,cosθ) on a sphere of radius rin the far field. As a first example, consider a plane electromagnetic wave, Ex=E0ei(kz−ωt)=E0ei(krcosθ−ωt), (15) where the second form holds in a spherical coordinate system (r,θ,φ) whereθis measured with respect to the zaxis. We take the point of observation to be ( x,y,z ) = (0,0,r0), and evaluate the diffraction integral (14) over a sphere of radiu sr≫r0. In the exponential factor in the Kirchhoff integral, we approximate r′as r′≈r−ˆr·robs=r−r0cosθ, (16) while in the denominator we approximate r′asr. Then, Ex(obs) ≈ik 2π/integraldisplay1 −1r2dcosθ/integraldisplay2π 0dφe−ik(r−r0cosθ) rE0eikrcosθ =r r+r0E0[eikr0−e−ik(2r+r0)] (17) ≈E0eikr0, 3where we ignore the rapidly oscillating term e−ik(2r+r0)as unphysical. This verifies that the time-reversed diffraction formula wor ks for a simple example. 4 The Transverse Field near a Laser Focus We now consider the far field of a laser beam whose optic axis is thezaxis with focal point at the origin. The polarization is along the xaxis, and the electric field has Gaussian dependence on polar angle with characteristic angle θ0≪1. Then, we can write Ex(r,θ,φ) =E(r)eikre−θ2/θ2 0, (18) whereE(r) is the magnitude of the electric field on the optic axis at dis tancerfrom the focus. In the exponential factor in the Kirchhoff integral (1 4),r′is the distance from the observation point r|rmobs= (x,y,z ) to a point r=r(sinθcosφ,sinθsinφ,cosθ) on the sphere. We approximate r′as r′≈r−ˆr·robs=r−xsinθcosφ−ysinθsinφ−zcosθ, (19) while in the denominator we approximate r′asr. Inserting eqs. (18) and (19) into (14), we find Ex(obs) =ikrE(r) 2π/integraldisplay1 −1eikzcosθe−θ2/θ2 0dcosθ/integraldisplay2π 0eikxsinθcosφ+ikysinθsinφdφ =ikrE(r)/integraldisplay1 −1eikzcosθe−θ2/θ2 0J0(kρsinθ)dcosθ, (20) where ρ=/radicalBig x2+y2, (21) andJ0is the Bessel function of order zero. Since we assume that the characteristic angle θ0of the laser beam is small, we can approximate cos θas 1−θ2/2 andkρsinθaskρθ. Then, we have Ex(obs) ≈ikrE(r)eikz/integraldisplay∞ 0e−(2/θ2 0+ikz)θ2/2J0/parenleftbigg√ 2kρ/radicalBig θ2/2/parenrightbigg d(θ2/2) =ikθ2 0rE(r)eikze−k2θ2 0ρ2/4(1+ikθ2 0z/2) 2(1 +ikθ2 0z/2), (22) where the Laplace transform, which is given explicitly in [2 ], can be evaluated using the series expansion for the Bessel function. This expression c an be put in a more familiar form by introducing the Rayleigh range (depth of focus), z0=2 kθ2 0, (23) and the so-called waist of the laser beam, w0=θ0z0=2 kθ0. (24) 4We define the electric field strength at the focus ( ρ= 0,z= 0) to be E0, so we learn that the far-field strength is related by E(r) =−iz0 rE0. (25) The factor −i=e−iπ/2is the 90◦Guoy phase shift between the focus and the far field. Then, the transverse component of the electric field near the focus is Ex(x,y,z )≈E0e−ρ2/w2 0(1+iz/z0)eikz (1 +iz/z0) =E0e−ρ2/w2 0(1+z2/z2 0)e−itan−1z/z0eiρ2z/w2 0z0(1+z2/z2 0)eikz /radicalBig 1 + (z/z0)2. (26) This is the usual form for the lowest-order mode of a linearly polarized Gaussian laser beam [3]. Figure 1 plots this field. Figure 1: The electric field Ex(x,0,z) of a linearly polarized Gaussian beam with diffraction angle θ0= 0.45. The Gaussian beam (26) could also be deduced by a similar argu ment using eq. (2), starting from the far field of the laser before the focus. The f orm (26) is symmetric in z 5except for a phase factor, and so is a solution to the problem o f transporting a wave from z=−rtoz= +rsuch that the functional dependence on ρandzis invariant up to a phase factor. One of the earliest derivations [4] of the Gaussian b eam was based on the formulation of this problem as an integral equation for the eigenfunctio n (26). 5 The Longitudinal Field Far from the focus, the electric field E(r) is perpendicular to the radius vector r. For a field linearly polarized in the xdirection, there must also be a longitudinal component Ezrelated by E·ˆr=Exsinθcosφ+Ezcosθ= 0. (27) Thus, far from the focus, Ez(r) =−Ex(r) tanθcosφ. (28) Then, similarly to eqs. (14) and (20), we have Ez(obs) =ik 2π/integraldisplaye−ikr′ r′Ez(r)dArea =−ikrE(r) 2π/integraldisplay1 −1eikzcosθe−θ2/θ2 0tanθdcosθ/integraldisplay2π 0eikxsinθcosφ+ikysinθsinφcosφdφ =−ikxz 0E0 ρ/integraldisplay1 −1eikzcosθe−θ2/θ2 0tanθJ1(kρsinθ)dcosθ, (29) using eq. (3.937.2) of [5]. We again note that the integrand is significant only for small θ, so we can approximate eq. (29) as the Laplace transform Ez(x,y,z )≈ −ik2xz0E0eikz√ 2/integraldisplay∞ 0e−(2/θ2 0+ikz)θ2/2/radicalBig θ2/2J1/parenleftbigg√ 2kρ/radicalBig θ2/2/parenrightbigg d(θ2/2) =−ik2θ4 0xz0E0eikze−ρ2/w2 0(1+iz/z0) 4(1 +iz/z0)2 =−iθ0x w0Ex(x,y,z ) (1 +iz/z0), (30) withExgiven by eq. (26). Figure 2 plots this field. Together, the electric field components given by eqs. (26) an d (30) satisfy the Maxwell equation ∇ ·E= 0 to order θ2 0[6, 7, 8]. 6 Oscillating Dipole at the Origin We cannot expect the Kirchhoff diffraction integral to apply t o the example of an oscillating dipole, if our bounding surface surrounds the dipole. Let us see what happens if we try to use eq. (3) anyway. 6Figure 2: The electric field Ez(x,0,z) of a linearly polarized Gaussian beam with diffraction angle θ0= 0.45. The dipole is taken to be at the origin, with moment palong thexaxis. Then, the x component of the radiation field is Ex=k2psinθxeikr r. (31) whereθxis the angle between the xaxis and a radius vector to the observer. We consider an observer near the origin at ( x,y,z ) = (0,0,r0), for which sin θx= 1, and so Ex(obs) =k2peikr0 r0. (32) We now attempt to reconstruct this field near the origin from i ts value on a sphere of radiusrusing the time-reversed Kirchhoff integral (3). We use a sphe rical coordinate system (r,θ,φ) that favors the zaxis. Then, the xcomponent of the radiation field on the sphere of radiusris Ex(r,θ,φ) =k2p/radicalBig 1−sin2θcos2φeikr r. (33) This form cannot be integrated analytically, so we use a Tayl or expansion of the square root, which will lead to an expansion in powers of 1 /r0. It turns out that the coefficient of the 71/r0term, which is our main interest, is very close to that if we si mply approximate the square root by unity. For brevity, we write Ex(r,θ,φ)≈k2peikr r. (34) In the time-reversed Kirchhoff integral (3), we make the usua l approximation that r′= r−r0cosθin the exponential factor, but r′=rin the denominator. Then, using eq. (34) we have Ex(obs) ≈ik3pe−ikr 2πr/integraldisplay1 −1r2dcosθ/integraldisplay2π 0dφeikr0cosθeikr r =k2peikr0 r0−k2pe−ikr0 r0 = 2ik3psinkr0 kr0. (35) The first, outgoing wave in middle line of eq. (35) is the desir ed form, but the second, incoming wave is of the same magnitude. Together, they lead t o the form sin( kr0)/kr0 which is nearly constant for kr0<∼1. The presence of outgoing as well as incoming waves is to be expected because dipole radiation is azimuthally symm etric about the xaxis. In the absence of a charged source at the origin, an outgoing wave at θ=πmust correspond to an incoming wave at θ= 0. The result that the reconstructed field is uniform for distan ces within a wavelength of the origin is consistent with the laws of diffraction that electr omagnetic waves cannot be focused to a region smaller than a wavelength. Far fields of the form (3 1) could only be propagated back to the form of dipole fields near the origin with the addit ion of nonradiation fields tied to a charge at the origin. Such a construction is outside the s cope of optics and diffraction. 7 References [1] J.D. Jackson, Classical Electrodynamics , 3d ed. (Wiley, New York, 1999). [2] W. Magnus and F. Oberhettinger, Functions of Mathematical Physics (Springer-Verlag, Berlin, 1943; reprinted by Chelsea Publishing Company, New York, 1949), pp. 131-132. [3] See, for example, sec. 14.5 of P.W. Milonni and J.H. Eberl y,Lasers (Wiley Interscience, New York, 1988). [4] G.D. Boyd and J.P. Gordon, Confocal Multimode Resonator for Millimeter Through Optical Wavelength Masers , Bell Sys. Tech. J. 40, 489-509 (1961). [5] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products , 5th ed. (Academic Press, San Diego, 1994). [6] M. Lax, W.H. Louisell and W.B. McKnight, From Maxwell to paraxial wave optics , Phys. Rev. A 11, 1365-1370 (1975). 8[7] L.W. Davis, Theory of electromagnetic beams , Phys. Rev. A 19, 1177-1179 (1979). [8] J.P. Barton and D.R. Alexander, Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam , J. Appl. Phys. 66, 2800-2802 (1989). 9

arXiv:physics/0003059v1 [physics.acc-ph] 23 Mar 2000Classical “Dressing” of a Free Electron in a Plane Electroma gnetic Wave Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princet on, NJ 08544 Konstantine Shmakov Department of Physics and Astronomy, University of Tenness ee, Knoxville, Tennessee 37996 (Feb. 28, 1998) The energy and momentum densities of the fields of a free electron in a plane electromagnetic wave include interfere nce terms that are the classical version of the “dressing” of the electron the arises in a quantum analysis. The transverse me - chanical momentum of the oscillating electron is balanced b y the field momentum resulting from the interference between the driving wave and the static part of the electron’s field. The interference between the wave and the oscillating part of the electron’s field leads to a longitudinal field momentum and a negative field energy that compensate for the longi- tudinal momentum and kinetic energy of the electron. The interference terms are dominated by the near zone, so that as the wave passes the electron by the latter reverts to its ener gy and momentum prior to the arrival of the wave. I. INTRODUCTION The behavior of a free electron in a electromagnetic wave is one of the most commonly discussed topics in classical electromagnetism. Yet, several basic issues re- main to be clarified. These relate to the question: to what extent can net energy be transferred from an elec- tromagnetic pulse (such as that of a laser) in vacuum to a free electron? These issues are made more complex by quantum con- siderations, including the role of the “quasimomentum” of an electron that is “dressed” by an electromagnetic wave [1]. As a small step towards understanding of the larger is- sues, we consider a simpler question here. The response of a free electron to a plane electromagnetic wave is oscil- latory motion in the plane perpendicular to the direction of the wave, in the first approximation. Thus, the elec- tron has momentum transverse to the direction of the wave. However, the wave contains momentum only in its direction, and the radiated wave contains no net mo- mentum (in the nonrelativistic limit). How is momentum conserved in this process? The general sense of the answer has been given by Poynting [2], who noted that an electromagnetic field can be said to contain a flux of energy (energy per unit area per unit time) given by S=cE×B 4π, (1) in Gaussian units, where Eis the electric field, Bis themagnetic field (taken to be in vacuum throughout this paper) and cis the speed of light. Poincar´ e [3] noted that this flow of energy can also be associated with a momentum density given by Pfield=S c2=E×B 4πc, (2) Hence, in the problem of a free electron in a plane electro- magnetic wave we are led to seek an electromagnetic field momentum that is equal and opposite to the mechanical momentum of the electron. In this paper we demonstrate that indeed the mechan- ical momentum of the oscillating electron is balanced by the field momentum in the interference term between the incident wave and the static field of the electron. We are left with some subtleties when we consider the interfer- ence between the incident wave and the oscillating field of the electron. II. GENERALITIES A. Motion of an Electron in a Plane Wave We consider a plane electromagnetic wave that prop- agates in the + zdirection of a rectangular coordinate system. A fairly general form of this wave is Ewave=ˆxExcos(kz−ωt)−ˆyEysin(kz−ωt), Bwave=ˆxEysin(kz−ωt) +ˆyExcos(kz−ωt),(3) where ω=kcis the angular frequency of the wave, k= 2π/λis the wave number and ˆxis a unit vector in the xdirection, etc.When either ExorEyis zero we have a linearly polarized wave, while for Ex=±Eywe have circular polarization. A free electron of mass moscillates in this field such that its average position is at the origin. This simple statement hides the subtlety that our frame of reference is not the lab frame of an electron that is initially at rest but which is overtaken by a wave [4–6]. If the velocity of the oscillating electron is small, we can ignore the v/c×B force and take the motion to be entirely in the plane z= 0. Then, (also ignoring radiation damping) the equation of motion of the electron is m¨x=eEwave(0, t) =e(ˆxExcosωt+ˆyEysinωt).(4) 1Using eq. (3) we find the position of the electron to be x=−e mω2(ˆxExcosωt+ˆyEysinωt), (5) and the mechanical momentum of the electron is pmech=m˙x=e ω(ˆxExsinωt−ˆyEycosωt). (6) The root-mean-square (rms) velocity of the electron is vrms=/radicalbig /an}b∇acketle{t˙x2+ ˙y2/an}b∇acket∇i}ht=e mω/radicalbigg E2x+E2y 2=eErms mωcc.(7) The condition that the v/c×Bforce be small is then η≡eErms mωc≪1, (8) where the dimensionless measure of field strength, η, is a Lorentz invariant. Similarly, the rms departure of the electron from the origin is xrms=eErms mω2=ηλ 2π. (9) Thus, condition (8) also insures that the extent of the mo- tion of the electron is small compared to a wavelength, and so we may use the dipole approximation when con- sidering the fields of the oscillating electron. In the weak-field approximation, we can now use (6) for the velocity to evaluate the second term of the Lorentz force: ev c×B=e2(E2 x−E2 y) 2mωcˆzsin 2ωt. (10) This term vanishes for circular polarization, in which case the motion is wholely in the transverse plane. However, for linear polarization the v/c×Bforce leads to oscilla- tions along the zaxis at frequency 2 ω, as first analyzed in general by Landau [7]. For polarization along the ˆx axis, the x-zmotion has the form of a “figure 8”, which for weak fields ( η≪1) is described by x=−eEx mω2cosωt, z =−e2E2 x 8m2ω3csin 2ωt. (11) If the electron had been at rest before the arrival of the plane wave, then inside the wave it would move with an average drift velocity given by vz=η2/2 1 +η2/2c, (12) along the direction of the wave vector, as first deduced by McMillan [8]. In the present paper we work in the frame in which the electron has no average velocity along thezaxis. Therefore, prior to its encounter with the plane wave the electron had been moving in the negative zdirection with speed given by (12).B. Field Momentum The fields associated with the electron can be regarded as the superposition of those of an electron at rest at the origin plus those of a dipole consisting of the actual oscillating electron and a positron at rest at the origin. Thus, we can write the electric field of the electron as Estatic+Eoscand the magnetic field as Bosc, where the oscillating fields have the pure frequency ωin the low- velocity limit. The entire electromagnetic momentum density can then be written Pfield=(Ewave+Estatic+Eosc)×(Bwave+Bosc) 4πc.(13) However, in seeking the field momentum that opposes the mechanical momentum of the electron, we should not include either of the self-momenta Ewave×Bwaveor (Estatic+Eosc)×Bosc. The former is independent of the electron, while the latter can be considered as a part of the mechanical momentum of the electron according to the concept of “renormalization”. We therefore restrict our attention to the interaction field momentum Pint=Pwave,static+Pwave,osc, (14) where Pwave,static=Estatic×Bwave 4πc. (15) and Pwave,osc=Ewave×Bosc+Eosc×Bwave 4πc. (16) We recall from eqs. (6) and (11) that the transverseme- chanical momentum of the oscillating electron has pure frequency ω. Since the wave and the oscillating part of the electron’s field each have frequency ω, the term Pwave,osccontains harmonic functions of ω2, which can be resolved into a static term plus ones in frequency 2ω. Hence we should not expect this term to cancel the mechanical momentum. Rather, we look to the term Pwave,static, since this has pure frequency ω. III. THE MOMENTUM P WAVE ,STATIC The static field of the electron at the origin is, in rect- angular coordinates, Estatic=e r3(xˆx+yˆy+zˆz), (17) where ris the distance from the origin to the point of observation. Combing this with eq. (3) we have 2Pwave,static=e 4πcr3{−ˆxzExcos(kz−ωt) +ˆyzEysin(kz−ωt) (18) +ˆz[xExcos(kz−ωt)−yEycos(kz−ωt)]}. When we integrate this over all space to find the total field momentum, the terms in ˆzvanish as they are odd in either xory. Likewise, after expanding the cosine and sine of kz−ωt, the terms proportional to zcoskzvanish on integration. The remaining terms are thus pwave,static=/integraldisplay VPwave,static (19) =e 4πc(−ˆxExsinωt+ˆyEycosωt)/integraldisplay Vzsinkz r3 =e ω(−ˆxExsinωt+ˆyEycosωt) =−pmech, after an elementary volume integration. It is noteworthy that the integration is independent of any hypothesis as to the size of a classical elec- tron. Indeed, the integrand of (19) can be expressed as cos θsin(krcosθ)/r2via the substitution z=rcosθ. Hence, the integral over a spherical shell is independent ofrforkr≪1, and significant contributions to the inte- gral occur for radii up to one wavelength of the electro- magnetic wave. This contrasts with the self-momentum density of the electron which is formally divergent; if the integration is cut off at a minimum radius (the classical electron radius), the dominant contribution occurs within twice that radius. Thus, we have demonstrated the principal result of this paper. IV. THE MOMENTUM P WAVE ,OSC Several subtleties in the argument appear when we consider the other interference term in the momentum density (14). For this we must first display the electro- magnetic fields of an oscillating electron. A. The Fields E oscand B osc Since we restrict our attention to an electron that os- cillates with amplitude much less than a wavelength of the driving wave, and the electron attains velocities that are much less than the speed of light, it is sufficient to use the dipole approximation to the fields of the electron. While these fields are well known, they are typically pre- sented in imaginary notation, of which only the real part has physical significance. This notation is very useful for discussions in which only time-averaged behavior is of interest. However, we wish to consider the details of momentum balance at an arbitrary moment, and it is preferable to use purely real notation.We begin by noting that the retarded vector potential of the oscillating electron at a point rat time tcan be written Aosc(r, t) =e c˙x(t′=t−r/c) r(20) =−e2 mωcr[ˆxExsin(kr−ωt) +ˆyEycos(kr−ωt)], using eq. (5) for the motion xof the electron. The oscil- lating part of the scalar potential is obtained by integra- tion of the Lorentz gauge condition: ∇ ·Aosc+1 c∂φosc ∂t= 0. (21) We find φosc=−e2 mω2/braceleftbigg Ex/bracketleftbiggkx r2sin(kr−ωt) +x r3cos(kr−ωt)/bracketrightbigg +Ey/bracketleftbiggky r2cos(kr−ωt)−y r3sin(kr−ωt)/bracketrightbigg/bracerightbigg .(22) The constant static potential is omitted in the above. The scalar potential could also be deduced from the retarded potential of a moving charge. Equation (22) results on expanding the retarded distance to first order in the field strength of the plane wave. The electric and magnetic fields are, of course, found from the potentials via B=∇ ×A and E=−∇φ−1 c∂A ∂t.(23) The lengthy expressions for the rectangular components of the fields are Bosc,x=−e2Ey mω2/bracketleftbiggk2z r2sin(kr−ωt) +kz r3cos(kr−ωt)/bracketrightbigg , Bosc,y=−e2Ex mω2/bracketleftbiggk2z r2cos(kr−ωt)−kz r3sin(kr−ωt)/bracketrightbigg , Bosc,z=e2Ex mω2/bracketleftbiggk2y r2cos(kr−ωt)−ky r3sin(kr−ωt)/bracketrightbigg (24) +e2Ey mω2/bracketleftbiggk2x r2sin(kr−ωt) +kx r3cos(kr−ωt)/bracketrightbigg , and Eosc,x=−e2Ex mω2/bracketleftbigg/parenleftbigg3kx2 r4−k r2/parenrightbigg sin(kr−ωt) +/parenleftbiggk2 r−k2x2 r3+3x2 r5−1 r3/parenrightbigg cos(kr−ωt)/bracketrightbigg −e2Ey mω2/bracketleftbigg3kxy r4cos(kr−ωt) +/parenleftbiggk2xy r3−3xy r5/parenrightbigg sin(kr−ωt)/bracketrightbigg , Eosc,y=−e2Ex mω2/bracketleftbigg3kxy r4sin(kr−ωt) 3−/parenleftbiggk2xy r3−3xy r5/parenrightbigg cos(kr−ωt)/bracketrightbigg −e2Ey mω2/bracketleftbigg/parenleftbigg3ky2 r4−k r2/parenrightbigg cos(kr−ωt) (25) −/parenleftbiggk2 r−k2y2 r3+3y2 r5−1 r3/parenrightbigg sin(kr−ωt)/bracketrightbigg , Eosc,z=−e2Ex mω2/bracketleftbigg3kxz r4sin(kr−ωt) −/parenleftbiggk2xz r3−3xz r5/parenrightbigg cos(kr−ωt)/bracketrightbigg −e2Ey mω2/bracketleftbigg3kyz r4cos(kr−ωt) +/parenleftbiggk2yz r3−3yz r5/parenrightbigg sin(kr−ωt)/bracketrightbigg . These expressions can also be deduced from the Li´ enard- Wiechert forms for the fields of an accelerated charge, keeping terms only to first order in the strength of the plane wave. B. Components of P wave ,osc Since the wave fields have no zcomponent, the xcom- ponent of Pwave,oscis given by Pwave,osc,x=Ewave,yBosc,z−Eosc,zBwave,y 4πc.(26) From eqs. (24) and (25) we see that both Bosc,zandEosc,z are odd in either xory. Therefore, the volume integral ofPwave,osc,xvanishes, and we do not consider it fur- ther. Likewise, Pwave,osc,yvanishes on integration. This confirms the claim made at the end of sec. II that the interference term Pwave,oscis not relevant to the balance of transverse momentum between the electron and the fields. However, the zcomponent of Pwave,oscdoes not vanish on integration, and requires further discussion. As the details include some surprises (to the author) I present them at length. Pwave,osc,z= Ew,xBo,y−Ew,yBo,x+Eo,xBw,y−Eo,yBw,x 4πc= −e2E2 xcos(kz−wt) 4πmω2c/bracketleftbiggk2z r2cos(kr−ωt)−kz r3sin(kr−ωt)/bracketrightbigg −e2E2 ysin(kz−wt) 4πmω2c/bracketleftbiggk2z r2sin(kr−ωt) +kz r3cos(kr−ωt)/bracketrightbigg −e2E2 xcos(kz−wt) 4πmω2c/bracketleftbigg/parenleftbigg3kx2 r4−k r2/parenrightbigg sin(kr−ωt) +/parenleftbiggk2 r−k2x2 r3+3x2 r5−1 r3/parenrightbigg cos(kr−ωt)/bracketrightbigg−e2ExEycos(kz−wt) 4πmω2c/bracketleftbigg3kxy r4cos(kr−ωt) +/parenleftbiggk2xy r3−3xy r5/parenrightbigg sin(kr−ωt)/bracketrightbigg (27) −e2ExEysin(kz−wt) 4πmω2c/bracketleftbigg −3kxy r4sin(kr−ωt) +/parenleftbiggk2xy r3−3xy r5/parenrightbigg cos(kr−ωt)/bracketrightbigg +e2E2 ysin(kz−wt) 4πmω2c/bracketleftbigg/parenleftbigg3ky2 r4−k r2/parenrightbigg cos(kr−ωt) −/parenleftbiggk2 r−k2y2 r3+3y2 r5−1 r3/parenrightbigg sin(kr−ωt)/bracketrightbigg . The terms of Pwave,osc,zthat are proportional to EyEy are odd on both xandy, and so will vanish on integra- tion. We now consider the implications of eq. (27) separately for waves of circular and linear polarization. C. Circular Polarization For a circularly polarized wave, we have E2 x=E2 y. Consequently the dimensionless measure of field strength isη=eEx/mωc =eEy/mωc, according to (8). The prefactors e2E2 x/4πmω2cande2E2 y/4πmω2ccan there- fore both be written η2mc/4π, and have dimensions of momentum. The terms of eq. (27) in E2 xandE2 ycan be combined in pairs via the identities cos(kz−ωt)cos(kr−ωt) + sin( kz−ωt)sin(kr−ωt) = cos kzcoskr+ sinkzsinkr, (28) and sin(kz−ωt)cos(kr−ωt)−cos(kz−ωt)sin(kr−ωt) = sinkzcoskr−coskzsinkr. (29) A detail: the second term of eq. (27) in E2 xcontains fac- tors of x2, while second term of in E2 ycontains factors ofy2. But during integration, we can replace y2byx2, after which the terms can be combined via (28-29). We see already that the volume integral of Pwave,osc,z will contain no time dependence! On integration, terms such as f(x, r)sinkzand g(x, r)zcoskzthat are odd in zwill vanish. The inte- grated field momentum is thus, pwave,osc,z=/integraldisplay VPwave,osc,z=−η2mc 4πI1=−4 3η2mc, (30) where I1is the volume integral whose integrand is 4k2z r2sinkzsinkr+kz r3sinkzcoskr +/parenleftbigg3kx2 r4−k r2/parenrightbigg coskzsinkr +/parenleftbiggk2 r−k2y2 r3+3y2 r5−1 r3/parenrightbigg coskzcoskr. (31) We return to the significance of eq. (30) after describ- ing the evaluation of integral I1. As seen from eq. (30), the integral I1must be dimen- sionless, although it is apparently a function of the wave number k. However, the form of (31) indicates that I1 is actually independent of the length scale, so we can set k= 1 during integration. To perform the integration we consider a volume el- ement r2dr dcosθ dφin a spherical coordinate system with angle θdefined relative to the zaxis. It is more convenient to keep z=rcosθas a variable of integra- tion, using dz=rdcosθ. Then the volume integration has the form /integraldisplay V=/integraldisplay∞ 0rdr/integraldisplayr −rdz/integraldisplay2π 0dφ. (32) Most terms of (31) are independent of φ, so their φ integral is just 2 π. For the terms in x2, we have /integraldisplay2π 0x2dφ=/integraldisplay r2sin2θcos2φ dφ=π(r2−z2).(33) While each of the four main terms of (31) diverges on integration, it turns out that the two terms in cos z taken together are finite (and likewise for the two terms in sin z). We find that I1=IA+IB=16π 3, (34) where IA= 2π/integraldisplay∞ 0drsinr r/integraldisplayr −rdz zsinz +2π/integraldisplay drcosr r2/integraldisplay dz zsinz = 4π, (35) and IB=π/integraldisplay drsinr r/integraldisplay dzcosz −3π/integraldisplay drsinr r3/integraldisplay dz z2cosz +π/integraldisplay drcosr/parenleftbigg 1 +1 r2/parenrightbigg/integraldisplay dzcosz +π/integraldisplay drcosr r2/parenleftbigg 1−3 r2/parenrightbigg/integraldisplay dz z2cosz =4π 3. (36)From detailed evaluation of the radial integral, we find that the integrand approaches a constant value as rgoes to zero, and that the contribution to the integral at large rdiminishes as 1 /r. That is, the principal contribution is from the region kr≈1. We are left with the result (30) that the integral of the interference term in the field momentum density has a constant longitudinal term for an electron oscillating in a circularly polarized wave. Recall that we have performed the analysis in a frame in which the electron has no longitudinal momentum. However, as remarked in sec. IIA, prior to its encounter with the wave, the electron had velocity vz=−η2c/2 (as- suming η2≪1), and therefore had initial mechanical mo- mentum pmech ,z=−η2mc/2. So, we would expect that this initial mechanical momentum had been converted to field momentum, if momentum is to be conserved. The result (30) can be described as a kind of “hidden momentum” [9], whose appearance can be surprising if one ignores the physical processes needed to arrive at the nominal conditions of the problem. We continue to be puzzled as to why the result (30) is 8/3 times larger than that required to satisfy momentum conservation. D. Linear Polarization Consider now the case of a linearly polarized wave with electric field along the xaxis. Then Erms=Ex/√ 2, and the prefactors in (27) can be written as η2mc/2π. The remaining terms in the momentum density Pwave,osc,zhave time dependences that can be expressed as sums of pure frequencies via the identities 2 cos(kz−ωt)cos(kr−ωt) = cos kzcoskr+ sinkzsinkr +(cos kzcoskr−sinkzsinkr)cos 2 ωt (37) +(cos kzsinkr+ sinkzsinkr)sin 2ωt, and 2 cos(kz−ωt)sin(kr−ωt) = cos kzsinkr−sinkzcoskr +(cos kzsinkr+ sinkzcoskr)cos 2 ωt (38) +(sin kzsinkr−coskzcoskr)sin 2ωt, Inserting these into eq. (27) and keeping only those terms that are even in z, we find the integrated field momentum to be pwave,osc,z=/integraldisplay VPwave,osc,z =−η2mc 4π(I1+I2cos2ωt+I3sin 2ωt),(39) where integral I1= 16π/3 has been discussed in (31-36), 5I2=−IA+IB=−8π 3, (40) and integral I3has the integrand, k2z r2sinkzsinkr−kz r3sinkzcoskr −/parenleftbigg3kx2 r4−k r2/parenrightbigg coskzsinkr +/parenleftbiggk2 r−k2y2 r3+3y2 r5−1 r3/parenrightbigg coskzcoskr. (41) On evaluation, I3= 0. Hence, the longitudinal component of the interference field momentum of a free electron in a linearly polarized wave is pwave,osc,z=−4 3η2mc+2 3ηmccos2ωt. (42) The constant term is the same as that found in eq. (30) for circular polarization, and represents the initial me- chanical momentum of the electron that became stored in the electromagnetic field once the electron became im- mersed in the wave. As for the second term of (42), recall from eq. (11) that for linear polarization the electron oscillates along thezaxis at frequency 2 ω. Hence the zcomponent of the mechanical momentum of the electron is pmech ,z=m˙z=−η2mc 2cos2ωt. (43) The term in pwave,osc,zat frequency 2 ωis−4/3 of the lon- gitudinal component of the mechanical momentum asso- ciated with the “figure 8” motion of the electron. Thus, we have not been completely successful in accounting for momentum conservation when the small, oscillatory lon- gitudinal momentum is considered. The factors of 4/3 and 8/3 are presumably not the same as the famous factor of 4 /3 that arise in analyses of the electromagnetic energy and momentum of the self fields of an electron [10,11]. A further appearance of a factor of 8/3 in the present example occurs when we consider the field energy of the interference terms. V. THE INTERFERENCE FIELD ENERGY It is also interesting to examine the electromagnetic field energy of an electron in a plane wave. As for the momentum density (13), we can write Utotal=(Ewave+Estatic+Eosc)2+ (Bwave+Bosc)2 8π, (44) for the field energy density. Again, we no not consider the divergent energies of the self fields, but only the in- terference terms,Uint=Uwave,static+Uwave,osc, (45) where Uwave,static=Ewave·Estatic 4π. (46) and Uwave,osc=Ewave·Eosc+Bwave·Bosc 4π. (47) In general, the interference field energy density is os- cillating. Here, we look for terms that are nonzero after averaging over time. We see at once that /an}b∇acketle{tUwave,static/an}b∇acket∇i}ht= 0, (48) since all terms have time dependence of cos ωtor sin ωt. In contrast, /an}b∇acketle{tUwave,osc/an}b∇acket∇i}htwill be nonzero as its terms are products of sines and cosines: Uwave,osc= −e2E2 xcos(kz−wt) 4πmω2/bracketleftbigg/parenleftbigg3kx2 r4−k r2/parenrightbigg sin(kr−ωt) +/parenleftbiggk2 r−k2x2 r3+3x2 r5−1 r3/parenrightbigg cos(kr−ωt)/bracketrightbigg −e2ExEycos(kz−wt) 4πmω2/bracketleftbigg3kxy r4cos(kr−ωt) +/parenleftbiggk2xy r3−3xy r5/parenrightbigg sin(kr−ωt)/bracketrightbigg , +e2ExEysin(kz−wt) 4πmω2/bracketleftbigg3kxy r4sin(kr−ωt) −/parenleftbiggk2xy r3−3xy r5/parenrightbigg cos(kr−ωt)/bracketrightbigg (49) +e2E2 ysin(kz−wt) 4πmω2/bracketleftbigg/parenleftbigg3ky2 r4−k r2/parenrightbigg cos(kr−ωt) −/parenleftbiggk2 r−k2y2 r3+3y2 r5−1 r3/parenrightbigg sin(kr−ωt)/bracketrightbigg −e2E2 ysin(kz−wt) 4πmω2/bracketleftbiggk2z r2sin(kr−ωt) +kz r3cos(kr−ωt)/bracketrightbigg −e2E2 xcos(kz−wt) 4πmω2/bracketleftbiggk2z r2cos(kr−ωt)−kz r3sin(kr−ωt)/bracketrightbigg . The terms in ExEywill vanish on integration over vol- ume. The various time averages are /an}b∇acketle{t2 cos(kz−ωt)cos(kr−ωt)/an}b∇acket∇i}ht = cos kzcoskr+ sinkzsinkr, /an}b∇acketle{t2 sin(kz−ωt)cos(kr−ωt)/an}b∇acket∇i}ht = sinkzcoskr−coskzsinkr, /an}b∇acketle{t2 cos(kz−ωt)sin(kr−ωt)/an}b∇acket∇i}ht = cos kzsinkr−sinkzcoskr, /an}b∇acketle{t2 sin(kz−ωt)sin(kr−ωt)/an}b∇acket∇i}ht = cos kzcoskr+ sinkzsinkr. (50) 6After performing the time average on eq. (49), we keep only terms that are even in z. These terms have the form (31), and so we find that uint=/integraldisplay V/an}b∇acketle{tUwave,osc/an}b∇acket∇i}ht=−e2(E2 x+E2 y) 8πmω2I1=−4 3η2mc2, (51) for waves of either linear or circular polarization. As with the case of the interference field momentum, this interference field energy is distributed over a volume of order a cubic wavelength around the electron. Being an interference term, its sign can be negative. We can interpret the quantity, uint c2=−4 3η2m, (52) as compensation for the relativistic mass increase of the oscillating electron, which scales as v2 rms/c2and hence as η2(for small η, recall eq. (7)). Indeed, a general result for the motion of an electron in a plane wave of arbitrary strength ηis that its rms relativistic mass, often called its effective mass, is [4,7] meff=m/radicalbig 1 +η2. (53) For small η, the increase in mass is ∆m≈1 2η2m. (54) Thus, the decrease in field energy due to the interfer- ence terms between the electromagnetic fields of the wave and electron is −8/3 times the mass increase it should compensate. VI. DISCUSSION A. Temporary Acceleration We remarked in sec. IIA that the preceding analysis holds in the average rest frame of the electron. If instead the electron had been at rest prior to the arrival of the plane wave, the velocity of the average rest frame would bevz= (η2/2)/(1+η2/2). For this, the amplitude of the plane wave is presumed to have a slow rise from zero to a long plateau at strength η, followed by a slow decline back to zero. Once the wave has passed by the electron, the inter- ference field energy, (51), goes to zero since the integral is dominated by the contribution at distances of order a wavelength from the electron. Hence, the electron’s ki- netic energy must return to zero (or to its initial value if that was nonzero). A plane wave, or more precisely, a long pulse that is very nearly a plane wave, cannot transfer net energy to an electron. The acceleration of the electron from zero velocity to vzis only temporary, i.e.,for the duration of the plane wave pulse. This result was first deduced by di Francia [12] and by Kibble [4] by different arguments.B. The Radiation Reaction Our analysis of the energy balance of an electron in a plane wave is not quite complete. We have neglected the energy radiated by the electron. Since the rate of radiation is constant (once the electron is inside the plane wave), the total radiated energy grows linearly with time, and eventually becomes large. The interference energy, (51), is constant in time, and hence cannot account for the radiated energy. More to follow..... VII. APPENDIX: LI ´ENARD-WIECHERT FIELDS As an alternative to the dipole approximation, we con- sider the use of the Li´ enard-Wiechert potentials and fields of a moving electron. We have limited our analysis to the case of a weak plane wave ( η≪1), for which the velocity of the electron is always small ( β=v/c≪1). In this case we may approximate the time-dependent part of the fields of the electron as proportional to the strength of the field of the plane wave (proportional to η. Then we find that the Li´ enard-Wiechert fields of the electron are the same as the fields in the dipole approximation. We can show this in two ways. First, we verify that the Li´ enard-Wiechert potentials reduce to eqs. (20) and (22). Second, we can verify directly that the Li´ enard-Wiechert fields are the same as eqs. (24) and (25). The Li´ enard-Wiechert potentials are φ=/bracketleftbigge R(1−β·ˆn/bracketrightbigg ,A=/bracketleftbiggeβ R(1−β·ˆn/bracketrightbigg ,(55) where the electron is at postion x, the observer is at r, their separation is R=r−x, the unit vector ˆnisR/R, and the brackets, [ ], indicate that quantities within are to be evaluated at the retarded time, t′=t−R/c. We work in the average rest frame of the electron. In the weak-field approximation we ignore the longitudinal motion of the electron, (11), which is quadratic in the strength of the plane wave. Then the velocity vector of the electron is β(t) =e mωc(ˆxExsinωt−ˆyEycosωt), (56) from eq. (6). The retarded velocity is thus, [β] =β(t′=t−R/c) (57) =−e mωc(ˆxExsin(kR−ωt) +ˆyEycos(kr−ωt)). Distance Rdiffers from rbecause the electron’s oscilla- tory motion takes it away from the origin. However, the amplitude of the motion is proportional to strength of the plane wave. Hence, we may replace Rbyrin eq. (57) with error only in the second order of field strength. Since the vector potential includes a factor βin the numerator, we can replace Rbyrand 1 −β·ˆnby 1 7in the first order in the field strength of the plane wave. Thus, A=−e2 mωcr(ˆxExsin(kr−ωt) +ˆyEycos(kr−ωt)), (58) in agreement with eq. (23). In the scaler potential, we first bring βto the numer- ator: φ≈e[1 +β·ˆn] [R]. (59) Unit vector [ ˆn] differs from unit vector ˆrdue to the oscil- lation of the electron, which is proportional to the field strength of the plane wave. For the scalar potential, how- ever, we must expand the factor 1 /[R] to first order in the field strength. Now, [R] =|r−x(t′)|=/radicalbig r2−2r·x(t′) +x2(t′),(60) with x(t′)≈ −e mω2(ˆxExcosωt′+Eycosωt′) (61) ≈ −e mω2(ˆxExcosω(kr−ωt)−ˆyEycos(kr−ωt)), again approximating Rbyrin the arguments of the co- sine and sine, accurate to first order in the field strength. Hence, 1 [R]≈1 r(1 +r·x(t′)) (62) ≈1 r/braceleftbigg 1−e(xExcos(kr−ωt)−yEysin(kr−ωt)) mω2r2/bracerightbigg . Altogether, φ≈e r−e2 mω2/braceleftbigg Ex/parenleftbiggkx r2sin(kr−ωt) +x r3cos(kr−ωt)/parenrightbigg +Ey/parenleftbiggky r2cos(kr−ωt)−y r3sin(kr−ωt)/parenrightbigg/bracerightbigg ,(63) in agreement with eq. (22). Similarly, we could proceed from the Li´ enard-Wiechert fields, E=e/bracketleftbiggˆn−β γ2(1−β·ˆn)3R2/bracketrightbigg +e c ˆn×/braceleftBig (ˆn−β)×˙β/bracerightBig (1−β·ˆn)3R , B= [ˆn×E]. (64) After some work, we find that these fields are the same as eqs. (24-25), to first order in the strength of the plane wave.[1] K.T. McDonald, “Comment on “Experimental Observa- tion of Electrons Accelerated in Vacuum to Relativistic Energies by a High-Energy Laser” by G. Malka, E. Lefeb- vre and J.L. Miquel, Phys. Rev. Lett. 78, 3314 (1997)”, Phys. Rev. Lett. 80, 1350 (1998). [2] J.H. Poynting, “On the Transfer of Energy in the Electro- magnetic Field” Phil. Trans. 175, 343-361 (1884); also, pp. 174-193, Collected Scientific Papers (Cambridge U. Press, 1920). [3] H. Poincar´ e, “Th´ eorie de Lorentz et le Principe de la R´ eaction”, Arch. Ne´ erl. 5, 252-278 (1900). [4] T.W.B. Kibble, “Frequency Shift in High-Intensity Compton Scattering”, Phys. Rev. 138, B740-753, (1965); “Radiative Corrections to Thomson Scattering from Laser Beams”, Phys. Lett. 20, 627-628 (1966); “Re- fraction of Electron Beams by Intense Electromagnetic Waves”, Phys. Rev. Lett. 16, 1054-1056 (1966); ‘Mu- tual Refraction of Electrons and Photons”, Phys. Rev. 150, 1060-1069 (1966); “Some Applications of Coher- ent States”, Carg` ese Lectures in Physics , Vol. 2, ed. by M. L´ evy (Gordon and Breach, New York, 1968), pp. 299- 345. [5] E.S. Sarachik and G.T.Schappert, “Classical Theory of the Scattering of Intense Laser Radiation by Free Elec- trons”, Phys. Rev. D 1, 2738-2753 (1970). [6] K.T. McDonald and K. Shmakov, “Temporary Accelera- tion of Electrons While Inside an Intense Electromagnetic Pulse”, Phys. Rev. ST Accel. Beams 2, 121301-121305 (1999). [7] L. Landau and E.M. Lifshitz, The Classical Theory of Fields , 4th ed. (Pergamon Press, Oxford, 1975), prob. 2, §47 and prob. 2, §49; p. 112 of the 1941 Russian edition. [8] E.M. McMillan, “The Origin of Cosmic Rays”, Phys. Rev.79, 498-501 (1950). [9] V. Hnizdo, “Hidden momentum and the electromagnetic mass of a charge and current carrying body”, Am. J. Phys.65, 55-65 (1997), and references therein. [10] J. Schwinger, “Electromagnetic Mass Revisited”, Foun d. Phys.13, 373-383 (1983). [11] P. Moylan, “An Elementary Account of the Factor of 4/3 in the Electromagnetic Mass”, Am. J. Phys. 63, 818-820 (1995). [12] G. Toraldo di Francia, Interaction of Focused Laser Ra- diation with a Beam of Charged Particles , Nuovo Cim. 37, 1553 (1965). 8

arXiv:physics/0003060v1 [physics.acc-ph] 23 Mar 2000Princeton University March 9, 1987 DOE/ER/3072-41 THE HAWKING-UNRUH TEMPERATURE AND QUANTUM FLUCTUATIONS IN PARTICLE ACCELERATORS K. T. McDonald Joseph Henry Laboratories, Princeton University, Princet on, New Jersey 08544 We wish to draw attention to a novel view of the effect of the quantum fluctuations during the radiation of accel- erated particles, particularly those in storage rings. Thi s view is inspired by the remarkable insight of Hawking1 that the effect of the strong gravitational field of a black hole on the quantum fluctuations of the surrounding space is to cause the black hole to radiate with a temperature T=¯hg 2πck, where gis the acceleration due to gravity at the surface of the black hole, cis the speed of light, and kis Boltz- mann’s constant. Shortly thereafter Unruh2argued that an accelerated observer should become excited by quan- tum fluctuations to a temperature T=¯ha⋆ 2πck, where a⋆is the acceleration of the observer in its instan- taneous rest frame. In a series of papers Bell and co- workers3−5have noted that electron storage rings provide a demonstration of the utility of the Hawking-Unruh tem- perature, with emphasis on the question of the incomplete polarization of the electrons due to quantum fluctuations of synchrotron radiation. Here we expand slightly on the results of Bell et al., and encourage the reader to consult the literature for more detailed understanding. Applicability of the Idea When an accelerated charge radiates, the discrete en- ergy and momentum of the radiated photons induce fluc- tuations on the motion of the charge. The insight of Un- ruh is that for uniform linear acceleration (in the absense of the fluctuations), the fluctuations would excite any in- ternal degrees of freedom of the charge to the temperature stated above. His argument is very general ( i.e., thermo- dynamic) in that it does not depend on the details of the accelerating force, nor of the nature of the accelerated particle. The idea of an effective temperature is strictly applicable only for uniform linear acceleration, but shoul d be approximately correct for other accelerations, such as that due to uniform circular motion. A charged particle whose motion is confined by the fo- cusing system of a particle accelerator exhibits transvers e and longitudinal oscillations about its ideal path. These oscillations are excited by the quantum fluctuations of the particle’s radiation, and thus provide an excellent physi- cal example of the viewpoint of Unruh. Further, the particles take on a thermal distribution of energies when viewed in the average rest frame of a bunch, which transforms to the observed energy spread in the laboratory. While classical synchrotron radiation would eventually polarize the spin-1 2particles completely, the thermal fluctuations oppose this, reducing the maximum beam polarization.It is suggestive to compare the excitation energy U⋆= kT, as would be observed in the particle’s rest frame, to the rest energy mc2when the acceleration is due to laboratory electromagnetic fields EandB. Noting that a⋆=eE⋆/mwe find U⋆ mc2=¯heE⋆ 2πm2c3=/bracketleftbig E/bardbl+γ(E⊥+βB⊥)/bracketrightbig 2πEcrit, where the particle’s laboratory momentum is γβmc , and Ecrit≡m2c3 e¯h. For an electron, Ecrit= 1.3×1016volts/cm = 4 .4×1013gauss. (Ecritis the field strength at which spontaneous pair pro- duction becomes highly probable, i.e., the field whose voltage drop across a Compton wavelength is the parti- cle’s rest energy.) We might expect that the fluctuations become noticeable when U⋆∼0.1 eV, and hence compa- rable to any other thermal effects in the system, such as the particle-source temperature. For linear accelerators E/bardbl∼106volts/cm at best, so U⋆<10−5eV. The effect of quantum fluctuations is of course negligible because the radiation itself is of little importance in a linear accelerator. For an electron storage ring such as LEP, γ∼105, andB⊥∼103gauss, so that U⋆∼0.2 eV. For the SSC proton storage ring, γ∼2×104, while B⊥∼6×104 gauss, so that U⋆∼2 eV. As is well known, in essentially all electron storage rings, and in future proton rings, the effect of quantum fluctuations is quite important. The remaining discussion is restricted to beams in stor- age rings (= transverse particle accelerators). Beam-Energy Spread An immediate application of the excitation energy U⋆is to the beam-energy spread. In the average rest frame of a bunch of particles, the distribution of energies is approx- imately thermal, with characteristic kinetic energy U⋆, and momentum p⋆=√ 2mU⋆. The spread in laboratory energies is then given by Ulab≈γ(mc2+U⋆±βp⋆c)≈U0/parenleftBigg 1±γ/radicalBigg λC πρ/parenrightBigg , where U0=γmc2is the nominal beam energy, ρ= U0/eB⊥is the radius of curvature of the central orbit, andλC= ¯h/mc is the Compton wavelength. Writing this as/parenleftbiggδU U0/parenrightbigg2 ≈γ2λC πρ, Submitted to the 1987 Particle Accelerator Conference 1Princeton University March 9, 1987 DOE/ER/3072-41 we obtain the standard result, as given by equation (5.48) of the review by Sands.6 Beam Height The quantum fluctuations of synchrotron radiation drive the oscillations of particles about the bunch cen- ter, and set lower limits on the transverse and longitudi- nal beam size. If we associate a harmonic oscillator with each component of the motion about the bunch center, then each oscillator will be excited to amplitudes whose corresponding energy is U⋆=kT⋆. For example, consider the vertical betatron oscillations which determine the beam height. The frequency of these oscillations is ω=νzω0=νzc/R, where νzis the vertical betatron number, and R=L/2πis the mean radius of the storage ring. In the average rest frame of a bunch the oscillation frequency appears to be ω⋆=γω, and the spring constant in this frame is given by k⋆=mω⋆2= γ2mω2. The typical amplitude of oscillation in this frame is then 1 2k⋆z⋆2≈U⋆=¯ha⋆ 2πc=¯hγ2a 2πc=¯hγ2c 2πρ, noting that in uniform circular motion the acceleration is transverse. For the vertical oscillation the lab frame amplitude zis the same as z⋆. Combining the above we find z2=λCR2 πν2zρ, which reproduces the standard result, such as equation (5.107) of Sands.6 An analogous argument is given in ref. 5 to derive the beam height in a weakly focused storage ring. Bunch Length and Beam Width A similar analysis can be given for oscillations in the plane of the orbit. However, radial and longitudinal ex- cursions are also directly coupled to energy excursions, which proves to be the stronger effect. As the present method finds the standard result for the beam-energy spread, the usual results for bunch length and beam width follow at once. [In ref. 6, use equations (5.64) and (5.93) to yield expressions (5.65) and (5.95).] Beam Polarization Sokolov and Ternov7predicted that quantum fluctua- tions in synchrotron radiation limit the transverse polar- ization of the beam to 92%. In the absense of quantum fluctuations the polarization should reach 100% after long times. Bell and Leinaas3realized that the thermal char- acter of the fluctuations provides an alternate view of the depolarizing mechanism. In ref. 5 they provide a detailed justification that the thermodynamic arguments are fully equivalent to the original QED calculation of Sokolov and Ternov. In the process they find that for circular motion in a weakly focused ring (betatron), the effective temper- ature due to quantum fluctuations is kT=13 96√ 3¯ha⋆ c,which is about 1.5 times Unruh’s result for linear accel- eration. Radiation Spectrum Because of the quantum fluctuations the motion of the particles departs from the central orbit, and a classical calculation of the synchrotron-radiation spectrum is in- correct in principle. The deviations become significant only when the characteristic energy of the radiation ap- proaches the beam energy, i.e., when γB⊥/Ecrit∼1, and the prominent effect is the cutoff at the high-energy end of the spectrum. In the regime where the quantum corrections to the radiation spectrum are small the author has given an estimate of their size.8For this we imagine the accel- erated charge is surrounded (in its rest frame) by a bath of photons with a Planck spectrum of temperature kT= ¯ha⋆/2πc. The correction to the classical spectrum is considered to arise from the Thomson scattering of these virtual photons off the charged particle. In the lab frame the spectral correction is proportional to the Lorentz transform of the Planck spectrum, whose peak photon energy is then 2 γkT= ¯hγ3c/πρ, essentially the same as that of the classical spectrum. On integrating over energy, the total rate of the correction term is the classical (Larmor) rate times α 60π/parenleftbiggγB⊥ Ecrit/parenrightbigg2 , which is indeed very small at present storage rings. Acknowledgements I would like to thank Ian Affleck and Heinrich Mitter for several discussions on this topic. This work was supported in part by the U.S. Department of Energy under contract DOE-AC02-76ER-03072. 1S.W. Hawking, “Black-Hole Explosions”, Nature 248, 30-31 (1974). 2W.G. Unruh, “Notes on Black-Hole Evaporation”, Phys. Rev. D 14, 870-892 (1976). 3J.S. Bell and J.M. Leinaas, “Electrons as Accelerated Thermometers”, Nucl. Phys. B212 , 131-150 (1983). 4J.S. Bell, R.J. Hughes and J.M. Leinaas, “The Unruh Effect in Extended Thermometers”, Z. Phys. C 28, 75-80 (1985). 5J.S. Bell and J.M. Leinaas, “The Unruh Effect and Quantum Fluctuations of Electrons in Storage Rings”, Nucl. Phys. B284 , 488-508 (1987). 6M. Sands, “The Physics of Electron Storage Rings”, SLAC-121, (1970); also in Proc. 1969 Int. School of Physics, ‘Enrico Fermi,’ ed. by B. Touschek (Academic Press, 1971), p. 257. 7A.A. Sokolov and I.M. Ternov, “On Polarization and Spin Effects in the Theory of Synchrotron Radiation”, Sov. Phys. Dokl. 8, 1203 (1964). 8K.T. McDonald, “Fundamental Physics During Vio- lent Acceleration”, in Laser Acceleration of Particles , AIP Conf. Proc. 13023-54 (1985). 2

arXiv:physics/0003061v1 [physics.acc-ph] 23 Mar 2000THE HAWKING-UNRUH TEMPERATURE AND DAMPING IN A LINEAR FOCUSING CHANNEL KIRK T. McDONALD Joseph Henry Laboratories, Princeton University, Princet on, NJ 08544 mcdonald@puphep.princeton.edu http://puhep1.princeton.edu/ ˜mcdonald/accel/ (January 30, 1998) The Hawking-Unruh effective temperature,¯ha⋆ 2πck, due to quantum fluctuations in the radiation of an accelerated charged-particle beam can be used to show that transverse oscillations of the beam in a practical linear focusing chan - nel damp to the quantum-mechanical limit. A comparison is made between this behavior and that of beams in a wiggler. I. INTRODUCTION Many of the effects of quantum fluctuations on the be- havior of charged particles can be summarized concisely by an effective temperature first introduced in gravita- tional fields by Hawking [1], and applied to accelerated particles (with the neglect of gravity) by Unruh [2]. Hawking argued that the effect of the strong gravita- tional field of a black hole on the quantum fluctuations of the surrounding space is to cause the black hole to radiate with a temperature T=¯hg 2πck, (1) where gis the acceleration due to gravity at the surface of the black hole, cis the speed of light, and kis Boltz- mann’s constant. Shortly thereafter, Unruh argued that an accelerated observer should become excited by quan- tum fluctuations to a temperature T=¯ha⋆ 2πck, (2) where a⋆is the acceleration of the observer in its instan- taneous rest frame. In a series of papers, Bell and co-workers [3], have noted that electron storage rings provide a demonstra- tion of the utility of the Hawking-Unruh temperature (2), with emphasis on the question of the incomplete po- larization of the electrons due to quantum fluctuations of synchrotron radiation. The author has commented on how the Hawking-Unruh temperature can be used to characterize quickly the limits on damping of the phase volume of beams in electron storage rings [4], leading to well-known results of Sands [5].II. QUANTUM ANALYSIS OF A LINEAR FOCUSING CHANNEL Recently, Chen, Huang and Ruth have discussed radi- ation damping in a linear focusing channel [6–8], finding that in such devices the beam can be damped to the quantum mechanical limit set by the uncertainty princi- ple. I show here how this result follows very quickly from an application of the Hawking-Unruh temperature. A linear focusing channel is a beam-transport system that confines the motion of a charged particle along a straight central ray via a potential that is quadratic in the transverse spatial coordinates. This potential can be characterized by a spring constant k, and hence the frequency ωof transverse oscillations (as observed in the laboratory frame) of a particle of mass mand Lorentz factor γis ω=/radicalBigg k γm. (3) If the amplitude of the oscillation in transverse coordi- natexis called x0, then the amplitude a0of the corre- sponding transverse acceleration is a0=x0ω2=kx0 γm. (4) To apply the Hawking-Unruh temperature, we con- sider the motion in the instantaneous rest frame of the particle. Supposing the transverse oscillations are small , the instantaneous rest frame is very nearly the frame in which the particle has no longitudinal motion. Quanti- ties measured in the instantaneous rest frame will by de- noted with the superscript ⋆. Thus, in the instantaneous rest frame the amplitude of the transverse acceleration as measured is a⋆ 0=γ2a0=γkx0 m, (5) the frequency of the oscillation is ω⋆=γω, (6) and hence the transverse spring constant of the focusing channel appears as k⋆=mω⋆2=γk. (7) 1In the instantaneous rest frame, the charge particle finds itself in a bath of radiation of characteristic tem- perature given by eq. (2) with acceleration a⋆given by eq. (5). This bath can be regarded as the effect of quantum fluctuations, which excite transverse oscilla- tions (having two degrees of freedom) to characteristic energy U⋆(as measured in the instantaneous rest frame) given by U⋆=kT=¯ha⋆ 0 2πc=¯hγkx 0 2πmc. (8) The energy of transverse oscillation can also be written in terms of the (invariant) transverse amplitude x0as U⋆=k⋆x⋆2 0 2=γkx2 0 2. (9) Hence, the amplitude of excitation of the transverse os- cillations is x0=¯h πmc=λC π, (10) where λCis the (reduced) Compton wavelength of the particle. The amplitude (10) must, however, be compared to the amplitude of the zero-point oscillations of the system, considered as a quantum oscillator: x0,zero point =/radicalBigg ¯h γmω=/radicalBigg λCλ γ, (11) where λ=c/ωis the laboratory (reduced) wavelength of the transverse oscillation as measured along the beam axis. In practical laboratory devices, we will have λ≫ γλC. Hence, the excitation of the transverse oscillations by fluctuations in the radiation of the oscillating charge, as are described by the Hawking-Unruh temperature, is negligible compared to the zero-point fluctuations of the transverse oscillations. In this sense, we can say along with Huang, Chen and Ruth that the radiation does not excite the transverse oscillations, and those oscillation s will damp to the quantum-mechanical limit. In futuristic devices, for which γ >λ/λC,i.e., when γ >mc2 kλC, (12) quantum excitations of oscillations in a linear focusing channel would become important. When (12) holds, the transverse oscillations would be relativistic even when their amplitude is only a Compton wavelength. The strength of the transverse fields in the channel would then exceed the QED critical field strength (in the average rest frame), Ecrit=m2c3 e¯h= 1.6×1016V/cm = 3 .3×1013Gauss , (13)and the beam energy would be rapidly dissipated by pair creation. Another way of viewing a practical linear focusing channel is that its Hawking-Unruh excitation energy, (8), is small compared to the zero-point energy, ¯ hω⋆/2 = γ¯hω/2 of transverse oscillations. The quantum-mechanical limit for transverse motion can, of course, also be deduced from the uncertainty prin- ciple: σxσpx>∼¯h, (14) which leads to a minimum normalized emittance of ǫN=σxσpx mc≈λC, (15) corresponding to geometric emittance of ǫx=ǫN γβz≈λC γ. (16) III. SEMICLASSICAL ANALYSIS In a quantum analysis of a linear focusing channel, we found that the transverse oscillations can damp to the limit set by the uncertainty principle. Hence, in a classical analysis we would expect the damping to be able to proceed until the transverse amplitude was zero. Indeed, a simple analysis confirms this. Transform to the longitudinal rest frame, in which the particle’s mo- tion is purely transverse. The particle has nonzero kinetic energy in this frame, but its average momentum is zero. The radiation due to the transverse oscillation is reflec- tion symmetric about the transverse plane in this frame, so the radiation carries away energy but not momentum. With time, all of the energy would be radiated away, and the particle would come to rest. The transverse oscil- lations will have damped to zero without affecting the longitudinal motion. If we add the concept of photons to the preceding anal- ysis, we can say that the radiated photons carry away momentum along the direction of emission, but the ra- diation pattern is symmetric, so the averaged radiated momentum is zero. Again, the radiation carries away energy, now in the form of photons. Back in the lab frame, we view the photons as car- rying away a small amount of longitudinal momentum on average, as a result of the Lorentz transformation of the energy radiated in the longitudinal rest frame. This momentum, however, is only that part of the particle’s longitudinal momentum associated with its transverse os- cillation; the longitudinal velocity of the particle is una f- fected. On average, the photons carry away no transverse mo- mentum in the lab frame, and the average momentum of the radiated photons is therefore parallel to the beam axis 2in lab frame. However, there is no need to argue that the momentum of individual radiated photons is parallel to the beam axis, nor to imply that the matter of the focus- ing channel absorbs transverse momentum in a manner than affects the kinematics of the radiation process [7]. IV. COMPARISON OF A LINEAR FOCUSING CHANNEL TO A WIGGLER A comparison with the behavior of particle beams in a wiggler is instructive. Here the transverse confinement of the beam motion is provided by a series of alternat- ing transverse magnetic fields. This has the notable ef- fect that even if a particle enters the wiggle parallel to the beam axis, transverse oscillations will result whose amplitude is independent of the initial transverse coordi- nate. In contrast, a particle that enters a linear focusing channel parallel to and along the axis undergoes no os- cillation, no matter what is the particle’s longitudinal momentum. We thereby see that radiation damping cannot reduce the oscillations in a wiggler to zero unless the longitudina l momentum falls to zero also, since the wiggler continu- ally re-excites transverse oscillations for any particle w ith nonzero kinetic energy. Another difference between a wiggler and a linear fo- cusing channel can be seen by going to the longitudinal rest frame. In the case of the wiggler, the alternating magnetic fields in the laboratory transform to fields that are very much like a plane wave propagating against the direction of the laboratory motion of the beam. The ra- diation induced by this effective plane wave is not sym- metric with respect to the transverse plane, but results in a net kick of the particle into the backward direction. Viewed in the lab frame, we find that along with the damping of their transverse oscillations, the parti- cles’ longitudinal momenta are significantly reduced. To maintain the initial longitudinal momentum, the beam must be reaccelerated. The momentum (and energy) added back into the beam then increases the amplitude of the transverse oscillations, and the damping cannot continue beyond some limit. In contrast, in a linear focusing channel, the transverse damping proceeds without significant reduction in the longitudinal momentum of the particle, and the trans- verse oscillations can damp to the quantum limit without the need of adding energy back into the beam. ACKNOWLEDGMENTS I wish to thank Pisin Chen, Ron Ruth and Max Zolo- torev for conversations on radiation damping. This work was supported in part by DoE grant DE-FG02- 91ER40671.REFERENCES [1] S.W. Hawking, Black Hole Explosions , Nature 248, 30-31 (1974); Particle Creation by Black Holes , Comm. Math.- Phys.43, 199-220 (1975). [2] W.G. Unruh, Notes on Black Hole Evaporation , Phys. Rev. D 14, 870-892 (1976); Particle Detectors and Black Hole Evaporation , Ann. N.Y. Acad. Sci. 302, 186-190 (1977). [3] J.S. Bell and J.M. Leinaas, Electrons as Accelerated Ther- mometers , Nuc. Phys. B212 , 131-150 (1983); The Unruh Effect and Quantum Fluctuations of Electrons in Storage Rings,B284 , 488-508 (1987); J.S. Bell, R.J. Hughes and J.M. Leinaas, The Unruh Effect in Extended Thermome- ters, Z. Phys. C28, 75-80 (1985); W.G. Unruh, Acceler- ation Radiation for Orbiting Electrons (hep-th/9804158, 23 Apr 98); J.M. Leinaas, Accelerated Electrons and the Unruh Effect (hep-th/9804179, 28 Apr 1998). [4] K.T. McDonald, The Hawking-Unruh Temperature and Quantum Fluctuations in Particle Accelerators , Proceed- ings of the 1987 IEEE Particle Accelerator Conference, E.R. Lindstrom and L.S. Taylor, eds. (Washington, D.C., Mar. 16-19, 1987) pp. 1196-1197. [5] M. Sands, The Physics of Electron Storage Rings , SLAC- 121, (1970); also in Proc. 1969 Int. School of Physics, ‘En- rico Fermi’, ed. by B. Touschek (Academic Press, 1971), p. 257. [6] Z. Huang, P. Chen and R.D. Ruth, Radiation Reaction in a Continuous Focusing Channel , Phys. Rev. Lett. 74, 1759-1762 (1995). [7] Z. Huang, P. Chen and R.D. Ruth, Radiation and Ra- diation Reactions in Continuous Focusing Channels , in Advanced Accelerator Concepts , AIP. Conf. Proc. 335, 646-658 (1995). [8] Z. Huang and R.D. Ruth, Effects of Focusing on Radia- tion Damping and Quantum Excitation in Electron Stor- age Rings , Phys. Rev. Lett. 80, 2318 (1998). 3

arXiv:physics/0003062v1 [physics.class-ph] 23 Mar 2000Limits on the Applicability of Classical Electromagnetic F ields as Inferred from the Radiation Reaction Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princet on, NJ 08544 (Jan. 29, 1998) Can the wavelength of a classical electromagnetic field be arbitrarily small, or its electric field strength be arbitra rily large? If we require that the radiation-reaction force on a charged particle in response to an applied field be smaller th an the Lorentz force we find limits on the classical electromag- netic field that herald the need for a better theory, i.e., one in better accord with experiment. The classical limitation s find ready interpretation in quantum electrodynamics. The examples of Compton scattering and the QED critical field strength are discussed. It is still open to conjecture wheth er the present theory of QED is valid at field strengths beyond the critical field revealed by a semiclassical argument. I. INTRODUCTION The ultimate test of the applicability of a physical the- ory is the accuracy with which it describes natural phe- nomena. Yet on occasion the difficulty of a theory in dealing with a “thought experiment” provides a clue as to limitations of that theory. It has long since been recognized that classical elec- trodynamics has been surplanted by quantum electrody- namics in some respects. But one doubts that quantum electrodynamics, or even its generalization, the Standard Model of elementary particles, is valid in all domains. To aid in the search for new physics, it is helpful to re- view the warning signs of the past transitions from one theoretical description to another. The debates as to the meaning of the classical radia- tion reaction for pointlike particles provide examples of such warning signs. One case is the “4/3 problem” of electromagnetic mass, where covariance does not imply uniqueness [1]. Such difficulties have often been inter- preted as suggesting that classical electrodynamics can- not be a complete description of matter on the scale of the classical electron radius, r0=e2/mc2in Gaussian units. It seems less appreciated that the part of the classi- cal radiation reaction that is independent of particle size provides clues as to the limits of applicability of classi- cal electromagnetic fields. For example, a recent article [2] ends with the sentence, “Only when all distances in- volved are in the classical domain is classical dynamics acceptable for electrons”. While this condition is neces- sary, it is not sufficient. For a classical description to be accurate, an electron can only be subject to fields that are not too strong. This paper seeks to illustrate what “not too strong” means.Considerations of strong fields have been very influen- tial in the development of other modern theories besides quantum electrodynamics. In classical gravity, i.e., gen- eral relativity, the strong-field problem is identified with black holes. One of the best known intersections between gravity and quantum electrodynamics is the Hawking ra- diation of black holes. In the case of the strong (nu- clear) interaction, the fields associated with nuclear mat- ter all appear to be strong, and weak fields are thought to exist only in the high-energy limit (asymptotic free- dom). Such considerations led to the introduction of non- Abelian gauge theories. These constructs, when applied to the weak interaction, led to the concept of a back- ground (Higgs) field that is strong in the sense of having a large vacuum expectation value, which in turn has the effect of generating the masses of the WandZgauge bosons. Most recently, considerations of the strong-field (strong-coupling) limit of string theories have led to the notion of “duality”, i.e., the various string theories of the 1980’s are actually different weak-field limits of a single strong-field theory. These string theories are noteworthy for suggesting that particles are to be considered as exci- tations of small, but extended quantum strings, thereby avoiding the infinite self energies that have appeared in theories of point particles since J.J. Thomson introduced the concept of electromagnetic mass in 1881 [3]. The main argument concerning classical electromag- netic fields is given in sec. 2, and is brief. This argument could have been given around 1900 by Lorentz [4–6] or by Planck [7]. who made remarks of a related nature. But the argument seems to have been first made in 1935 by Oppenheimer [8], and more explicitly by Landau and Lifshitz [9]. Additional historical commentary is given in sec. 1.1. Sections 2.1-2.5 comment on various aspects of the main argument, still from a classical perspective. A quantum view in introduced in sec. 3, and the important examples of Compton scattering and the QED critical field are discussed in secs. 3.1-2. The paper concludes in sec. 4 with remarks on the role of strong fields on the de- velopment of quantum electrodynamics, and presents two examples (secs. 4.1-2) of speculative features of strong- field QED and one of very short distance QED (sec. 4.3). A. Historical Introduction The relation between Newton’s third law and electro- magnetism has been of concern at least since the inves- tigations of Amp` ere, who insisted that the force of one 1current element on another be along their line of centers. See Part IV, Chap. II, especially sec. 527, of Maxwell’s Treatise for a review [10]. However, the presently used differential version of the Biot-Savart law does not satisfy Newton’s third law for pairs of current elements unless they are parallel. Perhaps discomfort with this fact contributed to the delay in acceptance of the concept of isolated electrical charges, in contrast to complete loops of current, until the late nineteenth century. A way out of this dilemma became possible after 1884 when Poynting [11] and Heaviside [12] argued that elec- tromagnetic fields (in suitable configurations) can be thought of as transmitting energy. The transmission of energy was then extended by Thomson [13], Poincar´ e [14] and Abraham [15] to include transmission (and storage) of momentum by an electromagnetic field. That a moving charge interacting with thermal radia- tion should feel a radiation pressure was anticipated by Stewart in 1871 [16], who inferred that both the energy and the momentum of the charge would be affected. In 1873, Maxwell discussed the pressure of light on conducting media at rest, and on “the medium in which waves are propagated” ( [10], secs. 792-793). In the for- mer case, the radiation of a reflected wave by a (perfectly conducting) medium in response to an incident wave re- sults in momentum, but not energy, being transferred to the medium. The energy for the reflected wave comes from the incident wave. The present formulation of the radiation reaction is due to Lorentz’ investigations of the self force of an extended electron, beginning in 1892 [4] and continuing through 1903 [5]. The example of dipole radiation of a single charge contrasts strikingly with Maxwell’s discussion of reaction forces during specular reflection. There is no net momentum radiated by an oscillating charge with zero average velocity, but energy is radiated. The external force alone can not account for the energy balance. An additional force is needed, and was identified by Lorentz as the net electromagnetic force of one part of an ex- tended, accelerated charge distribution on another. See eq. (1) below. In 1897, Planck [7,17] applied the radiation reaction force of Lorentz to a model of charged oscillators and noted the existence of what are sometimes called “run- away” solutions, which he dismissed as having no physical meaning (keine physikalishe Bedeutung). The basic concepts of the radiation reaction were brought essentially to their final form by Abraham [18,19], who emphasized the balance of both energy and momentum in the motion of extended electrons moving with arbitrary velocity. Important contributions to the subject in the early twentieth century include those of Sommerfeld [20], Poincar´ e [21], Larmor [22], Lindemann [23], Von Laue [24], Born [25], Schott [26–29], Page [30], Nordstr¨ om [31] , Milner [32], Fermi [33], Wenzel [34], Wesel [35] and Wil- son [36]. The main theme of these works was, however,models of classical charges and the related topic of elec- tromagnetic mass. The struggle to understand the physics of atoms led to diminished attention to classical models of charged parti- cles in favor of quantum mechanics and quantum electro- dynamics (QED). In 1935, there was apparent disagree- ment between QED and reported observations at energy scales of 10-100 MeV. Oppenheimer [8] then conjectured whether QED might fail at high energies and, in partial support of his view, invoked a classical argument con- cerning difficulties of interpretation of the radiation re- action at short distances. The present article illustrates an aspect of Oppenheimer’s argument that was developed further by Landau [9]. Another response to Oppenheimer’s conjecture was by Dirac [37] in 1938, when he deduced a covariant expres- sion for the radiation reaction force (previously given by Abraham, Lorentz and von Laue in noncovariant nota- tion) by an argument not based on a model of an ex- tended electron. Dirac also gave considerable discus- sion of the paradoxes of runaway solutions and pre- acceleration. This work of Dirac, and most subsequent work on the classical radiation reaction, emphasized the internal consistency of classical electromagnetism as a mathematical theory, rather than as a description of na- ture. But, as has been remarked by Schott [26], “there is considerable danger, in a purely mathematical investiga- tion, of losing touch with reality”. Quantum mechanics had triumphed. Research articles on the classical radiation reaction are still being produced; see, for example, Refs. [38]- [89]. Sarachik and Schappert ( [67], sec. IIID) present a brief version of the argument given below in sec. 2. Reviews of the subject include Refs. [90]- [124]. Most noteworthy in relation to the present article are the re- views by Lorentz [6], Erber [100] and Klepikov [109], which are the only ones that indicate an awareness of the problem of strong fields. The texts of Landau and Lifshitz [9], Jackson [106] and Milonni [111] briefly men- tion that issue. The radiation reaction has been a frequent topic of articles in the American Journal of Physics, including Refs. [2] and [113]- [129]. The article of Page and Adams [113] is noteworthy for illustrating how the concept of electromagnetic field momentum restores the full validity of Newton’s third law in an interesting example of the interaction of a pair of moving charges. II. A GENERAL RESULT FOR THE RADIATION REACTION Consider an electron of charge eand mass mmov- ing in electric and magnetic fields EandB. The mass mis the “effective mass” in the language of Lorentz [6], now called the “renormalized” mass, for which the di- vergent electromagnetic self energy of a small electron is 2cancelled in a manner beyond the scope of this article. Then the remaining leading effect of the radiation reac- tion is the “radiation resistance” which is independent of hypotheses as to the structure of the electron. Our argument emphasizes the effect of radiation resistance, since any deductions about properties of electromagnetic fields will then be as free as possible from controversy as to the nature of matter. The (nonrelativistic) equation of motion including ra- diation resistance is (in Gaussian units) m˙v=Fext+Fresist, (1) where Fext=eE+ev c×B (2) is the Lorentz force on the electron due to the external field, Fresist=2e2 3c3¨v+O(v/c) (3) is the force of radiation resistance, vis the velocity of the electron, cis the speed of light and the dot indicates differentiation with respect to time. Equation (3) is the form of the radiation reaction given in the original deriva- tions of Lorentz [4] and Planck [7,17], which is sufficient for the main argument of this paper. Some discussion of the larger context of the classical radiation reaction is given in secs. 2.1-5. If the second time derivative of the velocity is small we estimate it by taking the derivative of (1): ¨v≈e˙E m+e m˙v c×B+e mv c×˙B. (4) We further suppose that the velocity is small (without loss of generality according to the principle of relativity ; see sec. 2.4 for a relativistic discussion). so it suffices to approximate ˙vaseE/min (4). Hence, ¨v≈e˙E m+e2 m2cE×B. (5) The radiation resistance is now Fresist≈2e2 3c3/parenleftBigg e˙E m+e2 m2cE×B/parenrightBigg . (6) The first term in (6) contributes only for time-varying fields, which I take to have frequency ωand reduced wavelength λ; hence, ˙E∝ωE. The second term con- tributes only when E×B∝negationslash= 0, which is most likely to be in a wave (with E=B) if the fields are large. So, for an electron in an external wave field, the magnitude of the radiation-resistance force isFresist≈2 3eE/radicalBigg/parenleftbigge2 mc2ω c/parenrightbigg2 +/parenleftbigge3E m2c4/parenrightbigg2 ≈Fext/radicalBigg/parenleftBigr0 λ/parenrightBig2 +/parenleftbiggE e/r2 0/parenrightbigg2 , (7) where r0=e2/mc2= 2.8×10−13cm is the classical electron radius. Equation (7) makes physical sense only when the ra- diation reaction force is smaller than the external force. Here we don’t explore whether the length r0describes a physical electron; we simply consider it to be a length that arises from the charge and mass of an electron. Rather, we concentrate on the implication of eq. (7) for the electromagnetic field. Then we infer that a classical description becomes implausible for fields whose wave- length is small compared to length r0, or whose strength is large compared to e/r2 0. A. Commentary The argument related to eq. (7) is that there are clas- sical electromagnetic fields that lead to physically im- plausible behavior when radiation-reaction effects are in- cluded. This does not necessarily imply any mathemati- cal inconsistency in the theory. Indeed, various authors have displayed solutions for electron motion coupled to an oscillator of very high natural frequency [48,101]. Such solutions are well-defined mathematically but ap- pear “physically implausible”. Of course, the mathemat- ics might be correct in predicting the physical behavior in an unfamiliar situation. So it becomes a matter of exper- iment to decide whether the characterization “implausi- ble” corresponds to physical reality or not. The experi- ments that produce the most influential results are typ- ically those that reveal new phenomena in realms where prevailing theories are “implausible”. Thus far, there is no evidence for the behavior pre- dicted by the classical equations for electrons interact- ing with waves of frequencies greater than c/r0. Rather, quantum mechanics is needed for a good description of the phenomena observed in that case, Compton scatter- ing being an early example (sec. 3.1). Laboratory stud- ies of strong-field electrodynamics have been undertaking only recently (sec. 3.2), and deal primarily with effects not anticipated in a classical description. The argument of sec. 2 can also be considered as a model-independent version of a restriction that Lorentz placed on his derivation of eqs. (1-3) ( [6], sec. 37, eq. (73)). Namely, the derivation makes physical sense only if l ct≪1, (8) where lis a characteristic length of the problem, and tis a characteristic time “during which the state of motion is sensibly altered”. 3Lorentz would certainly have considered the classical electron radius, r0, as an example of a relevant character- istic length. Hence, for an electron in an electromagnetic wave of (reduced) wavelength λ, the characteristic time of the resulting motion is λ/c, and Lorentz’ condition (8) becomes r0 λ≪1, (9) A close variant of the above argument was also given by Planck [7]. In case of a strong field with a long (possibly infinite) wavelength, Lorentz’ condition (8) can be interpreted as requiring the change in the electron’s velocity to be small compared to the speed of light during the time it takes light to travel one classical electron radius. That is, we require ∆v=a∆t=eE mr0 c≪c, (10) and hence, E≪mc2 er0=e r2 0. (11) Thus, we arrive by another (although closely related) path to the conclusion drawn previously from eq. (7). Perhaps because the limiting field strength implied by (11) is extraordinarily large by practical standards, nei- ther Lorentz nor Planck mentioned it explicitly. In the first sentence of his 1938 article, Dirac [37] stated that “the Lorentz model of the electron...has proved very valuable...in a certain domain of problems, in which the electromagnetic field does not vary too rapidly and the accelerations of the electrons are not too great”. How- ever, he does not elaborate on the meaning of “not too great”. Dirac’s derivation of the radiation-reaction 4-force was not based on a model of an extended electron, and so the derivation was not subject to Lorentz’ restriction (8). But as a co-inventor of quantum mechanics, Dirac cannot have expected his classical results to have unrestricted validity in the physical world. In the decade after Dirac’s 1938 paper, a few works [35,40,42,43,45,49] appeared that commented on the con- cept of a limiting field strength, typically in classical dis - cussions of electron-positron pair creation. In sec. 3.2 we return to the issue of pair creation, but in a quantum context. After the discovery of pulsars in 1967 there was a burst of interest in the behavior of electrons in very strong mag- netic fields. Several papers appeared in which classical electrodynamics was applied [66,70,71,73,74,76,77,81], of- ten with the intent of clarifying the boundary between the classical and quantum domains. For very large fields, classical solutions to the motion were obtained in which the electron has a damping time constant that is small compared to the period of cyclotron motion. Whetheror not such highly damped solutions are “implausible”, they are outside ordinary experience. Again, one must perform experiments to decide whether the classical the- ory is valid in this domain. If such experiments had been possible prior to the development of quantum mechan- ics, they would have revealed deviations from the classi- cal theory that would have encouraged development of a new theory. Arguments such as those leading to eqs. (7), (9) and (11) would have motivated the experiments. B. Another Strong-Field Regime Are there any other domains in which classical electro- dynamics might be called into question? Another interpretation of Lorentz’ criterion (8) is that the amplitude of the oscillatory motion of an electron in a wave of frequency ωshould be small compared to the wavelength. As is well known (see prob. 2, sec. 47 of Ref. [9]), this leads to the condition that the dimension- less, Lorentz-invariant quantity, η=eErms mωc, (12) should be small compared to one. Parameter ηcan ex- ceed unity for waves of very low field strength if the fre- quency is low enough. An interesting result is that the electron can be said to have an effective mass, meff=m/radicalbig 1 +η2, (13) when inside a wave field [134]. An electron in a spatially varying wave experiences a force F=−∇meffc2which is often called the “ponderomotive force”, but which can be regarded as a kind of radiation pressure for a case where the “reflected” wave cannot be distinguished from the incident wave, and hence as a kind of radiation reaction force in its broadest meaning. Debates continue regarding energy-transfer mecha- nisms between electrons and strong classical waves (as represented by a laser beam with η>∼1). To what extent can net energy be exchanged between a free electron and a laser pulse in vacuum? Does a classical discussion suf- fice? Our understanding suggests that quantum aspects should be unimportant even for η≫1 so long as con- dition (11) is satisfied, but full understanding has been elusive. Detailed discussion of this matter is deferred to a future article. C. Utility of the Classical Radiation Reaction Besides provoking extensive discussion on the valid- ity of classical electrodynamics, the radiation reaction has enjoyed some well-known success in classical phe- nomenology. In particular, the topics of linewidth of ra- diation by a classical oscillator and resonance width in 4scattering of waves off such an oscillator show how par- tial understanding of atomic systems can be obtained in a classical context. Also, the radiation reaction is very important in antenna engineering where the power source must provide for the energy (and momentum, if any) ra- diated as well as that consumed in Joule losses. It is worth noting that these successes hold where the elec- tron is part of an extended system. In contrast, the radiation reaction has been almost completely negligible in descriptions of the radiation of free electrons for practical parameters in the classical do - main ( i.e., outside the domain of quantum mechanics). That this might be the case is the main argument of sec. 2. Section 3 discusses effects of the radiation reac- tion in the quantum domain. D. Relativistic Radiation Reaction For purposes of additional commentary, it is useful to record relativistic expressions for the radiation reactio n. The relativistic version of (1) in 4-vector notation is mc2duµ ds=Fµ ext+Fµ resist, (14) with external 4-force Fµ ext=γ(Fext·v/c,Fext), and radiation-reaction 4-force given by Fµ resist=2e2 3d2uµ ds2−Ruµ c, (15) where R=−2e2c 3duν dsduν ds=2e2γ6 3c3/bracketleftbigg ˙v2−(v×˙v)2 c2/bracketrightbigg ≥0 (16) is the invariant rate of radiation of energy of an ac- celerated charge, uµ=γ(1,v/c) is the 4-velocity, γ= 1//radicalbig 1−v2/c2,ds=cdτis the invariant interval and the metric is (1 ,−1,−1,−1). The time component of eq. (14) can be written dγmc2 dt=Fext·v+d dt/parenleftbigg2e2γ4v·˙v 3c2/parenrightbigg −R, (17) and the space components as dγmv dt=Fext (18) +2e2γ2 3c3/bracketleftbigg ¨v+3γ2 c2(v·˙v)˙v+γ2 c2(v·¨v)v+3γ4 c4(v·˙v)2v/bracketrightbigg . Keeping terms only to first order in velocity, eqs. (17-18) become dmv2/2 dt=Fext·v+2e2v·¨v 3c3, (19) anddmv dt=Fext+2e2¨v 3c3+2e2(v·˙v)˙v c3. (20) Equations (17-18) were first given by Abraham [19]. Von Laue [24] was the first to show that these equa- tions can be obtained by a Lorentz transformation of the nonrelativistic results (19-20). The covariant notation o f eqs. (14-16) was first applied to the radiation reaction by Dirac [37]. An interesting discussion of the development of eqs. (17-18) has been given recently by Yaghjian [110]. E. Terminology During a century of discussion of the radiation reac- tion a variety of terminology has been employed. In this article I use the phrase “radiation reaction” to cover all aspects of the physics of “R¨ uckwirkung der Strahlung” as introduced by Lorentz and Abraham. This usage con- trasts with a proposed narrow interpretation discussed at the end of this section. “Æthereal friction” was the first description by Stewart [16] in 1871, which he used in only a qualitative manner. In 1873, Maxwell wrote on the “pressure exerted by light” in secs. 792-793 of his Treatise [10]. Lorentz used the French word “r´ esistance” in describ- ing eq. (3) when he presented it in 1892, and used the English equivalent “resistance” in his 1906 Columbia lec- tures [6]. Planck [7,17] also discussed eq. (3), which he de- scribed as “D¨ ampfung” (damping) and “D¨ ampfung durch Strahlung” (literally, “damping by radiation” but translated more smoothly as “radiation damping”). The term “Strahlungsd¨ ampfung” (radiation damping) does not, however, appear in the German literature until 1933 [34]. Around 1900, Larmor [22] used the terms “frictional resistance” and “ray pressure” to describe a result meant to quantify Stewart’s insight, but which analysis has not stood the test of time. The massive analyses of Abraham were accompanied by the introduction of several new terms. The ti- tle of Abraham’s 1904 article [19] included the term “Strahlungsdruck” (radiation pressure). This use of the phrase “radiation pressure” can, however, be confused with the simpler concept of the pressure that results when a wave is reflected from a conducting surface [10]. Per- haps for this reason, Abraham also introduced the phrase “Reaktionskraft der Strahlung”, which I translate as “ra- diation reaction force”. This appears to be the origin of the phrase “radiation reaction”, although in German that phrase remained a qualifier to “Kraft” (force) for many years. The variant “Strahlungsreaktion” (radiation reac- tion) appeared for the first time in 1933 [35]. Lorentz’ 1903 Encyklop¨ adie article [5] introduced the topic of the radiation reaction with the phrase “R¨ uckwirkung des ¨Athers” (back interaction of the æther). In his 1905 monograph [90], Abraham used the 5variant “R¨ uckwirkung der Strahlung” (back interaction of radiation, which could also be translated agreeably as “radiation reaction”). In England in 1908, the Adams Prize examiners chose the topic of the radiation reaction, suggesting the cum- bersome title “The Radiation from Electric Systems or Ions in Accelerated Motion and the Mechanical Reactions on their Motion which arise from it”. The winning essay by Schott [26] adopted much of this title, but in the text Schott refers to “radiation pressure” and indicates that he follows Abraham in this. In his 1915 article, Schott [27] also used the phrase “reaction due to radiation” and indicated that it was equivalent to his use of the phrase “radiation pressure”. Schott also introduced other terms that seem less than ideal descriptions of the phenomena associated with the radiation reaction. His argument of 1912 [26] is less crisp than one he gave in 1915 [27], so I follow the latter here. Schott considered the rate at which a radiating charge loses energy, and deduced eq. (17) in essentially that form. Schott noted that term Ris just the rate of ra- diation of energy by an accelerated charge, which he de- scribed as an “irreversible” process. He then interpreted the term Q=2γ4e2v·˙v 3c3, (21) as an energy stored “in the electron in virtue of its ac- celeration” and gave it the name “acceleration energy”. Schott considered the term ˙Qin eq. (17) to be a “re- versible” loss of energy. Insights related to the concept of the “acceleration energy” have been useful in resolving the paradox of whether a charge radiates if its acceleration is uniform, i.e., if¨v= 0. In this case the radiation reaction force (3) vanishes and many people have argued that this means there is no radiation [25,31,92,131]. But as first argued by Schott [27], in the case of uniform acceleration “the en- ergy radiated by the electron is derived entirely from its acceleration energy; there is as it were internal compensa- tion amongst the different parts of its radiation pressure, which causes its resultant effect to vanish”. This view is somewhat easier to follow if “acceleration energy” means energy stored in the near and induction zones of the elec- tromagnetic field [52,99]. Schott’s use of the word “irreversible” to describe the process of radiation seems inapt. He may have meant that in a classical universe containing only one electron and an external force field, the radiated energy can never return to the electron. But as noted by Planck [132], “the fundamental equations of mechanics as well as those of electrodynamics allow the direct reversal of every pro- cess as regards time”. For example, “if we now con- sider any radiation processes whatsoever, taking place in a perfect vacuum enclosed by reflecting walls, it is found that, since they are completely determined by the prin- ciples of classical electrodynamics, there can be in their case no question of irreversibility of any kind”. However,“an irreversible element is introduced by the addition of emitting and absorbing substance”. Thus, consistent use of the word “irreversible” goes beyond classical electron theory. These views of Planck were seconded by Ein- stein [133] and elaborated upon in the absorber theory of radiation of Wheeler and Feynman [95]. As another counterexample to the view that radia- tion is irreversible, a theme of contemporary accelerator physics is that every radiation process can be inverted to produce energy gain, not loss. Hence, there are now de- vices that accelerate electrons based on inverse ˇCerenkov radiation, inverse free-electron radiation, inverse Smit h- Purcell radiation, inverse transition radiation, etc.Uni- form acceleration is the inverse of uniform deceleration, and the inverse transformation is especially simple here: sinceFresistvanishes, it suffices to reverse the sign of the external force. These inverse radiation processes will be the subject of a future paper. Schott’s use of “irreversible” as applied to the term −Ruµ/cof the radiation reaction has not been followed in the German literature. See Ref. [99] for an interesting contrast. The English phrase “radiation reaction” appears to have been first used by Page in 1918 [30]. In his 1938 paper, Dirac [37] used the phrase “the effect of radiation damping on the motion of the elec- tron”. As a consequence, most subsequent papers use “radiation damping” interchangeably with “radiation re- action” as a general description of the subject. Thus, in German there appeared the use of “Strahlungsd¨ amfung” [34] (already in 1933), in French, “force de freinage” [43] (braking force, compare “rayonnement freinage” = Bremsstrahlung), and in Russian the equivalent of “radi- ation damping” must have been used as well [9]. Dirac seconded Schott’s use of the terms “irreversible” and “ac- celeration energy”, and these become fairly common in the English literature thereafter. Indeed, “acceleration energy” becomes “Schott acceleration energy”, or just “Schott energy”. The terminology of Schott and Dirac was taken a step further by Rohrlich in 1961 [53] and 1965 [104], who pro- posed that only the second term in the covariant expres- sion (15) is entitled to be called “the radiation reaction”. The first term of (15) is to be called the “Schott term”. A motivation for this terminology appears to be that in the case of uniform acceleration, expression (15) vanishes by virtue of cancellation of its two nonzero terms. Then the broadly defined “radiation reaction” ( i.e., eq. (15)) vanishes, but the radiation does not (although it takes considerable effort to demonstrate this [52]). The termi- nology of Rohrlich has the merit that the paradox “how can there be radiation if there is no radiation reaction” is avoided in this case since only the (nonvanishing) term −Ruµ/cis called the “radiation reaction”. However, this terminology is at odds with the origins of the subject, which emphasize the low-velocity limit, eqs. (19-20). Here, the radiated momentum enters only in terms of order v2/c2, so the direct back reaction of 6the radiated momentum ( i.e.,−Ruµ/c) plays no role in the nonrelativistic limit. Thus, according to Rorhlich’s terminology there is no “radiation reaction” in the non- relativistic limit. But the original, and continuing, purpose of the con- cepts of the radiation reaction is to describe how a charge reacts to the radiation of energy when it does not radiate net momentum. To define the “radiation reaction” to be zero in this circumstance is counterproductive. It appears that the terminology of Rohrlich has been adopted only by three subsequent workers [65,67,108]. III. A QUANTUM INTERPRETATION To go further, we pass beyond the realm of classical electromagnetism. The remainder of this paper is not a direct consequence of that theory, but considers how only a modest admixture of quantum concepts greatly clarifies the hints deduced by classical argument. A simple device is to multiply and divide eq. (7) by Planck’s constant ¯ h, which was introduced by him [130] shortly after his work on eq. (1) [7]. Then we can write Fresist≈Fext/radicalBigg/parenleftbigge2 ¯hc¯h mcω c/parenrightbigg2 +/parenleftbigge2 ¯hce¯h m2c3E/parenrightbigg2 ≈αFext/radicalBigg/parenleftbiggλC λ/parenrightbigg2 +/parenleftbiggE Ecrit/parenrightbigg2 , (22) where α=e2/¯hcis the QED fine structure constant, λC= ¯h/mc is the reduced Compton wavelength of an electron and Ecrit=m2c3 e¯h= 1.6×1016V/cm = 3 .3×1013Gauss (23) is the QED critical field strength, discussed in sec. 3.2 below. Thus, our na¨ ıve quantum theory (classical electromag- netism plus ¯ h) leads us to expect important departures from classical electromagnetism for waves of wavelength much shorter than the Compton wavelength of the elec- tron, and for fields of strength larger than the QED crit- ical field strength. A. The Radiation Reaction and Compton Scattering Compton scattering [135] was one of the earlier pre- dictions of quantum theory and its observation had an important historical role in widespread acceptance of photons as quanta of light. Compton scattering is dis- tinguished from Thomson scattering of classical electro- magnetism in that wavelengths of the photons involved in Compton scattering are not small compared to theCompton wavelength of the electron, when measured in the frame in which the electron is initially at rest. Hence Compton scattering appears to be exactly the kind of example discussed above in which the radiation reaction should be important. A description of a quantum scattering experiment re- lates the energy and momentum (plus relevant internal quantum numbers) of the initial state to those of the final state without discussion of forces. Yet, we can identify various correspondences between the quantum and clas- sical descriptions. In the case of Compton scattering, the initial photon corresponds to the external force field on the electron, while the final photon corresponds to the radiated wave. The quantum changes in momentum (and energy) of the electron in the scattering process can be said to corre- spond to classical time integrals of force (and of F·v). Conservation of momentum (and energy) is described in the scattering process by including the back reaction of the final photon on the electron as well as the direct reaction of the initial photon. Thus, the quantum de- scription, which incorporates conservation of momentum (and energy), can be said to include automatically the (time-integrated) effects of the radiation reaction. Compton scattering is an electromagnetic scattering process in which large changes in momentum (and en- ergy) of the electron are observed (in the frame in which the electron is initially at rest). It can therefore be said t o correspond to a situation in which the radiation reaction is large, in agreement with the semiclassical inferences of secs. 2 and 3. The correspondence between quantum conservation of energy and the classical radiation reaction appears to in- volve only the second term, −Ruµ/c, in expression (15) for the radiation-reaction force. Since the electron has constant (though different) initial and final velocities in a scattering experiment, the “acceleration energy” Qof eq. (21) is zero both before and after the scattering, and the equivalent of ˙Qcannot be expected to appear in the quantum description (at “tree level”, in the technical jar- gon) of Compton scattering. Effects corresponding to the near-field “acceleration energy” can be said to occur in quantum electrodynamics in the case of so-called vertex corrections and propagator (mass) corrections, in which a virtual photon is emitted and absorbed by the same electron. These “loop correc- tions” to the behavior of a quantum point charge imple- ment the equivalent of the self interaction of an extended charge, but diverge when the emission and absorption occur at the same spacetime point. They are the source of the famous infinities of QED that are dealt with by “renormalization”. See also sec. 4.1 below. In the early 1940’s, Heitler [136,93] and coworkers [137,138] formulated a version of QED in which radia- tion damping played a prominent role. Following the suggestion of Oppenheimer [8], they hoped that this the- ory would provide a general method of dealing with the 7divergences of QED. By selecting a subset of “loop cor- rections”, they deduced an expression for Compton scat- tering that corresponds to classical Thomson scattering plus classical radiation damping. While this result is sug- gestive, it does not appear to be endorsed in detail by subsequent treatments of “renormalization” in QED. B. The Critical Field The second term under the radical in eq. (22) may be less familiar. The concept of a critical field in quantum mechanics began with Klein’s paradox [139]: an electron that encounters an (electric) potential step appears to be reflected with greater than unit probability in Dirac’s theory. Sauter [140] noted that this effect arises only when the potential gradient is larger than the critical field, m2c3/e¯h. The resolution of the paradox is due to Heisen- berg and Euler [141], who noted that electrons and positrons can be spontaneously produced in critical fields – a very extreme form of the radiation reaction. The crit- ical field has been discussed at a sophisticated level by Schwinger [142] and by Brezin and Itzykson [143], among many others. An electron that encounters an electromagnetic wave of critical strength produces not only Compton scatter- ing of the wave photons but also electron-positron pairs. These effects have recently been observed in experiments in which the author participated [144,145]. There is speculation that critical magnetic fields exist at the surface of neutron stars [146–148], and may be responsible for some aspects of pulsar radiation. Pomeranchuk [40] noted that the Earth’s magnetic field appears to have critical strength from the point of view of an electron of energy 1019eV, which energy is at the upper limit of observation of cosmic rays. The critical field arises in discussion of the radiation, commonly called synchrotron radiation, of electrons mov- ing in circular orbits under the influence of a magnetic fieldB. If an electron of laboratory energy E ≫ mc2 moves in an orbit with angular velocity ω0, then the char- acteristic frequency of the synchrotron radiation is ω≈γ3ω0, (24) where γ=E/mc2is the Lorentz boost to the rest frame of the electron. For motion in a magnetic field B, the cyclotron frequency ω0can be written ¯hω0=mc2B γBcrit, (25) where Bcrit=m2c3/e¯h. Thus, the characteristic energy of synchrotron-radiation photons (often called the critic al energy) is ¯hω≈ EγB Bcrit. (26)Hence an electron radiates away roughly 100% of its en- ergy in a single synchrotron-radiation photon if the mag- netic field in the electron’s rest frame, B⋆=γB, has critical field strength. In this regime a classical theory of synchrotron radiation is inadequate [149,150]. Critical electric fields can be created for short times in the collision of nonrelativistic heavy ions, resulting i n positron production [151–153]. As a final example of the inapplicability of classical electromagnetism in strong fields, the performance of fu- ture high-energy electron-positron colliders will be lim- ited by the disruptive (quantum) effect of the critical fields experienced by one bunch of charge as it passes through the oncoming bunch [154]. IV. DISCUSSION In this paper I have followed the example of Landau in using the argument of sec. 2 to suggest limitations to the concepts of classical electrodynamics. However, this line of argument appears to have played no role in the early development of quantum mechanics. Rather, the argument was used in the 1930’s to suggest that quan- tum electrodynamics might have conceptual limitation when carried beyond the leading order of approximation [8,141,155–159]. The history of this era has been well reviewed in the recent book by Schweber [160]. While the program of renormalization, of which Lorentz was an early advocate in the classical context [6], appears to have been successful in eliminating the formal divergences that were so troublesome in the 1930’s, quan- tum electrodynamics is still essentially untested for field s in excess of the critical field strength (23) [145]. It still may be the case that this realm contains new physical phenomena that will validate the cautionary argument of Oppenheimer [8]. We close with three examples to stimulate additional discussion. Two are from strong-field electrodynamics; while not necessarily suggesting defects in the theory, they indicate that not all aspects of QED are integrated in the most familiar presentations. The third example considers the case of extraordinarily short wavelengths. A. The Mass Shift of an Accelerated Charge We can rewrite the nonrelativistic expressions (1-3) for the radiation reaction as d dt/parenleftbigg mv−2e2˙v 3c3/parenrightbigg =Fext, (27) and the relativistic expressions (14-16) as d ds/parenleftbigg mc2uµ−2e2 3duµ ds/parenrightbigg =Fµ ext−Ruµ c. (28) 8These forms suggest the interpretation that the momen- tum, mv, of a moving charge is decreased by amount 2e2˙v/3c2if that charge is accelerating as well [52,53]. If we take mcas the scale of the ordinary momen- tum, then the effect of acceleration, eE/m , due to an electric field Ebecomes large in eq. (27) only when E>∼m2c4/e3=e/r2 0,i.e., when the electric field is large compared to the classical critical field found in sec. 2. This interpretation has been seconded by Ritus [161] based on a semiclassical analysis (classical electromag- netic field, quantum electron) of the behavior of electrons in a strong, uniform electric field. He finds that the mass of an electron (= eigenvalue of the mass operator) obeys m=m0/parenleftbigg 1−αE 2Ecrit+O(E2/E2 crit)/parenrightbigg , (29) and remarks on the relation between this result and the classical interpretations (27-28). The mass shift of an accelerated charge becomes large when E>∼Ecrit/α= e/r2 0, as found above. The physical meaning of Ritus’ result remains some- what unclear. For example, a mass shift of the form (29) does not appear in Ritus’ treatment of Compton scatter- ing in intense wave fields [162] (which treatment agrees with other works), although the effective mass (13) does appear. B. Hawking-Unruh Radiation According to Hawking [163], an observer outside a black hole experiences a bath of thermal radiation of tem- perature T=¯hg 2πck, (30) where gis the local acceleration due to gravity and kis Boltzmann’s constant. In some manner, the background gravitational field interacts with the quantum fluctua- tions of the electromagnetic field with the result that en- ergy can be transferred to the observer as if he(she) were in an oven filled with black-body radiation. Of course, the effect is strong only if the background field is strong. An extreme example is that if the temperature is equiv- alent to 1 MeV or more, virtual electron-positron pairs emerge from the vacuum into real particles. As remarked by Unruh [164], this phenomenon can be demonstrated in the laboratory according to the princi- ple of equivalence: an accelerated observer in a gravity- free environment experiences the same physics (locally) as an observer at rest in a gravitational field. There- fore, an accelerated observer (in zero gravity) should find him(her)self in a thermal bath of radiation characterized by temperature T=¯ha⋆ 2πck, (31)where a⋆is the acceleration as measured in the observer’s instantaneous rest frame. The Hawking-Unruh temperature finds application in accelerator physics as the reason that electrons in a stor- age ring do not reach 100% polarization despite emitting polarized synchrotron radiation [166]. Indeed, the vari- ous limiting features of performance of a storage ring that arise due to quantum fluctuations of the synchrotron ra- diation can be understood quickly in terms of eq. (31) [167]. Here we consider a more speculative example. Suppose the observer is an electron accelerated by an electromag- netic field E. Then, scattering of the electron off photons in the apparent thermal bath would be interpreted by a laboratory observer as an extra contribution to the ra- diation rate of the accelerated charge [168]. The power of the extra radiation, which I call Unruh radiation, is given by dUUnruh dt= (energy flux of thermal radiation) ×(scattering cross section) . (32) For the scattering cross section, we use the well-known result for Thomson scattering, σThomson = 8πr2 0/3. The energy density of thermal radiation is given by the usual expression of Planck: dU dν=8π c3hν3 ehν/kT−1, (33) where νis the frequency. The flux of the isotropic radi- ation on the electron is just ctimes the energy density. Note that these relations hold in the instantaneous rest frame of the electron. Then dUUnruh dtdν=8π c2hν3 ehν/kT−18π 3r2 0. (34) On integrating over νwe find dUUnruh dt=8π3¯hr2 0 45c2/parenleftbiggkT ¯h/parenrightbigg4 =¯hr2 0a⋆4 90πc6, (35) using the Hawking-Unruh relation (31). The presence of ¯hin eq. (35) reminds us that Unruh radiation is a quantum effect. This equals the classical Larmor radiation rate, dU/dt = 2e2a⋆2/3c3, when E⋆=/radicalbigg 60π αEcrit≈Ecrit α, (36) where Ecritis the QED critical field strength introduced in eq. (23). In this case, the acceleration a⋆=eE⋆/mis about 1031Earth g’s. The physical significance of Unruh radiation remains unclear. Sciama [169] has emphasized how the apparent temperature of an accelerated observed should be inter- preted in view of quantum fluctuations. Unruh radiation 9is a quantum correction to the classical radiation rate that grows large only in situations where quantum fluc- tuations in the radiation rate become very significant. This phenomenon should be contained in the standard theory of QED, but a direct demonstration of this is not yet available. Likewise, the relation between Unruh ra- diation and the mass shift of an accelerated charge, both of which become prominent at fields of strength Ecrit/α, is not yet evident. The existence of Unruh radiation provides an interest- ing comment on the “perpetual problem” of whether a uniformly accelerated charge emits electromagnetic radi- ation [63]; this issue has been discussed briefly in sec. 2.5. The interpretation of Unruh radiation as a measure of the quantum fluctuations in the classical radiation implies that the classical radiation exists. It is noteworthy that while discussion of radiation by an accelerated charge is perhaps most intricate classically in case of uniform ac- celeration, the discussion of quantum fluctuations is the most straightforward for uniform acceleration. In addition, Hawking-Unruh radiation helps clarify a residual puzzle in the discussion of the equivalence be- tween accelerated charges and charges in a gravitational field. Because of the difficulty in identifying an unam- biguous wave zone for uniformly accelerated motion of a charge (in a gravity-free region) and also in the case of a charge in a uniform gravitational field, there remains some doubt as to whether the ‘radiation’ deduced by classical arguments contains photons. Thus, on p. 573 of the article by Ginzburg [63] we read: “neither a ho- mogeneous gravitational field nor a uniformly accelerated reference frame can actually “generate” free particles, ex - pecially photons”. We now see that the quantum view is richer than anticipated, and that Hawking-Unruh radia- tion provides at least a partial understanding of particle emission in uniform acceleration or gravitation. Hence, we can regard the concerns of Bondi and Gold [50], Ful- ton and Rohrlich [52], the DeWitt’s [58] and Ginzburg [63] on radiation and the equivalence principle as precur- sors to the concept of Hawking radiation. C. Can a Photon Be a Black Hole? While quantum electrodynamics appears valid in all laboratory studies so far, which have explored photons energies up to the TeV energy scale, will this success con- tinue at arbitrarily high energies ( i.e., arbitrarily short wavelengths)? Consider a photon whose (reduced) wavelength λis the so-called Planck length [170], LP=/radicalbigg ¯hG c3≈10−33cm, (37) where Gis Newton’s gravitational constant. The gravi- tational effect of such a photon is quite large. A measure of this is the “equivalent mass”,mequiv=¯hω c2=¯h cλ=¯h cLP. (38) The Schwarzschild radius corresponding to this equiva- lent mass is R=2Gmequiv c2=2¯hGω c3LP= 2LP= 2λ. (39) A na¨ ıve interpretation of this result is that a photon is a black hole if its wavelength is less than the Planck length. 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arXiv:physics/0003063v1 [physics.acc-ph] 23 Mar 2000Energy Balance in an Electrostatic Accelerator Max S. Zolotorev Center for Beam Physics, Lawrence Berkeley National Labora tory, Berkeley, CA 94720 Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princet on, NJ 08544 (Feb. 1, 1998) 1 Problem The principle of an electrostatic accelerator is that when a charge eescapes from a conducting plane that supports a uniform electric field of strength E0, then the charge gains energy eE0d as it moves distance dfrom the plane. Where does this energy come from? Show that the mechanical energy gain of the electron is balan ced by the decrease in the electrostatic field energy of the system. 2 Solution Once the charge has reached distance dfrom the plane, the static electric field Eeat an arbitrary point rdue to the charge can be calculated by summing the field of the c harge plus its image charge, Ee(r, d) =er1 r3 1−er2 r3 2, (1) where r1(r2) points from the charge (image) to the observation point r, as illustrated in Fig. 1. The total electric field is then E0ˆz+Ee. Figure 1: The charge eand its image charge −eat positions ( r, θ, z) = (0,0,±d) with respect to a conducting plane at z= 0. Vectors r1andr2are directed from the charges to the observation point ( r,0, z). It turns out to be convenient to use a cylindrical coordinate system, where the observation point is r= (r, θ, z) = (r,0, z), and the charge is at (0 ,0, d). Then, r2 1,2=r2+ (z∓d)2. (2) 1The part of the electrostatic field energy that varies with th e position of the charge is the interaction term (in Gaussian units), Uint=/integraldisplayE0ˆz·Ee 4πdVol =eE0 4π/integraldisplay∞ 0dz/integraldisplay∞ 0πdr2/parenleftiggz−d [r2+ (z−d)2]3/2−z+d [r2+ (z+d)2]3/2/parenrightigg =eE0 4/integraldisplay∞ 0dz   2 if z > d −2 ifz < d  −2  =−eE0/integraldisplayd 0dz=−eE0d. (3) When the particle has traversed a potential difference V=E0d, it has gained energy eVand the electromagnetic field has lost the same energy. In a practical “electrostatic” accelerator, the particle i s freed from an electrode at po- tential −Vand emerges with energy eVin a region of zero potential. However, the particle could not be moved to the negative electrode from a region of z ero potential by purely electrostatic forces unless the particle lost energy eVin the process, leading to zero overall energy change. An “electrostatic” accelerator must have an essential component (such as a battery) that provides a nonelectrostatic force that can ab sorb the energy extracted from the electrostatic field while moving the charge from potenti al zero, so as to put the charge at rest at potential −Vprior to acceleration. 2

arXiv:physics/0003064v1 [physics.class-ph] 23 Mar 2000A bounded source cannot emit a unipolar electromagnetic wave Kwang-Je Kim Argonne National Laboratory, Argonne, IL 60439 Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princet on, New Jersey 08544 Gennady V. Stupakov Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309 Max S. Zolotorev Center for Beam Physics, Lawrence Berkeley National Labora tory, Berkeley, CA 94720 (May 1, 1999) 1 Problem Show that a bounded source cannot produce a unipolar electro magnetic pulse. Equivalently, show that there are no three-dimensional ele ctromagnetic solitons in vac- uum. 2 Solution This problem was first discussed by Stokes over 150 years ago [ 1, 2, 3, 4], who noted that three-dimensional sound waves from a bounded source cannot be unipolar. While his ar- gument applies to electromagnetic waves as well, this is lit tle recognized in the literature. One-dimensional electromagnetic (and sound) waves can be u nipolar. A plane wave can have any pulseform, but, strictly speaking, a plane wave can only be generated by an unbounded source. A well-known pedagogic example given in sec. II-20 o f [5] is based on this case. Hence, this old problem still has a new flavor. Here, we offer tw o new solutions, followed by discussion. 2.1 Via Conservation of Energy and Fourier Analysis If the source is bounded, it appears pointlike when viewed fr om a great enough distance. Then, energy conservation requires that the pulse energy de nsity fall off as 1 /r2, for distance rmeasured from some characteristic point within the source. Since the energy density is proportional to the square of the the electromagnetic fields , we have the well-known result that the radiation fields from a bounded source fall off as 1 /rfar from the source. This is in contrast to the static fields, which must fall off at l east as quickly at 1 /r2, far from a bounded source. Now, consider the possibility of a unipolar pulse, i.e., one for which the electric field components Ei(r, t) have only one sign. At a fixed point t, the time integral of at least one component of such a pulse would be nonzero: /integraldisplay Ei(r, t)dt/negationslash= 0. (1) 1Then a Fourier analysis of this component, Ei(r, ω) =/integraldisplay Ei(r, t)eiωtdt, (2) would have a nonzero value at zero frequency, Ei(r, ω= 0) /negationslash= 0. However, the quantity Ei(r, ω= 0) would then be a static solution to Maxwell’s equation, and so must fall off like 1 /r2. This contradicts the hypothesis that Ei(r, t) represented an electromagnetic pulse from a bounded source, which must fal l off as 1 /r. Thus, a bounded source cannot emit a unipolar electromagnet ic pulse. 2.2 Via the Fields of an Accelerated Charge The electric field vector radiated by a charge is opposite to t he transverse component of the acceleration. In the case of a bounded source, the accelerations of the char ges cannot impart a nonzero average velocity to any charge; otherwise the source would n ot remain bounded. Hence, any accelerations must include both positive and neg ative components such that their time integral vanishes. Therefore, the radiated electric field must also include bot h positive and negative com- ponents. A bounded source cannot emit a unipolar electromagnetic pul se. 2.3 Three-Dimensional Unipolar Radiation from an Unbounde d Source In the case of two nonrelativistic, unbound charged particl es that interact only via the Coulomb force q1q2/r, the component of the acceleration of one of the charges alon g the axis of its hyperbolic trajectory always has the same sign. H ence, the radiated electric field component along that axis is unipolar, and a Fourier analysi s of the field has a zero-frequency component (see sec. 70 of [6]). Thus, a three-dimensional un ipolar pulse can be emitted by a system whose motion is unbounded. 2.4 Unipolarlike Pulses It is possible for a bounded system to produce an electromagn etic pulse that consists almost entirely of a single central pulse of one sign. But according to the argument above, this pulse must include long tails of the opposite sign so that the time i ntegral of the fields vanish at any point far from the source. This behavior has been observe d in several recent reports on subcycle electromagnetic pulses [7, 8, 9, 10]. 3 Can the Far-Zone Radiation From a Bounded Source Transfer Energy to a Charged Particle? The present considerations can be extended to comment on thi s topical question. 2If a unipolar electromagnetic pulse existed far from its bou nded source, the corresponding vector potential Awould have different values at asymptotically early and late times. Then, as argued by Lai [11], an interaction with a charge eand mechanical momentum Pthat conserves the canonical momentum P+eA/ccould produce a net change in the magnitude of the mechanical momentum, i.e., provided a transfer of energy between the particle and the pulse. Such far-zone unipolar particle acceleration is desirable, but is not consistent with Maxwell’s equations. Near-zone particle acceleration by time-dependent electr omagnetic fields can be accom- plished by passing a particle through a bounded field region, such as an rf cavity, during a short interval when the fields have only one sign. While the el ectromagnetic fields are not unipolar in this case, their interaction with the charged pa rticle is effectively so. A variant on such considerations is the fact that a bounded el ectrostatic field cannot exchange net energy with a charged particle that begins and e nds its history at large distances from the source. So-called electrostatic particle acceler ators all contain a nonelectrostatic component that can move the charge to a region of nonzero elec tric potential and leave it there with effectively zero electrical and mechanical energ y. Then, the charge can extract energy from the field as it is expelled to large distances. Returning to the case of far-zone radiation, we give a brief a rgument based the well- known relation for the time rate of change of energy Uof a particle of charge eand velocity vin an electromagnetic field E(see, for example, eq. (17.7) of [6]): dU dt=eE·v (3) This expression holds for particles of any velocity less tha n the speed of light. Of course, the magnetic field cannot change the particle’s energy. In the approximation that the particle’s velocity is essent ially unchanged by its interaction with the electromagnetic field, we have ∆U=ev·/integraldisplay Edt. (4) To perform the integral, we can use Feynman’s expression for the far-zone radiated electric field of an accelerated charge (sec. I-34 of [5]): Erad=e c2d2ˆn dt2, (5) in Gaussian units, where ˆnis a unit vector from the retarded position of the source char ge to the observer. Then,/integraltextEraddtis the difference between dˆn/dtat early and late times. Since dˆn/dtis the angular velocity of the relative motion between the so urce and charge, this vanishes at both early and late times as the moving charg e is then arbitrarily far from the bounded source. Hence, in the constant-velocity approx imation, the far-zone fields from a bounded source cannot transfer energy to a particle. While the constant-velocity approximation is not necessar ily good for a particle whose initial velocity is nonrelativistic, it is an excellent app roximation for a relativistic particle. It is possible for the far-zone fields of a bounded source (in v acuum and far from any other material) to transfer energy to a nonrelativistic charge, a s recently observed [12], but this 3energy transfer becomes increasingly inefficient as the part icle’s velocity approaches that of light. The above considerations have ignored energy transfers due to scattering, which can be significant if the energy of a photon of the electromagnetic w ave is large compared to the total energy of the charged particle in the center of mass fra me. In this case, a quantum description is more appropriate. In the regime in which the e nergy transfer from a single scattering process is small, the classical idea of a “radiat ion pressure” associated with the Poynting vector S= (c/4π)E×Hcan be formalized by including the radiation reaction in the equation of motion of the charged particle. See, for exam ple, eq. (75.10) of [6]. However, this effect is always very small in the classical regime. 4 References [1] G.G. Stokes, On Some Points in the Received Theory of Sound , Phil. Mag. 34, 52 (1849); Mathematical and Physical Papers (Johnson Reprint Co., New York, 1996), pp. 82-88c. [2] Lord Rayleigh, The Theory of Sound , (Macmillan, London, 1894; Dover, New York, 1945), sec. 279. [3] B.B. Baker and E.T. Copson, The Mathematical Theory of Huygens’ Principle , 2nd ed. (Clarendon Press, Oxford, 1950) sec. 3.3. [4] L.D. Landau and E.M. Lifshitz, Fluid Mechanics , 2nd ed. (Pergamon Press, Oxford, 1987), sec. 70. [5] R.P. Feynman, R.B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, MA, 1964). [6] L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields , 2nd ed. (Pergamon Press, Oxford, 1975). [7] B.I. Greene et al.,Interferometric characterization of 160 fs far-infrared l ight pulses , Appl. Phys. Lett. 59, 893-895 (1991). [8] D. You et al.,Generation of high-power sub-single-cycle 500-fs electro magnetic pulses , Opt. Lett. 18, 290-292 (1993). [9] C.W. Dormier et al.,Ultrashort-pulse reflectometry , Rev. Sci. Instrum. 66, 399-401 (1995). [10] A. Bonvalet et al.,Generation of ultrabroadband femtosecond pulses in the mid - infrared by optical rectification of 15 fs light pulses at 100 MHz repetition rate , Appl. Phys. Lett. 67, 2907-2909 (1995). [11] H.M. Lai, Particle acceleration by an intense solitary electromagne tic wave , Phys. Fluids 23, 2373-2375 (1980). 4[12] G. Malka et al.,Experimental observation of electrons accelerated in vacu um to rela- tivistic energies by a high-energy laser , Phys. Rev. Lett. 78, 3314-17 (1997). 5

arXiv:physics/0003065v1 [physics.acc-ph] 23 Mar 2000The Grating Accelerator Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princet on, NJ 08544 (Sept. 14, 1984) 1 Problem In optics, a reflective grating is a conducting surface with a ripple. For example, consider the surface defined by z=asin2πx d. (1) The typical use of such a grating involves an incident electr omagnetic wave with wave vector kin the x-zplane, and interference effects lead to a discrete set of refle cted waves also with wave vectors in the x-zplane. Consider, instead, an incident plane electromagnetic wave with wave vector in the y-z plane and polarization in the xdirection: Ein=E0ˆxei(kyy−kzz−ωt), (2) where ky>0 and kz>0. Show that for small ripples ( a≪d), this leads to a reflected wave as if a= 0, plus two surface waves that are attenuated exponentiall y with z. What is the relation between the grating wavelength dand the optical wavelength λsuch that the x component of the phase velocity of the surface waves is the sp eed of light, c? In this case, a charged particle moving with vx≈ccould extract energy from the wave, which is the principle of the proposed “grating accelerator ” [1, 2, 3]. 2 Solution The interaction between particle beams and diffraction grat ings was first considered by Smith and Purcell [4], who emphasized energy transfer from the par ticle to free electromagnetic waves. The excitation of surface waves by particles near con ducting structures was first discussed by Pierce [5], which led to the extensive topic of w akefields in particle accelerators. The presence of surface waves in the Smith-Purcell effect was noted by di Francia [6]. A detailed treatment of surface waves near a diffraction grati ng was given by van den Berg [7]. Here, we construct a solution containing surface waves by starting with only free waves, then adding surface waves to satisfy the boundary condition at the grating surface. If the (perfectly) conducting surface were flat, the reflecte d wave would be Er=−E0ˆxei(kyy+kzz−ωt). (3) However, the sum Ein+Erdoes not satisfy the boundary condition that Etotalmust be perpendicular to the wavy surface (1). Indeed, [Ein+Er]surface = 2iE0ˆxei(kyy−ωt)sinkzz≈2iakzE0ˆxei(kyy−ωt)sinkxx, (4) 1where the approximation holds for a≪d, and we have defined kx= 2π/d. Hence, we require additional fields near the surface to cance l that given by (4). For z≈0, these fields therefore have the form E=−akzE0ˆxei(kyy−ωt)/parenleftBig eikxx−e−ikxx/parenrightBig . (5) This can be decomposed into two waves E±given by E±=∓akzE0ˆxei(±kxx+kyy−ωt). (6) Away from the surface, we suppose that the zdependence of the additional waves can be described by including a factor eik′ zz. Then, the full form of the additional waves is E±=∓akzE0ˆxei(±kxx+kyy+k′ zz−ωt). (7) The constant k′ zis determined on requiring that each of the additional waves satisfy the wave equation, ∇2E±=1 c2∂2E± ∂t2. (8) This leads to the dispersion relation k2 x+k2 y+k′2 z=ω2 c2. (9) The component kyof the incident wave vector can be written in terms of the angl e of incidence θinand the wavelength λas ky=2π λsinθin. (10) Combining (9) and (10), we have k′ z= 2πi/radicaltp/radicalvertex/radicalvertex/radicalbt1 d2−/parenleftBiggcosθin λ/parenrightBigg2 . (11) For short wavelengths, k′ zis real and positive, so the reflected wave (3) is accompa- nied by two additional plane waves with direction cosines ( kx, ky, k′ z). But for long enough wavelengths, k′ zis imaginary, and the additional waves are exponentially at tenuated in z. When surface waves are present, consider the fields along the liney= 0,z=π/2kz. Here, the incident plus reflected fields vanish (see the first form of (4)), and the surface waves are E±=∓akze−π|k′ z|/2kzE0ˆxei(±kxx−ωt). (12) The phase velocity of these waves is vp=ω kx=d λc. (13) 2When d=λ, the phase velocity is c, and k′ z=ikyaccording to (11). The surface waves are then, E±=∓2πacosθin de−(π/2) tan θinE0ˆxei(±kxx−ωt). (14) A relativistic charged particle that moves in, say, the + xdirection remains in phase with the wave E+, and can extract energy from that wave for phases near π. On average, the particle’s energy is not affected by the counterpropagating waveE−. In principle, significant particle acceleration can be achieved via this technique. F or a small angle of incidence, and witha/d= 1/2π, the accelerating field strength is equal to that of the incid ent wave. 3 References [1] Y. Takeda and I. Matsui, Laser Linac with Grating , Nucl. Instr. and Meth. 62, 306-310 (1968). [2] K. Mizuno, S. Ono and O. Shimoe, Interaction between coherent light waves and free electrons with a reflection grating , Nature 253, 180-181 (1975). [3] R.B. Palmer, A Laser-Driven Grating Linac , Part. Accel. 11, 81-90 (1980). [4] S.J. Smith and E.M. Purcell, Visible Light from Localized Surface Charges Moving across a Grating , Phys. Rev. 62, 1069 (1953). [5] J.R. Pierce, Interaction of Moving Charges with Wave Circuits , J. Appl. Phys. 26, 627-638 (1955). [6] G.T. di Francia, On the Theory of some ˇCerenkovian Effects , Nuovo Cim. 16, 61-77 (1960). [7] P.M. van den Berg, Diffraction Theory of a Reflective Grating , Appl. Sci. Res. 24, 261-293 (1971). 3

arXiv:physics/0003066v1 [physics.acc-ph] 23 Mar 2000The Laser Driven Vacuum Photodiode Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princet on, NJ 08544 (Sept. 26, 1986) 1 Problem A vacuum photodiode is constructed in the form of a parallel p late capacitor with plate separation d. A battery maintains constant potential Vbetween the plates. A short laser pulse illuminates that cathode at time t= 0 with energy sufficient to liberate all of the surface charge density. This charge moves across the capaci tor gap as a sheet until it is collected at the anode at time T. Then another laser pulse strikes the cathode, and the cycle repeats. Estimate the average current density /angbracketleftj/angbracketrightthat flows onto the anode from the battery, ignoring the recharing of the cathode as the charge sheet mov es away. Then calculate the current density and its time average when this effect is inclu ded. Compare with Child’s Law for steady current flow. You may suppose that the laser photon energy is equal to the wo rk function of the cathode, so the electrons leave the cathode with zero veloci ty. 2 Solution The initial electric field in the capacitor is E=−V/dˆx, where the xaxis points from the cathode at x= 0 to the anode. The initial surface charge density on the cat hode is (in Gaussian units) σ=E/4π=−V/4πd. (1) The laser liberates this charge density at t= 0. The average current density that flows onto the anode from the battery is /angbracketleftj/angbracketright=−σ T=V 4πdT, (2) where Tis the transit time of the charge across the gap d. We first estimate Tby ignoring the effect of the recharging of the cathode as the charge sheet moves away from it. In this approximation, the field on the charge sheet is always E=−V/d, so the acceleration of an electron is a=−eD/m =eV/dm , where eandmare the magnitudes of the charge and mass of the electron, respectively. The time to travel dista ncedisT=/radicalBig 2d/a=/radicalBig 2d2m/eV . Hence, /angbracketleftj/angbracketright=V3/2 8πd2/radicalBigg 2e m. (3) This is quite close to Child’s Law for a thermionic diode, jsteady=V3/2 9πd2/radicalBigg 2e m. (4) 1We now make a detailed calculation, including the effect of th e recharging of the cathode, which will reduce the average current density somewhat. At some time t, the charge sheet is at distance x(t) from the cathode, and the anode and cathode have charge densities σAandσC, respectively. All the field lines that leave the anode terminate on either the charge sheet or on the cathode, so σ+σC=−σA, (5) where σAandσCare the charge densities on the anode and cathode, respectiv ely. The the electric field strength in the region I between the anode and t he charge sheet is EI=−4πσA, (6) and that in region II between the charge sheet and the cathode is EII= 4πσC. (7) The voltage between the capacitor plates is therefore, V=−EI(d−x)−EIIx= 4πσAd−Vx d, (8) using (1) and (5-7), and taking the cathode to be at ground pot ential. Thus, σA=V 4πd/parenleftbigg 1 +x d/parenrightbigg , σ C=−V x 4πd2, (9) and the current density flowing onto the anode is j(t) = ˙σA=V˙x 4πd2. (10) This differs from the average current density (2) in that ˙ x/d/negationslash=T, since ˙ xvaries with time. To find the velocity ˙ xof the charge sheet, we consider the force on it, which is due t o the field set up by charge densities on the anode and cathode, Eonσ= 2π(−σA+σC) =−V 2d/parenleftbigg 1 +2x d/parenrightbigg . (11) The equation of motion of an electron in the charge sheet is m¨x=−eEonσ=eV 2d/parenleftbigg 1 +2x d/parenrightbigg , (12) or ¨x−eV md2x=eV 2md. (13) With the initial conditions that the electrons start from re st,x(0) = 0 = ˙ x(0), we readily find that x(t) =d 2(coshkt−1), (14) 2where k=/radicalBigg eV md2. (15) The charge sheet reaches the anode at time T=1 kcosh−13. (16) The average current density is, using (2) and (16), /angbracketleftj/angbracketright=V 4πdT=V3/2 4πcosh−1(3)d2/radicalbigge m=V3/2 9.97πd2/radicalBigg 2e m. (17) The electron velocity is ˙x=dk dsinhkt, (18) so the time dependence of the current density (10) is j(t) =1 8πV3/2 d2/radicalbigge msinhkt (0< t < T ). (19) A device that incorporates a laser driven photocathode is th e laser triggered rf gun [1]. 3 References [1] K.T. McDonald, Design of the Laser-Driven RF Electron Gun for the BNL Accele rator Test Facility , IEEE Trans. Electron Devices, 35, 2052-2059 (1988). 3

arXiv:physics/0003067v1 [physics.acc-ph] 23 Mar 2000The Helical Wiggler Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princet on, NJ 08544 Heinrich Mitter Institut f¨ ur Theoretische Physik, Karl-Franzens-Univer sit¨ at Graz, A-8010 Graz, Austria (Oct. 12, 1986) 1 Problem A variant on the electro- or magnetostatic boundary value pr oblem arises in accelerator physics, where a specified field, say B(0,0, z), is desired along the zaxis. In general there exist static fields B(x, y, z ) that reduce to the desired field on the axis, but the “boundar y condition” B(0,0, z) is not sufficient to insure a unique solution. For example, find a field B(x, y, z ) that reduces to B(0,0, z) =B0coskzˆx+B0sinkzˆy (1) on the zaxis. In this, the magnetic field rotates around the zaxis as zadvances. The use of rectangular or cylindrical coordinates leads “na turally” to different forms for B. One 3-dimensional field extension of (1) is the so-called he lical wiggler [1], which obeys the auxiliary requirement that the field at z+δbe the same as the field at z, but rotated by angle kδ. 2 Solution 2.1 Solution in Rectangular Coordinates We first seek a solution in rectangular coordinates, and expe ct that separation of variables will apply. Thus, we consider the form Bx=f(x)g(y) coskz, (2) Bx=F(x)G(y) sinkz, (3) Bz=A(x)B(y)C(z). (4) Then ∇ ·B= 0 = f′gcoskx+FG′sinkx+ABC′, (5) where the′indicates differentiation of a function with respect to its a rgument. Equation (5) can be integrated to give ABC =−f′g ksinkz+FG′ kcoskx. (6) Thezcomponent of ∇ ×B= 0 tells us that ∂Bx ∂y=fg′coskz=∂By ∂x=F′Gsinkz, (7) 1which implies that gandFare constant, say 1. Likewise, ∂Bx ∂z=−fksinkz=∂Bz ∂x=A′BC=−f′′ ksinkz, (8) using (6-7). Thus, f′′−k2f= 0, so f=f1ekx+f2e−kx. (9) Finally, ∂By ∂z=Gkcoskz=∂Bz ∂y=AB′C=G′′ ksinkz, (10) so G=G1eky+G2e−ky. (11) The “boundary conditions” f(0) = B0=G(0) are satisfied by f=B0coshkx, G =B0coshky, (12) which together with (6) leads to the solution Bx=B0coshkxcoskz, (13) By=B0coshkysinkz, (14) Bz=−B0sinhkxsinkz+B0sinhkycoskz, (15) This satisfies the last “boundary condition” that Bz(0,0, z) = 0. However, this solution does not have helical symmetry. 2.2 Solution in Cylindrical Coordinates Suppose instead, we look for a solution in cylindrical coord inates ( r, θ, z). We again expect separation of variables, but we seek to enforce the helical s ymmetry that the field at z+δ be the same as the field at z, but rotated by angle kδ. This symmetry implies that the argument kzshould be replaced by kz−θ, and that the field has no other θdependence. We begin constructing our solution with the hypothesis that Br=F(r) cos(kz−θ), (16) Bθ=G(r) sin(kz−θ). (17) To satisfy the condition (1) on the zaxis, we first transform this to rectangular components, Bz=F(r) cos(kz−θ) cosθ+G(r) sin(kz−θ) sinθ, (18) By=−F(r) cos(kz−θ) sinθ+G(r) sin(kz−θ) cosθ, (19) from which we learn that the “boundary conditions” on FandGare F(0) = G(0) = B0. (20) 2A suitable form for Bzcan be obtained from ( ∇ ×B)r= 0: 1 r∂Bz ∂θ=∂Bθ ∂z=kGcos(kz−θ), (21) so Bz=−krGsin(kz−θ), (22) which vanishes on the zaxis as desired. From either ( ∇ ×B)θ= 0 or ( ∇ ×B)z= 0 we find that F=d(rG) dr. (23) Then, ∇ ·B= 0 leads to (kr)2d2(krG) d(kr)2+krd(krG) d(kr)−[1 + (kr)2](krG) = 0. (24) This is the differential equation for the modified Bessel func tion of order 1 [2]. Hence, G=CI1(kr) kr=C 2/bracketleftBigg 1 +(kr)2 8+· · ·/bracketrightBigg , (25) F=CdI1 d(kr)=C/parenleftbigg I0−I1 kr/parenrightbigg =C 2/bracketleftBigg 1 +3(kr)2 8+· · ·/bracketrightBigg . (26) The “boundary conditions” (20) require that C= 2B0, so our second solution is Br= 2B0/parenleftBigg I0(kr)−I1(kr) kr/parenrightBigg cos(kz−θ), (27) Bθ= 2B0I1 krsin(kz−θ), (28) Bz=−2B0I1sin(kz−θ), (29) which is the form discussed in [1]. 3 References [1] J.P. Blewett and R. Chasman, Orbits and fields in the helical wiggler , J. Appl. Phys. 48, 2692-2698 (1977). [2] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964), sec. 9.6 3

arXiv:physics/0003068v1 [physics.chem-ph] 24 Mar 2000Phase changes in 38 atom Lennard-Jones clusters. I: A parall el tempering study in the canonical ensemble J. P. Neirotti Department of Chemistry, University of Rhode Island 51 Lower College Road, Kingston, RI 02881-0809 F. Calvo D´ epartement de Recherche sur la Mati` ere Condens´ ee Service des Ions, Atomes et Agr´ egats CEA Grenoble F38054 Grenoble Cedex, France David L. Freeman Department of Chemistry, University of Rhode Island 51 Lower College Road, Kingston, RI 02881-0809 and J. D. Doll Department of Chemistry, Brown University Providence, RI 02912 (January 14, 2014) Abstract The heat capacity and isomer distributions of the 38 atom Len nard-Jones cluster have been calculated in the canonical ensemble usin g parallel temper- ing Monte Carlo methods. A distinct region of temperature is identified that corresponds to equilibrium between the global minimum stru cture and theicosahedral basin of structures. This region of temperatur es occurs below the melting peak of the heat capacity and is accompanied by a peak in the deriva- tive of the heat capacity with temperature. Parallel temper ing is shown to introduce correlations between results at different temper atures. A discussion is given that compares parallel tempering with other relate d approaches that ensure ergodic simulations. Typeset using REVT EXI. INTRODUCTION Because the properties of molecular aggregates impact dive rse areas ranging from nu- cleation and condensation1to heterogeneous catalysis, the study of clusters has conti nued to be an important part of modern condensed matter science. C lusters can be viewed as an intermediate phase of matter, and clusters can provide in formation about the transfor- mation from finite to bulk behavior. Furthermore, the potent ial surfaces of clusters can be complex, and many clusters are useful prototypes for studyi ng other systems having complex phenomenology. The properties of small clusters can be unusual owing to the d ominance of surface rather than bulk atoms. A particularly important and well studied e xample of a property that owes its behavior to the presence of large numbers of surface atoms is cluster structure.2–4 The structure of clusters can differ significantly from the st ructure of the corresponding bulk material, and these differences in structure have impli cations about the properties of the clusters. For example, most small Lennard-Jones (LJ) cl usters have global potential surface minima that are based on icosahedral growth pattern s. The five-fold symmetries of these clusters differ substantially from the closest-pac ked arrangements observed in bulk materials. While most small Lennard-Jones clusters have geometries ba sed on icosahedral core structures, there can be exceptions.2,5–7A notable example is the 38-atom Lennard-Jones cluster [LJ 38]. This cluster is particularly interesting owing to its com plex potential surface and associated phenomenology. The potential surface for LJ 38has been described in detail by Doye, Miller and Wales5who have carefully constructed the disconnectivity graph8,9for the system using information garnered from basin hopping an d eigenvector following studies of the low energy potential minima along with examinations o f the transition state barriers. The general structure of this potential surface can be imagi ned to be two basins of similar energies separated by a large energy barrier with the lowest energy basin being significantly narrower than the second basin. Striking is the global minim um energy structure for LJ 38which, unlike the case for most small Lennard-Jones cluster s, is not based on an icosahedral core, but rather is a symmetric truncated octahedron. The ve rtices defined by the surface atoms of LJ 38have a morphology identical to the first Brillouin zone of a fa ce centered cubic lattice,10and the high symmetry of the cluster may account for its stabi lity. It is interesting to note that recent experimental studies11of nickel clusters using nitrogen uptake measurements have found the global minimum of Ni 38to be a truncated octahedron as well. The basin of energy minima about the global minimum of LJ 38is narrow compared to the basin about the next highest energy isomer which does hav e an icosahedral core. The difference in energy between the global minimum and the lowes t minimum in the icosahedral basin is only 0.38% of the energy of the global minimum.5 Characteristic of some thermodynamic properties of small c lusters are ranges of temper- ature over which these properties change rapidly in a fashio n reminiscent of the divergent behavior known to occur in bulk phase transitions at a single temperature. The rapid changes in such thermodynamic properties for clusters are n ot divergent and occur over a range of temperatures owing to the finite sizes of the systems . In accord with the usage introduced by Berry, Beck, Davis and Jellinek12we refer to the temperature ranges where rapid changes occur as “phase change” regions, rather than u sing the term “phase transi- tion,” that is reserved for systems at the thermodynamic lim it. As an example LJ 55displays a heat capacity anomaly over a range in temperatures often as sociated with what has been termed “cluster melting.”13Molecular dynamics and microcanonical simulations perfor med at kinetic temperatures in the melting region of LJ 55exhibit van der Waals type loops in the caloric curves and coexistence between solid-like and l iquid-like forms. In recent studies, Doye, Wales and Miller14and Miller, Doye and Wales15have examined the phase change behavior of LJ 38. These authors have calculated the heat capacity and isomer distributions as a function of temperature using the superposition method.16,17In the superposition method the microcanonical density of states is calculated for each potential minimum, and the total density of states is then constructed by summation with respect to each local density of states. Because it is not possible to find all potential minima fora system as complex as LJ 38, the summation is augmented with factors that represent the effective weights of the potential minima that are included i n the sum. The superposition method has also been improved to account for anharmonicitie s and stationary points.17For LJ38Doyeet al.14have identified two phase change regions. The first, accompan ied by a heat capacity maximum, is associated with a solid-to-solid phas e change between the truncated octahedral basin and the icosahedral basin. A higher temper ature heat capacity anomaly represents the solid-liquid coexistence region, similar t o that found in other cluster systems. The heat capacity anomaly associated with the melting trans ition in LJ 38is steeper and more pronounced than the heat capacity peak that Doye et al.14have associated with the solid- solid transition. Because the weights that enter in the sum t o construct the microcanonical density of states are estimated, it is important to confirm th e findings of Doye et al.14by detailed numerical simulation. Such simulations are a goal of the current work and its companion paper. As is found in Section III, the simulations provide a heat capacity curve for LJ 38that has some qualitative differences with the curve reporte d by Doye et al.14 Owing to the complex structure of the potential surface of LJ 38, the system represents a particularly challenging case for simulation. It is well k nown that simulations of systems having more than one important region of space separated by s ignificant energy barriers can be difficult. The difficulties are particularly severe if any of the regions are either narrow or reachable only via narrow channels. The narrow basin abou t the global minimum makes simulations of LJ 38especially difficult. There are several methods that have bee n developed that can prove to be useful in overcoming such ergodicity diffi culties in simulations. Many of these methods use information about the underlying potenti al surface generated from simu- lations on the system using parameters where the various reg ions of configuration space are well-connected. One of the earliest of these methods is J-wa lking18where information about the potential surface is obtained from simulations at high t emperatures, and the information is passed to low temperature walks by jumping periodically t o the high temperature walk. Closely allied with J-walking is the parallel tempering met hod19–23where configurations are exchanged between walkers running at two differing temperat ures. A related approach,24similar in spirit to J-walking, uses Tsallis distributions that are sufficiently broad to cover much of configuration space. Another recent addition25to these methods is the use of multi- canonical distributions26in the jumping process. Multicanonical walks are performed using the entropy of the system, and multicanonical distribution s are nearly independent of the energy thereby allowing easy transitions between energy ba sins. As we discuss in the cur- rent work, we have found the parallel tempering method to be m ost useful in the context of simulations of LJ 38. A comparative discussion of some of the methods outlined ab ove is given later in this paper. In the current work we apply parallel tempering to the calcul ation of the thermodynamic properties of LJ 38in the canonical ensemble. In the paper that follows27we again use parallel tempering to study LJ 38, but using molecular dynamics methods along with microcano nical Monte Carlo simulations. Our goals are to understand better this complex system and to determine the best simulation method for systems of compara ble complexity. The contents of the remainder of this first paper are as follows. In Section II we discuss the methods used with particular emphasis on the parallel tempering app roach and its relation to the J-walking method. In Section III we present the results incl uding the heat capacity as a function of temperature and identify the phase change behav iors of LJ 38. In Section IV we present our conclusions and describe our experiences with a lternatives to parallel tempering for insuring ergodicity. II. METHOD For canonical simulations we model a cluster with Natoms by the standard Lennard- Jones potential augmented by a constraining potential Ucused to define the cluster U(r) = 4εN/summationdisplay i<j /parenleftBiggσ rij/parenrightBigg12 −/parenleftBiggσ rij/parenrightBigg6 +Uc, (1) where σandεare respectively the standard Lennard-Jones length and ene rgy parameters, andrijis the distance between particles iandj. The constraining potential is necessarybecause clusters at defined temperatures have finite vapor pr essures, and the evaporation events can make the association of any atom with the cluster a mbiguous. For classical Monte Carlo simulations, a perfectly reflecting constraining pot ential is most convenient Uc=N/summationdisplay i=1u(ri), (2) with u(r) =  ∞ |r−rcm|> R c 0|r−rcm|< R c(3) where rcmis the center of mass of the cluster, and we call Rcthe constraining radius. Thermodynamic properties of the system are calculated with Monte Carlo methods using the parallel tempering technique.19–23To understand the application of the parallel temper- ing method and to understand the comparison of parallel temp ering with other related methods, it is useful to review the basic principles of Monte Carlo simulations. In the canonical ensemble the goal is the calculation of cano nical expectation values. For example, the average potential energy is expressed /an}bracketle{tU/an}bracketri}ht=/integraltextd3Nr U(r)e−βU(r) /integraltextd3Nr e−βU(r), (4) where β= 1/kBTwithTthe temperature and kBthe Boltzmann constant. In Monte Carlo simulations such canonical averages are determined b y executing a random walk in configuration space so that the walker visits points in space with a probability proportional to the canonical density ρ(r) =Z−1exp[−βU(r)], where Zis the configurational integral that normalizes the density. After generating Msuch configurations in a random walk, the expectation value of the potential energy is approximated b y /an}bracketle{tU/an}bracketri}htM=1 MM/summationdisplay i=1U(ri). (5) The approximate expectation value /an}bracketle{tU/an}bracketri}htMbecomes exact in the limit that M→ ∞. A sufficiency condition for the random walk to visit configurat ion space with a probability proportional to the density ρ(r) is the detailed balance condition28,29ρ(ro)K(ro→rn) =ρ(rn)K(rn→ro), (6) whereroandrnrepresent two configurations of the system and K(ro→rn) is the conditional probability that if the system is at configuration roit makes a transition to rn. In many Monte Carlo approaches, the conditional probability is not known and is replaced by the expression K(ro→rn) =T(ro→rn) acc(ro→rn), (7) where T(ro→rn) is called the trial probability and acc( ro→rn) is an acceptance proba- bility constructed to ensure K(ro→rn) satisfies the detailed balance condition. The trial probability can be any normalized density function chosen f or convenience. A common choice for the acceptance probability is given by28,29 acc(ro→rn) = min/bracketleftBigg 1,ρ(rn)T(rn→ro) ρ(ro)T(ro→rn)/bracketrightBigg . (8) The Metropolis method,30obtained from Eq.(8) by choosing T(ro→rn) to be a uniform distribution of points of width ∆ centered about ro, is arguably the most widely used Monte Carlo method and the basis for all the approaches discussed i n the current work. The Metropolis method rigorously guarantees a random walk visi ts configuration space propor- tional to a given density function asymptotically in the lim it of an infinite number of steps. In practice when configuration space is divided into importa nt regions separated by signif- icant energy barriers, a low temperature finite Metropolis w alk can have prohibitively long equilibration times. Such problems in attaining ergodicity in the walk do not occu r at temperatures suffi- ciently high that the system has significant probability of fi nding itself in the barrier regions. In both the J-walking and parallel tempering methods, infor mation obtained from an ergodic Metropolis walk at high temperatures is passed to a low tempe rature walker periodically to enable the low temperature walker to overcome the barriers b etween separated regions. In the J-walking method18the trial probability at inverse temperature βis taken to be a high temperature Boltzmann distributionT(ro→rn) =Z−1e−βJU(rn)(9) where βJrepresents the jumping temperature that is sufficiently high that a Metropolis walk can be assumed to be ergodic. Introduction of Eq.(9) into Eq. (8) results in the acceptance probability acc(ro→rn) = min {1,exp[−(β−βJ)(U(rn)−U(ro))]}. (10) In practice at inverse temperature βthe trial moves are taken from the Metropolis distri- bution about 90% of the time with jumps attempted using Eq.(9 ) about 10% of the time. The jumping configurations are generated with a Metropolis w alk at inverse temperature βJ, and jump attempts are accepted using Eq.(10). The acceptan ce expression [Eq.(10)] is correct provided the configurations chosen for jumping ar e a random representation of the distribution e−βJU(r). The Metropolis walk that is used to generate the configurati ons at inverse temperature βJis correlated,28and Eq.(10) is inappropriate unless jumps are attempted sufficiently infrequently to break the correlatio ns. In practice Metropolis walks are still correlated after 10 steps, and it is not possible to use Eq.(10) correctly if jumps are attempted 10% of the time. In J-walking the difficulty with correlations is overcome in two ways. In the first method, often called serial J-walking,18a large set of configurations is stored to an external distribution with the configurations g enerated with a Metropolis walk at inverse temperature βJ, and configurations stored only after sufficient steps to brea k the correlations in the Metropolis walk. Additionally, the con figurations are chosen from the external distribution at random. This external distributi on is made sufficiently large that the probability of ever choosing the same configuration more than once is small. In this method detailed balance is strictly satisfied only in the lim it that the external distribution is of infinite size. In the second method, often called parall el J-walking,31,32the walks at each temperature are made in tandem on a parallel machine. Ma ny processors, randomly initialized, are assigned to the jumping temperature, and e ach processor at the jumping temperature is used to donate a high temperature configurati on to the low temperaturewalk sufficiently infrequently that the correlations in the M etropolis walk at inverse tem- perature βJare broken. In this parallel method, configurations are neve r reused, but the acceptance criterion [Eq.(10)] is strictly valid only in th e limit of an infinite set of processors at inverse temperature βJ. In practice both serial and parallel J-walking work well fo r many applications with finite external distributions or with a fin ite set of processors.18,31–38 In parallel tempering19–23configurations from a high temperature walk are also used to make a low temperature walk ergodic. In contrast to J-walk ing rather than the high temperature walk feeding configurations to the low temperat ure walk, the high and low temperature walkers exchange configurations. By exchangin g configurations detailed balance is satisfied, once the Metropolis walks at the two temperatur es are sufficiently long to be in the asymptotic region. To verify detailed balance is sati sfied by the parallel tempering procedure we let ρ2(r,r′) =Z−1e−βU(r)e−βJU(r′)(11) be the joint density that the low temperature walker is at con figuration rand the high temperature walker is at configuration r′. When configurations between the two walkers are exchanged, the detailed balance condition is ρ2(r,r′)K(r→r′,r′→r) =ρ2(r′,r)K(r′→r,r→r′) (12) By solving for the ratio of the conditional transition proba bilities K(r→r′,r′→r) K(r′→r,r→r′)= exp[ −(β−βJ)(U(r′)−U(r))], (13) it is evident that if exchanges are accepted with the same pro bability as the acceptance criterion used in J-walking [see Eq.(10)], detailed balanc e is satisfied. Although the basic notions used by both J-walking and parall el tempering are similar, the organization of a parallel tempering calculation can be significantly simpler than the organization of a J-walking calculation. In parallel tempe ring no external distributions are required nor are multiple processors required at any temper ature. Parallel tempering canbe organized in the same simple way that serial tandem J-walk ing is organized as discussed in the original J-walking reference.18Unlike serial tandem J-walking where detailed balance can be attained only asymptotically, parallel tempering sa tisfies detailed balance directly. For a problem as difficult as LJ 38where very long simulations are required, the huge exter- nal distributions needed in serial J-walking, or the large s et of jumping processors needed in parallel J-walking, make the method prohibitive. As disc ussed in Section III, parallel tempering can be executed for arbitrarily long simulations making the method suitable at least for LJ 38. In the current calculation parallel tempering is used not ju st to simulate the system at some low temperature using high temperature information, b ut simulations are performed for a series of temperatures. As is the case for J-walking18and as discussed elsewhere for parallel tempering,22the gaps between adjacent temperatures cannot be chosen arb itrarily. Temperature gaps must be chosen so that exchanges are accept ed with sufficient frequency. If the temperature gap is too large, the configurations impor tant at the two exchanging temperatures can be sufficiently dissimilar that no exchange s are ever accepted. Preliminary calculations must be performed to explore the temperature d ifferences needed for acceptable exchange probabilities. In practice we have found at least 1 0% of attempted exchanges need to be accepted for the parallel tempering procedure to be use ful. In general the temperature gaps must be decreased near phase change regions or when the t emperature becomes low. By exchanging configurations between temperatures, correl ations are introduced at dif- ferent temperature points. For example, the average heat ca pacities at two temperatures may rise or fall together as each value fluctuates statistica lly. In some cases the values of the heat capacities or other properties at two temperatures can be anti-correlated. The magnitude of these correlations between temperatures are m easured and discussed in Sec- tion III. As discussed in Section III the correlations betwe en differing temperatures imply that the statistical fluctuations must be sufficiently low to e nsure any features observed in a calculation as a function of temperature are meaningful.III. RESULTS Forty distinct temperatures have been used in the parallel t empering simulations of LJ 38 ranging from T= 0.0143ε/kBtoT= 0.337ε/kB. The simulations have been initiated from random configurations of the 38 atoms within a constrain ing sphere of radius 2.25 σ. We have chosen Rc= 2.25σ, because we have had difficulties attaining ergodicity with larger constraining radii. With large constraining radii, the system has a significant boil- ing region at temperatures not far from the melting region, a nd it is difficult to execute an ergodic walk with any method when there is coexistence bet ween liquid-like and vapor regions. Constraining radii smaller than 2 .25σcan induce significant changes in thermody- namic properties below the temperature of the melting peak. Using the randomly initialized configurations the initialization time to reach the asympto tic region in the Monte Carlo walk has been found to be long with about 95 million Metropolis Mon te Carlo points followed by 190 million parallel tempering Monte Carlo points includ ed in the walk prior to data accumulation. This long initiation period can be made signi ficantly shorter by initializing each temperature with the structure of the global minimum. W e have chosen to initialize the system with random configurations to verify the parallel tempering method is able to equilibrate this system with no prior knowledge about the st ructure of the potential surface. Following this initiation period, 1 .3×1010points have been included with data accumula- tion. Parallel tempering exchanges have been attempted eve ry 10 Monte Carlo passes over the 38 atoms in the cluster. In an attempt to minimize the correlations in the data at diffe ring temperatures, an exchange strategy has been used that includes exchanges bet ween several temperatures. To understand this strategy, we let the set of temperatures be p ut into an array. One-half of the exchanges have been attempted between adjacent tempera tures in the array, one-fourth have been attempted between next near neighboring temperat ures, one-eighth between every third temperature, one-sixteenth between every fourth tem perature and one-thirty second between every fifth temperature in the array. We have truncat ed this procedure at fifth nearneighboring temperatures, because exchanges between temp eratures differing by more than fifth neighbors are accepted with frequencies of less than te n per cent. The data presented in this work have been generated using the procedure outline d above. In retrospect, we have found exchanges are only required between adjacent tempera tures. We have also performed the calculations where exchanges are included only between adjacent temperatures, and we have seen no significance differences either in the final res ults or in the correlations between different temperatures. Using the random initializ ations of the clusters, after the initialization period the lowest temperature walks are dom inated by configurations well represented by small amplitude oscillations about the glob al minimum structure. For all data displayed in this work, the error bars represent two standard deviations of the mean. The heat capacity, calculated from the standard flu ctuation expression of the energy CV=kBβ2[/an}bracketle{tE2/an}bracketri}ht − /an}bracketle{tE/an}bracketri}ht2], (14) is displayed in the upper panel of Fig. 1. In agreement with th e heat capacity for LJ 38 reported by Doye et al.,14the heat capacity displayed in Fig. 1 has a melting maximum centered at about T= 0.166ε/kB. In contrast to the results of Doye et al.14we find no maximum associated with the solid-solid transition betwee n the two basins in the potential surface. Rather, we see a small change in slope at about T= 0.1ε/kB. To characterize this region having a change in slope, in the lower panel of Fig . 1 we present a graph of (∂CV/∂T)Vcalculated from the fluctuation expression /parenleftBigg∂CV ∂T/parenrightBigg V=−2CV T+1 k2 BT4[/an}bracketle{tE3/an}bracketri}ht+ 2/an}bracketle{tE/an}bracketri}ht3−3/an}bracketle{tE2/an}bracketri}ht/an}bracketle{tE/an}bracketri}ht] (15) The small low temperature maximum in ( ∂CV/∂T)Voccurs within the slope change region. To interpret the configurations associated with the various regions of the heat capacity, we use an order parameter nearly identical to the order paramet er introduced by Steinhardt, Nelson and Ronchetti39to distinguish face centered cubic from icosahedral struct ures in liquids and glasses. The order parameter has been used by Doy eet al.5to monitor phase changes in LJ 38. The order parameter Q4is defined by the equationQ4= 4π 94/summationdisplay m=−4|Q4,m|2 1/2 , (16) where Q4,m=1 Nb/summationdisplay rij<rbY4,m(θij, φij). (17) To understand the parameters used in Eq.(17), it is helpful t o explain how Q4,mis evaluated. The center of mass of the full 38 atom cluster is located and th e atom closest to the center of mass is then identified. The atom closest to the center of mass plus the 12 nearest neighbors of that atom define a “core” cluster of the 38 atom cluster. The center of mass of the core cluster is then calculated. The summation in Eq.(17) is perf ormed over all vectors that point from the center of mass of the core cluster to all Nbbonds formed from the 13 atoms of the core cluster. A bond is assumed to be formed between two at oms of the core cluster if their internuclear separation rijis less than a cut-off parameter rb, taken to be rb= 1.39σ in this work. In Eq.(17) θijandφijare respectively the polar and azimuthal angles of the vector that points from the center of mass of the core cluster to the center of each bond, andY4,m(θ, φ) is a spherical harmonic. To verify that the optimal value of Q4is obtained, the procedure is repeated by choosing the second closest ato m to the center of mass of the whole cluster to define the core cluster. The value of Q4obtained from this second core cluster is compared with that obtained from the first core clu ster, and the smallest resulting value of Q4is taken to be the value of Q4for the entire cluster. In the work of Steinhardt et al.39fewer bonds are included in the summation appearing in Eq.(17) than in the current work. In the definition used by S teinhardt et al.,39the only bonds that contribute to the sum in Eq.(17) are those involvi ng the central atom of the core cluster. In the definition used in this work, at low tempe ratures the sum includes all the bonds included by Steinhardt et al.39in addition to vectors that connect the center of mass of the core cluster with the centers of bonds that conn ect atoms at the surface of the core cluster with each other. For a perfect and undisto rted icosahedral or truncated octahedral cluster, the current definition and the definitio n of Steinhardt et al.39are identicalnumerically owing to the rotational symmetry of the spheric al harmonics. However, for distorted clusters the two definitions differ numerically. F or perfect, undistorted icosahedral clusters Q4= 0 whereas for perfect, undistorted truncated octahedral c lusters, Q4∼=0.19, and both definitions of the order parameter are able to distin guish configurations from the truncated octahedral basin and other basins at finite temper atures. However, we have found the definition introduced by Steinhardt et al.39is unable to distinguish structures in the icosahedral basin from liquid-like structures. This same i ssue has been discussed previously by Lynden-Bell and Wales.40In contrast, we have found that liquid-like structures have larger values of Q4than icosahedral structures when the present definition of Q4[i.e the definition that includes additional bonds in Eq.(17)], is us ed. Consequently, as discussed shortly, the current definition of Q4enables an association of each configuration with either the icosahedral basin, the truncated octahedral basin, or s tructures that can be identified as liquid-like. The average of Q4as a function of temperature is plotted in the upper panel of F ig. 2. Again the error bars represent two standard deviations of th e mean. At the lowest calculated temperatures /an}bracketle{tQ4/an}bracketri}htis characteristic of the global truncated octahedral minim um. As the temperature is raised to the point where the slope change beg ins in the heat capacity, /an}bracketle{tQ4/an}bracketri}ht begins to drop rapidly signifying the onset of transitions b etween the structures associated with the global minimum and icosahedral structures. We then have the first hint that the slope change in CVis associated with a analogue of a solid-solid transition fr om the truncated octahedron to icosahedral structures. To clarify the transition further, the data plotted in the lo wer panel of Fig. 2 represent the probability of observing particular values of Q4as a function of temperature. The probabilities have been calculated by tabulating the frequ ency of observing particular values ofQ4for each configuration generated in the simulation. Differen t values of Q4are then assigned to either icosahedral structures (labeled IC in th e graph), truncated octahedral structures (labeled FCC) or liquid-like structures (label ed LIQ). below. By comparing the lower panel of Fig. 2 with the derivative of the heat capacity plotted in the lower panel ofFig. 1, it is evident that icosahedral structures begin to be occupied and the probability of finding truncated octahedral structures begins to fall wh en the derivative in the heat capacity begins to rise. Equilibrium between the truncated octahedral structures and the icosahedral structures continues into the melting region, and truncated octahedral structures only disappear on the high temperature side of the melting pe ak of the heat capacity. Doye et al.14and Miller et al.15have generated data analogous to that depicted in the lower panel of Fig. 2 using the superposition method, and the data o f Miller et al.15are in qualitative agreement with the present data. A more direct c omparison with the data of these authors can be made by performing periodic quenching a long the parallel tempering trajectories. We then use an energy criterion similar to tha t of Doye et al.14to distinguish the three categories of geometries and to generate the respe ctive probabilities P. For a given total cluster energy E, a truncated octahedron is associated with E < −173.26ε, icosahedral-based structures with −173.26ε≤E <−171.6ε, and liquid-like structures with E≥ −171.6ε. The quenches have been performed every 104MC steps for each temperature, and the results of these quenches are plotted in Fig. 3. Using the energy criterion, the behavior we observe is qualitatively similar to the data of D oyeet al..14However, the largest probability of observing icosahedral structures is found h ere to be substantially lower than Doyeet al.14The data accumulated more recently by Miller et al.15using the superposition method include contributions from more stationary points t han in the previous work of Doye et al.,14but no reweighting has been performed. As a result, the distr ibutions of isomers look quite different, especially at high temperatures.14 The assignment of a particular value of Q4to a structure as displayed in Fig. 2, is made by an analysis of the probability distribution PQ(T, Q 4) of the order parameter displayed in Figs. 4 and 5. Figure 4 is a representation of the three-dimen sional surface of PQ(T, Q 4) as a function of temperature and order parameter. A projecti on of this surface onto two dimensions is given in Fig. 5. The probability density in Fig . 5 is represented by the shading so that the brighter the area the greater the probability. Th e horizontal white lines in Fig. 5 define the regions of the heat capacity curve. The lowest tem perature horizontal linerepresents the temperature at which the slope of the heat cap acity first changes rapidly, the middle temperature horizontal line represents the lowest t emperature of the melting peak and the highest temperature horizontal line represents the end of the melting region. An additional representation of the data is given in Fig. 6, whe re the probability of observing particular values of Q4is given as a function of Q4at a fixed temperature of 0 .14ε/kB. In Fig. 6 three regions are evident for PQ(T= 0.14ε/kB, Q4) with Q4ranging from 0.13 to 0.19. Although the presence of three regions seems to indica te three distinct structures, all three regions correspond to the truncated octahedral globa l minimum. We have verified this assignment by quenching the structures with Q4ranging from 0.13 to 0.19 to their nearest local minima, and we have found all such structures quench to the truncated octahedron. To explain the three regions, we have found that there are sma ll distortions of LJ 38about the truncated octahedral structure where both the energy an dQ4increase together. These regions where both the energy and Q4increase above Q4∼=0.13 have low probability and account for the oscillations observed in Figs. 4–6. In the lo wer panel of Fig. 2, all structures having Q4>0.13 have been identified as truncated octahedra. Quench studi es of the broad region visible in Fig. 5 at the lowest values of Q4, or equivalently in the first low Q4peak in Fig. 6 find all examined structures to belong to the icosahedr al basin. To determine if a given configuration is associated with the icosahedral basin, one -dimensional cross sectional plots are made from Fig. 4 at each temperature used in the calculati on. Figure 6 is a particular example of such a cross sectional plot. The maximum present a t low Q4represents the center for structures in the icosahedral basin. The next two maxima at higher Q4represents the midpoint of the liquid region. Consequently, in generat ing the lower panel of Fig. 2, all configurations with Q4between Q4= 0 and the first minimum in Fig. 6 have been identified as icosahedral structures. All other values of Q4, represented by the broad intermediate band in Fig. 5 (or the region about the second two maxima in Fig . 6), have been identified as liquid-like structures. To make these identifications, s eparate cross sections of Fig. 4 must be made at each temperature. Of course, it is impossible to ve rify that the identification of all values of Q4with a particular structure as discussed above would agree w ith the result ofquenching the structure to its nearest potential minimum. T he differences found by defining icosahedral, truncated octahedral or liquid-like structu res using either an energy criterion orQ4is clarified by comparing Fig. 3 and the lower panel of Fig. 2. B oth definitions are arbitrary, and the information carried by the two classifica tion methods complement each other. Figure 5 also provides additional evidence that the peak in ( ∂CV/∂T)Vis associated with the equilibrium between the truncated octahedral structur es and the icosahedral structures. There is significant density for both kinds of structures in t he region between the lowest two parallel lines that define the region with the slope change. A dditionally, both icosahedral structures and truncated octahedral structures begin to be in equilibrium with each other at the beginning of the slope change region. This equilibrium c ontinues to temperatures above the melting region. Another identification of the slope change region with a tran sition between truncated octahedral and icosahedral forms can be made by defining PR(T, R)dRto be the probability that an atom in the cluster is found at location RtoR+dRfrom the center of mass of the cluster at temperature T. A projection of PR(T, R) onto the RandTplane is depicted in Fig. 7. The solid vertical lines represent the location of at oms from the center of mass of the truncated octahedral structure (the lower set of vertic al lines), and the lowest energy icosahedral structure (the upper set of vertical lines). As in Fig. 5, increased probability is represented by the lighter shading. At the lowest tempera tures PR(T, R) is dominated by contributions from the truncated octahedron as is eviden t by comparing the shaded regions with the lowest set of vertical lines. As the tempera ture is increased, contributions toPR(T, R) begin to appear from the icosahedral structures. The shade d region at R= 0.45 does not match any of the vertical lines shown, but correspon ds to atoms in the third lowest energy isomer, which like the second lowest energy isomer, c omes from the icosahedral basin. The equilibrium between the icosahedral and truncated octa hedral structures observed in Fig. 7 matches the regions of temperature observed in Fig. 5. We have mentioned previously that parallel tempering intro duces correlations in the dataaccumulated at different temperatures, and it is important t o ensure the statistical errors are sufficiently small that observed features are real and not artifacts of the correlations. To measure these correlations we define a cross temperature c orrelation function for some temperature dependent property gby γ(T1, T2) =/an}bracketle{t(g(T1)− /an}bracketle{tg(T1)/an}bracketri}ht)(g(T2)− /an}bracketle{tg(T2)/an}bracketri}ht)/an}bracketri}ht [/an}bracketle{t(g(T1)− /an}bracketle{tg(T1)/an}bracketri}ht)2/an}bracketri}ht/an}bracketle{t(g(T2)− /an}bracketle{tg(T2)/an}bracketri}ht)2/an}bracketri}ht]1/2. (18) A projection of γ(T1, T2) when g=CVis given in Fig. 8. In Fig. 8 white represents γ= 1 and black represents γ=−1 with other shadings representing values of γbetween these two extremes. The white diagonal line from the lower le ft hand corner to the upper right hand corner represents the case that T1=T2so that γ= 1. The light shaded areas near this diagonal represent cases where T1andT2are adjacent temperatures in the parallel tempering simulations, and we find γto be only slightly less than unity. More striking are the black regions off the diagonal where γis nearly −1. These black regions correspond to anti-correlations between results at temperatures near th e heat capacity maximum in the melting peak and temperatures near the center of the slope ch ange region associated with the transition between icosahedral and truncated octahedr al structures. These correlations imply the importance of performing sufficiently long simulat ions to ensure that statistical fluctuations of the data are small compared to important feat ures in the data as a function of temperature. IV. CONCLUSIONS Using parallel tempering methods we have successfully perf ormed ergodic simulations of the equilibrium thermodynamic properties of LJ 38in the canonical ensemble. As discussed by Doye et al.5the potential surface of this system is complex with two sign ificant basins; a narrow basin about the global minimum truncated octahedral structure, and a wide icosa- hedral basin. These two basins are separated both by structu re and a large energy barrier making simulations difficult. In agreement with the results o f Doye et al.14we find clearevidence of equilibria between structures at the basin of th e global minimum and the icosa- hedral basin at temperatures below the melting region. Unli ke previous work we find no heat capacity maximum associated with this transition, but rather a region with a change in the slope of the heat capacity as a function of temperature . We have found parallel tempering to be successful with this s ystem, and have noted correlations in our data at different temperatures when the p arallel tempering method is used. These correlations imply the need to perform long simu lations so that the statistical errors are sufficiently small that the correlations do not int roduce artificial conclusions. We believe that the methods used in this work could be applied to a variety of other systems including clusters of complexity comparable to LJ 38. For instance, the 75-atom Lennard-Jones cluster is known to share many features with t he 38-atom cluster investi- gated here. LJ 75is also characterized by a double funnel energy landscape, o ne funnel being associated with icosahedral structures, and the other funn el being associated with the dec- ahedral global minimum. The landscape of LJ 75has been recently investigated by Doye, Miller and Wales6who have used Q6as the order parameter. In another paper,2Wales and Doye have predicted that the temperature where the decahedr al/icosahedral equilibrium takes place should be close to 0 .09ε/kB. This prediction is made by using the superposi- tion method, but no caloric curves have yet been reported for LJ75. The parallel tempering Monte Carlo method can be expected to work well for LJ 75, and such a parallel tempering study would be another good test case for theoretical method s discussed in this work. A useful enhancement of parallel tempering Monte Carlo is th e use of multiple histogram methods13,41that enables the calculation of thermodynamic functions in both the canonical and microcanonical ensembles by the calculation of the micr ocanonical entropy. In practice the multiple histogram method requires the generation of hi stograms of the potential energy at a set of temperatures such that there is appreciable overl ap of the potential energy distributions at adjacent temperatures. This overlap requ irement is identical to the choice of temperatures needed in parallel tempering. In performing simulations on LJ 38we have tried other methods to reduce ergodicityerrors, and we close this section by summarizing the difficult ies we have encountered with these alternate methods. It is important to recognize that t he parallel tempering simulations include in excess of 1010Monte Carlo points, and most of our experience with these alt ernate methods have come from significantly shorter simulations. O ur ability to include this large number of Monte Carlo points with parallel tempering is an im portant reason why we feel parallel tempering is so useful. ¿From experience with other smaller and simpler clusters, f or a J-walking simulation to include 1010points, an external distribution containing at least 109points is required to prevent oversampling of the distribution. Such a large di stribution is prohibitive with current computer technology. Our J-walking simulations co ntaining about 107Monte Carlo points have resulted in data that have not been internally re producible, and data that are not in good agreement with the parallel tempering data. Many lon g J-walking simulations with configurations initiated at random only have icosahedral st ructures at the lowest calculated temperatures. To stabilize the J-walking method with respe ct to the inclusion of truncated octahedral structures at low temperatures, we have attempt ed to generate distributions using the modified potential energy function Um(r, λ) =U(r)−λQ4. In this modified potential λis a parameter chosen to deepen the octahedral basin without significantly distorting the cluster. While this modified potential has led to more stable results than J-walking using the bare potential, the results with 108Monte Carlo points have not been reproducible in detail. The application of Tsallis distributions24has not improved this situation. We have also tried to apply the multicanonical J-walking app roach recently introduced by Xu and Berne.25While this multicanonical approach has been shown to improv e the original J-walking strategy for other cluster systems, in t he case of LJ 38the iterations needed to produce the external multicanonical distribution have n ot produced truncated octahedral structures. The iterations have produced external distrib utions having either liquid-like structures or structures from the icosahedral basin. The mu lticanonical distribution is known to have deficiencies at low energies, and this low energy diffic ulty appears to be problematic for LJ 38. We have attempted to solve these deficiencies by including p rior information aboutthe thermodynamics of the system. In this attempt we have cho sen the multicanonical weight to be wmu(U) = exp[ −SPT(U)] where SPT(U) is the microcanonical entropy extracted from a multihistogram analysis13,41of a parallel tempering Monte Carlo simulation. In several attempts using this approach we have not observed either the truncated octahedral structure nor structures from the icosahedral basin with significant p robability. The multicanonical distribution so generated is dominated by liquid-like stru ctures, and the distribution appears to be incapable of capturing the solid-to-solid transition that leads to the low temperature peak in ( ∂CV/∂T)V. Whether there are other approaches to generate a multicano nical distribution that are more successful in capturing low temp erature behaviors is unknown to us. Much insight about phase change behaviors can be obtained fr om simulations in the microcanonical ensemble or using molecular dynamics metho ds. For example, the van der Waals loops observed in LJ 5513complement the interpretation of the canonical caloric cur ves. In the next paper27we present parallel tempering results for LJ 38using both molecular dynamics and microcanonical Monte Carlo methods. ACKNOWLEDGMENTS Some of this work has been motivated by the attendance of two o f us (DLF and FC) at a recent CECAM meeting on ‘Overcoming broken ergodicity i n simulations of condensed matter systems.’ We would like to thank CECAM, J.E. Straub an d B. Smit who organized the meeting, and those who attended the workshop for stimula ting discussions, particularly on the connections between J-walking and parallel temperin g. Two of us (DLF and JPN) would also like to thank Professor M.P. Nightingale for help ful discussions concerning the parallel tempering method. This work has been supported in p art by the National Science Foundation under grant numbers CHE-9714970 and CDA-972434 7. This research has been supported in part by the Phillips Laboratory, Air Force Mate rial Command, USAF, through the use of the MHPCC under cooperative agreement number F296 01-93-0001. The viewsand conclusions contained in this document are those of the a uthors and should not be interpreted as necessarily representing the official polici es or endorsements, either expressed or implied, of Phillips Laboratory or the U.S. Government.REFERENCES 1F.F. Abraham, Homogeneous Nucleation Theory (Academic, New York, 1974). 2D.J. Wales and J.P.K. Doye, J. Phys. Chem. A 101, 5111 (1997). 3D.L. Freeman and J.D. Doll, Ann. Rev. Phys. Chem. 47, 43 (1980). 4J. Xie, J.A. Northby, D.L. Freeman and J.D. Doll, J. Chem. Phy s.91, 612 (1989). 5J.P.K. Doye, M.A. Miller and D.J. Wales, J. Chem. Phys. 110, 6896 (1999). 6J.P.K. Doye, M.A. Miller and D.J. Wales, J. Chem. Phys. 111, 8417 (1999) 7R.H. Leary and J.P.K. Doye, Phys. Rev. E 60, R6320 (1999). 8R. Czerminski and R. Elber, J. Chem. Phys. 92, 5580 (1990). 9O.M. Becker and M. Karplus, J. Chem. Phys. 106, 1495 (1997). 10C. Kittel, Introduction to Solid State Physics , 4’th Edition (John Wiley, New York, 1971) 11E.K. Parks, G.C. 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Caballero and G.E. Lopez, J. Chem. Phys. 106, 7257 (1997).39P.J. Steinhardt, D.R. Nelson and M. Ronchetti, Phys. Rev. B 28, 784 (1983). 40R.M. Lynden-Bell and D.J. Wales, J. Chem. Phys. 101, 1460 (1994). 41A.M. Ferrenberg and R.H. Swendsen, Phys. Rev. Lett. 61, 2635 (1988).FIGURES FIG. 1. The heat capacity CVper particle of LJ 38in units of kB(upper panel) and ( ∂CV/∂T)V per particle (lower panel) as a function of reduced temperat ure. The small low temperature maxi- mum in the derivative associated with a change in slope of the heat capacity identifies the transition region between the truncated octahedral basin and the icosa hedral basin. The large heat capacity peak identifies the melting region. FIG. 2. The expectation value of the order parameter (upper p anel) and the order parameter probability distribution (lower panel) as a function of red uced temperature. In the lower panel FCC labels the truncated octahedron, IC labels structures f rom the icosahedral basin and LIQ labels structures from the liquid region. The transition be tween FCC and IC occurs at the same temperature as the low temperature peak in ( ∂CV/∂T)Vin Fig. 1. FIG. 3. The probability distributions of observing differen t structures as a function of temper- ature using the energy criterion. The labels are the same as t hose defined in the lower panel of Fig. 2, and the data complements the interpretation of the lo wer panel of Fig. 2 FIG. 4. The probability of observing configurations with par ticular values of Q4withQ4dis- played along one axis and the reduced temperature displayed along the other axis. The large peak at low temperatures comes from the truncated octahedral str uctures and the broad region with small Q4at intermediate temperatures represents structures in the icosahedral basin. FIG. 5. A projection of Fig. 4 onto the T-Q4plane. The probability is measured by the shading with increasing probability represented by lighter shadin g. The lowest temperature horizontal white line represents the temperature at which transitions betwe en the icosahedral and lowest energy basins begin. The second lowest temperature horizontal whi te line represents the beginning of the melting region, and the highest temperature horizontal whi te line represents the end of the melting peak of the heat capacity. The coexistence of icosahedral an d octahedral structures continues into the melting region.FIG. 6. The probability of observing configurations with par ticular values of Q4as a function ofQ4atT= 0.14ε/kB. The region from Q4= 0 through the first maximum to the first minimum defines the icosahedral basin at T= 0.14ε/kB, the region from the first minimum to the third defines liquid-like structures, and the region about the thr ee maxima having the highest values of Q4define the truncated octahedral basin. The oscillations in t he truncated octahedral basin arise from distorted structures of low probability where both the energy and Q4rise together. FIG. 7. The projected probability of observing particles a d istance Rfrom the center of mass of LJ38as a function of Rand reduced temperature. As in Fig. 5 increased probability is represented by the lightest shading. The lower vertical lines represent the location of atoms in the fully relaxed truncated octahedron and the upper vertical lines represen t the location of atoms in the fully relaxed icosahedral structure that is lowest in energy. Equ ilibrium between the icosahedral and octahedral forms are observed in the same temperature range as found in Figs. 2 and 5. FIG. 8. γ(T1,T2) for the heat capacity as defined in Eq.(18) [with g=CV] as a function of reduced temperature along two axes. White shading represen tsγ= 1 and black represents γ=−1. The white diagonal line connecting the lower left hand corne r with the upper right hand corner indicates T1=T2so that γ= 1. The black areas show anti-correlation from parallel tem pering between the heat capacity calculated at the maximum of the he at capacity and the center of the change in slope region.0.0 0.1 0.2 0.3 kBT / ε−100−50050100(1/NkB) ( ∂〈CV 〉 / ∂T )V 3.04.05.06.0〈 CV 〉 / NkB0.0 0.1 0.2 0.3 kBT / ε0.00.20.40.60.81.0PQ4 IC LIQ FCC0.050.100.150.20〈 Q4 〉0.0 0.1 0.2 0.3 kBT / ε0.00.20.40.60.81.0PIC LIQ FCC0 0.45 0.9 1.35 1.8 2.25 Q40.337 0.273 0.21 0.14 0.071 0.002kBT 0.00 0.05 0.10 0.15 0.20 Q41030507090PQ (T = 0.14 ε / kB , Q4 )0.061 0.12 0.171 0.227 0.282 0.337 kBT0.0610.120.1710.2270.2820.337kBT

arXiv:physics/0003069v1 [physics.acc-ph] 24 Mar 2000TheStern-Gerlach interaction between a traveling particle anda timevarying magnetic field M. Conte1, M.Ferro1,G.Gemme1,W.W.MacKay2, R.Parodi1andM.Pusterla3 1)Dipartimento di Fisica dell’Universit `a di Genova and INFNSezione di Genova, ViaDodecaneso 33, 16146 Genova, Ita ly 2)C-A Dept.,Brookhaven National Laboratory, Upton, NY11973 , USA 3)Dipartimento di Fisica dell’Universit `a di Padova and INFNSezione di Padova, Via Marzolo 8, 35131 Padova, Italy Abstract ThegeneralexpressionoftheStern-Gerlachforceisdeduce dforacharged particle,endowedwithamagneticmoment,whichtravelsins ideatimevary- ing magnetic field. Then, the energy integral of the Stern-Ge rlach force is evaluated in the case of a particle crossing a TE rf cavity wit h its magnetic moment oriented in different ways with respect as the cavity axis. We shall demonstrate that appropriate choices of the cavity charact eristics and of the spin orientation confirm the possibility of separating in en ergy the opposite spin states of a fermion beam circulating in a storage ring an d, in addition, make feasible an absolute polarimeter provide that a parame tric converter acting between twocoupled cavities isimplemented. Report no.: INFN/TC-00/03, March, 22, 2000 PACS:29.27.Hj; 03.65 11 Introduction The Stern-Gerlach force acts on particles, carrying a magne tic moment, which cross inhomogeneousmagneticfields. In a reference frame wh ere particles are at rest,theexpressionofthisforce is /vectorfSG=−∇U (1) where U=−/vector µ·/vectorB (2) isthemagneticpotentialenergy,and /vector µ=ge 2m/vectorS (3) is the magnetic moment. Here e=±1.602×10−19Cis the elementary charge with+forp, e+and−for¯p, e−, making /vector µand/vectorSeither parallel or antiparallel, respectively. Therestmass, m,is1.67×10−27kgforp,¯pand9.11×10−31kgfor e±, and therelation betweenthegyromagneticratio gand theanomaly ais a=g−2 2=/braceleftbigg1.793 ( g= 5.586) for p,¯p 1.160×10−3(g= 2.002) for e± (4) In the rest system, the quantum vector /vectorS, named spin, has modulus |/vectorS|=/radicalig s(s+ 1)¯h, and its component parallel to the magnetic field lines can ta ke only thefollowingvalues: Sm= (−s,−s+ 1, ...., s−1, s)¯h, (5) where ¯h= 1.05×10−34Jsthe reduced Planck’s constant. Combining Eqs. (3) and (5)weobtainforagenericspin-1 2fermion µ=|/vector µ|=g|e|¯h 4m(6) or µ=/braceleftbigg1.41×10−26JT−1 9.28×10−24JT−1(7) Takenotethat theBohrmagnetonis µB= 2 [µ/g]electron = 9.27×10−24JT−1(8) 2Aiming to have the expression of the Stern-Gerlach force in t he laboratory frame, we have first to carry out the Lorentz transformation o f the electric and magnetic field from the laboratory frame, where we are at rest , to the center-of- massframe,whereparticlesareatrestandwecancorrectlye valuatesuchaforce. Then this force must be boosted back to the laboratory frame. All of these rather cumbersomeoperationswillbediscussedin thenextSection . 2 LorentzBoostofa Force In order to accomplish the sequence of Lorentz boosts more ea sily, we choose a Cartesian 4-dimensional Minkowski metric [1] (x1, x2, x3, x4) = (x, y, z, ict ), where i=√−1. Therefore, the back-and-forth Lorentz transformations b etween laboratory frame and particle’s rest frame (usually labele d with a prime) are the following:  x′ y′ z′ ict′ =M x y z ict = 1 0 0 0 0 1 0 0 0 0 γ iβγ 0 0 −iβγ γ  x y z ict ⇒  x′=x y′=y z′=γ(z−βct) t′=γ/parenleftig t−β cz/parenrightig (9)/braceleftigg β=|/vectorβ|=|/vector v| c, γ=1√1−β2/bracerightigg and  x y z ict =M−1 x′ y′ z′ ict′ = 1 0 0 0 0 1 0 0 0 0 γ−iβγ 0 0 iβγ γ  x′ y′ z′ ict′ ⇒  x=x′ y=y′ z=γ(z′+βct′) t=γ/parenleftig t′+β cz′/parenrightig (10) Moreover,combiningbotheqs. (9)and (10), weobtainthefol lowingexpressions forthepartialderivatives: ∂ ∂x′=∂ ∂x,∂ ∂y′=∂ ∂y(11) ∂ ∂z′=γ/parenleftigg∂ ∂z+β c∂ ∂t/parenrightigg (12) 3The4-vectorformalismisstillappliedforundergoingtheL orentztransforma- tionofaforce. First ofall,let usdefine as4-velocitythequ antity uµ=dxµ dτ(13) where dτ=ds c=dt γ(14) is the differential of the proper time. We define the 4-moment um as the product oftherest mass mtimesthe4-velocity,i.e. Pµ=m uµ= (/vector p, iγmc ) (15) The 4-force is the derivative of the 4-momentum (15) with res pect to the proper time,thatis Fµ=dPµ dτ=/parenleftigg γd/vector p dt, iγ cd(γmc2) dt/parenrightigg =/parenleftigg γ/vectorf, iγ cdEtot dt/parenrightigg (16) where /vectorfistheordinaryforce. Inthec.m. systemeq. (16)reduces to F′ µ= (/vectorf′,0) (17) sinceγ′= 1andE′ tot=mc2is a constant. Bearing in mind the last step of the whole procedure, i.e. the boost of any force from rest to labo ratory frame, we haveto usetherelation Fµ=M−1F′ µ= γfx γfy γfz F4 = 1 0 0 0 0 1 0 0 0 0 γ−iβγ 0 0 iβγ γ  f′ x f′ y f′ z 0 = f′ x f′ y γf′ z iβγf′ z (18) or /vectorf⊥=1 γ/vectorf′ ⊥ (19) /vectorf/bardbl=/vectorf′ /bardbl(fz=f′ z) (20) 43 Stern-GerlachForce TheStern-Gerlachforce,asdescribedbyeq. (1),mustbeeva luatedintheparticle rest framewhereittakes theform /vectorf′ SG=∇′(/vector µ∗·/vectorB′) =∂ ∂x′(/vector µ∗·/vectorB′)ˆx+∂ ∂y′(/vector µ∗·/vectorB′)ˆy+∂ ∂z′(/vector µ∗·/vectorB′)ˆz(21) havingdefined themagneticmomentas µ∗, ratherthan µ′, foropportunereasons. By applying the transformations (11), (19) and (20), the for ce (21) is boosted to thelaboratory systembecoming /vectorfSG=1 γ∂ ∂x(/vector µ∗·/vectorB′)ˆx+1 γ∂ ∂y(/vector µ∗·/vectorB′)ˆy+∂ ∂z′(/vector µ∗·/vectorB′)ˆz(22) Bearing in mindtheLorentztransformation[2]ofthefields /vectorE,/vectorBand/vectorE′,/vectorB′ /vectorE′=γ(/vectorE+c/vectorβ×/vectorB)−γ2 γ+ 1/vectorβ(/vectorβ·/vectorE) (23) /vectorB′=γ /vectorB−/vectorβ c×/vectorE −γ2 γ+ 1/vectorβ(/vectorβ·/vectorB) (24) theenergy (/vector µ∗·/vectorB′) =µxB′ x+µyB′ y+µzB′ zbecomes (/vector µ∗·/vectorB′) =γµ∗ x/parenleftigg Bx+β cEy/parenrightigg +γµ∗ y/parenleftigg By−β cEx/parenrightigg +µ∗ zBz(25) If we introduce eq. (25) into eq. (22) and take into account eq . (12), we can finallyobtaintheStern-Gerlach forcecomponentsinthelab oratoryframe: fx=µ∗ x/parenleftigg∂Bx ∂x+β c∂Ey ∂x/parenrightigg +µ∗ y/parenleftigg∂By ∂x−β c∂Ex ∂x/parenrightigg +1 γµ∗ z∂Bz ∂x(26) fy=µ∗ x/parenleftigg∂Bx ∂y+β c∂Ey ∂y/parenrightigg +µ∗ y/parenleftigg∂By ∂y−β c∂Ex ∂y/parenrightigg +1 γµ∗ z∂Bz ∂y(27) fz=µ∗ xCzx+µ∗ yCzy+µ∗ zCzz (28) with Czx=γ2/bracketleftigg/parenleftigg∂Bx ∂z+β c∂Bx ∂t/parenrightigg +β c/parenleftigg∂Ey ∂z+β c∂Ey ∂t/parenrightigg/bracketrightigg (29) 5zy xa b dBeam Figure 1: Sketch ofthe rectangular cavity; take notethat co ordinates ofthe beam axisare x=a/2and y=b/2. Czy=γ2/bracketleftigg/parenleftigg∂By ∂z+β c∂By ∂t/parenrightigg −β c/parenleftigg∂Ex ∂z+β c∂Ex ∂t/parenrightigg/bracketrightigg (30) Czz=γ/parenleftigg∂Bz ∂z+β c∂Bz ∂t/parenrightigg (31) 4 TheRectangularCavity Inordertosimplifyourcalculationswithoutloosingthege neralphysicalmeaning, we shall consider a rectangular resonator, as the one shown i n Fig.1, which is characterized [3]bythefollowingfield components: 6Bx=−B0 K2c/parenleftbiggmπ a/parenrightbigg/parenleftbiggpπ d/parenrightbigg sin/parenleftbiggmπx a/parenrightbigg cos/parenleftbiggnπy b/parenrightbigg cos/parenleftbiggpπz d/parenrightbigg cosωt(32) By=−B0 K2 c/parenleftbiggnπ b/parenrightbigg/parenleftbiggpπ d/parenrightbigg cos/parenleftbiggmπx a/parenrightbigg sin/parenleftbiggnπy b/parenrightbigg cos/parenleftbiggpπz d/parenrightbigg cosωt(33) Bz=B0cos/parenleftbiggmπx a/parenrightbigg cos/parenleftbiggnπy b/parenrightbigg sin/parenleftbiggpπz d/parenrightbigg cosωt (34) Ex=−B0/parenleftbiggnπ b/parenrightbiggω K2ccos/parenleftbiggmπx a/parenrightbigg sin/parenleftbiggnπy b/parenrightbigg sin/parenleftbiggpπz d/parenrightbigg sinωt(35) Ey=B0/parenleftbiggnπ b/parenrightbiggω K2 csin/parenleftbiggmπx a/parenrightbigg cos/parenleftbiggnπy b/parenrightbigg sin/parenleftbiggpπz d/parenrightbigg sinωt(36) Ez= 0 (as typical for a TE mode) (37) where B0istheamplitudeofthe Bz-componentand Kc=/radicaligg/parenleftbiggmπ a/parenrightbigg2 +/parenleftbiggnπ b/parenrightbigg2 (38) ω c=K=2π λ=/radicaligg/parenleftbiggmπ a/parenrightbigg2 +/parenleftbiggnπ b/parenrightbigg2 +/parenleftbiggpπ d/parenrightbigg2 (39) Thewave’sphasevelocityis vph=βphcwhere βph=K/radicalig K2−K2c=/radicaltp/radicalvertex/radicalvertex/radicalbt1 +/parenleftiggmd pa/parenrightigg2 +/parenleftiggnd pb/parenrightigg2 (40) We have to recall that the polarization of a beam, revolving i n a ring whose guidefield is /vectorBring, can bedefined as P=N↑−N↓ N↑+N↓(41) where N↑=No. ParticlesSpin Up (e.g. parallelto /vectorBring) N↓=No. ParticlesSpin Down(antiparallelto /vectorBring) andPindicatesthemacroscopicaverageovertheparticledistri butioninthebeam, whichisequivalenttothequantummechanicalexpectationv aluefoundbymeans 7of the quantum statistical matrix. Obviously, an unpolariz ed beam has P= 0or N↑=N↓. A quick comparison among the SG-force components, given by t he set of equations (26)-(31), suggests that fzwill dominate at high energy, since it con- tains terms proportional to γ2, whereas the transverse components have terms independentof γ,not tomentionthe γ−1terms. Themostappropriatechoiceofthespinorientationseemsto betheoneparallel toˆyi.e. to /vectorBring, i.e. the force component is the one given by eq. (28) with the insertion of eq. (30). This means that particles undergoing energy gain (or loss) don’t need any spin rotation while entering and leaving the r f cavity, beyond the advantageofhavingtodealwithaforcecomponentproportio naltoγ2. Choosing thesimplest TE011mode,thequantities(38), (39)and (40)reduceto kc=π b(42) ω=c/radicaligg/parenleftbiggπ b/parenrightbigg2 +/parenleftbiggπ d/parenrightbigg2 (43) βph=/radicaltp/radicalvertex/radicalvertex/radicalbt1 +/parenleftiggd b/parenrightigg2 (44) Setting x=a 2andy=b 2thefield componentsalong thebeam axisbecome Bx=Bz= 0 (45) By=−B0b dcos/parenleftbiggπz d/parenrightbigg cosωt (46) Ex=−ω B0b πsin/parenleftbiggπz d/parenrightbigg sinωt (47) Ey=Ez= 0 (48) thereforetheforcecomponent fzcan bewrittenas fz=µ∗γ2B0b  1 π /parenleftbiggπ d/parenrightbigg2 +/parenleftiggβω c/parenrightigg2 sin/parenleftbiggπz d/parenrightbigg cosωt+2 d/parenleftiggβω c/parenrightigg cos/parenleftbiggπz d/parenrightbigg sinωt   (49) For completeness, we shall also analyze the possibility of u sing a spin orien- tationparallelto ˆz, i.e. to themotiondirection,eventhoughthisoptionrequi resa systemofspinrotators andloosesafactor of γintheforcecomponent. 85 InvolvedEnergy The energy gained, or lost, by a particle with a magnetic mome nt after having crossed a rf cavity can be evaluated by integrating the Stern -Gerlach force (22) overthecavity length,namely: ∆U=/integraldisplayd 0dU=/integraldisplayd 0/vectorf·d/vector r=/integraldisplayd 0fzdz=/integraldisplayd 0µ∗Czydz (50) Bearing in mind eq. (49) and carrying out the trivial substit utionωt=ωz βc, the integral(50)becomes ∆U=µ∗γ2B0b  1 π /parenleftbiggπ d/parenrightbigg2 +/parenleftiggβω c/parenrightigg2 I1+2 d/parenleftiggβω c/parenrightigg I2   with I1=/integraldisplayd 0sin/parenleftbiggπz d/parenrightbigg cos/parenleftiggωz βc/parenrightigg dz=π d/parenleftig π d/parenrightig2−/parenleftig ω βc/parenrightig2/bracketleftigg 1 + cos/parenleftiggωd βc/parenrightigg/bracketrightigg I2=/integraldisplayd 0cos/parenleftbiggπz d/parenrightbigg sin/parenleftiggωz βc/parenrightigg dz=−ω βc/parenleftig π d/parenrightig2−/parenleftig ω βc/parenrightig2/bracketleftigg 1 + cos/parenleftiggωd βc/parenrightigg/bracketrightigg or ∆U=µ∗γ2B0b d/parenleftig π d/parenrightig2+/parenleftigβω c/parenrightig2−2/parenleftig ω c/parenrightig2 /parenleftig π d/parenrightig2−/parenleftig ω βc/parenrightig2/bracketleftigg 1 + cos/parenleftiggωd βc/parenrightigg/bracketrightigg (51) Takingintoaccountthestationarywaveconditions(eqs. 43 and44)pertaining tothe TE011mode,thelengthofthecavitycan beexpressed as d=1 2βphλ (52) whichallowsusto writeeq. (51)as ∆U=γ2β2µ∗B0b d1 +β2 ph(β2−2) β2−β2 ph/parenleftigg 1 + cosβph βπ/parenrightigg (53) 9In theultrarelativisticlimit( γ≫1andβ≃1), ∆U≃µ∗B0b dγ2(1 + cos βphπ) = 2µ∗B0b dγ2(βph=eveninteger )(54) As hinted before, let us evaluate the work-energy integral w hen the particle entersintothecavitywithitsspinparallelto ˆz. Inthisexamplewemustchoosethe mode TE021as thelowestone;then wehavefrom eqs. (34)and (31)respect ively Bz=−B0sin/parenleftbiggπz d/parenrightbigg cosωt (55) fz=µ∗Czz=−µ∗γB0/bracketleftiggπ dcos/parenleftbiggπz d/parenrightbigg cosωt−/parenleftiggβω c/parenrightigg sin/parenleftbiggπz d/parenrightbigg cosωt/bracketrightigg (56) and proceedingas aboveweobtain ∆U=µ∗B0γπ dω βc−βc ω/parenleftig π d/parenrightig2−/parenleftig ω βc/parenrightig2sin/parenleftiggωd βc/parenrightigg (57) and ∆U=µ∗B0 γβphβ β2 ph−β2sin/parenleftiggβph βπ/parenrightigg (58) orultrarelativistically ∆U≃µ∗B0 γβph β2 ph−1sinβphπ, ∆Umax∼ −1.62µ∗B0 γ(whenβph∼1.13) (59) confirmingaresult[4]already achieved. Before making up our mind, we need to compare the energy gain/ loss due to the Stern-Gerlach interaction with the same quantity cause d by the electric field. Tothisaim, weemphasizethat dUE=/vectorfE·d/vector r=eExdx (60) ascanbeeasilyunderstoodlookingateqs. (47)and(48). Sin cethecarrierparticle travels from 0 to dalong the z-axis, the only integral which makes sense is the following: ∆UE=/integraldisplayd 0eExdx=/integraldisplayd 0eExdx dzdz=/integraldisplayd 0eExx′dz (61) 10or ∆UE=−x′eωB 0b π/integraldisplayd 0sin/parenleftbiggπz d/parenrightbigg sin/parenleftiggωz βc/parenrightigg dz=−x′eωB 0b dsin/parenleftig ωd βc/parenrightig /parenleftig π d/parenrightig2−/parenleftig ω βc/parenrightig2 or ∆UE=/bracketleftigg eωB 0bd π2β2 β2 ph−β2sinβph βπ/bracketrightigg x′=κx′(62) havingproceeded as before. We recall that the Stern-Gerlach interaction in the realm of particle acceler- ators has been proposed either for separating in energy part icles with opposite spin states, the well known [5] spin-splitter concept, or fo r settling an absolute polarimeter[6]. As far as the spin-splitter is concerned, we quickly recall t hat spin up parti- cles receive (or loose) that amount of energy given by eq. (54 ) at each rf cavity crossing, and this will take place all overthe timerequired . Simultaneously, spin down particles behave exactly in the opposite way, i.e. they loose (or gain) the sameamountofenergy turnafterturn. Theactualmostimport antissueisthatthe energyexchangessumupcoherently. Morequantitatively,w emayindicateasthe final energy separationafter Nrevolutions: ∆↑↓=/summationdisplay {∆↑−(−∆↓)}= 4b dN µ∗B0γ2≃4N µ∗B0γ2(63) Instead,theaddingup oftheenergy contribution(62)dueto theelectricfield is (∆UE)tot=/summationdisplay ∆UE=κ/summationdisplay x′= 0 (64) sincex′changes continuously its sign with a periodicity related to the period of thebetatron oscillations. The result (63), together with the demonstration (64), woul d seem to provide verygoodnews forthespin-splittermethod! As far as the polarimeter is concerned, we have to bear in mind that we are interested in the instantaneous interaction between magne tic moment and the rf fields: thereforethezero-averagingduetotheincoherence ofthebetatronoscilla- tions would not help us. Notwithstanding,if we set βphequal to an integerin eq. (62), wehaveforU.R. particles: ∆UE=x′eωB 0bd π2(β2 ph−1)sin/parenleftigg βphπ+βphπ 2γ2/parenrightigg ≃ ±x′bd 2πβph β2 ph−1eωB 0 γ2(65) 11Thenthis 1/γ2dependenceofthespurioussignal,comparedtothe γ2dependence ofthesignal(54)tobemeasured,soundsinterestingforthe feasibilityofthiskind ofpolarimeter;however,onemustrealizethatif βphisnotexactlyaninteger,then eq. (65)wouldbecome ∆UE∼ ±x′bd 2πeωB 0 β2 ph−1/parenleftigg ǫ+βph γ2/parenrightigg (66) where ǫis theerror in βph. 6 AFew NumericalExamples Thespin-splitterprinciplerequiresarepetitivecrossin gofNcavcavitiesdistributed alongthering,eachofthemresonatingintheTEmode. Aftere achrevolution,the particleexperiences avariation,or kick, ofitsenergy orofitsmomentumspread ζ=δp p=1 β2δE E≃Ncav∆U E≃2√ 3 3NcavB0 B∞γ (67) havingmadeuseofeq. (54),furthersimplifiedbyreasonably setting βph= 2,and with B∞=mc2 µ∗=1.503×10−10J 1.41×10−26JT−1≃1016T (68) for (anti)protons. From eq. (67) we may find as the number of tu rns needed for attaininga momentumseparation equalto 2/parenleftig ∆p p/parenrightig NSS=/parenleftig ∆p p/parenrightig ζ=√ 3 2NcavγB∞ B0/parenleftigg∆p p/parenrightigg (69) Multiplying NSSby therevolutionperiod τrevweobtain ∆t=NSSτrev (70) as the actual time spent in this operation. For the sake of hav ing some data, we considerRHIC [7] and HERA [8] whoseessentialparameters ar e shownin Table I together with what can be found by making use of eqs. (69) and (70) where B0≃0.1TandNcav= 200arechosen as realisticvalues. 12TableI:RHICandHERA parameters RHIC HERA E(GeV) 250 820 γ 266.5 874.2 τrev(µs) 12.8 21.1 ∆p p4.1×10−35×10−5 NSS 6.67×1092.48×107 ∆t8.52×104s≃23.7h523 s In the example of the polarimeter we have to pick up a signal ge nerated at each cavitycrossing. Therefore, makinguseofeq. (54)weha veforabunchtrain madeup of Nparticles thetotalenergy transfer ∆U≈2NPµ∗B0b dγ2(71) where Pisthebeampolarizationslightlymodifiedwithrespectthed efinition(41) P=N→−N← N→+N←(72) Theaveragepowertransferred willbe W=∆U τrev(73) If we operate our cavity as a parametric converter [9][10], w ith an initially emptylevel,wehaveforthepowertransferred to thisemptyl evel W2=ωrf ωrevW=νrf νrevW (74) where νrfis the working frequency of the resonant cavity (typically i n the GHz range), and νrevis therevolutionfrequency. Puttingall togetherwehave W2≃2Pνrf νrevµ∗B0b dγ2(75) Afeasibilitytestofthepolarimeterprinciplehasbeenpro posed[6]andstudied [11]tobecarriedoutinthe500MeVelectronring[12]ofMIT- Bates,whosemain characteristics are 13TableII:MIT-Bates parameters τrev 634 nsec νrev 1.576MHz Nelectrons 3.6×108·225 = 8 .1×1010 γ ≃103 b/d√ 3/3 B0 ≃0.1T νrf/νrev ≈103 µ∗9.27×10−24JT−1 and, since polarized electrons can be injected into this rin g but precessing on a horizontal plane, the TE101mode is more appropriate than the TE011as we shall have to use Bxrather than By: a choice that does not make any substantial difference! From theabovedataweobtain W2≃137Pwatts (76) Paradoxically, even for an almost unpolarized beam with N→−N←= 1 and, as a consequence of eq. (72), with P≃1.23×10−11, we should obtain W2≈1.7 nW, which can beeasilymeasured. As a last check, let us compare the energy exchanges ( /vector µ⇔/vectorB) and ( e⇔/vectorE). Taking into account eqs. (52), (54) and (65), and setting x′≃1 mrad,βph= 2 andλ= 10 cm, wehavefortheBates-MIT ring: r=∆UE ∆U=x′ 8β3 ph β2 ph−1λec µ∗1 γ4= 1.72×10−4(77) i.e. the spurious signal, depending upon the electric inter action between eand /vectorE, is absolutely negligible with respect the measurable sign al generated by the magneticinteraction. 7 Conclusions There is not too much to add to what has been found in the previo us Sections, asidefromperformingmoreaccuratecalculationsandnumer icalsimulations. The Stern-Gerlachinteractionseemsverypromisingeitherfor attainingtheselfpolar- izationofa p(¯p)beamorforrealizing an absolutepolarimeter. 14In the first example the problem raised [13] by the rf filamenta tion still holds on, although some tricks can be conceived: the extreme one co uld be the imple- mentationofatriangularwaveform intheTMcavity whichbun chesthebeam. The second example requires nothing but to implement that ex perimental test at theBates-MIT electron ring. References [1] Synge, Relativity: The Special Theory , North Holland Publ. Co., Amster- dam,1956. [2] J. D. Jackson, Classical Electrodynamics , John Wiley and & Sons, New York,1975. [3] S. Ramo, J. R. Whinnery and T. Van Duzer, Fieldsand Waves in Communi- cationElectronics ,JohnWileyand& Sons,New York, 1965. [4] W. W. MacKay, “Notes on a Generalization of the Stern-Ger lach Force”, RHIC/AP/153, April 6 1998, and W. W. MacKay, “Converging tow ards a Solutionof γvs1/γ”RHIC/AP/175, June1999. [5] M.Conte, A.Penzo and M.Pusterla, Il NuovoCimento, A108(1995)127. [6] P. Cameron et al., “An RF Resonant Polarimeter Phase 1 Pro of-of-Principle Experiment”,RHIC/AP/126, January 6 1998. [7] M.A.Harrison, “TheRHIC Project”, Proceedings ofEPAC9 6,p. 13,Sitges (Barcelona), 1996. [8] E. Gianfelice-Wendt, “HERA Upgrade Plans”, Proc. of EPAC98 , p. 118, Stolkholm(1998). [9] J.M.Manley,H.E.Rowe,“SomeGeneralPropertiesofNonl inearElements -Part I. General Energy Relations”,Proceedings oftheIRE, 44(1956)904. [10] W. H. Louisell, Coupled Modes and Parametric Electronics , John Wiley & Sons,New York,1965. [11] M.Ferro, ThesisofDegree, GenoaUniversity,April22 1 999. 15[12] K. D. Jacobs et al., “Commissioning the MIT-Bates South Hall Ring”, Pro- ceedingsPAC1, 1995. [13] M.Conte, W.W. MacKay and R. Parodi, “An OverviewoftheL ongitudinal Stern-Gerlach Effect”, BNL-52541,UC-414, November17199 7. 16

arXiv:physics/0003070v1 [physics.gen-ph] 24 Mar 2000HIERARCHIC THEORY OF CONDENSED MATTER AND ITS INTERACTION WITH LIGHT: New Theories of Light Refraction, Brillouin Scattering and M¨ ossbauer effect Alex Kaivarainen JBL, University of Turku, Finland, FIN-20520 URL: http://www.karelia.ru/˜alexk H2o@karelia.ru Materials, presented in this article are the part of new quan tum theory of condensed matter, described in: [1]. Book by A. Kaivarainen: Hierarchic Concept of Matter an d Field. Water, biosystems and elementary particles. New Yor k, 1995; ISBN 0-9642557-0-7 and two articles: [2]. ”New Hierarchic Theory of Matter General for Liquids an d Solids: dynamics, thermodynamics and mesoscopic structur e of water and ice” (see URL: http://www.karelia.ru/˜alexk [New arti cles]) and: [3]. Hierarchic Concept of Matter, General for Liquids and So lids: Water and ice (see Proceedings of the Second Annual Advanced Wa- ter Sciences Symposium, October 4-6, 1966, Dallas, Texas. Computerized verification of new described here theories ar e pre- sented on examples of WATER and ICE, using special computer program: ”Comprehensive Analyzer of Matter Properties (CA MP)” (copyright, 1997, A. Kaivarainen). CONTENTS Summary of ”New Hierarchic Theory of Condensed Matter.” 1: New approach to theory of light refraction 1.1. Refraction in gas 1.2. Light refraction in liquids and solids 2: Mesoscopic theory of Brillouin light scattering 2.1. Traditional approach 2.2. Fine structure of scattering 2.3. Mesoscopic approach 2.4. Quantitative verification of hierarchic theory of Bril louin scat- tering 3: Mesoscopic theory of M¨ ossbauer effect 3.1. General background 3.2. Probability of elastic effects 3.3. Doppler broadening in spectra nuclear gamma-resonanc e (NGR) 13.4. Acceleration and forces, related to thermal dynamics o f molecules and ions. Vibro-gravitational interaction =================================================== =========== Summary of: New Hierarchic Theory of Condensed Matter by: A. Kaivarainen A basically new hierarchic quantitative theory, general fo r solids and liquids, has been developed. It is assumed, that unharmonic oscillations of particles in any con- densed matter lead to emergence of three-dimensional (3D) s uperpo- sition of standing de Broglie waves of molecules, electroma gnetic and acoustic waves. Consequently, any condensed matter could b e con- sidered as a gas of 3D standing waves of corresponding nature . Our approach unifies and develops strongly the Einstein’s and De bye’s models. Collective excitations, like 3D standing de Broglie waves o f molecules, representing at certain conditions the molecular mesoscop ic Bose con- densate, were analyzed, as a background of hierarchic model of con- densed matter. The most probable de Broglie wave (wave B) length is deter- mined by the ratio of Plank constant to the most probable impu lse of molecules, or by ratio of its most probable phase velocity to fre- quency. The waves B are related to molecular translations (t r) and librations (lb). As the quantum dynamics of condensed matter does not follow i n general case the classical Maxwell-Boltzmann distribution, the re al most probable de Broglie wave length can exceed the classical thermal de Brog lie wave length and the distance between centers of molecules many times. This makes possible the atomic and molecular Bose condensat ion in solids and liquids at temperatures, below boiling point. It is one o f the most important results of new theory, which we have confirmed by computer sim ulations on examples of water and ice. Four strongly interrelated new types of quasiparticles (collective excita- tions) were introduced in our hierarchic model: 1.Effectons (tr and lb) , existing in ”acoustic” (a) and ”optic” (b) states represent the coherent clusters in general case ; 2.Convertons , corresponding to interconversions between trandlbtypes of the effectons (flickering clusters); 23.Transitons are the intermediate [ a⇋b] transition states of the trandlb effectons; 4.Deformons are the 3D superposition of IR electromagnetic or acoustic waves, activated by transitons andconvertons. Primary effectons (tr and lb) are formed by 3D superposition of the most probable standing de Broglie waves of the oscillating ions, atoms or molecules. The volume of effectons (tr and lb) may contain fro m less than one, to tens and even thousands of molecules. The first condition m eans validity ofclassical approximation in description of the subsystems of the effect ons. The second one points to quantum properties of coherent clusters due to molecular Bose condensation . The liquids are semiclassical systems because their primar y (tr) effectons contain less than one molecule and primary (lb) effectons - mo re than one molecule. The solids are quantum systems totally because both kind of t heir primary effectons (tr and lb) are molecular Bose condensates .These conse- quences of our theory are confirmed by computer calculations . The 1st order [ gas→liquid ] transition is accompanied by strong decreasing of rotational (librational) degrees of freedom due to emerg ence of primary (lb) effectons and [ liquid →solid] transition - by decreasing of translational degrees of freedom due to Bose-condensation of primary (tr) effecton s. In the general case the effecton can be approximated by par- allelepiped with edges corresponding to de Broglie waves le ngth in three selected directions (1, 2, 3), related to the symmetry of the molecular dynamics. In the case of isotropic molecular moti on the effectons’ shape may be approximated by cube. The edge-length of primary effectons (tr and lb) can be consid ered as the ”parameter of order”. The in-phase oscillations of molecules in the effectons corr espond to the effecton’s (a) - acoustic state and the counterphase oscillations correspond to their (b) - optic state. States (a) and (b) of the effectons differ in potential energy only, however, their kinetic energies, impulses and spatial dimensions - are the same. The b-state of the effectons has a common feature with Fr¨ olich’s polar mode. The(a→b)or(b→a)transition states of the primary effectons (tr and lb), defined as primary transitons, are accompanied b y a change in molecule polarizability and dipole moment withou t density fluctuations. At this case they lead to absorption or radiati on of IR photons, respectively. Superposition (interception) of three internal standing I R pho- tons of different directions (1,2,3) - forms primary electro magnetic deformons (tr and lb). On the other hand, the [lb ⇋tr]convertons andsecondary transitons are accompanied by the density fluctuations, leading to absorption or radiation of phonons . 3Superposition resulting from interception of standing phonons in three direc- tions (1,2,3), forms secondary acoustic deformons (tr and lb). Correlated collective excitations of primary and secondary effectons and deformons (tr and lb) ,localized in the volume of primary trandlb electromag- netic deformons ,lead to origination of macroeffectons, macrotransitons andmacrodeformons (tr and lb respectively) . Correlated simultaneous excitations of tr and lb macroeffec tonsin the vol- ume of superimposed trandlbelectromagnetic deformons lead to origination ofsupereffectons. In turn, the coherent excitation of both: tr andlb macrodeformons and macroconvertons in the same volume means creation of superdeformons. Su- perdeformons are the biggest (cavitational) fluctuations, leading to microbub- bles in liquids and to local defects in solids. Total number of quasiparticles of condensed matter equal to 4!=24, reflects all of possible combinations of the four basic ones [ 1-4], intro- duced above. This set of collective excitations in the form o f ”gas” of 3D standing waves of three types: de Broglie, acoustic and el ectro- magnetic - is shown to be able to explain virtually all the pro perties of all condensed matter. The important positive feature of our hierarchic model of ma tter is that it does not need the semi-empiric intermolecular potentials f or calculations, which are unavoidable in existing theories of many body systems. T he potential energy of intermolecular interaction is involved indirectly in di mensions and stability of quasiparticles, introduced in our model. The main formulae of theory are the same for liquids and solid s and include following experimental parameters, which take into ac- count their different properties: [1]- Positions of (tr) and (lb) bands in oscillatory spectra; [2]- Sound velocity; [3]- Density; [4]- Refraction index (extrapolated to the infinitive wave leng th of photon ). The knowledge of these four basic parameters at the same temp erature and pressure makes it possible using our computer program, to ev aluate more than 300 important characteristics of any condensed matter. Amo ng them are such as: total internal energy, kinetic and potential energies, heat-capacity and ther- mal conductivity, surface tension, vapor pressure, viscos ity, coefficient of self- diffusion, osmotic pressure, solvent activity, etc. Most of calculated parameters are hidden, i.e. inaccessible to direct experimental measu rement. The new interpretation and evaluation of Brillouin light sc attering and M¨ ossbauer effect parameters may also be done on the basis of h ierarchic the- ory. Mesoscopic scenarios of turbulence, superconductivi ty and superfluity are elaborated. Some original aspects of water in organization and large-sc ale dynamics of biosystems - such as proteins, DNA, microtubules, membrane s and regulative 4role of water in cytoplasm, cancer development, quantum neu rodynamics, etc. have been analyzed in the framework of Hierarchic theory. Computerized verification of our Hierarchic concept of matt er on examples of water and ice is performed, using special comput er pro- gram: Comprehensive Analyzer of Matter Properties (CAMP, c opy- right, 1997, Kaivarainen). The new opto-acoustical device (CAMP), based on this program, with possibilities much wider, than t hat of IR, Raman and Brillouin spectrometers, has been proposed (see URL: http://www.karelia.ru/˜alexk). This is the first theory able to predict all known experimenta l temperature anomalies for water and ice. The conformity bet ween theory and experiment is very good even without any adjustab le pa- rameters. The hierarchic concept creates a bridge between micro- and m acro- phenomena, dynamics and thermodynamics, liquids and solid s in terms of quantum physics. 51: New approach to theory of light refraction 1.1. Refraction in gas If the action of photons onto electrons of molecules is consi dered as a force, activating a harmonic oscillator with decay, it l eads to the known classical equations for a complex refraction inde x (Vuks, 1984). The Lorentz-Lorenz formula obtained in such a way is conveni ent for practical needs. However, it does not describe the depen dence of refraction index on the incident light frequency and did n ot take into account the intermolecular interactions. In the new th eory pro- posed below we have tried to clear up the relationship betwee n these parameters. Our basic idea is that the dielectric penetrability of matte rǫ, (equal in the optical interval of frequencies to the refract ion index squared n2), is determined by the ratio of partial volume energies of photon in vacuum to similar volume energy of photon in matter : ǫ=n2=[E0 p] [Emp]=mpc2 mpc2m=c2 c2m(1.1) where mp=hνp/c2is the effective photon mass, cis the light velocity in vacuum, cmis the effective light velocity in matter. We introduce the notion of partial volume energy of a photon in vacuum [E0 p]and in matter [Em p]as a product of photon energy (Ep=hνp)and the volume (Vp)occupied by 3D standing wave of photon in vacuum and in matter, correspondingly: [E0 p] =EpV0 p [Em p] =EpVm p (1.2) The 3D standing photon volume as an interception volume of 3 d ifferent standing photons normal to each other was termed in our mesos copic model as a primary electromagnetic deformon (see Introduction of [1 ,2,3]). In vacuum, where the effect of an excluded volume due to the spatial incompatibility of electron shells of molecules and photon is absent, the volume of 3 Dphoton standing wave (primary deformon) is: V0 p=1 np=3λ2 p 8π(1.3) We will consider the interaction of light with matter in this meso- scopic volume, containing a thousands of molecules of conde nsed mat- ter. It is the reason why we titled this theory of light refrac tion as mesoscopic one. 6Putting (1.3) into (1.2), we obtain the formula for the parti al volume energy of a photon in vacuum: [E0 p] =EpV0 p=hνp9λ2 p 8π=9 4/planckover2pi1cλ2 p (1.4) Then we proceed from the assumption that waves B of photons ca n not exist with waves B of electrons, forming the shells of atoms a nd molecules in the same space elements. Hence, the effect of excluded volume appears during the propagation of an external electromagnetic wave throug h the matter. It leads to the fact that in matter the volume occupied by a photo n, is equal to Vm p=V0 p−Vex p=V0 p−np M·VM e (1.5) where Vex p=np MVM eis the excluded volume which is equal to the product of the number of molecules in the volume of one photon standing w ave (np M) and the volume occupied by the electron shell of one molecule ( VM e). np Mis determined by the product of the volume of the photons 3D st anding wave in the vacuum (1.3) and the concentration of molecules ( nM=N0/V0): np M=9λ3 p 8π/parenleftbiggN0 V0/parenrightbigg (1.6) In the absence of the polarization by the external field and in termolecular in- teraction, the volume occupied by electrons of the molecule : VM e=4 3πL3 e (1.7) where Leis the radius of the most probable wave B(Le=λe/2π) of the outer electron of a molecule. As it has been shown in (7.5) that the m ean molecule polarizability is: α=L3 e (1.8) Then taking (1.7) and (1.6) into account, the excluded volum e of primary elec- tromagnetic deformon in the matter is: Vex p=9λ3 p 8πnM4 3πα=3 2λ3 pnMα (1.9) Therefore, the partial volume energy of a photon in the vacuu m is determined by eq.(1.4), while that in matter, according to (1.5): 7[Em p] =Ep·Vm p=Ep·[V0 p−Vex p] (1.10) Putting (1.4) and (1.10) into (1.1) we obtain: ǫ=n2=EpV0 p Ep(V0p−Vexp)(1.11) or 1 n2= 1−Vex p V0(1.12) Then, putting eq.(1.9) and (1.3) into (1.12) we derive new equation for refraction index, leading from our mesoscopic theory: 1 n2= 1−4 3πnMα (1.13) or in another form: n2−1 n2=4 3πnMα=4 3πN0 V0α (1.14) where: nM=N0/V0is a concentration of molecules; In this equation α=L3 eis the average static polarizability of molecules for the case when the external electromagnetic fie lds as well as intermolecular interactions inducing the addition al polariza- tion are absent. This situation is realized at Ep=hνp→0andλp→ ∞ in the gas phase. As will be shown below the value of resulting α∗in condensed matter is bigger. 1.2. Light refraction in liquids and solids According to the Lorentz classical theory, the electric com ponent of the outer electromagnetic field is amplified by the additional inner fie ld (Ead), related to the interaction of induced dipole moments in composition of condensed matter with each other: Ead=n2−1 3E (1.15) 8The mean Lorentz acting field ¯Fcan be expressed as: F=E+Ead=n2+ 2 3E(atn→1,F→E) (1.16) ¯F- has a dimensions of electric field tension and tends to E in th e gas phase when n→1. In accordance with our mesoscopic model, except the Lorentz act- ing field, the total internal acting field, includes also two a nother con- tributions, increasing the molecules polarizability ( α) in condensed matter: 1. Potential intermolecular field, including all the types o f Van- der-Waals interactions in composition of coherent collect ive excita- tions, even without external electromagnetic field. Like to tal po- tential energy of matter, this contribution must be depende nt on temperature and pressure; 2. Primary internal field, related with primary electromagn etic deformons (tr and lib). This component of the total acting fie ld also exist without external fields. Its frequencies corresponds to IR range and its action is much weaker than the action of the external v isible light. Let us try to estimate the energy of the total acting field and i ts effective frequency ( νf) and wavelength ( λf), that we introduce as: Af=hνf=hc λf=AL+AV+AD (1.17) where: AL, AVandADare contributions, related with Lorentz field, po- tential field and primary deformons field correspondingly. When the interaction energy of the molecule with a photon ( Ep=hνp) is less than the energy of the resonance absorption, then it l eads to elastic polarization of the electron shell and origination of secon dary photons, i.e. light scattering. We assume in our consideration that the increme nt of polarization of a molecule ( α) under the action of the external photon ( hνp) and the total active field ( Af=hνf) can be expressed through the increase of the most probable radius of the electron’s shell ( Le=α1/3), using our (eq. 7.6 from [1]): ∆Le=ωpme 2/planckover2pi1α (1.18) where the resulting increment: ∆L∗= ∆Le+ ∆Lf=(hνp+Af)me 2/planckover2pi12α (1.18a) 9where: α=L3 eis the average polarizability of molecule in gas phase at νf→ 0. For water molecule in the gas phase: Le=α1/3= 1.13·10−10m is a known constant, determined experimentally [4]. The total increment of polarizability radius (∆L∗)and resulting polarizability of molecules ( α∗)in composition of condensed matter affected by the acting field α∗= (L∗)3 can be find from the experimental refraction index (n) using o ur formula (1.14): L∗=/bracketleftbigg3 4πV0 N0·n2−1 n2/bracketrightbigg1/3 (1.19) ∆L∗=L∗−Le (1.20) where: L∗= (α∗)1/3 From (1.18) we get a formula for the increment of radius of pol arizability (∆ Lf), induced by the total internal acting field: ∆Lf= ∆L∗−∆Le=Af·me 2/planckover2pi12α (1.21) Like total internal acting field energy (1.17), this total ac ting increment can be presented as a sum of contributions, related to Lorent z field (∆ LF), potential field (∆ LV) and primary deformons field (∆ LD): ∆Lf= ∆LL+ ∆LV+ ∆LD (1.22) Increment ∆ Le, induced by external photon only, can be calculated from the known frequency ( νp) of the incident light (see 1.18a): ∆Le=hνp·me 2/planckover2pi12α (1.23) It means that ∆ Lfcan be found from (1.21) and (1.17), using (1.23). Then from (1.21) we can calculate the energy ( Af), effective frequency ( νf) and wave length ( λf) of the total acting field like: 10Af=hνf=hc/λ f= 2∆Lf·/planckover2pi12 meα(1.24) The computer calculations of α∗;L∗=Le+ ∆L∗= (α∗)1/3andAfin the temperature range (0 −950) are presented on Fig.1.1. One must keep in mind that in general case αandLare tensors. It means that all the increments, calculated on the base of eq.(1.18a ) must be considered as the effective ones. Nevertheless, it is obvious that our ap proach to analysis of the acting field parameters can give useful additional inf ormation about the properties of transparent condensed matter. Fig. 1.1. (a)- Temperature dependencies of the most probable outer electron shell radius of H2O(L∗) and the effective polarizabil- ityα∗= (L∗)3in the total acting field; (b)- Temperature dependence of the total acting field ( Af) en- ergy in water at the wavelength of the incident light λp= 5.461· 10−5cm−1. The experimental data for refraction index n(t) were used in calculations. The initial electron shell radius is: Le= α1/3 H2O= 1.13·10−8cm[4]. In graphical calculations in Fig.1.1a, the used experimental temperature dependence of the water r efrac- tion index were obtained by Frontas’ev and Schreiber [5]. The temperature dependencies of these parameters were com- puted using the known experimental data on refraction index n(t) for water and presented in Fig.1.1a. The radius L∗in the range 0−950Cincreases less than by 1% at constant incident light wavelen gth (λ= 546 .1nm). The change of ∆Lfwith temperature is determined by its potential field component change ∆LV. 11The relative change of this component: ∆∆LV/∆Lf(t= 00C)is about 9%. Corresponding to this change the increasing of the acting field energy Af(eq.1.23) increases approximately by 8kJ/M (Fig 1.1 b) due to its potential field contribution. It is important that the total potential energy of water in th e same temperature range, according to our calculations, inc rease by the same magnitude (Fig.5b in [1] or Fig.3b in [ 3]). This fact points to the strong correlation between potential intermolecula r interaction in matter and the value of the acting field energy. It was calculated that, at constant temperature (200)the energy of the acting field (Af),(eq.1.23)in water practically does not depend on the wavelength of incident light (λp). At more than three time alterations of λp: from 12.56·10−5cmto3.03·10−5cmand the water refraction index (n)from 1.320999 to 1.358100 [6] the value of Af changes less than by 1%. At the same conditions the electron shell radius L∗and the acting polarizability α∗thereby increase from (1.45 to 1.5) ·10−10mand from (3.05 to 3.274)·10−30m3respectively (Fig.1.2). These changes are due to the incident photons action only. For water molecules in t he gas phase and λp→ ∞the initial polarizability (α=L3 e)is equal to 1.44· 10−24cm3[4], i.e. significantly less than in condensed matter under the action of external and internal fields. Obviously, the temperature change of energy Af(Fig.1.1b) is de- termined by the internal pressure increasing (section 11.2 of [1]), re- lated to intermolecular interaction change, depending on m ean dis- tances between molecules and, hence, on the concentration (N0/V0)of molecules in condensed matter. 12Fig. 1.2. Dependencies of the acting polarizability α∗= (L∗)3 and electron shell radius of water in the acting field ( L∗) on incident light wavelength ( λp), calculated from eq. (1.14) and experimental datan(λp) [6]. The initial polarizability of H2Oin the gas phase at λp→ ∞ is equal to α=L3 e= 1.44·10−24cm3. The corresponding initial radius of the H2Oelectron shell is Le= 1.13·10−8cm. On the basis of our data, changes of Af,calculated from (1.24) are caused mainly by the heat expansion of the matter. The pho ton induced increment of the polarizability (α→α∗)practically do not change Af. The ability to obtain new valuable information about change s of molecule polarizability under the action of incident light and about temperature dependent molecular interaction in condensed medium markedly reinforce such a widely used method as refractomet ry. The above defined relationship between the molecule polariz abil- ity and the wave length of the incident light allows to make a n ew endeavor to solve the light scattering problems. 2. Mesoscopic theory of Brillouin light scattering in condensed matter 2.1. Traditional approach According to traditional concept, light scattering in liqu ids and crystals as well as in gases takes place due to random heat fluctuations. I n condensed media the fluctuations of density, temperature and molecule orien tation are possible. Density ( ρ) fluctuations leading to dielectric penetrability ( ǫ) fluctuations are of major importance. This contribution is estimated by m eans of Einstein formula for scattering coefficient of liquids [7]: R=Ir2 I0V=π 2λ4kTβ T/parenleftbigg ρ∂ǫ ∂ρ/parenrightbigg T(2.1) where βTis isothermal compressibility. Many authors made attempts to find a correct expression for th e variable (ρ∂ǫ ∂ρ). The formula derived by Vuks [8, 9] is most consistent with exp erimental data: ρ∂ǫ ∂ρ= (n2−1)3n2 2n2−1(2.2) 132.2. Fine structure of scattering The fine structure - spectrum of the scattering in liquids is r epresented by two Brillouin components with frequencies shifted relativ ely from the incident light frequency: ν±=ν0±∆νand one unshifted band like in gases ( ν0). The shift of the Brillouin components is caused by the Dopple r effect result- ing from a fraction of photons scattering on phonons moving a t sound speed in two opposite directions [8]. This shift can be explained in different way as well. If in the a ntinodes of the standing wave the density oscillation occurs at frequen cy (Ω): ρ=ρ0cosΩt, (2.3) then the scattered wave amplitude will change at the same fre quency. Such a wave can be represented as a superposition of two monochroma tic waves having the frequencies:( ω+ Ω) and ( ω−Ω), where Ω = 2 πf (2.4) is the elastic wave frequency at which scattering occurs whe n the Wolf-Bragg condition is satisfied: 2Λ sin ϕ= 2Λ sinθ 2=λ′(2.5) or Λ =λ′/(2 sinθ 2) =c nν(2 sinθ 2) =vph/f (2.6) where Λ is the elastic wave length corresponding to the frequ encyf;λ′= λ/n=c/nν(λ′andλare the incident light wavelength in matter and vacuum, respectively); ϕis the angle of sliding; θis the angle of scattering; n is the refraction index of matter; cis the light speed. The value of Brillouin splitting is represented as: ±∆νM−B=f=Vph Λ= 2νVph cnsinθ 2(2.7) where: νn/c= 1/λ;nis the refraction index of matter; νis incident light frequency; vph=vS (2.8) 14is the phase velocity of a scattering wave equal to hypersoni c velocity. The formula (2.7) is identical to that obtained from the anal ysis of the Doppler effect: ∆ν ν=±2VS cnsinθ 2(2.9) According to the classical theory, the central line, which i s analogous to that observed in gases, is caused by entropy fluctuations in l iquids, without any changes of pressure [8]. On the basis of Frenkel theory of liquid state, the central line can be explained by fluctuations of ”hole” numbe r - cavitational fluctuations [10]. The thermodynamic approach of Landau and Plachek leads to th e formula, which relates the intensities of the central (I) and two late ral (IM−B) lines of the scattering spectrum with compressibility and heat capa cities: I 2IM−B=Ip Iad=βT−βS βS=Cp−Cv Cv(2.10) where: βTandβSare isothermal and adiabatic compressibilities; CpandCv are isobaric and isohoric heat capacities. In crystals, quartz for example, the central line in the fine s tructure of light scattering is usually absent or very small. However, instea d of one pair of shifted components, observed in liquids, there appear three Brillouin components in crystals. One of them used to be explained by scattering on th e longitudinal phonons, and two - by scattering on the transversal phonons. 2.3. New mesoscopic approach to problem In our hierarchic theory the thermal ”random” fluctuations a re ”organized” by different types of superimposed quantum exci tations. According to our Hierarchic model, including microscopic, meso- scopic and macroscopic scales of matter (see Introduction o f [1,2,3]), the most probable (primary) and mean (secondary) effectons, trans- lational and librational are capable of quantum transition s between two discreet states: (a⇔b)tr,lband(¯a⇔¯b)tr,lbrespectively. These transitions lead to origination/annihilation of photons a nd phonons, forming primary and secondary deformons. The mean heat energy of molecules is determined by the value of 3kT, which as our calculations show, has the intermediate value between the discreet energies of a and b quantum states of pri mary effectons (Fig.19 of [1]), making, consequently, the non equ ilibrium conditions in condensed matter. Such kind of instability is a result of ”competition” between classical and quantum distribution s of energy . 15The maximum deviations from thermal equilibrium and that of the dielec- tric properties of matter occur when the same states of prima ry and secondary quasiparticles, e.g. a,¯ aandb,¯boccur simultaneously. Such a situation corre- sponds to the A and B states of macroeffectons. The ( A⇔B)tr,lbtransitions represent thermal fluctuations. The big density fluctuation s are related to ”flick- ering clusters” (macroconvertions between librational an d translational primary effectons) and the maximum fluctuations correspond to Superd eformons. Only in the case of spatially independent fluctuations the in ter- ference of secondary scattered photons does not lead to thei r total compensation. The probability of the event that two spatially uncorrelate d events coincide in time is equal to the product of their independent probabil ities [10]. Thus, the probabilities of the coherent ( a,¯ a) and ( b,¯b) states of primary and secondary effectons, corresponding to A and B states of the ma croeffectons (tr and lib), independent on each other, are equal to: /parenleftbigPA M/parenrightbigind tr,lb=/parenleftbigPa ef¯Pa ef/parenrightbigS tr,lb·/parenleftbigg1 Z2/parenrightbigg =/parenleftbiggPA M Z2/parenrightbigg tr,lb(2.11) /parenleftbig PB M/parenrightbigind tr,lb=/parenleftbigPb ef¯Pb ef/parenrightbigS tr,lb·/parenleftbigg1 Z2/parenrightbigg =/parenleftbiggPB M Z2/parenrightbigg tr,lb(2.12) where 1 Z/parenleftbig Pa ef/parenrightbig tr,lband1 Z/parenleftbig¯Pa ef/parenrightbig tr,lb(2.13) are the independent probabilities of aand¯ astates determined according to for- mulae (4.10 and 4.18 of [2,3]), while probabilities/parenleftBig Pb ef/Z/parenrightBig tr,lband/parenleftBig ¯Pb ef/Z/parenrightBig tr,lb are determined according to formulae (4.11 and 4.19 of [2,3] ); Zis the sum of probabilities of all types of quasiparticles st ates - eq.(4.2 of [2, 3]). The probabilities of molecules being involved in the spatia lly independent translational and librational macrodeformons are express ed as the products (2.11) and (2.12): /parenleftbigPM D/parenrightbigind tr,lb=/bracketleftBig/parenleftbigPA M/parenrightbigind·/parenleftbigPB M/parenrightbigind/bracketrightBig tr,lb=PM D Z4(2.14) Formulae (2.11) and (2.12) may be considered as the probabil ities of space- independent but coherent macroeffectons in A and B states, re spectively. For probabilities of space-independent supereffectons in A∗andB∗states we have: 16/parenleftbig PA∗ S/parenrightbigind=/parenleftbigPA M/parenrightbigind tr·/parenleftbigPA M/parenrightbigind lb=PA∗ S Z4(2.15) /parenleftbig PB∗ S/parenrightbigind=/parenleftbigPB M/parenrightbigind tr·/parenleftbigPB M/parenrightbigind tr=Pb∗ S Z4(2.15a) In a similar way we get from (2.14) the probabilities of spati ally independent superdeformons: /parenleftbig PD∗ S/parenrightbigind=/parenleftbig PD M/parenrightbig tr·/parenleftbig PD M/parenrightbig lb=PD∗ S Z4(2.16) The concentrations of molecules, the states of which marked ly differ from the equilibrium one and which cause light scattering in composi tion of spatially independent macroeffectons and macrodeformons, are equal t o: /bracketleftbigg NA M=N0 Z2V0/parenleftbigPA M/parenrightbig/bracketrightbigg tr,lb;/bracketleftbigg NB M=N0 Z2V0/parenleftbigPB M/parenrightbig/bracketrightbigg tr,lb(2.17) /bracketleftbigg ND M=N0 Z4V0/parenleftbigPD M/parenrightbig/bracketrightbigg tr,lb The concentrations of molecules, involved in a-convertons , b- convertons and Macroconvertons or c-Macrotransitons (see Introduction) are correspondingly: Nac M=N0 Z2V0Pac;Nbc M=N0 Z2V0Pbc;NC M=N0 Z4V0PcMt (2.18) The probabilities of convertons-related excitations are t he same as used in Chap- ter 4 of book [1]. The concentration of molecules, participating in the indep endent supereffec- tons and superdeformons: NA∗ M=N0 Z4V0PA∗ s;NB∗ M=N0 Z4V0PB∗ S (2.19) ND∗ M=N0 Z8V0PD∗ S (2.20) where N0andV0are the Avogadro number and the molar volume of the matter. 17Substituting (2.17 - 2.20) into well known Raleigh formula for scattering coefficient, measured at the right angle between incident and scat- tered beams: R=I I0r2 V=8π4 λ4α2nM(cm−1) (2.20a) we obtain the values of contributions from different states o f quasiparticles to the resulting scattering coefficient: /parenleftbig RM A/parenrightbig tr,lb=8π4 λ4(α∗)2 Z2N0 V0/parenleftbigPA M/parenrightbig tr,lb;Rs A=8π4 λ4(α∗)2 Z4N0 V0PA∗ s (2.21) /parenleftbig RM B/parenrightbig tr,lb=8π4 λ4(α∗)2 Z2N0 V0/parenleftbigPB M/parenrightbig tr,lb;Rs B=8π4 λ4(α∗)2 Z4N0 V0PB∗ s (2.22) /parenleftbig RM D/parenrightbig tr,lb=8π4 λ4(α∗)2 Z2N0 V0/parenleftbig PD M/parenrightbig tr,lb;Rs D=8π4 λ4(α∗)2 Z4N0 V0PD∗ s (2.23) The contributions of excitations, related to [ tr/lb] convertons are: Rac=8π4 λ4(α∗)2 Z2N0 V0Rbc=8π4 λ4(α∗)2 Z2N0 V0Pbc Rabc=8π4 λ4(α∗)2 Z4N0 V0PcMt where: α∗is the acting polarizability determined by eq.(1.24) and (1 .25). The resulting coefficient of the isotropic scattering ( Riso) is defined as the sum of contributions (2.21-2.23) and is subdivided into thr ee kinds of scattering: caused by translational quasiparticles, caused by librati onal quasiparticles and by the mixed type of quasiparticles: Riso= [RM A+RM B+RM D] + [RM A+RM B+RM D] + [Rac+Rbc+Rabc] + [Rs A+Rs B+Rs D] (2.24) Total contributions, related to convertons and superexcit ations are corre- spondingly: RC=Rac+Rbc+Rabcand RS=Rs A+Rs B+Rs D The polarizability of anisotropic molecules having no cubi c symmetry is a tensor. In this case, total scattering (R) consists of scatt ering at density fluctua- tions ( Riso) and scattering at fluctuations of the anisotropy/parenleftBig Ran=13∆ 6−7∆Riso/parenrightBig : 18R=Riso+13∆ 6−7∆Riso=Riso6 + 6∆ 6−7∆=RisoK (2.25) where R isocorresponds to eq.(2.24); ∆ is the depolarization coefficient. The factor:/parenleftBig 6+6∆ 6−7∆/parenrightBig =Kwas obtained by Cabanne and is called after him. In the case of isotropic molecules when ∆ = 0, the Cabanne fact or is equal to 1. The depolarization coefficient (∆) could be determined exper imentally as the ratio: ∆ =Ix/Iz, (2.26) where IxandIzare two polarized components of the beam scattered at right angle with respect to each other in which the electric vector is directed parallel and perpendicular to the incident beam, respectively. For e xample, in water ∆ = 0 .09 (Vuks, 1977). According to the proposed theory of light scattering in liqu ids the central un- shifted (like in gases) component of the Brillouin scatteri ng spectrum, is caused by fluctuations of concentration and self-diffusion of molec ules, participating in the convertons, macrodeformons (tr and lib) and superdefor mons. The scatter- ing coefficients of the central line ( Rcentr) and side lines (2 Rside) in transparent condensed matter, as follows from (2.24) and (2.25), are equ al correspondingly to: Rcent=K/bracketleftbig/parenleftbig RM D/parenrightbig tr+/parenleftbig RM D/parenrightbig lb/bracketrightbig +K(RC+RS) (2.27) and 2Rside=/parenleftbigRM A+RM B/parenrightbig tr+/parenleftbigRM A+RM A/parenrightbig lb(2.27a) where Kis the Cabanne factor. The total coefficient of light scattering is: Rt=Rcent+ 2Rside (2.28) In accordance with our model the fluctuations of anisotropy ( Cabanne factor) should be taken into account for calculations of the central component only. The orientations of molecules in composition of A and B states of Macroeffectons are correlated and their coherent oscillations are not accompa nied by fluctuations of anisotropy of polarizability (see Fig.2.1). 19The probabilities of the convertons, macrodeformons and su perdeformons excitations (eqs.2.14, 4.16, 4.27 in [1]) are much lower in c rystals than in liquids and hence, the central line in the Brillouin spectra of cryst als is not usually observed. The lateral lines in Brillouin spectra are caused by the scat tering on the molecules forming (A) and (B) states of spatially independe nt macroeffectons, as it was mentioned above. The polarizabilities of the molecules forming the independ ent macroeffec- tons, synchronized in (A) tr,lband (B)tr,lbstates and dielectric properties of these states, differ from each other and from that of transiti on states (macrode- formons). Such short-living states should be considered as the non equilibrium ones. In fact we must keep in mind, that static polarizabilities in the more stable ground A state of the macroeffectons are higher than in B state , because the energy of long-term Van der Waals interaction between molec ules of the A state is bigger than that of B-state. If this difference may be attributed mainly to the difference i n the long-therm dispersion interaction, then from (1.33) we obtain: EB−EA=VB−VA=−3 2E0 r6/parenleftbig α2 B−α2 A/parenrightbig (2.29) where polarizability of molecules in A-state is higher, tha n that in B-state: α2 A>/bracketleftBig/parenleftbigα∗/parenrightbig2≈α2 D/bracketrightBig > α2 B The kinetic energy and dimensions of ”acoustic” and ”optic” states of macroef- fectons are the same: TA kin=TB kin. In our present calculations of light scattering we ignore th is difference (2.29) between polarizabilities of molecules in A and B states. But it can be taken into account if we assume, that polarizabi lities in (A) and (a), (B) and (b) states of primary effectons are like: αA≃αa≃α∗;αB≃αb and the difference between the potential energy of (a) and (b) states is deter- mined mainly by dispersion interaction (eq.2.28). Experimental resulting polarizability ( α∗≃αa) can be expressed as: αa=faαa+fbαb+ftα (2.29a) where αt≃αis polarizability of molecules in the gas state (or transiti on state); 20fa=Pa Pa+Pb+Pt;fb=Pb Pa+Pb+Pt; and ft=fd=Pt Pa+Pb+Pt are the fractions of (a), (b) and transition (t) states (equa l to 2.66) as far Pt=Pd=Pa·Pb. On the other hand from (1.33) at r=const we have: ∆Vb→a dis=−3 4(2α∆α) r6·I0(ra=rb;Ia 0≃Ib 0) and ∆Vb→a dis Vb=hνp hνb=∆αa αor ∆αa=αaνp νb(2.29b) αb=αa−∆αa=αa(1−νp/νb) where: ∆ αais a change of each molecule polarizability as a result of the primary effecton energy changing: Eb→Ea+hνpwith photon radiation; νbis a frequency of primary effecton in (b)- state (eq.2.28). Combining (2.29) and (2.29b) we derive for αaandαbof the molecules composing primary translational or librational effectons: αa=ftα 1−/parenleftBig fa+fb+fb·νp νb/parenrightBig (2.30) αb=αa/parenleftbigg 1−νp νb/parenrightbigg (2.30a) The calculations by means of (2.30) are approximate in the fr amework of our assumptions mentioned above. But they correctly reflect the tendencies of αa andαbchanges with temperature. The ratio of intensities or scattering coefficients for the ce ntral component to the lateral ones previously was described by Landau- Plac hek formula (2.10). According to our mesoscopic theory this ratio can be calcula ted in another way leading from (2.27) and (2.28): Icentr 2IM−B=Rcent 2Rside(2.30b) Combining (2.30) and Landau- Plachek formula (2.10) it is po ssible to cal- culate the ratio ( βT/βS) and ( CP/CV) using our mesoscopic theory of light scattering. 212.4. Factors that determine the Brillouin line width The known equation for Brillouin shift is (see 2.7): ∆νM−B=ν0= 2vs λnsin(θ/2) (2.31) where: vsis the hypersonic velocity; λis the wavelength of incident light, nis the refraction index of matter, and θ- scattering angle. The deviation from ν0that determines the Brillouin side line half width may be expressed as the result of fluctuations of sound velocity vsand n related to A and B states of tr and lib macroeffectons: ∆ν0 ν0=/parenleftbigg∆vs vs+∆n n/parenrightbigg (2.32) ∆ν0is the most probable side line width, i.e. the true half width of Brillouin line. It can be expressed as: ∆ν0= ∆νexp−F∆νinc where ∆ νexpis the half width of the experimental line, ∆ νinc- the half width of the incident line, F- the coefficient that takes into account apparatus effects. Let us analyze the first and the second terms in the right part o f (2.32) separately. Thevssquared is equal to the ratio of the compressibility modulus (K) and density ( ρ): v2 s=K2/ρ (2.33) Consequently, from (2.33) we have: ∆vs vs=1 2/parenleftbigg∆K K−∆ρ ρ/parenrightbigg (2.34) In the case of independent fluctuations of K and ρ:: ∆vs vs=1 2/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆K K/vextendsingle/vextendsingle/vextendsingle/vextendsingle−/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆ρ ρ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg (2.35) From our equation (1.14) we obtain for refraction index: n2=/parenleftbigg 1−4 3Nα∗/parenrightbigg−1 , (2.36) 22where N=N0/V0is the concentration of molecules. From (2.36) we can derive: ∆n n=1 2/parenleftbig n2−1/parenrightbig/parenleftbigg∆α∗ α∗+∆N N/parenrightbigg (2.37) where: (∆N/N) = (∆ ρ/ρ) (2.38) and /parenleftbigg∆α∗ α∗/parenrightbigg ≃/parenleftbigg∆K K/parenrightbigg (2.39) we can assume eq.(2.39) as far both parameters: polarizabil ity (α∗) and com- pressibility models (K) are related with the potential ener gy of intermolecular interaction. For the other hand one can suppose that the following relatio n is true: ∆α∗ α∗≃|¯Ea ef−3kT| 3kT=∆K K(2.40) where: ¯Ea efis the energy of the secondary effectons in (¯ a) state; E0= 3kT is the energy of an ”ideal” quasiparticle as a superposition of 3D standing waves. The density fluctuations can be estimated as a result of the fr ee volume ( vf) fluctuations (see 2.45): /parenleftbigg∆vf vf/parenrightbigg tr,lb=1 Z/parenleftbig PM D/parenrightbig tr,lb≃(∆N/N)tr,lb(2.41) Now, putting (2.40) and (2.41) into (2.37) and (2.34) and the n into (2.32), we obtain the semiempirical formulae for the Brillouin line ha lf width calculation: ∆νf νf≃n2 2/bracketleftBigg |¯Ea ef−3kT| 3kT+1 Z/parenleftbig PM D/parenrightbig/bracketrightBigg tr,lb(9.42) Brillouin line intensity depends on the half-width ∆ νof the line in following ways: for a Gaussian line shape: I(ν) =Imax 0exp/bracketleftBigg −0.693/parenleftbiggν−ν0 1 2∆ν0/parenrightbigg2/bracketrightBigg ; (2.43) 23for a Lorenzian line shape: I(ν) =Imax 0 1 +/bracketleftbig(ν−ν0)/1 2∆ν0/bracketrightbig2(2.44) The traditional theory of Brillouin line shape gives a possibility for calculation of ∆ν0taking into account the elastic (acoustic) wave dissipatio n. The fading out of acoustic wave amplitude may be expressed as : A=A0e−αxorA=A0e−αvs(2.45) where αis the extinction coefficient; x=vst- the distance from the source of waves; vsandt- sound velocity and time, correspondingly. The hydrodynamic theory of sound propagation in liquids lea ds to the fol- lowing expression for the extinction coefficient: α=αs+αb=Ω2 2ρv3s/parenleftbigg4 3ηs+ηb/parenrightbigg (2.46) where: αsandαbare contributions to α, related to share viscosity ( ηs) and bulk viscosity ( ηb), respectively; Ω = 2 πfis the angular frequency of acoustic waves. When the side lines in Brillouin spectra broaden slightly, t he following rela- tion between their intensity (I) and shift (∆ ω=|ω−ω0|) from frequency ω0, corresponding to maximum intensity ( I=I0) of side line is correct: I=I0 1 +/parenleftbigω−ω0 a/parenrightbig, (2.47) where: a=αvs. One can see from (2.46) that at I(ω) =I0/2, the half width: ∆ω1/2= 2π∆ν1/2=αvsand ∆ ν1/2=1 2παvs (2.48) It will be shown in Chapter 12 how one can calculate the values ofηsand consequently αson the basis of the mesoscopic theory of viscosity. 2.5. Quantitative verification of mesoscopic theory of Brillouin scattering 24The calculations made according to the formula (2.21 - 2.27) are presented in Fig.2.1-2.7. The proposed theory of scatterin g in liquids, based on our hierarchic concept, is more adequate than the tr aditional Einstein, Mandelschtamm-Brillouin, Landau-Plachek theo ries based on classical thermodynamics. It describes experimental te mperature dependencies and the Icentr/2IM−Bratio for water very well (Fig.2.3). The calculations are made for the wavelength of incident lig ht:λph= 546.1nm= 5.461·10−5cm. The experimental temperature dependence for the refraction index (n) at this wavelength was taken from th e Frontas’ev and Schreiber paper (1965). The rest of data for calculating of v arious light scat- tering parameters of water (density the location of transla tional and librational bands in the oscillatory spectra) are identical to those use d above in Chapter 6. Fig. 2.1. Theoretical temperature dependencies of the total scattering coefficient for water without taking into account the anisotropy of water molecules polarizability fluctuations in the volum e of macroef- fectons, responsible for side lines: [ R(tot)] - eq.(2.27a; 2.28) and tak- ing them into account: [ KR(tot)], where Kis the Cabanne factor (eq.2.25). 25Fig. 2.2. Theoretical temperature dependencies of contributions to the total coefficient of total light scattering (R) caused b y transla- tional and librational macroeffectons and macrodeformons ( without taking into account fluctuations of anisotropy). Fig. 2.3. Theoretical temperature dependencies of central to side bands intensities ratio in Brillouin spectra (eq.2.30 ). Mesoscopic theory of light scattering can be used to verify t he correctness of our formula for refraction index of condensed matter we go t from our theory 26(eq. 1.14): n2−1 n2=4 3πN0 V0α∗(2.48a) and to compare the results of its using with that of the Lorent z-Lorenz formula: n2−1 n2+ 1=4 3πN0 V0α (2.49) From formula (2.48a) the resulting or effective molecular po larizability squared ( α∗)2used in eq.(2.21-2.23) is: (α∗)2=/bracketleftbigg(n2−1)/n2 (4/3)π(N0/V0)/bracketrightbigg2 (2.50) On the other hand, from the Lorentz-Lorenz formula (2.49) we have another value of polarizability: α2=/bracketleftbigg(n2−1)/(n2+ 2) (4/3)π(N0/V0)/bracketrightbigg2 (2.51) It is evident that the light scattering coefficients (eq.2.28 ), calculated using (2.50) and (2.51) taking refraction index: n= 1.33 should differ more than four times as far: R(α∗) R(α)=(α∗)2 (α)2=(n2−1)/n2 (n2−1)/(n2+ 2)=/parenleftbiggn2+ 2 n2/parenrightbigg2 = 4.56 (2.52) At 250andλph= 546 nmthe theoretical magnitude of the scattering coefficient for water, calculated from our formulae (2.28) is equal (see Fig.2.1) to: R= 11.2·10−5m−1(2.53) This result of our theory coincides well with the most reliab le experimental value (Vuks, 1977): Rexp= 10.8·10−5m−1 Multiplication of the side bands contribution (2 Rside) to Cabanne factor in- creases the calculated total scattering to about 25% and mak es the correspon- dence with experiment worse. This fact confirms our assumpti on that fluctua- tions of anisotropy of polarizability in composition of A an d B states of macroef- fectons should be ignored in light scattering evaluation du e to correlation of molecular dynamics in these states, in contrast to that of ma crodeformons. 27Fig. 2.4. Theoretical temperature dependencies of the contribu- tions of A and B states of translational Macroeffectons to the total scattering coefficient of water (see also Fig.2.2); Fig. 2.5. Theoretical temperature dependencies of the contri- butions of the A and B states of librational Macroeffectons to the coefficient of light scattering (R). It follows from the Fig.2.4 and 2.5 that the light scattering depends on (A⇔B) equilibrium of macroeffectons because ( RA)>(RB), i.e. scattering onAstates is bigger than that on Bstates. 28Fig. 2.6. Theoretical temperature dependencies of the contribu- tions to light scattering (central component), related to t ranslational (RD)trand librational ( RD)lbmacrodeformons. Comparing Figs. 2.1; 2.3, and 2.6 one can see that the main con tribution to central component of light scattering is determined by [ lb/tr] convertons Rc(see eq.2.27). 29Fig. 2.7. Theoretical temperature dependences for temperature derivative ( dR/dT ) of the total coefficient of light scattering of water. Nonmonotonic deviations of the dependencies dR/dT (Fig.2.7) reflect the nonmonotonic changes of the refraction index for water nH2O(T), as indicated by available experimental data (Frontas’ev and Schreiber, 1965). The deviations of dependence nH2O(t) from the monotonic way in accordance with hierarchic theory, are a consequence of the nonmonotonic change in the s tability of water structure, i.e. nonlinear change of ( A⇔B)tr,lbequilibrium. Some possible reasons of such equilibrium change were discussed in Chapte r 6. It is clear from (2.52) that the calculations based on the Lor entz- Lorentz formula (2.51) give scattering coefficient values of about 4.5 times smaller than experimental ones. It means that the true α∗value can be calculated just on the basis of our mesoscopic theory o f light refraction (eq.2.50). The traditional Smolukhovsky-Einstein theory, valid for t he integral light scattering only (eq. 2.1), yield values in the range of R= 8.85·10−5m−1to R= 10.5·10−5m−1[4, 8]. All the results, discussed above, mean that our mesoscopic t heory of light scattering works better and is much more informativ e than the conventional one. 2.6. Light scattering in solutions If the guest molecules are dissolved in a liquid and their siz es are much less than incident light wavelength, they do not radically alter the solvent properties. For this case the described above mechanism of light scatter ing of pure liquids does not changed qualitatively. For such dilute solutions the scattering on the fluctuations of concentration of dissolved molecules ( Rc) is simply added to the scattering on the density fluctuations of molecules of the host solvent (eq.2.28). Tak ing into account the fluctuations of molecule polarizability anisotropy (see 2. 25) the total scattering coefficient of the solution ( RS) is: RS=Rt+Rc (2.54) Eqs. (2.21 - 2.28) could be used for calculating Rtuntil critical concentra- tions ( Ccr) of dissolved substance when it start to destroy the solvent structure, so that the latter is no longer able to form primary libration al effectons. Pertur- bations of solvent structure will induce low-frequency shi ft of librational bands in the oscillatory spectrum of the solution until these band s totally disappear. If the experiment is made with a two-component solution of li quids, soluble in each other, e.g. water-alcohol, benzol-methanol etc., a nd the positions of translational and librational bands of solution component s are different, then 30at the concentration of the dissolved substance: C > C cr, the dissolved sub- stance and the solvent (the guest and host) can switch their r oles. Then the translational and librational bands pertinent to the guest subsystem start to dominate. In this case, Rtis to be calculated from the positions of the new bands corresponding to the ”new” host-solvent. The total ”m elting” of the pri- mary librational ”host effectons” and the appearance of the d issolved substance ”guest effectons” is like the second order phase transition and should be accom- panied by a heat capacity jump. The like experimental effects take place indeed [8]. According to our concept, the coefficient R cin eq.(2.54) is caused by the fluc- tuations of concentration of dissolved molecules in the vol ume of translational and librational macro- and superdeformons of the solvent. I f the destabilization of the solvent is expressed in the low frequency shift of librational bands, then the coefficients ( RAandRB)lbincrease (eq.2.21 and 2.22) with the prob- ability of macro-excitations.. The probabilities of conve rtons and macro- and superdeformons and the central component of Brillouin spec tra will increase also. Therefore, the intensity of the total light scatterin g increases correspond- ingly. The fluctuations of concentration of the solute molecules, i n accordance with our model, occur in the volumes of macrodeformons and superd eformons. Con- sequently, the contribution of solute molecules in scatter ing (Rcvalue in eq.2.54) can be expressed by formula, similar to (2.23), but containi ng the molecule po- larizability of the dissolved substance (”guest”) ,equal to ( α∗ g)2instead of the molecule polarizability ( α∗) of the solvent (”host”), and the molecular con- centration of the ”guest” substance in the solution ( ng) instead of the solvent molecule concentration ( nM=N0/V0). For this case Rccould be presented as a sum of the following contributions: (Rc)tr,lb=8π4 λ4(α∗ g)2ng·/bracketleftBig (PD M)tr,lb+PD∗ S/bracketrightBig (2.55) RD∗ c=8π4 λ4(α∗ g)2ng·(PD∗ S) (2.55a) The resulting scattering coefficient ( Re) on fluctuations of concentration in (2.54) is equal to: Rc= (Rc)tr+ (Rc)lb+RD∗ c (2.56) Ifseveral substances are dissolved with concentrations lower than ( Ccr), then their Rcare summed up additively. Formulae (2.55) and (2.56) are valid also for the dilute solu tions. Eqs.(2.21-2.28) and (2.54-2.56) should, therefore, be use d for calculating the resulting coefficient of light scattering in solutions ( RS). 31The traditional theory represents the scattering coefficient at fluctuations of concentration as (Vuks, 1977): Rc=π2 2λ4/parenleftbigg∂ǫ ∂x/parenrightbigg2 ∆x2v (2.57) where ( ∂ǫ/∂x ) is the dielectric penetrability derivative with respect t o one of the components: ∆¯ x2is the fluctuations of concentration of guest molecules squared in the volume element v. The transformation of (2.57) on the basis of classical therm odynamics [8] leads to the formula: Rc=π2 2λ4N0/parenleftbigg 2n∂n ∂x/parenrightbigg/parenleftbigg9n2 (2n2+ 1)(n2+ 2)/parenrightbigg2 x1x2V12f, (2.58) where N0is the Avogadro number, x1andx2are the molar fractions of the first and second components in the solution, V12is the molar volume of the solution, fis the function of fluctuations of concentration determined exper- imentally from the partial vapor pressures of the first ( P1) and second ( P2) solution components [8]: 1 f=x1 P1∂P1 ∂x1=x2 P2∂P2 ∂x2(2.59) In the case of ideal solutions ∂P1 ∂x1=P1 x1;∂P2 ∂x2=P2 x2; and f= 1. For application the mesoscopic theory of light scattering t o study of crystals, liquids and solutions, the following informatio n is needed: 1. Positions of translational and librational band maxima i n oscil- latory spectra; 2. Concentration of all types of molecules in solutions; 3. Refraction index or polarizability in the acting field of e ach component of solution at given temperature. Application of our theory to quantitative analysis of trans parent liquids and solids yields much more information about prope rties of matter, its mesoscopic and hierarchic dynamic structure th an the traditional one . 3. Mesoscopic theory of M¨ ossbauer effect 3.1. General background 32When the atomic nucleus with mass (M) in the gas phase irradia tesγ- quantum with energy of E0=hν0=mpc2(3.1) where: mpis the effective photon mass, then according to the law of impu lse conservation, the nuclear acquires additional velocity in the opposite direction: v=−E0 Mc(3.2) The corresponding additional kinetic energy ER=Mv2 2=E2 0 2Mc2(3.3) is termed recoil energy. When an atom which irradiates γ-quantum is in composition of the solid body, then three situations are possible: 1. The recoil energy of the atom is higher than the energy of at om - lattice interaction. In this case, the atom irradiating γ-quantum would be knocked out from its position in the lattice. That le ads to defects origination; 2. Recoil energy is insufficient for the appreciable displace ment of an atom in the structure of the lattice, but is higher than the en- ergy of phonon, equal to energy of secondary transitons and p honons excitation. In this case, recoil energy is spent for heating the lattice; 3. Recoil energy is lower than the energy of primary transito ns, related to [emission/absorption] of IR translational and l ibrational photons (hνp)tr,lband phonons (hνph)tr,lb. In that case, the probability (f) of γ-quantum irradiation without any the losses of energy appea rs, termed the probability (fraction) of a recoilless processe s. For example, when ER<< hν ph(νph- the mean frequency of phonons), then the mean energy of recoil: ER= (1−f)hνph (3.4) Hence, the probability of recoilless effect is f= 1−ER hνph(3.5) According to eq.(3.3) the decrease of the recoil energy ERof an atom in the structure of the lattice is related to increase of its effecti ve mass ( M). In our model Mcorresponds to the mass of the effecton. 33The effect of γ-quantum irradiation without recoil was discovered by M¨ os sbauer in 1957 and named after him. The value of M¨ ossbauer effect is determined by the value of f≤1. The big recoil energy may be transferred to the lattice by por tions that areresonant to the frequency of IR photons (tr and lb) and phonons. The possibility of superradiation of IR quanta stimulation as a result of such recoil process is a consequence of our mesoscopic model . The scattering of γ-quanta without lattice excitation, when ER<< hν ph, is termed the elastic one.The general expression [11, 12] for the probability of such phononless elastic γ-quantum radiation acts is equal to: f= exp/parenleftbigg −4π < x2> λ2 0/parenrightbigg (3.6) where λ0=c/ν0is the real wavelength of γ-quantum; <x2>- the nucleus oscillations mean amplitude squared in the direction of γ-quantum irradiation. Theγ-quanta wavelength parameter may be introduced like: L0=λ0/2π, (3.7) where: L0= 1.37·10−5cmforFe57, then eq.(3.6) could be written as follows: f= exp/parenleftbigg −< x2> L2 0/parenrightbigg (3.8) It may be shown [12], proceeding from the model of crystal as a system of 3N identical quantum oscillators, that when temperature (T) i s much lower than the Debye one ( θD) then: < x2>=9/planckover2pi12 4Mkθ D/braceleftbigg 1 +2/planckover2pi12T2 3θ2 D/bracerightbigg , (3.9) where θD=hνD/kandνDis the Debye frequency. From (3.1), (3.3) and (3.7) we have: 1 L=E0 /planckover2pi1c(3.10) where: E0=hν=c(2ME R)1/2is the energy of γ-quantum Substituting eqs.(3.9 and 3.10) into eq.(3.8), we obtain th e Debye-Valler formula: f= exp/bracketleftbigg −ER kθD/braceleftbigg3 2+π2T2 θD/bracerightbigg/bracketrightbigg (3.11) 34when T→0, then f→exp/parenleftbigg −3ER 2kθD/parenrightbigg (3.12) 3.2. Probability of elastic effects Mean square displacements <x2>of an atoms or molecules in condensed matter (eq. 3.8) is not related to excitation of thermal phot ons or phonons (i.e. primary or secondary transitons). According to our co ncept, < x2>is caused by the mobility of the atoms forming effectons and diffe rs for primary and secondary translational and librational effectons in ( a,¯a)tr,lband (b, b)tr,lb states. We will ignore below the contributions of macro- and supereff ectons in M¨ ossbauer effect as very small. Then the resulting probabil ity of elastic effects atγ-quantum radiation is determined by the sum of the following contributions (seeeqs.4.2−4.4 of [1, 2]): f=1 Z/summationdisplay tr,lb/bracketleftbig/parenleftbigPa effa ef+Pb effb ef/parenrightbig +/parenleftbig¯Pa ef¯fa ef+¯Pb ef¯fb ef/parenrightbig/bracketrightbig tr,lb(3.13) where: Pa ef, Pb ef,¯Pa ef,¯Pb efare the relative probabilities of the acoustic and optic states for primary and secondary effectons; Z is the total par tition function. These parameters are calculated as described in Chapter 4 of book [1] and in papers cited in the Summary of this article. Each of contribu tions to resulting probability of the elastic effect can be calculated separate ly as: /parenleftbigfa ef/parenrightbig tr,lb= exp −</parenleftbigxa/parenrightbig2 tr,lb> L2 0  (3.14) /parenleftbigfa ef/parenrightbig tr,lbis the probability of elastic effect, related to dynamics of p rimary translational and librational effectons in a-state; /parenleftbigfb ef/parenrightbig tr,lb= exp −</parenleftbig xb/parenrightbig2 tr,lb> L2 0  (3.15) /parenleftbigfb ef/parenrightbig tr,lbis the probability of elastic effect in primary translationa l and libra- tional effectons in b-state; 35/parenleftbig¯fa ef/parenrightbig tr,lb= exp/bracketleftbigg −</parenleftBig ¯xa/parenrightBig2 tr,lb> L2 0/bracketrightbigg (3.16) /parenleftbig¯fa ef/parenrightbig tr,lbis the probability for secondary effectons in ¯ a-state; /parenleftbig¯fb ef/parenrightbig tr,lb= exp −</parenleftbig ¯xb/parenrightbig2 tr,lb> L2 0  (3.17) /parenleftbig¯fb ef/parenrightbig tr,lbis the probability of elastic effect, related to secondary eff ectons in ¯b-state. Mean square displacements within different types of effecton s in eqs.(3.14- 3.17) are related to their phase and group velocities. At firs t we express the displacements using group velocities of the waves B(vgr) and periods of corre- sponding oscillations ( T) as: </parenleftbigxa/parenrightbig2 tr,lb>=<(va gr)2 tr,lb> < ν2a>tr,lb=</parenleftbigva grTa/parenrightbig2 tr,lb> (3.18) where ( Ta)tr,lb= (1/νa)tr,lbis a relation between the period and the frequency of primary translational and librational effectons in a-state; (va gr=vb gr)tr,lbare the group velocities of atoms forming these effectons equal in (a) and (b) states. In a similar way we can express the displacements of atoms for ming (b) state of primary effectons (tr and lib): </parenleftbig xb/parenrightbig2 tr,lb>=<(vb gr)2 tr,lb> < ν2 b>tr,lb(3.19) where νbis the frequency of primary translational and librational effectons in b-state. The mean square displacements of atoms forming secondary translational and librational effectons in ¯ aand¯bstates: </parenleftbig¯xa/parenrightbig2 tr,lb>=<(va gr)2 tr,lb> <¯ν2a>tr,lb(3.20) </parenleftbig ¯xb/parenrightbig2 tr,lb>=<(vb gr)2 tr,lb> <¯ν2 b>tr,lb(3.21) where: (¯ va gr= ¯vb gr)tr,lb 36Group velocities of atoms in primary and secondary effectons may be expressed using corresponding phase velocities ( vph) and formulae for waves B length as follows: /parenleftbig λa/parenrightbig tr,lb=h m < v gr>tr,lb=/parenleftbiggva ph νa/parenrightbigg tr,lb= (3.22) =/parenleftbigλb/parenrightbig tr,lb=/parenleftBigg vb ph νb/parenrightBigg tr,lb hence for the group velocities of the atoms or molecules form ing primary effec- tons (tr and lb ) squared we have: /parenleftbigva,b gr/parenrightbig2 tr,lb=h2 m2/parenleftBigg νa,b va,b ph/parenrightBigg2 tr,lb(3.23) In accordance with mesoscopic theory, the wave B length, imp ulses and group velocities in aandbstates of the effectons are equal. Similarly to (3.23), we obtain the group velocities of particles, composing second ary effectons: /parenleftbig ¯va,b gr/parenrightbig2 tr,lb=h2 m2/parenleftBigg ¯νa,b ¯va,b ph/parenrightBigg2 tr,lb(3.24) Substituting eqs.(3.23) and (3.24) into (3.18-3.21), we fin d the important ex- pressions for the average coherent displacements of partic les squared as a result of their oscillations in the volume of the effectons ( tr, lib) in both discreet states (acoustic and optic): <(xa)2 tr,lb>= (h/mva ph)2 tr,lb (3.25) <(xb)2 tr,lb>= (h/mvb ph)2 tr,lb (3.26) <(xa)2 tr,lb>= (h/mva ph)2 tr,lb (3.27) <(xb)2 tr,lb>= (h/mvb ph)2 tr,lb (3.28) Then, substituting these values into eqs.(3.14-3.17) we ob tain a set of different contributions to the resulting probability of effects witho ut recoil: 37/parenleftbigfa f/parenrightbig tr.lb= exp/bracketleftbigg −/parenleftBig h mL0va ph/parenrightBig2/bracketrightbigg tr,lb; /parenleftbigfb f/parenrightbig tr.lb= exp/bracketleftbigg −/parenleftBig h mL0vb ph/parenrightBig2/bracketrightbigg tr,lb;  (3.29) /parenleftbig¯fa f/parenrightbig tr.lb= exp/bracketleftbigg −/parenleftBig h mL0¯va ph/parenrightBig2/bracketrightbigg tr,lb; /parenleftbig¯fb f/parenrightbig tr.lb= exp/bracketleftbigg −/parenleftBig h mL0¯vb ph/parenrightBig2/bracketrightbigg tr,lb;  (3.30) where the phase velocities ( va ph, vb ph,¯va ph,¯vb ph)tr,lbare calculated from the resulting sound velocity and the positions of translationa l and librational bands in the oscillatory spectra of matter at given temperature us ing eqs.2.69-2.75. The wavelength parameter: L0=c 2πν0=hc 2πE0= 1.375·10−11m for gamma-quanta, radiated by nuclear of Fe57, with energy: E0= 14.4125 kev = 2 .30167·10−8erg Substituting eqs.(3.29) and (3.30) into (3.13), we find the t otal probability of recoilless effects ( ftot) in the given substance. Corresponding computer calcu- lations for ice and water are presented on Figs.3.1 and 3.2. As far the second order phase transitions in general case are accompanied by the alterations of the sound velocity and the positions of translational and librational bands, they should also be accompanied by alter ations of f totand its components. 38Fig. 3.1. Temperature dependences of total probability ( f) for elastic effect without recoil and phonon excitation: (a) in i ce; (b) in water; (c)-during phase transition. The calculations were performed using eq.(3.13). Fig. 3.2. (a) - The contributions to probability of elastic effect (f) (see Fig.3.1) for primary ( fa,b ef)trand secondary ( ¯f)trtransla- 39tional effectons and (b)and those of librational effectons ( fa,b ef)lb and ( ¯f)lbnear the temperature of [ice ⇔water] phase transition. The total probability ( f) and its components, caused by primary and sec- ondary quasiparticles were calculated according to formul a (3.13). The value of (f) determines the magnitude of the M¨ ossbauer effect register ed by γ-resonance spectroscopy. The band width caused by recoilless effects is determined by t he uncertainty principle and expressed as follows: Γ =h τ≈10−27 1.4·10−7= 7.14·10−21erg = 4 .4·10−9eV (3.31) where τis the lifetime of nucleus in excited state (for Fe57τ= 1.4·10−7s). The position of the band depends on the mean square velocity o f atoms, i.e. on second order Doppler effect. In the experiment, such an effe ct is compen- sated by the velocity of γ-quanta source motion relative to absorbent. In the framework of our model this velocity is interrelated with th e mean velocity of the secondary effectons diffusion in condensed matter. 3.3. Doppler broadening in spectra of nuclear gamma-resona nce (NGR) M¨ ossbauer effect is characterized by the nonbroadened comp onent of NGR spectra only, with probability of observation determined b y eq.(3.13). When the energy of absorbed γ-quanta exceeds the energy of thermal IR pho- tons (tr,lib) orphonons excitation, the absorbance band broadens as a result of Doppler effect. Within the framework of our mesoscopic conce pt the Doppler broadening is caused by thermal displacements of the partic les during [ a⇔b and ¯a⇔¯b]tr,lbtransitions of primary and secondary effectons, leading to o rig- ination/annihilation of the corresponding type of deformo ns (electromagnetic and acoustic). Theflickering clusters : [lb/tr] convertons ( aandb), can contribute in the NGR line broadening also. In that case, the value of Doppler broadening (∆Γ) of the band in the NGR spectrum could be estimated from corresponding kinetic ene rgies of these ex- citations, related to their group velocities (see eq. 4.31) . In our consideration we take into account the reduced to one molecule kinetic energies of pri- mary and secondary translational and librational transito ns,a-convertons and b-convertons. The contributions of macroconvertons, macro - and superdefor- mons are much smaller due to their small probability and conc entration: ∆Γ =V0 N0Z/summationdisplay tr,lb/parenleftbigntPtTt+ ¯nt¯Pt¯Tt/parenrightbig tr,lb+ (3.32) +V0 N0Z(nef)lb[PacTac+PbcTbc] 40where: N0andV0are the Avogadro number and molar volume; Z is the total partition function ( eq.4.2);ntand ¯ntare the concentrations of primary and secondary transitons (eqs.3.5 and 3.7); (nef)lb=nconis a concentration of primary librational effectons, equal t o that of the convertons; Ptand¯Ptare the relative probabilities of primary and secondary transitons (eqs. 4.26 and 4 .27);PacandPbcare relative probabilities of (aandb) -convertons (see Chapter 4 of [1]); Ttand¯Ttare the kinetic energies of primary and secondary transiton s, re- lated to the corresponding total energies of these excitati ons (Etand¯Et), their masses ( MtandMt) and the resulting sound velocity ( vs, see eq.2.40) in the following form: (Tt)tr,lb=/summationtext3 1/parenleftBig E1,2,3 t/parenrightBig tr,lb 2Mt(vress)2(3.33) (Tt)tr,lb=/summationtext3 1/parenleftBig ¯E1,2,3 t/parenrightBig tr,lb 2¯Mt(vress)2(3.34) The kinetic energies of (a and b) convertons are expressed in a similar way: (Tac) =/summationtext3 1/parenleftbig E1,2,3 ac/parenrightbig tr,lb 2Mc(vress)2 (Tbc) =/summationtext3 1/parenleftBig E1,2,3 bc/parenrightBig tr,lb 2Mc(vress)2 where: E1,2,3 acandE1,2,3 bcare the energies of selected states of corresponding convertons; Mcis the mass of convertons, equal to that of primary libration al effectons. The broadening of NGR spectral lines by Doppler effect in liqu ids is generally expressed using the diffusion coefficient (D) at the assumptio n that the motion of M¨ ossbauer atom has the character of unlimited diffusion [ 13]: ∆Γ =2E2 0 /planckover2pi1c2D (3.35) where: E0=hν0is the energy of gamma quanta; c is light velocity and D=kT 6πηa(3.36) 41where: ηis viscosity, (a) is the effective Stokes radius of the atom Fe57 The probability of recoilless γ-quantum absorption by the matter containing for example Fe57, decreases due to diffusion and corresponding Doppler broad - ening of band (∆Γ): fD=Γ Γ + ∆Γ(3.37) where ∆Γ corresponds to eq.(3.32).The formulae obtained he re make it possible to experimentally verify a set of consequences of our mesosc opic theory using the gamma- resonance method. A more detailed interpretation of the data obtained by this method also becomes possible. The magnitude of (∆Γ) was calculated according to formula (3 .32). It cor- responds well to experimentally determined Doppler wideni ng in the nuclear gamma resonance (NGR) spectra of ice. Fig. 3.3. The temperature dependences of the parameter ∆Γ, characterizing the nonelastic effects and related to the exc itation of thermal phonons and IR photons: a) in ice; b) in water; c) near phase transition. 423.4. Acceleration and forces, related to thermal dynamics o f molecules and ions. Hypothesis of Vibro-gravitational interaction During the period of particles thermal oscillations (tr and lb), their in- stant velocity, acceleration and corresponding forces alt ernatively and strongly change. The change of wave B instant group velocity, averaged during the molecule oscillation period in composition of the (a) and (b) states o f the effectons, de- termines the average acceleration: /bracketleftBigg aa,b gr=dva,b gr dt=va,b gr T=vgrνa,b/bracketrightBigg1,2,3 tr,lb(3.38) We keep in mind that group velocities, impulses and wave B len gth in (a) and (b) states of the effectons are equal, in accordance with our m odel. Corresponding to (3.38) forces: /bracketleftbig Fa,b=maa,b gr/bracketrightbig1,2,3 tr,lb(3.39) The energies of molecules in (a) and (b) states of the effecton s also can be expressed via accelerations: /bracketleftBig Ea,b=hνa,b=Fa,bλ=maa,b·λ=maa,b(va,b ph/νa,b)/bracketrightBig1,2,3 tr,lb(3.40) From (3.40) one can express the accelerations of particles i n the primary effec- tons of condensed matter, using their phase velocities as a w aves B: /bracketleftBigg aa,b gr=h(νa,b)2 mva,b ph/bracketrightBigg1,2,3 tr,lb(3.41) The accelerations of particles in composition of secondary effectons have a sim- ilar form: /bracketleftBigg ¯aa,b gr=h(¯νa,b)2 m¯va,b ph/bracketrightBigg1,2,3 tr,lb(3.42) These parameters are important for understanding the dynam ic properties of condensed systems. The accelerations of the atoms, forming primary and sec- ondary effectons can be calculated, using eqs.(2.74-2.75 of [1]) to determine phase velocities and eqs. (2 .27,2.28,2.54,2.55 [1]) to find a frequencies. 43Multiplying (3.41) and (3.42) by the atomic mass m, we derive the most probable and mean forces acting upon the particles in both st ates of primary and secondary effectons in condensed matter: /bracketleftBigg Fa,b gr=h(νa,b)2 va,b ph/bracketrightBigg1,2,3 tr,lb/bracketleftBigg ¯Fa,b gr=h(¯νa,b)2 ¯va,b ph/bracketrightBigg (3.43) The comparison of calculated accelerations with empirical data of the M¨ ossbauer effect - supports the correctness of our approach. According to eq.(2.54) in the low temperature range, when hνa<< kT , the frequency of secondary tr and lb effectons in the (a) state can be estimated as: νa=νa exp/parenleftbighνa kT/parenrightbig −1≈kT h(3.44) For example, at T= 200 Kwe have ¯ νa≈4·1012s−1. If the phase speed in eq.(3.42) is taken equal to ¯ va ph= 2.1·105cm/s (see Fig.2a) and the mass of water molecule: m= 18·1.66·10−24g= 2.98·10−23g, then from (3.42) we get the acceleration of molecules in comp osition of secondary effectons of ice in (a) state: ¯aa gr=h(¯νa)2 m¯va ph= 1.6·1016cm/s2 This value is about 1013times more than that of free fall acceleration ( g= 9.8·102cm/s2), which agrees well with experimental data, obtained for so lid bodies [11]. Accelerations of H2Omolecules in composition of primary librational effec- tons ( aa gr) in the ice at 200K and in water at 300K are equal to: 0 .6·1013cm/s2 and 2·1015cm/s2, correspondingly. They also exceed to many orders the free fall acceleration. It was shown experimentally (Sherwin, 1960), that heating o f solid body leads to decreasing of gamma-quanta frequency (red Doppler shift) i.e. increas- ing of corresponding quantum transitions period. This can b e explained as the relativist time-pace decreasing due to elevation of averag e thermal velocity of atoms [11]. The thermal vibrations of particles (atoms, molecules) in c omposition of primary effectons as a partial Bose-condensate are coherent . The increasing of such clusters dimensions, determined by most probable wa ve B length, as a result of cooling, pressure elevation or under magnetic fiel d action (see section 14.6 of [1]) leads to enhancement of coherent regions. 44Each coherently vibrating cluster of particles with big alt ernating accelerations, like librational and translational effecto ns is a source of coherent gravitational waves. The frequency of these vibro-gravitational waves (VGW) is e qual to fre- quency of particles vibrations (i.e. frequency of the effect ons in aorbstates). The amplitude of VGW ( AG) is proportional to the number of vibrating coher- ently particles ( NG) in composition of primary effectons: AG∼NG∼Vef/(V0/N0) = (1 /nef)/(V0/N0) The resonant long-distance gravitational interaction bet ween coher- ent clusters of the same body or between that of different bodi es is possible .The formal description of this vibro-gravitational intera c- tion (VGI) could be like that of distant macroscopic Van der W aals interaction. Different patterns of nonlocal Bose-condensate of standing grav- itational waves in vacuum represent the field-informationa l copy of local Bose- condensate of the effectons of condensed matter. Very important role of proposed here distant resonant VIBRO - GRAVITATIONAL INTERACTION (VGI) in elementary acts of perception and memory can be contributed by coherent primar y li- brational water effectons in microtubules of the nerve cells (see paper ”Hierarchic Model of Consciousness” in URL: http://www.ka relia.ru/˜alexk and http://arXiv.org/abs/physics/0003045). References [1]. Kaivarainen A. Hierarchic Concept of Matter and Field. Water, biosystems and elementary particles. New York, 1995, pp. 485. [2]. Kaivarainen A. New Hierarchic Theory of Matter General for Liquids and Solids: dynamics, thermodynamics and mesoscopic structure of water and ice (see URL: http://www.karelia.ru/˜alexk) [3]. Kaivarainen A. Hierarchic Concept of Matter, Gen- eral for Liquids and Solids: Water and ice (see Proceedings of the Second Annual Advanced Water Sciences Sympo- sium, October 4-6, 1966, Dallas, Texas. [4]. Eisenberg D., Kauzmann W. The structure and properties of water. Oxford University Press, Oxford, 1969 . [5]. Frontas’ev V.P., Schreiber L.S. J. Struct. Chem. (USSR )6, (1966), 512 . [6]. Kikoin I.K. (Ed.) Tables of physical values. Atom- izdat, Moscow, 1976 (in Russian). 45[7]. Einstein A. Collection of works. Nauka, Moscow, 1965. [8]. Vuks M.F. Light scattering in gases, liquids and solutions. Leningrad University Press, Leningrad. 1977. [9]. Vuks M.F. Electrical and optical properties of molecul es and condensed matter. Leningrad University Press, Leningr ad, 1984. [10]. Theiner O., White K.O. J.Opt. Soc.Amer. 1969,59,181. [11]. Wertheim G.K. M¨ ossbauer effect. Academic Press, N.Y. and London. 1964. [12]. Shpinel V.C. Gamma-ray resonance in crystals. Nauka, Moscow, 1969. [13]. Singvi K., Sielander A. In book: M¨ ossbauer effect. Ed. Kogan Yu. Moscow 1962. 46

arXiv:physics/0003071v1 [physics.gen-ph] 24 Mar 2000Hierarchic Theory of Complex Systems (biosystems, colloids): self-organization & osmos Alex Kaivarainen JBL, University of Turku, FIN-20520, Turku, Finland URL: http://www.karelia.ru/˜alexk H2o@karelia.ru Materials, presented in this original article are based on following publications: [1]. A. Kaivarainen. Book: Hierarchic Concept of Mat- ter and Field. Water, biosystems and elementary particles. New York, NY, 1995 and new version of this book. (see URL: http://www.karelia.ru/˜alexk [Book prospect and New articles]). [2]. A. Kaivarainen. New Hierarchic Theory of Matter General for Liquids and Solids: dynamics, thermodynamics and mesoscopic structure of water and ice (see URL: http://www.karelia.ru/˜alexk [New articles]). [3]. A. Kaivarainen. Hierarchic Concept of Condensed Matter and its Interaction with Light: New Theories of Light Refraction, Brillouin Scattering and M¨ ossbauer ef- fect (see URL: http://www.karelia.ru/˜alexk [New articles]). [4]. A. Kaivarainen. Hierarchic Theory of Condensed Matter: Interrelation between mesoscopic and macroscopic properties (see URL: http://www.karelia.ru/˜alexk [New articles]). Computerized verification of described here new theo- ries is presented, using special computer program, based on Hierarchic Theory of Condensed Matter (copyright, 1997, A. Kaivarainen). CONTENTS OF ARTICLE Summary to Part I of book: ”Hierarchic theory of con- densed matter” Introduction 11. Protein domain mesoscopic organization 2. Quantum background of lipid domain organization in biome m- branes 3. Hierarchic approach to theory of solutions and colloid sy stems 4. Distant solvent-mediated interaction between macromol ecules 5. Spatial self-organization in the water-macromolecular systems 6. Properties of [bisolvent - polymer system] 7. Osmosis and solvent activity. Traditional and mesoscopi c ap- proach =================================================== =========== Summary of New Hierarchic Theory of Condensed Matter (see http://arXiv.org/abs/physics/0003044) A basically new hierarchic quantitative theory, general fo r solids and liquids, has been developed. It is assumed, that anharmonic oscillations of particles in any con- densed matter lead to emergence of three-dimensional (3D) s uperpo- sition of standing de Broglie waves of molecules, electroma gnetic and acoustic waves. Consequently, any condensed matter could b e con- sidered as a gas of 3D standing waves of corresponding nature . Our approach unifies and develops strongly the Einstein’s and De bye’s models. Collective excitations, like 3D standing de Brogli e waves of molecules, representing at certain conditions the mesosco pic molec- ular Bose condensate, were analyzed, as a background of hier archic model of condensed matter. The most probable de Broglie wave (wave B) length is deter- mined by the ratio of Plank constant to the most probable impu lse of molecules, or by ratio of its most probable phase velocity to fre- quency. The waves B are related to molecular translations (t r) and librations (lb). As the quantum dynamics of condensed matter does not follow i n general case the classical Maxwell-Boltzmann distribution, the re al most probable de Broglie wave length can exceed the classical thermal de Brog lie wave length and the distance between centers of molecules many times. This makes possible the atomic and molecular Bose condensat ion in solids and liquids at temperatures, below boiling point. It is one o f the most important results of new theory, which we have confirmed by computer sim ulations on examples of water and ice. Four strongly interrelated new types of quasiparticles (collective excita- tions) were introduced in our hierarchic model: 21.Effectons (tr and lb) , existing in ”acoustic” (a) and ”optic” (b) states represent the coherent clusters in general case ; 2.Convertons , corresponding to interconversions between trandlbtypes of the effectons (flickering clusters); 3.Transitons are the intermediate [ a⇋b] transition states of the trandlb effectons; 4.Deformons are the 3D superposition of IR electromagnetic or acoustic waves, activated by transitons andconvertons. Primary effectons (tr and lb) are formed by 3D superposition of the most probable standing de Broglie waves of the oscillating ions, atoms or molecules. The volume of effectons (tr and lb) may contain fro m less than one, to tens and even thousands of molecules. The first condition m eans validity ofclassical approximation in description of the subsystems of the effect ons. The second one points to quantum properties of coherent clusters due to molecular Bose condensation . The liquids are semiclassical systems because their primar y (tr) effectons contain less than one molecule and primary (lb) effectons - mo re than one molecule. The solids are quantum systems totally because both kind of t heir primary effectons (tr and lb) are molecular Bose condensates .These conse- quences of our theory are confirmed by computer calculations . The 1st order [ gas→liquid ] transition is accompanied by strong decreasing of rotational (librational) degrees of freedom due to emerg ence of primary (lb) effectons and [ liquid→solid] transition - by decreasing of translational degrees of freedom due to Bose-condensation of primary (tr) effecton s. In the general case the effecton can be approximated by par- allelepiped with edges corresponding to de Broglie waves le ngth in three selected directions (1, 2, 3), related to the symmetry of the molecular dynamics. In the case of isotropic molecular moti on the effectons’ shape may be approximated by cube. The edge-length of primary effectons (tr and lb) can be consid ered as the ”parameter of order”. The in-phase oscillations of molecules in the effectons corr espond to the effecton’s (a) - acoustic state and the counterphase oscillations correspond to their (b) - optic state. States (a) and (b) of the effectons differ in potential energy only, however, their kinetic energies, impulses and spatial dimensions - are the same. The b-state of the effectons has a common feature with Fr¨ olich’s polar mode. The(a→b)or(b→a)transition states of the primary effectons (tr and lb), defined as primary transitons, are accompanied b y a change in molecule polarizability and dipole moment withou t density fluctuations. At this case they lead to absorption or radiati on of IR photons, respectively. Superposition (interception) of three internal standing I R pho- tons of different directions (1,2,3) - forms primary electro magnetic deformons (tr and lb). 3On the other hand, the [lb ⇋tr]convertons andsecondary transitons are accompanied by the density fluctuations, leading to absorption or radiation of phonons . Superposition resulting from interception of standing phonons in three direc- tions (1,2,3), forms secondary acoustic deformons (tr and lb). Correlated collective excitations of primary and secondary effectons and deformons (tr and lb) ,localized in the volume of primary trandlb electromag- netic deformons ,lead to origination of macroeffectons, macrotransitons andmacrodeformons (tr and lb respectively) . Correlated simultaneous excitations of tr and lb macroeffec tonsin the vol- ume of superimposed trandlbelectromagnetic deformons lead to origination ofsupereffectons. In turn, the coherent excitation of both: tr andlb macrodeformons and macroconvertons in the same volume means creation of superdeformons. Su- perdeformons are the biggest (cavitational) fluctuations, leading to microbub- bles in liquids and to local defects in solids. Total number of quasiparticles of condensed matter equal to 4!=24, reflects all of possible combinations of the four basic ones [ 1-4], intro- duced above. This set of collective excitations in the form o f ”gas” of 3D standing waves of three types: de Broglie, acoustic and el ectro- magnetic - is shown to be able to explain virtually all the pro perties of all condensed matter. The important positive feature of our hierarchic model of ma tter is that it does not need the semi-empiric intermolecular potentials f or calculations, which are unavoidable in existing theories of many body systems. T he potential energy of intermolecular interaction is involved indirectly in di mensions and stability of quasiparticles, introduced in our model. The main formulae of theory are the same for liquids and solid s and include following experimental parameters, which take into ac- count their different properties: [1]- Positions of (tr) and (lb) bands in oscillatory spectra; [2]- Sound velocity; [3]- Density; [4]- Refraction index (extrapolated to the infinitive wave leng th of photon ). The knowledge of these four basic parameters at the same temp erature and pressure makes it possible using our computer program, to ev aluate more than 300 important characteristics of any condensed matter. Amo ng them are such as: total internal energy, kinetic and potential energies, heat-capacity and ther- mal conductivity, surface tension, vapor pressure, viscos ity, coefficient of self- diffusion, osmotic pressure, solvent activity, etc. Most of calculated parameters are hidden, i.e. inaccessible to direct experimental measu rement. The new interpretation and evaluation of Brillouin light sc attering and M¨ ossbauer effect parameters may also be done on the basis of h ierarchic the- ory. Mesoscopic scenarios of turbulence, superconductivi ty and superfluity are 4elaborated. Some original aspects of water in organization and large-sc ale dynamics of biosystems - such as proteins, DNA, microtubules, membrane s and regulative role of water in cytoplasm, cancer development, quantum neu rodynamics, etc. have been analyzed in the framework of Hierarchic theory. Computerized verification of our Hierarchic concept of matt er on examples of water and ice is performed, using special comput er pro- gram: Comprehensive Analyzer of Matter Properties (CAMP, c opy- right, 1997, Kaivarainen). The new opto-acoustical device (CAMP), based on this program, with possibilities much wider, than t hat of IR, Raman and Brillouin spectrometers, has been proposed (see U RL: http://www.karelia.ru/˜alexk). This is the first theory able to predict all known experimenta l temperature anomalies for water and ice. The conformity bet ween theory and experiment is very good even without any adjustab le pa- rameters. The hierarchic concept creates a bridge between micro- and m acro- phenomena, dynamics and thermodynamics, liquids and solid s in terms of quantum physics. 5Introduction Domain granular structure is pertinent to solid bodies, liq uid crys- tals, and polymers of artificial and biological origin. In li quids, as is seen from the X-ray data, the local order is also kept like in s olid bodies. Just like in the case of solids, local order in liquid s can be caused by the most probable - primary effectons (see Introduc tion to [1] and [2]), but smaller in size. In water, the relatively stable clusters of molecules are re vealed by the quasielastic neutron scattering method. The diameter of th ese clusters are (20- 30)˚A and the lifetime is of the order of 10−10s(Gordeyev and Khaidarov, 1983). These parameters are close to those we have calculate d for librational water effectons (Fig. 7a of [1] or Fig4a of [2]). A coherent-inelastic neutron scattering, performed on liq uidD2Oat room temperature revealed collective high-frequency sound mod e. Observed collective excitation has a solid-like character with dimension aroun d 20˚A, resembling water clusters with saturated hydrogen bonds. The observed sound velocity in these clusters is about 3300 m/s, i.e. close to velocity of sound in ice and much bigger than that in liquid water: 1500 m/s (Teixeira et a l., 1985). Such data confirm the existence of primary librational effectons a s molecular Bose condensate, leading from our hierarchic theory. Among the earlier theoretical models of water the model of ”fl ickering clus- ters” proposed by Frank and Wen (1957) is closer to our model t han others. The ”flickering” of a cluster consisting of water molecules is ex pressed by the fact that it dissociates and associates with a short period (10−10−10−11) s. Near the non-polar molecules this period grows up (Frank and Wen, 1957, Frank and Evans, 1945) and ”icebergs” appear. The formation of hydrog en bonds in water is treated as a cooperative process. Our [ lb/tr] convertons, i.e. interconversions between primary lb and tr effectons, reflect the properties of flickering clusters better than other quasiparticles of hierarchic model. Proceeding from the flickering cluster model, Nemethy and Sc heraga (1962), using the methods of statistical thermodynamics, calculat ed a number of pa- rameters for water (free energy, internal energy, entropy) and their temperature dependences, which agree with the experimental data in the l imits of 3%. How- ever, calculations of heat capacity were less successful. T he quantity of water molecules decreases from 91 at 00Cto 25 at 700C(Nemethy and Scheraga, 1962). It is in rather good agreement with our results (Fig. 7 a of [1] or Fig.4a of [2]) on the change of the number of water molecules in a prim ary librational effecton with temperature. The stability of primary effectons (clusters, domains), for ming the condensed media, is determined by the coherence of heat motions, the eq uality of the most probable 3D standing waves B of atoms (molecules), increasi ng distant Van der Waals interaction in the volume of the effectons. It could be possible that molecules, atoms, or ions of different molecular masses belong to the same effectons. The equality of wave B lengths fo r such 6different particles, forming the effectons: λ1=h m1v1=λ2=h m2v2=...=λi=h mivi(1) means that differences in masses are compensated by differenc es in the group velocities of these particles so that their most probable im pulses are equal: Pi=m1v1=m2v2=...=mivi (2) The domains or the crystallites in solid bodies, which could be con- sidered as a primary effectons, can contain a big number of ele men- tary cells. Transitions between the different types of eleme ntary cells (second order phase transitions) means cooperative redist ribution in the positions and dynamics of atoms, leading to origination of new primary effectons. In accordance with our model, the second o rder phase transitions are related sometimes also with emergenc y of con- ditions for primary effectons polymerization or distant ass ociation in coherent superclusters, and concomitant shifting of the ir(a⇔b) equilibrium to the left. The mesoscopic theory could be used to describe a wide range o f physico-chemical and biological phenomena. 1. Protein domain mesoscopic organization If the geometry of cavities of protein surface are complemen tary to the geometry of water librational effectons, then the latt er are sta- bilized. In the opposite case, the water effectons in cavitie s are either not realized, or unstable. In that case, the probability of t he water cluster dissociation in the cavity, related to [lb→tr]conversion, in- creases. The evolution of biological macromolecules could have gone in such a way that they ”learned” to use the cooperative prope r- ties of water clusters and their dissociation for regulatio n of their large-scale dynamics and signal transmission. Calculated frequency of[lb/tr]convertons (106−107)c−1coincide with frequency of protein cavities large-scale pulsations, accompanied by domains r elative fluc- tuations . All sufficiently large globular proteins consist of domains w hose dimensions under normal conditions vary in the narrow limits: (10-20) ˚A (K¨ aiv¨ ar¨ ainen, 1985). This value is close to dimensions of librational wate r effecton (Fig. 7a of [1] or Fig. 4a of [2]) that confirms the important role of water in evolution of biopolymers. We may predict that the lower is the physiologi cal temperature of given organism the larger are the interdomain cavities an d domains of its proteins. It is known that the water in pours or cavities with diameter less than 50˚A freeze out at very low temperatures (about −600C) and its viscosity is 7high (Martini et al., 1983). For the other hand, our calculat ions shows (Fig.17b of [1]), that freezing in normal conditions should be accomp anied by increasing the linear size of primary librational effectons just till 50 ˚A. As far the condition for librational effectons growth in narr ow pours is absent, the formation of sufficiently big primary tra nslational effectons is also violated. As a result of that, condition (6. 6 of [1]) for [liquid →solid] phase transition occurs in such cavities at much lower temperature than in bulk water. If the sound velocity in proteins and the positions of maxima in their oscil- latory spectra, which characterize the librations of atoms are known , then the most probable wave B length of aminoacids groups and atoms, f orming the do- mains ( λ1, λ2, λ3) can be estimated. If the volume of an effecton is approximate d by a sphere: Vef=9(λ1λ2λ3) 4π=4 3πr3, (3) then its radius: r=/bracketleftbigg27(λ1λ2λ3) 16π2/bracketrightbigg1/3 = 0.555λres, (4) where: λres= (λ1λ2λ3)1/3, and the diameter: d= 2r= 1.11λres. The collective properties of protein’s primary effectons pr esented by α-structures, β-sheets and whole domains can determine the cooperative pro perties of biopoly- mers. Heat oscillations of atoms and atom groups, forming the prot ein effectons must be coherent, like in any other condensed matte r. Such ideas agree with the Fr¨ ochlich hypothesis about the po ssibility of Bose- condensation in biological systems (Fr¨ ohlich, 1975). The notion of ”knots” in proteins was introduced by R.Lumry a nd B. Gre- gory (1986). The knots are regions, containing very slow H⇔Dexchangeable protons in composition of compact cooperative structures. The dimensions of knots are less than dimensions of domains. It looks that knots could represent a translational effecton s, as far their (a⇔b)trtransitions, in contrast to ( a⇔b)lbones of librational effectons do not accompanied by reorganization of hydrogen bonds. Conseque ntly, the possibility forH⇔Dexchange in knots is more limited. The molecular dynamic computer simulations of proteins rev eal, indeed, a highly correlated collective motion of groups of atoms, inh omogeneously dis- tributed in proteins structure (Swaminathan et al., 1982). It looks that the traditional theory of protein tertiary nat ive structure self- organization from primary one (Cantor and Shimmel, 1980) is not totally suc- cessful as far it does not take into account quantum process, related to formation the protein effectons as 3D standing waves B of protein atoms. 8The change of interdomain interactions, the stabilization of its small-scale dynamics of proteins by ligands leading to the increase of λres(see eq.4) as a measure of cooperativeness, can provide the long- distance signal transmission in macromolecules and allosteric effects in oligomeric prot eins. The [ lb/tr] con- vertons, i.e. dissociation of librational water effectons i n the protein cavities is the key phenomenon in the above mentioned processes. The described events interrelate the small-scale dynamics of atoms and the large-scale dynamics of domains and subunits to the dynamic s of water clusters in protein interdomain cavities, dependent in turn on the pr operties of bulk water (K¨ aiv¨ ar¨ ainen, 1985,1989b, K¨ aiv¨ ar¨ ainen et al. , 1993). Our mesoscopic mechanism of signal transmission in protein s is alternative toDavidov ’s soliton mechanism. The latter is good only for highly ordered systems with small dissipation.. 2. Quantum background of lipid domain organization in biomembranes The importance of lipid domains in membranes for their funct ioning is known, but the physical background for domains origination remains unclear. Our mesoscopic theory was used for computer simulations of l ipid domain dimensions in model membranes. The known data on the positio n of IR bands, corresponding to asymmetric [ N−(CH3)3] stretching in holine for trance (920cm−1) andgauche (900 and 860 cm−1) conformations where taken for calculations. The knowledge of sound velocity and its changes as a result of pha se transitions: (1.97·105cm/s)38o→(1.82·105cm/s)42ois also necessary for calculations of the domain dimensions, using eq.(2.59 of [1]). The result s of calculations are presented on Fig.1. In our approach the lipid domains in b iomembranes and their artificial models are considered as quasiparticle s-primary librational -effectons , formed by 3D superposition of the most probable de Broglie w aves (λ=h/p)1,2,3, determined by coherent thermal oscillations of lipid mole cules. The lesser the value of the most probable impulse ( p=mv)1,2,3of lipid, the bigger is corresponding λ1,2,3and the effecton volume (eq.3). According to our calculations, a rise in temperature from 0 t o 700leads to a decrease in the most probable λfrom 88 to 25 ˚A. In the phase transition region (38−420)λdecreases from 46 to 37 ˚A(Fig.1). The former process corresponds to change of the lipid domains volume from (50 to 2) ·104˚A3and the latter one from (7.5 to 5) ·104˚A3, respectively (Fig. 2). The values of these changes coincid e with available experimental data. Like the calculations we made earlier for ice and water, these results provide further support of our theo retical approach. 9Fig. 1. Temperature dependencies of the most probable de Broglie wave length of lipids ( λ1, λ2, λ3), related to their stretching . The values of λ1, λ2, λ3determine the spatial dimensions of lipid domains. Domains are considered as quasiparticles (primar y effec- tons), formed by 3D superposition of the most probable de Bro glie waves of lipids. Fig. 2. Temperature dependence of the volume of lipid domains: V=9 4πλ1λ2λ3. The volume is determined by 3D superposition of the most prob - able de Broglie waves of lipid molecules or their fragments. Phase transition occurs in the region of 312 K(390) and accompanies by trance - gauche change of lipid conformation. As discussed in Chapter 17 of [1] and my paper: Hierarchic Mod el of Consciousness in URL: [http://kftt.karelia.ru/˜alexk /papers] the mesoscopic organization of biomembranes, related to their dynamics 10and that of microtubules system - may play an important role i n elementary act of consciousness. 3. Mesoscopic approach to theory of solutions The action of the dissolved molecules can lead to the shift of (a⇔b) equilibrium to the right or to the left for effectons (tr and lb ) of solvent. In the former case the lifetime of unstable state fo r primary effectons increases, and in the latter case the stabilizatio n of molecular associates (clusters) takes place. The same is true for convertons equilibrium: [lb⇔tr], reflecting [association ⇔dissociation] of water clusters (primary librational ef- fectons). The effects of stabilization of clusters can be reinforced at such con- centration of dissolved molecules, when the mean distances between them (r) coincide with one of the primary effectons ribs (λ1, λ2, λ3)or their integer number: r=11.8 (CM)1/3(˚A) (5) where C Mis the molar concentration. So the concentration dependence of the stabilizing action b rought by the dissolved substance upon the solvent can be non-monotonic a nd periodic. Such data have been published indeed (Tereshkevitch et al., 1974 ). If the wave B length of the dissolved molecule or atoms exceed s the dimen- sions of primary effectons, then it must increase the degree o f liquid associa- tion. In the opposite case the ordering of liquid structure d ecreases. To prove the aforementioned, it should be noted that the structure-forming ions, with a positive hydration ( Li+, Na+, F−), as a rule, have a lesser mass and lesser im- pulse (i.e. a larger value of λB=h/mv gr) than those with a negative hydration (Rb+, Cs+, Br−, I−) with the same charges. In accordance with such ideas, nonpolar atoms, minimally di storting the properties of a pure liquid, e.g. He, Ne, Ar, have a maximal st ructuring action, stabilizing a-states of primary librational effectons. Our Hierarchic model may be useful for elaboration a general theory of solutions. In host-guest systems a following situations are possible: 1) guest molecules stabilize (a)-states of host effectons (t r and lb) and increases their dimensions. The (a⇔b)equilibrium of the effec- tons and [lb⇔tr]equilibrium of convertons becomes shifted leftward decreasing potential energy of a system, corresponding to i ts stabi- lization effect; 112) guest molecules destabilize (a)-states of host effectons . The (a⇔b)and[lb⇔tr]equilibriums of the primary effectons and conver- tons correspondingly are shifted rightward, inducing gene ral destabi- lization effect of the system; 3) guest and host molecules form separate individual effecto ns (mesophase) without separation in two macrophases; With the increasing concentration of guest in solution of tw o molecular liq- uids (for example, water - ethanol) the roles of guest and hos t may change. It was shown that conductivity of aqueous solutions of NaCl, containing ions: Na+, Cl−, H3O+and OH−show to vary in a different linear fashion over two ranges of temperature: 273 ≤T≤323Kand 323 ≤T≤360K. The change in slope of the plot shows transition in the charac ter of water-ions interaction near 323 K (Roberts et al.1994). In the same work was revealed, that the above aqueous systems exhibited some ”memory” of th e temperature effects after changing the temperature from low to high and th en from high to low values. Such memory could mean a slow relaxation process , accompanied by redistribution between populations of different excitatio ns and their equilibrium constants, i.e. the new type of self-organization in soluti on. In accordance with our theory, all reorganizations of liqui d’s prop- erties must be accompanied by correlated changes in the foll owing parameters of solutions: 1) sound velocity; 2) positions of translational and librational bands in osci llatory spectra; 3) density; 4) refraction index; 5) share and bulk viscosities; 6) coefficient of self-diffusion 7) light scattering; 8) heat capacity and thermal conductivity; 9) vapor pressure and surface tension. There are also a lot of other parameters and properties that s hould change in solutions as a result of solute-solvent interaction. It c ould be revealed by computer simulations, using software, elaborated. The first four of the above listed parameters are present in th e main for- mulas of Hierarchic theory and must be determined under simi lar conditions (temperature, pressure, etc.). There are some experimental data which are in general agreem ent with the consequences of our theory, pointing to interrelation betw een the above listed parameters. The changes of sound velocity in different water - ethanol mixtures as well as that of light and neutron scattering were studied i n detail by D’Arrigo and Paparelli (1988 a,1988b,1989), Benassi et al. (1988), D’Arrigo and Teixeira (1990). 12Correlations between density, viscosity, the refractive i ndex, and the dielec- tric constant of mixed molecular liquids at different temper atures were investi- gated by D’Aprano’s group in Rome (D’Aprano et al., 1989 ,1990a,1990b). The interaction of a solute (guest) molecule with librational solvent effectons can be subdivided into two cases : when the rotational correlation time of a guest molecule ( τrot M) isless (a) andmore (b) than the rotational correlation time of librational effectons (τlb ef): a)τrot M< τlb ef (6) and b)τrot M> τlb ef (7) In accordance with the Stokes-Einstein formula the corresp onding rotational correlation times: τrot M=VM kη Tand τlb ef=Vlb ef k·η T(8) where: ηand T are the share viscosity and absolute temperature of the solvent; τrot Mandτlb efare dependent on the effective volumes of a guest molecule ( VM) and the volume of primary librational effecton: Vlb ef=nlb M(V0/N0) =9 4π(λ1λ2λ3) (9) λ1,2,3are most probable wave B length in 3 selected directions; nlib M- number of molecules in a librational primary effecton, depending on te mperature: in water it decreases from 280 till to 3 in the temperature interval 0 −1000C(Fig. 7a of [1] or Fig. 4a of [2]). When the condition (6) is realized, small and neutral guest m olecules affect presumably only the translational effectons. In the second case (7) guest macromolecules can change the pr operties of both types of effectons: translational and librational and s hift the equilibrium [lb⇔tr] of convertons to the left, stimulating the cluster-format ion. In accordance with our model, hydrophilic interaction is related to the shift of the ( a⇔b) equilibrium of translational effectons to the left. As far the potential energy of the ( a) state ( Va) is less than that in the ( b) state (Vb), it means that such solvent-solute (host- guest) interact ion will decrease the potential and free energy of the solution. Hydrophilic i nteraction does not need the realization of condition (7). Hydrophobic interaction is related to the shift of the ( a⇔b)lbequilib- rium of librational effectons to the right. Such a shift results in the increased 13potential energy of the system. The dimensions of coherent w ater clusters rep- resenting librational effectons under condition (7) may eve n increase. However, the decrease in of entropy (∆ S) in this case is more than that in enthalpy (∆ H) and, consequently, free energy will increase: ∆ G= ∆H−T∆S >0. This is a source of hydrophobic interaction, leading to aggregat ion of hydrophobic particles. Clusterphilic interaction was introduced by the author in 1980 already (K¨ aiv¨ ar¨ ainen, 1980, 1985) to describe the cooperative w ater cluster interaction with nonpolar protein cavities. This idea has got support in the framework of our hierarchic concept. Clusterphilic interaction is related to the left- ward shift of (a⇔b)lbequilibrium of primary librational effectons under condition (7), similar shift of the equilibrium of [lb⇔tr]of convertons and increasing of lb effectons dimensions due to w ater immobilization (eqs.. 1 and 9) The latter effect is a result of decreasing of the rotational c orrelation time of librational effectons and decreasing of the most probable impulses of water molecules (2), related to librations under the effect of gues t particles. Cluster- philic interactions can be subdivided into: 1.Intramolecular -when water cluster is placed in the ”open” states of big interdomain or intersubunit cavities and 2. Intermolecular clusterphilic interaction . Intermolecular cluster- philic interactions can be induced by very different sufficiently big macromolecul es. Clusterphilic interactions can play an important role in th e self-organization of biosystems, especially multiglobular allosteric enzym es, microtubules and the actin filaments. Cooperative properties of the cytoplasm, formation of thixotropic structures, signal transmission in biopoly mers, mem- branes and distant interactions between different macromol ecules can be mediated by both types of clusterphilic interaction s. From (4.4 of [1, 2]) the contributions of primary translatio nal and librational effectons to total internal energy are: Utr,lb ef=V0 Z/bracketleftbig nef/parenleftbig Pa efEa ef+Pb efEb ef/parenrightbig/bracketrightbig tr,lb(10) The contributions of this type of effectons to total kinetic e nergy (see 4.33) are: Ttr,lb ef=V0 Z/bracketleftBigg nef/summationtext3 1(Ea)2 1,2,3 2m(Va ph)2/parenleftbig Pa ef+Pb ef/parenrightbig/bracketrightBigg tr,lb(10a) Subtracting (10a) from (10), we get the potential energy of p rimary effectons: Vef=Vtr+Vlb= (Uef−Tef)tr+ (Uef−Tef)lb (11) 14Clusterphilic interaction and possible self-organization is promoted mainly bydecreasing of V lbin the presence of macromolecules. Hydrophilic interaction, in accordance with our model, is a result of Vtr. decreasing. On the other hand, hydrophobic interaction is a consequence of VlbandVtr increasing in the presence of guest molecules. Clusterphilic interaction has been revealed, for example, in dependencies of freezing temperature ( Tf) for buffer solutions of polyethyleneglycol (PEG) on its molecular mass and concentration (K¨ aiv¨ ar¨ ainen, 198 5). The anomalous in- creasing of Tfin the presence of PEG with molecular mass more than 2000 D and at concentration less than 30 mg/ml , pointing to increasing water activity, were registered by the cryoscopy method. It may be explained as a result of clusterphilic interaction increasing, when the fraction o f ice-like water structures with saturated hydrogen bonds, presented by primary librat ional effectons, in- creases. Big macromolecules and small ions should have the o pposite: positive and negative effects on the stability and volume of primary li brational effectons. Macromolecules or polymers with molecular mass less than 20 00 do not sat- isfy the condition (7) and can not stimulate the growth of lib rational effectons. On the other hand, a considerable increase in the concentrat ions of even big polymers, when the average distance between them (eq.5) com es to be less than the dimensions of a librational water effecton, perturbs clu sterphilic interactions and decreases freezing temperature, reducing water activi ty (K¨ aiv¨ ar¨ ainen, 1985, Fig.82). In general case each guest macromolecule has two opposite eff ects on clusterphilic interactions. The equilibrium between th ese tenden- cies depends on the temperature, viscosity, concentration of a guest macromolecule, its dynamics and water activity. When solute particles are sufficiently small they can associa te due to distant Van der Waals interaction, forming coherent p rimary effectons, when [solute-solvent] interaction is more prefe rable than [solvent-solvent] and the conditions (1 and 2) are fulfilled . Important confirmation of this consequence of mesoscopic th eory of complex systems is the observation of compact clusters of ions even in dilute salt solutions. For example, the extended x-ray ab sorption fine structure data showed that the average distance between Zn2+and Br−is2.37˚Ain 0.089M ZnBr 2aqueous solutions and 2.30˚Ain 0.05 M solutions (Lagarde et al., 1980). These values are very close to the inter ionic distance obser ved in the crys- talline state (2 .40˚A). The same conclusions was reached for NiBr 2ethyl acetate solutions (Sadoc et al., 1981) and aqueous CuBr 2solutions (Fontaine et al.,1978). The average distance between ions for the case of monotonic spatial dist ribution (see eq.5) are much bigger than in ionic clusters under experimental co nditions. Our theory of solutions consider formation of crystallites (inor- ganic ionic clusters), as a solute coherent primary effecton s self- organization. It is more favorable process, than separate i on-water 15interactions. For quantitative application of hierarchic theory to the de scription of the processes in different solutions, using our special so ftware, it is necessary to have four experimental parameters, obtained u nder the same conditions (temperature and pressure): - density, - sound velocity, - positions of translational and librational bands in oscil latory spec- tra, - refraction index. The Combined analyzer of matter properties (CAMP), propose d by us (see URL: http://kftt.karelia.ru/˜alexk), could be t he main tool for such a measurements and study of complicated proper ties of solutions and colloid systems. The multi-fractional nature and properties of interfacial water, leading from hierarchic theory We can present here a classification and description of four i nter- facial water fraction properties, based on our hierarchic m odel: 1. The first fraction - primary hydration shell with maximum energy of interaction with surface. The structure and dynam ics of this 1st fraction can differ strongly from those of bulk water . Its thickness: (3-10 ˚A) corresponds to 1-3 solvent molecule; 2. The second fraction - vicinal water (VW) is formed by elong ated primary lb effectons with saturated hydrogen bonds. It is a re sult of lb effecton adsorption on the primary hydration shell from th e bulk volume. The thickness of this fraction of interfacial water : (30-75 ˚A) corresponds to 10-25 molecules and is dependent on temper ature, dimensions of colloid particles and their surface mobility . VW can exist in rigid pores of corresponding dimensions; 3. The third fraction of interfacial water - the surface-sti mulated Bose-condensate (SSBC), represented by 3D network of prima ry lb effectons has a thickness of (50-300 ˚A). It is a next hierarchical level of interfacial water self-organization on the surface of se cond fraction (VW). The time of gradual formation of this 3D net of linked to each other coherent clusters (strings of polyeffectons), is much longer than that of VW and it is more sensitive to temperature and oth er perturbations. The second and third fractions of interfaci al water can play an important role in biological cells activity; 4. The biggest and most fragile forth fraction of interfacia l water is a result of slow orchestration of bulk primary effectons in th e volume of primary (electromagnetic) lb deformons. The primary def ormons 16appears as a result of three standing IR photons (lb) interce ption. Corresponding IR photons are superradiated by the enlarged lb effec- tons of vicinal water .The linear dimension of primary IR deformons is about half of librational IR photons, i.e. 5 microns (5 ·104˚A). This ”superradiation orchestrated water (SOW)” fraction easil y can be destroyed not only by temperature increasing, ultrasound a nd Brow- nian movement, but even by mechanical shaking. The time of sp on- taneous reassembly of this fraction after destruction has a n order of hours and is dependent strongly on temperature, viscosity a nd dimen- sions of colloid particles. The processes of self-organiza tion of third (SSBC) and forth (SOW) fractions can be interrelated by feed back interaction. Consequences and predictions of new model of interfacial so lvent structure In accordance to generally accepted and experimentally pro ved models of hydration of macromolecules and colloid particles, we assu me that the first 2-3 layers of strongly bound water molecules, serves like inter mediate shell, neutral- izing or ”buffering” the specific chemical properties of surf ace (charged, polar, nonpolar, etc.). Such strongly bound water can remain parti ally untouched even after strong dehydration of samples in vacuum. This first fraction of interfacial water serves like a matrix for second fraction - vicinal water shell formation. The therm ’paradoxal effect’ , introduced by Drost-Hansen (1985) means that the properties of vicinal wa ter are independent on specific chemical structure of the surface from quartz pla tes, mineral grains and membranes to large macromolecules (Clegg and Drost-Han sen, 1991). This can be a result of ”buffering” effect of primary hydration shel l. Vicinal water (VW) is defined as a water the structure of which is mod- ified in the volume of pores, by proximity to curved and plate i nterfaces and interaction with strongly ’bound’ water. For discussion of Vicinal water (VW) properties we proceed f rom the statement that if the rotational correlation time of hyd rated macromolecule is less than that of primary librational effec ton of bulk water, corresponding to condition (1.3 and 1.4), these effectons should have a tendency to ”condensate” on their hydration sh ell. It is a condition of their ”surviving” and life-time increasin g, because the resulting impulse of the primary effectons is close to zer o, in accordance to our model. The decreasing of most probable lb thermal impulses of water molecules, es- pecially in direction, normal to the surface of macromolecu le or colloid particle, should lead to increasing of corresponding edges of the 1st ” ground” layer of primary lb effectons as compared to the bulk ones: [λ=h/mv gr]vic lb>[λ=h/mv gr]bulk lb(20b) 17This turns the cube-like shape of effectons of the bulk water t o shape of elongated parallelepiped.. Consequently, the volume of these waves B three-dimensiona l (3D) super- position, representing the vicinal lb andtr primary effectons - is bigger than that of bulk liquid effectons. As far the stability and life-t ime of these enlarged primary lb effectons are increased, it means the increasing o f their concentration as well. As far we assume in our multifractional model, that VW is a result of ”condensation” of primary librational effectons on primary hydration shell and their elongation in direction, normal to surface, we can mak e some predictions, related to properties of this 2nd interfacial water fractio n: 1. The thickness of VW can be about 30-75 ˚A, depending on properties of surface (geometry, polarity), temperature, pressure and p resence the perturbing solvent structure agents; 2. This water should differ by number of physical parameters f rom bulk water. It can be characterized by: a) lower density; b) bigger heat capacity; c) bigger sound velocity d) bigger viscosity; e) smaller dielectric relaxation frequency, etc.). These differences should be enhanced in a course of third frac tion of interfacial water formation: surface-stimulated Bose- condensate (SSBC) as far the concentration of primary librational effec tons in this fraction is bigger than that in bulk water. The time, necessary for SSBC three-dimensional structure self-organization can h ave an order of minutes or hours, depending on temperature, geometry of surface and average distance between interacting surfaces. From Fig.4b we can see that the linear dimension of primary li brational ef- fecton of bulk water at 250Cis about [ λ]bu lb˜ 15˚A. The lower mobility of water molecules of vicinal water is confirmed directly by almost 10 -times difference of dielectric relaxation frequency (2 ·109Hz) as respect to bulk one (19 ·109Hz) (see Clegg and Drost-Hansen, 1991). The consequence of less mobi lity and most probable impulse (momentum) of water molecules should be th e increasing of most probable wave B length and dimensions of primary effecto ns.The en- hancement of lb primary effecton edge should be more pronounc ed in the direction, normal to the interface surface. In turn, s uch elon- gation of coherent cluster can be resulted in increasing the intensity of librational IR photons superradiation. The increasing of temperature should lead to decreasing the vicinal libra- tional effectons dimensions. The dimensions of primary translational effectons of water i s much less than of librational ones (see Fig. 4a). The contribution of trans lational effectons in vicinal effects is correspondingly much smaller than of libr ational. 18Our model predicts that not only dimensions but as well conce ntrations of primary librational effectons should increase near rigid su rfaces. The system of vicinal 3D polyclustrons can emerge. In accordance to our model the Drost-Hansen thermal anomali es of vicinal water behavior near 150,300,450and 600has the same explana- tion as presented in comments to Fig. 4a for bulk water. Becau se the dimensions, stability and concentration of vicinal librat ional effectons are bigger than that of bulk effectons the temperature of anom alies for these two water fraction can differ also. As far the positions of disjoining pressure sharp maxima of w ater between quartz plates did not shift markedly when their sepa ration change from 100 ˚A to 500 ˚A it points that the main contribution in temperature anomalies is related to bulk librational effect ons. The same is true for viscosity measurements of water between pla tes with separation: 300-900 ˚A. The possible explanation of Drost-Hansen temperature valu es sta- bility is that the vicinal water layer (50-70 ˚A) has the bigger dimen- sions of primary librational effectons edge, than that of bul k water, presumably only in the direction, normal to the surface of th e in- terface. Such first layer of surface - stimulated Bose-conde nsation induces the formation of string-like librational polyeffec tons, com- posed from elongated in the same direction bulk primary effec tons and stabilized by Josephson’s junctions and distant Van-de r-Waals interactions. These strings can be cross-linked by short ”c hains” of primary lb effectons, making such 3D polyeffecton net more sta ble as respect to Brownian motion. We can term this fraction of in terfa- cial water as ”Surface-stimulated Bose-condensate (SSBC) ”. As far in such mesophase the linear dimensions of two of three edges of primary lb effectons, approximated by parallelepiped, remains the s ame as in the bulk water, it explains the relative stability of therma l anomalies at temperature scale. It means the similar conditions of max imum stability for primary lb effectons (see eq.20c), in composit ion of SSBC and calculated for bulk water. For the other hand, the elongated structure of primary lb effe ctons, com- posing SSBC, should increase all the effects, related to inte nsive and coherent superradiation of lb IR photons in directions of the longest edge of the effec- tons (see Introduction). This largest and most subtle fraction of inter- facial water also has more orchestrated cooperative proper ties than the unperturbed by superradiation bulk water due to stronge r dis- tant Van-der-Waals interaction between primary lb effecton s. Such a ”superradiation-orchestrated water (SOW)” is even less o rdered and stable than the second fraction of interfacial water: su rface- stimulated Bose-condensate (SSBC). Its 3D structure of SSBC and SOW could be easily destroyed by mechanical perturbation or heating. The relaxation time of ”regener- ation” or self-organization of these water fractions can ta ke an hours 19and depends on temperature and mobility of the surface. The sharp conditions of maximum stability of librational primary effectons at certain temperatures - the integer number of molecules in the edge of lb effecton: κ=6, 5, 4, 3, 2 (see comments to Fig.4a) (20c) are responsible for deviations of temperature dependencie s for many parameters of water from monotonic ones. Vicinal water (VW) and surface-stimulated Bose-condensat ion of water near biological membranes and filaments (microtubule s, actin polymers) can play an important regulation role in cells and their compartments dynamics and function. Its highly cooperativ e prop- erties and thermal sensitivity near Drost-Hansen temperat ures can be used effectively in complicated processes, related to cel ls prolifer- ation, differentiation and migration. Changing of water act ivity: its increasing as a result of VW and SSBC deassembly and decreasi ng in a course of it 3D structure self-organization, can affect str ongly the dynamic equilibrium [association ⇋dissociation ]of oligomeric pro- teins, their allosteric properties and osmotic processes i n cells. COMPARISON OF EXPERIMENTAL DATA WITH THEORETICAL PREDICTIONS It will be shown below that our hierarchic model of interfaci al water explains the comprehensive and convincing experimen tal data, available on subject. The following experimental results, illustrating the diffe rence be- tween the second (VW) and third (SSBC) fractions of interfac ial and bulk water will be discussed: 1. The lower density of vicinal water near plates and in pores (˜0.970g/cm3)as compared to bulk one (˜1.000 g/cm) (Etzler and Fagundus, 1983; Low, 1979; Clegg 1985); 2. Different selectivity of vicinal water in pores to structu re- breaking and structure-making inorganic ions; 3. Volume contraction on sedimentation of particles disper sed in water (Braun et al. 1987); 4. Higher heat capacity of vicinal water as compared to bulk w ater (Etzler 1988); 5. Higher ultrasound velocity in the interfacial water; 6. Higher ultrasound absorption in the interfacial water; 7. Higher viscosity of interfacial water (Peshel and Adlfing er 1971) and its dependence on shearing rate; 208. Sharp decreasing of the effective radius of dilute solutio n of polysterone spheres, in a course of temperature increasing . In accordance to our model, the water molecules in compositi on of primary librational effectons (see Fig. 4a) are four-coordinated li ke in ideal ice structure with lowest density. In contrast to that, the water in the vol umes of translational effectons, [lb/tr] convertons and lbandtrmacroeffectons and supereffectons has the nonsuturated hydrogen bonds and a higher density. The co mpressibility of primary [lb] effectons should be lower and sound velocity - hi gher than that of bulk water. It is confirmed by results of Teixeira et. al. (198 5), obtained by coherent- inelastic- neutron scattering. They point on exi stance in heavy water at 250C the solid-like collective excitations with bigger sound v elocity than in bulk water. These experimental data can be considered as a di rect confirmation of our primary librational effectons existence.. As far the fraction of water involved in primary librational effectons in vicinal water is much higher than in the bulk one, this explains the re sult [1] at the list above. The biggest decreasing of density occur in pores, con taining enlarged primary librational effectons, due to stronger water molecu les immobilization. Different selectivity of vicinal water in pores of silica gel (result [2]) to structure-breaking and structure-making inorganic ions ( Wiggins 1971, 1973), leading to higher concentration of the former (like K+) as respect to latter (like Na+) ones in pores was revealed. It is in total accordance with ou r model as far for penetration into the pore the ion have at first to break the ordered struc- ture of enlarged librational effectons in the volume of pore. Such kind of Na/K selectivity can be of great importance in the passive transp ort of ions throw the pores of biological membranes. Result [3] of volume contraction of suspension of 5- µmsilica particles in a course of their sedimentation - is a consequence of mechanic al perturbation of cooperative and unstable 3D fraction: surface - stimulated Bose condensation (SSBC), its partial ’melting’ and increasing of water fract ion with nonsuturated hydrogen bonds and higher density. The available experimental data of the vicinal water thickn ess evaluate it as about 50-70 ˚A (Drost-Hansen, 1985). In totally or partly closed volumes like in silica pores the vicinal effect must be bigger than near the pl ain surface. This explains the maximum heat capacity of water at 250(result [4]) in silica pores with radii near 70 ˚A (Etzler and White, 1988). As far the cooperative properties of 2nd and 3d fractions of i nterfacial water, corresponding to vicinal water and surface - stimulated Bos e condensation are higher than that of bulk water, it explains the bigger heat ca pacity of both of these fractions. The higher sound velocity in the VW and SSBC fractions as comp ared to bulk water (result [5]) is a consequence of higher concentra tion of primary li- brational effectons with low compressibility due to saturat ed H-bonds. 21Higher absorption of ultrasound by interfacial water (resu lt [6]) can be a con- sequence of dissipation processes, accompanied the destru ction of water fraction, corresponding to SSBC by ultrasound. The higher viscosity of vicinal water (result [7]) is a resul t of higher activation energy of librational macrodeformons excitation/parenleftbig εM D/parenrightbig lbin a more stable system of vicinal polyclustrons (see our mesoscopic theory of visc osity). Sharp decreasing of the effective dimensions of dilute solut ion of polysterone (PS) spheres (0.1%), in a course of temperature increas- ing (result [8]) - is a consequence of cooperative destructi on (melting) of 3d fraction of interfacial water (SSBC). The corresponding transition occur at 30-340C, as registered by Photon correlation spectroscopy and is accompanied by the effectiv e Stocks radius of PS spheres decreasing onto 300 ˚A. The ”regeneration time” of this process is about 20 hours. It may include both: time of SAPS self-organization and self-orga nization of the less stable forth interfacial fraction: ”Water, orches trated in the volume of IR primary deformons”. The frequent non-reproducibility of results, related to pr operties of interfa- cial water, including Drost-Hansen temperature anomalies , can be resulted from different methods of samples preparation and experimental c onditions. For example, if samples where boiling or strongly heating ju st before the experiment, the 3d and 4th fractions of interfacial water ca n not be observable. The same negative result is anticipated if the colloid syste m in the process of measurement is under stirring or intensive ultrasound radi ation influence. Nonobserved yet predictions, related with third (SSBC) and forth (water, orchestrated by IR standing photons) fractions of n ew interfacial model 1. Gradual increasing of pH of distilled water in a course of t hese fractions formation near nonpolar surface - due to enhancement of prob ability of superde- formons excitation. Corresponding increasing of concentr ation of cavitational fluctuations are accompanied by dissociation of water molec ules: H2O⇋H++HO− and the protons concentration elevation; 2. Increasing the UV and visible photons spontaneous emissi on near nonpo- lar surface as a result of increasing of the frequency of wate r molecules recom- bination: H++HO−→H2O+photons These experiments should be performed in dark box, using sen sitive photon counter or photo-film. 22The both predicted effects should be enhanced in system, cont aining parallel nonpolar multi-layers, with distance ( l) between them, corresponding to condi- tions of librational IR photons standing waves formation: l= 5,10,15,20 microns. 3. More fast and ordered spatial self-organization of macro molecules, like described in the next section is anticipated also in the volu me of forth fraction of interfacial water. The dynamics of such process can be registered by optical con focal or tun- neling microscope. 4. Our model predicts also that the external weak coherent el ectromagnetic field, generated by IR laser, like the internal one, also can s timulate a process of self-organization in colloid systems. 4. Distant solvent-mediated interaction between macromol ecules The most of macromolecules like proteins can exist in dynami c equilibrium between two conformers (A and B) with different hydration ( nH2O) and flexi- bility: A+nH2O⇔B (11 a) Usually correlation time of more hydrated B- conformer ( τB) and its effec- tivevolume are less than that of A-conformer ( τA) : τA,B=VA,B k(η/T) τA> τBand VA> VB This means that flexibility, determined by large-scale dyna mics of B-conformer is more than that of A-conformer. For such case a changing of the bulk water activity ( aH2O) in solution by addition of another type of macromolecules or by inorganic i ons induce the change of the equilibrium constant: KA⇔B= (KB⇔A)−1and the dynamic behavior of macromolecules (K¨ aiv¨ ar¨ ainen, 1985, 1986) : ∆lnKB⇔A=nH2O·∆lnaH2O (12) In mixed systems: [PEG + spin-labeled antibody] the depende nce of large- scale (LS) dynamics of antibody on the molecular mass of polyethylengl ycol (PEG) is similar to dependence of water activity and freezin g temperature ( Tf) of PEG solution, discussed above (K¨ aiv¨ ar¨ ainen, 1985, Fi g. 82). The presence of PEG with mass and concentration increasing ( aH2O) and (Tf) stimulate the LS-dynamics of proteins decreasing their eff ective volume V and correlation time τMin accordance to eq.(12). 23If ∆Tf=T0 f−Tfis the difference between the freezing point of a solute (T0 f) and solution ( Tf), then the relation between water activity in solution and ∆ Tfis given by the known relation: lnaH2O=−/bracketleftBig ∆H/R/parenleftbigT0 f/parenrightbig2/bracketrightBig ·∆Tf (13) where ∆H is the enthalpy of solute (water) melting; R is the ga s constant. In our experiments with polymer solutions 0.1 M phosphate bu ffer (pH7.3+ 0.3MNaCl) was used as a solvent (K¨ aiv¨ ar¨ ainen, 1985, Fig. 82). One can see from (13) that the negative values of ∆ Tfin the presence of certain polymers means increasing water activity in three c omponent [water - ions - polymer] system (∆ln aH2O>0). In turn, it follows from (12) that the [A⇔B] equilibrium of guest proteins in the same system will shift to the right. Consequently, the flexibility of the proteins will increase . Correlation between Tf, water activity ( aH2O) and immunoglobulin flexibility ( τM), corresponding to (11 - 13) was confirmed (K¨ aiv¨ ar¨ ainen, 1985, Table 13). It was shown in our experiments that protein-protein distan t inter- action depends on their large-scale (LS) dynamics and confo rmational changes induced by ligand binding or temperature (K¨ aiv¨ ar ¨ ainen, 1985). Such distant solvent-mediated effects may be explained usin g our idea of inter- and intramolecular clusterphilic interacti ons, discussed above. We use the assumption that between inter- and intra-cluster philic interac- tion the positive correlation exist. It means that increase of dimensions and stabilization of the librational bulk (inter) water effecto ns induce an increase of the water clusters dimensions in protein cavities and shift A⇔Bequilibrium of the cavities to the right, i.e. to the more flexible conform er. This leads to decreasing of resulting correlation time and effective volu me of protein. Our interpretation is confirmed by fact that a decline in temp erature, in- creasing the dimensions of the bulk librational effectons ha s the stimulating influence on the flexibility of protein like the presence of gu est macromolecules. Decreasing of the temperature shifts the equilibrium betwe en hydrophobic and clusterphilic interaction to the latter one. Atlow concentration of macromolecules (CM), when the average distance (r) between them (eq.5) is much more than dimensions of prima ry librational water effecton ( r≫λlb), the intermolecular clusterphilic interaction does not work effectively. In this case the large-scale [ A⇔B] pulsations of proteins, accompanied by acoustic impulses in solvent can enhance the activity of water. This dynamic effect of proteins on solvent can be responsible for the dis- tant solvent- mediated interaction between macromolecule s at low concentration (K¨ aiv¨ ar¨ ainen, 1985, 1987). Acoustic impulses in protein solutions are result of the jum p-way B→A transition of interdomain or intersubunit cavities with ch aracteristic time about 2410−10sec. This rapid transition follows the cavitational fluctua tion of a wa- ter cluster formed by 30 - 50 water molecules. The fluctuation is a result of conversion of ( lb) primary effecton to ( tr) one: [ lb/tr] converton excitation. The hydrodynamic Bjorkness interaction between different type and identical proteins can be induced by acoustic wave packets i n solvent, stimu- lated by large-scale pulsations of proteins (K¨ aiv¨ ar¨ ain en et al., 1988). This new approach was used for estimation of frequency of LS- pulsati ons of interacting proteins like (immunoglobulins) as/parenleftbig 104−105/parenrightbig s−1. In very concentrated solutions of macromolecules, when the distance between macromolecules starts to be less than linear dimension of pr imary librational effecton of water: r≤λlb,the trivial aggregation process begins to dominate. It is re lated to the decrease of water activity in solutions. Let us analyze in more detail the new effect - increasing water activity ( aH2O) under the effect of macromolecules in a three component [wate r - salt - macro- molecules] system. The Gibbs-Duhem law for this case can be p resented as (K¨ aiv¨ ar¨ ainen, 1988): XH2O∆lnaH2O+XM∆¯µM RT+Xi∆lnai= 0 (14) where XH2O, XM, Xiare the molar fractions of water, macromolecules and ions in the system; aj=yjXj= exp/parenleftbigg −µ0−µj RT/parenrightbigg = exp/parenleftbigg −∆µj RT/parenrightbigg (15) is the activity of each component related to its molar fracti on (X j) and coeffi- cients of activity (y j); ∆µM≃(GB−GA)∆fB (16) - the change of the mean chemical potential (¯ µM) of a macromolecule (protein) pulsing between A and B conformers with corresponding parti al free energies ¯GAand¯GB, when the change of B-fraction is ∆ fBand µM∼=fBGB+fAGA (17) ∆lnai= (∆ai/ai)≃ −∆κi (18) where κi= 1−yi (19) 25is the fraction of thermodynamically excluded ions (for exa mple, due to ionic pair formation). One can see from (15) that when aH2O<1, it means that µ0 H2O> µS H2O=HS H2O−TSS H2O (20) It follows from (20) that the decrease of water entropy ( ¯S) in solution related to hydrophobic and clusterphilic interactions may lead to i ncreased µS H2Oand water activity. It is easy to see from (14) that the elevation of concentratio n and XMof macromolecules in a system at constant temperature and ∆¯ µMmay induce a rise in water activity ( aH2O) only if the activity of ions ( ai=yiXi) is decreased. The latter could happen due to increasing of fraction of ther modynamically excluded ions ( κ) (eqs. 18 and 19). There are two processes which may lead to increasing the probability of ionic pair formation and fraction κelevation. Thefirstone is the forcing out of the ions from the ice-like structure of en- larged librational effectons, stimulated by the presence of macromolecules. This exclusion volume phenomenon increases the effective concen tration of inorganic ions and their association probability. Thesecond one dominates at the low concentration of [ A⇔B] pulsing macromolecules, when thixotropic structure fail to form ( r= 11.8/C1/3 M≫λlb). Acoustic impulses in solvent generated by pulsing proteins stimulate the fluc- tuation of ion concentration (K¨ aiv¨ ar¨ ainen, 1988) incre asing the probability of ionic pairs formation. 5. Spatial self-organization in the water-macromolecular systems The new type of self-organization in aqueous solutions of bi opolymers was revealed in Italy (Giordano et al., 1981). The results obtai ned from viscosity, acoustic and light-scattering measurements showed the exi stence of long-range structures that exhibit a thixotropic behavior . This was shown for solutions of lysozyme, bovine serum albumin (BSA), hemoglobin and DNA. O rdered struc- ture builds up gradually in the course of time to become fully developed after more than 10-15 hours. When a sample is mechanically shaken this type of self-organ ization is de- stroyed. The ”preferred distance” between macromolecules in such an ordered system is about L≃50˚Aas revealed by small angle neutron scattering (Gior- dano et al., 1991). It is important that this distance can be much less than the average statistical distance between proteins at l ow molar concentrations (CM) (see eq.5). This fact points to attraction force between macromolecule s. In accordance with our mesoscopic model, this attraction is a r esult of external intermolecular clusterphilic interaction. It has been shown ex- perimentally that hypersonic velocity in the ordered thixo tropic structures of 2610% lysozyme solutions is about 2500 m/s, i.e. 60% higher than that in pure wa- ter (1500 m/s) (Aliotta et. al., 1990) Quasielastic and elastic neutron s cattering in 10% lysozyme aqueous system at 200shows that the dynamic properties of the ”bound” water in a thixotropic system are similar to the p roperties of pure water at 3 −40C, i.e. 160Cbelow actual temperature (Giordano et al., 1991). Experimental evidence for heat capacity increasing in lyso zyme solutions dur- ing 10-15 hours of self-organization was obtained by adiaba tic microcalorimetry (Bertolini et al., 1992). The character of this process is pr actically independent on pH and disappears only at the very low concentration of pro tein (<0.2%), when the average distance between macromolecules becomes t oo big. Increasing the temperature above 400also inhibits thixotropic-type of self-organization in water-macromolecular systems. In the series of experiments with artificial polymers - polye thyleneglycols (PEG) with decreasing molecular mass it was shown that self- organization (SO) in 0.1 M phosphate buffer ( pH7.3,250) exists in the presence of PEG with a weight of 20.000, 10.000 and 8.000 daltons, but disappears a t a molecular mass of 2.000 and lower (Salvetti et al., 1992). These data are in good agreement with the influence of PEG on th e freezing temperature and intramolecular interaction described abo ve. Independently of Italian group, similar ordering process w ere observed in Japan (Ise and Okubo, 1980) for aqueous solutions of macroio ns. The process of ”compactization” or ”clusterization” of sol ute (guest) molecules in one volumes of solution or colloid system, should lead to e mergency of voids in another ones. Such a phenomena were revealed by means of co nfocal laser scanning microscope, ultramicroscope and video image anal yzer (Ito et al.,1994; Yoshida et al.,1991; Ise et al.,1990). Inhomogeneity of guest particles distribution were reveal ed in different ionic systems, containing ionic polymers or macroions, like sodi um polyacrilate, the colloid particles, like polysterone latex ( N300,1.3µC/cm2) and Langmuir-Blodgett films. The time evolution of the numbers of different clusters from such particles were followed during few hours. The colloid crystal growth at 250C inH2Oand(H2O−D2O)systems can be divided on four stages (Yoshida et all.,1991). In the first stage the particles were diffusing freely. In the second stage the local concentration of the particles took place. In the third stage clusters from 3-10 particles were formed. In the last fourth stage the smaller clusters turns to the big ger ones and macroscopically well ordered structures were form ed. Si- multaneously the huge voids as large as 50−150µmwere observed. Such an ordering can be immediately destroyed by mechanical shaking or by adding of inorganic salts (NaCl), even in such relatively sm all concentrations as 10−4mol/dm3. Authors conclude that the repulsion as only one assumption i s not enough for explanation of the phenomena observed. The model, consi dering a short- 27range repulsive interaction and long-range attraction between particles should be used. They try to explain attraction between similar char ged macroions by presence of small inorganic counterions. Authors believ e that short range repulsion can be overwhelms by attraction. But such a simple model does not explain ”clusterization” of electrically neutral gues t macromolecules and acceleration of this process due to decreasing the concentr ation of inorganic salts in presence of ionic sorbents. Despite a large amount of different experimental data and gre at importance, the mechanism of spontaneous type of self-organization in c olloid systems re- mains unknown. It is evident, however, that it can not be attr ibuted to trivial aggregation. Our explanation of distant attraction between macromolecu les, by means of clusterphilic interaction and polymerization of clustron s in the volume of pri- mary electromagnetic deformons (see below) is more adequat e for description of self-organization phenomena in aqueous colloid systems , than counterion hy- pothesis. Just clusterphilic interactions determines the attractio n between large guest molecules or colloid particles on mesoscopic sc ale. They are responsible for the lot of vicinal water effects describe d below. Our computer calculations of water properties based on the m esoscopic the- ory show that the volume of primary librational effecton of wa ter at 250includes about 100 water molecules (Fig. 7a of [1] or Fig. 4a of [2]). In accordance with condition (7) for clusterphilic interac tions, the macro- molecules with volume VM>100·(V0/N0) can decrease the librational mobility of H2Omolecules, their impulses and, consequently, (eqs. 1 and 9) increase the dimensions of the l ibrational effectons of water. Obviously, there must exist a direct correlation b etween the effective Stokes radius of a macromolecule (i.e. its molecular mass) a nd the process of self-organization. The mass of a lysozyme (Lys) is about 13.000 D, and available experimental data (Giordano et al., 1991) show that the mobi lity of water in a hydration shell in composition of thixotropic s ystem at250is about 3 times less than that of pure water. This mobility is directly related to the most probable group velocity and i mpulse ofH2O(mvgr). It means a three-fold increase in the dimensions of librational effectons in the presence of lysozyme (see eq. 1) . [λH2O≈15˚A]→[L(H2O≈45˚A] (20a) This value is quite close to the experimental preferred distance (50o A) between proteins after self-organization (SO). Because the shape o f the effectons can be approximated by parallelepiped or cube, we suggest that eac h of its 6 sides can 28be bordered and stabilized by at least one macromolecule (Ta ble). We termed corresponding type of quasiparticles ”clustrons”. Enlarged librational effectons serve as a ”glue”, promoting interaction between surroundi ng macromolecules or colloid particles. Probability of librational effectons di ssociation to translational ones, i.e. [ lb/tr] convertons excitation,- is decreased in composition of cl ustrons. Cooperative water clusters in the volume of clustrons are ve ry sensitive to perturbation by temperature or by ion-dipole interactions . When the colloid particles have their own charged groups on the surface, very small addition of inorganic salts can influence on clustrons formation. Our model can answer the following questions: 1. Why polymers with a molecular mass less than 2000 D do not stimulate self-organization ? 2. Why is self-organization inhibited at sufficiently high te mpera- ture(>400)? 3. What causes the repulsive hydration force? The answer to the first question is clear from condition (7) and eqs.(8). Small polymers can not stimulate the growth of librational e ffectons and clus- trons formation due to their own high mobility. The response to the second question is that a decrease in the dimensions of librational effectons with temperature (Fig. 7 of [1]) lea ds to deterioration of their cooperative properties and stability. The probabili ty of clustrons formation drops and thixotropic structures can not develop. Therepulsion, like attraction between colloid particles can be explained by clusterphilic ineraction.. The increasing of external p ressure should shift the (a⇔b)lbequilibrium of librational water effectons to the left, like temperature decreasing. This means the enhancement of librational wate r effecton (and clustron) stability and dimensions. For this reason a repulsive or disjoining hydration force arises in different colloid systems. The swelling of clays ag ainst imposed pressure also is a consequence of clusterphilic int eractions enhancement, i.e. enlargement of water librational effectons due to H2Oimmobilization. In accordance with our model, effectons and their derivative s,clustrons , are responsible for self-organization in colloid systems on a mesoscopic level (Stage II in Table). Themacroscopic level of self-organization (polyclustrons), responsible for the increase of viscosity and long-distance interaction be tween clustrons origi- nates due to interaction between the electric and magnetic d ipoles of clustrons (Stage III, Table). Because the thermal movement of the H2Omolecules, com- posing clustrons is coherent, the dipole moment of clustron is proportional to the number of H2Oin its volume. 29Table. Schematic representation of three stages ( I→II→III) of gradual spatial self-organization in aqueous solutions of macro- molecules ( M). Formation of polyclustrons (Stage III, Table) as the space- time correlated large group of clustrons can be stimulated in the volume of primary electromagnetic deformons (tr and lib), having a lmost macroscopic dimensions. Clustrons (stage II) are complexes of enlarged primary libr ational effectons, bordered by each of 6 side with macromolecules. Th e in- crease of librational effecton dimensions (λ→L)is related to de- creased water molecule mobility due to external clusterphi lic inter- action. Polyclustrons (stage III) are space and time correlated sys tems of a large group of clustrons in the volume of electromagnetic de formons. 30Their linear dimensions (0.01−0.001cm)correspond to translational and librational wave numbers: ˜νtr≃60cm−1and ˜ νlb≃700cm−1 We suppose that for optimal process of self-organization, t he molar concentra- tion of macromolecules ( CM) must satisfy the condition: 10L >[r= 11.8·C−1/3 M]≥L, where: L∼=qλlb=h/m(vlb/q) is the dimension of a clustron ( λ1,2,3 lb= h/mv1,2,3 lbis the dimension of primary librational effecton in pure wate r;r- statistically most probable distance between macromolecu les (eq.5);qis the lib effecton magnification number, reflecting the effect of water m olecules immo- bilization (decreasing of their group velocity) in presenc e of macromolecules or colloid particles. Under condition r >> 10L, formation of clustrons can be accompanied by big voids in colloid system. At the opposite limit condition r < L , the triv- ial aggregation will dominate, accompanied by ”melting” of vicinal librational effectons. The librational IR photon wave length, calculated from the o scillatory spec- tra of water, is about (eq. 1.37): λp= (n·˜ν)−1≈(1.33·700)−1≈0.001cm where ˜ ν≃700cm−1is the wave number of librational band in the oscillatory spectra of water; n= 1.33 is the water refraction index. The 3D superposition (interception) of such three photons f orms primary electromagnetic deformons, stimulating development of po lyclustron system. Our model predicts that the external electromagnetic field, like internal one, also can stimulate a process of self-organiza tion in col- loid systems. On the other hand, the agents perturbing the st ructure of librational effectons (temperature, ions, organic solve nts) should have the opposite effect. The slowest stage of self-organiza tion, poly- clustrons and polyclustron net formation (thixotropic -li ke structure), is also sensitive to mechanical shaking. Due to the collecti ve effect the energy of interaction between clustrons in polyclustro ns is higher than kT at sufficiently low temperatures. The model presented explains the increasing of viscosity, h eat ca- pacity and sound velocity in an aqueous system by enhancing t he cooperative units with ice-like water structure - clustron s and poly- clustrons (Table) due to self-organization in water -macro molecular systems. 31The density of ice-like structures in clustrons and polyclu strons is lower than in usual bulk liquid water. Consequently, one can predict that the free volume of water w ill increase as the result of self-organization in aqueous syst ems of macro- molecules. Such a type of effect was revealed in our microcalorimetry stu dy of large-scale and small-scale protein dynamics contribut ions in the resulting heat capacity of solvent (K¨ aiv¨ ar¨ ainen et al., 1993). The ad- ditional free volume (vf)in [0.2−0.3%] (w/v)concentration of different proteins is quite close to the volume occupied by macromolec ules in the solution itself. The vfis dependent on the active site state (ligand- dependent) and large-scale dynamics of proteins. The more i ntensive the large-scale pulsations of proteins and their flexibilit y and the less its effective volume, the smaller the additional free volume solvent. This correlation could be resulted from acoustic impulses g enerated by pulsing protein. This additional acoustic noise can stim ulate dis- sociation (melting) of water clusters - librational effecto ns with sat- urated hydrogen bonds in a system (K¨ aiv¨ ar¨ ainen et al., 19 93). In therms of mesoscopic model it means increasing the probabil ity of [lb/tr]convertons excitation. The vicinal water in combination with osmotic processes cou ld be responsible for coordinated intra-cell spatial and dynami c reorgani- zations. 7. Osmosis and solvent activity. Traditional and mesoscopi c approach It was shown by Van’t Hoff in 1887 that osmotic pressure (Π) in t he dilute concentration of solute (c) follows a simple expression: Π = RTc (24) This formula can be obtained from an equilibrium condition between a solvent and an ideal solution after saturation of diffusion process o f the solvent through a semipermeable membrane: µ0 1(P) =µ1(P+ Π, Xi) where µ0 1andµ1are the chemical potentials of a pure solvent and a solvent in solution; P- external pressure; Π - osmotic pressure; X1is the solvent fraction in solution. At equilibrium dµ0 1=dµ1= 0 and dµ1=/bracketleftbigg∂µ1 ∂P1/bracketrightbigg X1dP1+/bracketleftbigg∂µ1 ∂X1/bracketrightbigg P1dX1= 0 (24a) 32Because µ1=/parenleftbig ∂G/∂n 1/parenrightbig P,T=µ0 1+RTlnX1 (25) then /parenleftbigg∂µ1 ∂P1/parenrightbigg X1=/parenleftbigg∂2G ∂P∂n 1/parenrightbigg P,T,X=/parenleftbigg∂V ∂n1/parenrightbigg =V1 (26) where V1is the partial molar volume of the solvent. For dilute soluti on:¯V1≃ V0 1(molar volume of pure solvent). From (25) we have: ∂µ1 ∂X1=RT/parenleftbigg∂lnX1 ∂X1/parenrightbigg P,T(27) Putting (26) and (27) into (24a) we obtain: dP1=−RT V0 1X1dX1 Integration: p+π/integraldisplay PdP1=−RT V0 1x1/integraldisplay 1dlnX1 (28) gives: Π =−RT V0 1lnX1=−RT V0 1ln(1−X2) (29) and for the dilute solution ( X2≪1) we finally obtain Van’t Hoff equation: Π =RT V0 1X2∼=RTn2/n1 V0 1= RTc (30) where X2=n2/(n1+n2)∼=n2/n1 (31) and 33n2/n1 V0 1=c (32) Considering a real solution, we only substitute solvent fra ction X1in (29) by solvent activity: X1→a1. Then taking into account (25), we can express osmotic pressure as follows: Π =−RT ¯V1lna1=∆µ1 ¯V1(33) where: ∆ µ1=µ0 1-µ1is the difference between the chemical potentials of a pure solvent and the one perturbed by solute at the starting m oment of osmotic process, i.e. the driving force of osmose; ¯V1∼=V1is the molar volume of solvent at dilute solutions. Although the osmotic effects are widespread in Nature and are very impor- tant, especially in biology, the physical mechanism of osmo se remains unclear (Watterson, 1992). The explanation following from Van’t Hoff equation (30) and p ointing that osmotic pressure is equal to that induced by solute molecule s, if they are consid- ered as an ideal gas in the same volume at a given temperature i s not satisfactory. The osmose phenomenon can be explained quantitatively on th e basis of our mesoscopic theory and state equation (11.7). To this end, we have to introduce the rules of conservation of the main int ernal parameters of solvent in the presence of guest (solute) mole cules or particles: 1. Internal pressure of solvent: Pin= const 2. The total energy of solvent: Utot= const/bracerightbigg (34) This conservation rules can be considered as the consequenc e of Le Chatelier principle. Using (11.6), we have for the pure solvent and the solvent per turbed by a solute the following two equations, respectively: Pin=Utot V0 fr/parenleftbigg 1 +V Tk/parenrightbigg −Pext (35) P1 in=U1 tot V1 fr/parenleftbigg 1 +V1 T1 k/parenrightbigg −P1 ext, (36) where: V0 fr=V0 n2and V1 fr=V0 n2 1(37) 34are the free volumes of pure solvent and solvent in presence o f solute (guest) molecules as a ratio of molar volume of solvent to correspond ent value of refrac- tion index. The equilibrium conditions after osmotic process saturation , leading from our conservation rules (34) are Pin=Pinwhen Pext=Pext+ Π (38) Utot=V+Tk=V1+T1 k=U1 tot (39) From (39) we have: Dif = Tk−T1 k=V1−V (40) The index (1) denote perturbed solvent parameters. Comparing (35) and (36) and taking into account (37 - 39), we o btain a new formula for osmotic pressure: Π =n2 V0Utot/bracketleftbiggn2 1Tk−n2T1 k TkT1 k/bracketrightbigg (41) where: n, V0, UtotandTkare the refraction index, molar volume, total energy and total kinetic energy of a pure solvent, respectiv ely;T∗ kandn1are the total kinetic energy and refraction index of the solvent in t he presence of guest (solute) molecules, that can be calculated from the mesosco pic theory (eq.4.33). For the case of dilute solutions, when TkT1 k∼=T2 kandn∼=n1,the eq.(41) can be simplified: Π =n2 V0/parenleftbiggUtot Tk/parenrightbigg2/parenleftbig Tk−T1 k/parenrightbig (42) or using (40): Π =n2 V0/parenleftbiggUtot Tk/parenrightbigg2 (V1−V) (43) The ratio: S=Tk/Utot (44) is generally known as a structural factor (see eq. 2.46): We can see from (42) and (43) that osmotic pressure is proport ional to the difference between total kinetic energy of a free solvent ( Tk) and that of the solvent perturbed by guest molecules: 35∆Tk=Tk−T1 k or related difference between the total potential energy of p erturbed and pure solvent: ∆V=V1−Vwhere: ∆ Tk= ∆V≡Dif (see Fig. 3). As far ∆T k>0 and ∆ V >0, it means that: Tk> T1 k or V1> V(45) Theoretical temperature dependence of the difference Dif= ∆Tk= ∆V calculated from (42) or (43) at constant osmotic pressure: Π ≡Pos= 8 atm, pertinent to blood is presented on Fig. 3. The next Fig. 4 illustrate theoretical temperature depende nce of osmotic pressure (43) in blood at the constant value of Dif= 6.7·10−3(J/M), corre- sponding on Fig. 3 to physiological temperature (370). The ratios of this Difvalue to total potential (V) and total kinetic energy (Tk) of pure water at 370(see Fig. 5b) are equal to: (Dif/V)≃6.7·10−3 1.3·104∼=5·10−7and (Dif/Tk)≃6.7·10−3 3.5·102∼=2·10−5 i.e. the relative changes of the solvent potential and kinet ic energies are very small. 36Fig. 3. Theoretical temperature dependence of the difference: Dif=V1−V=Tk−T1 kat constant osmotic pressure: Π ≡Pos= 8 atm, characteristic for blood. The computer calculations were performed using eqs. (42) or (43). For each type of concentrated macromolecular solutions the optimum amount of water is needed to minimize the potential energy of the sys temdetermined mainly by clusterphilic interactions. The conservation ru les (34) and self-organization in solutions of macromolecules (clustron formation) may be responsible for the driving force of osmos e in the different compartments of biological cells. Comparing (43) and (33) and assuming equality of the molar vo lumes V0= ¯V1, we find a relation between the difference in potential energi es and chemical potentials (∆ µ) of unperturbed solvent and that perturbed by the solute: ∆µ=µ0 1−µ1=n2/parenleftbiggUtot Tk/parenrightbigg2 (V1−V) (46) Fig. 4. Theoretical temperature dependence of osmotic pressure (eq. 43) in blood at constant value of difference: Dif = ∆ T= ∆V= 6.7·10−3J/M. This value in accordance with Fig.3 corresponds to physiological temperature (370). The results obtained above mean that solvent activity (a1)and a lot of other thermodynamic parameters for solutions can be calc ulated on the basis of our hierarchic concept: a1= exp/parenleftbigg −∆µ RT/parenrightbigg = exp/bracketleftbigg −/parenleftBign S/parenrightBig2V1−V RT/bracketrightbigg (47) where: S=Tk/Utotis a structural factor for the solvent. 37The molar coefficient of activity is: yi=ai/ci, (48) where ci=ni/V (49) is the molar quantity of i-component ( ni) in of solution (V - solution volume in liters). The molar activity of the solvent in solution is related to it s vapor pressure (Pi) as: ai=Pi/P0 i (50) where: P0 iis the vapor pressure of the pure solvent. Theoretical tempe rature dependence of water activity ( a1) in blood at constant difference: Dif = ∆ T= ∆V= 6.7·10−3J/M is presented on Fig. 5. Fig. 5. Theoretical temperature dependence of water activity (a1) (eq.47) in blood at constant difference: Dif= ∆T= ∆V= 6.7·10−3J/M. Another colligative parameter such as low temperature shift of freezing tem- perature of the solvent (∆ Tf) in the presence of guest molecules also can be calculated from (47) and (13): ∆Tf=−R(T0 f)2 ∆Hlna1(T0 f)2 ∆H·T/parenleftBign S/parenrightBig2 (V1−V) (51) 38where: T0 fis the freezing temperature of the pure solvent; T is the temp er- ature corresponding to the conditions of calculations of V1(T) and V(T) from (eqs. 4.33 and 4.36 of [1,2]). The partial molar enthalpy ( ¯H1) of solvent in solution are related to solvent activity like: H1=H0 1−RT2∂lna1 ∂T=H0 1+L0 1 (52) where H0 1is the partial enthalpy of the solvent at infinitive dilution ; ¯L1=−RT2∂lna1 ∂T=T2∂ ∂T/bracketleftbigg/parenleftBign S/parenrightBig2V1−V T/bracketrightbigg (53) is the relative partial molar enthalpy of solvent in a given solution. 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arXiv:physics/0003072v1 [physics.chem-ph] 24 Mar 2000Phase changes in 38-atom Lennard-Jones clusters. II: A para llel tempering study of equilibrium and dynamic properties in th e molecular dynamics and microcanonical ensembles F. Calvo D´ epartement de Recherche sur la Mati` ere Condens´ ee, Service des Ions, Atomes et Agr´ egats, CEA Grenoble F38054 Grenoble Cedex, France J. P. Neirotti and David L. Freeman Department of Chemistry, University of Rhode Island 51 Lower College Road, Kingston, RI 02881-0809 and J. D. Doll Department of Chemistry, Brown University Providence, RI 02912 (January 11, 2014) Abstract We study the 38-atom Lennard-Jones cluster with parallel te mpering Monte Carlo methods in the microcanonical and molecular dynamics ensembles. A new Monte Carlo algorithm is presented that samples rigorou sly the molecular dynamics ensemble for a system at constant total energy, lin ear and angular momenta. By combining the parallel tempering technique wit h molecular dy- namics methods, we develop a hybrid method to overcome quasi -ergodicity and to extract both equilibrium and dynamical properties fr om Monte Carlo and molecular dynamics simulations. Several thermodynami c, structural anddynamical properties are investigated for LJ 38, including the caloric curve, the diffusion constant and the largest Lyapunov exponent. Th e importance of insuring ergodicity in molecular dynamics simulations i s illustrated by com- paring the results of ergodic simulations with earlier mole cular dynamics sim- ulations. Typeset using REVT EXI. INTRODUCTION The simulation of systems having complex potential energy s urfaces (PES) is often diffi- cult owing to the problem of quasi-ergodicity. Quasi-ergod icity can arise in systems having several energy minima separated by high energy barriers. Wh en such situations occur, as for example in proteins, glasses or clusters, the system can become trapped in local basins of the energy landscape, and the ergodic hypothesis fails on the time scale of the simulation. In the canonical ensemble, the high energy regions of the PES are exponentially suppressed and barrier crossings become rare events. Calculations of e quilibrium properties when phase space is thus partitioned require methods that overcome qua si-ergodicity by enhanced bar- rier crossing. Many techniques have been proposed to addres s this problem, including the use of generalized ensembles such as multicanonical1or Tsallisian,2,3simulated tempering,4 configurational5or force bias6Monte Carlo, or various versions of the jump-walking7–11al- gorithm. Most of these techniques have been introduced for M onte Carlo (MC) simulations rather than molecular dynamics (MD) simulations. These tec hniques have been applied to a variety of sampling and optimization problems, and phase c hanges in clusters have often been considered as a benchmark to test these methods.2,10,11 The double-funnel energy landscape of the 38-atom Lennard- Jones (LJ) cluster has been investigated in detail by Doye, Miller and Wales,12–15who recently also estimated the inter- funnel rate constants using master equation dynamics.13This landscape is challenging for simulation because of the high free-energy barrier separat ing the two funnels.14In the preced- ing paper (hereafter referred to as I),16we have shown how the parallel tempering algorithm can be used to deal with this particularly complex system for Monte Carlo simulations in the canonical ensemble. Achieving ergodicity in microcano nical simulations is much harder than in the canonical ensemble, because the system is unable to cross any energy barrier higher than the total energy available. The 38-atom Lennard -Jones cluster is fundamentally non-ergodic in a range of energies. This non-ergodicity may not be a serious problem when considering one particular cluster on a short time scale. Ho wever, in a statistical sample ofsuch systems it is important to have ergodic results. To allow MD simulations to cross the high energy barriers, on e may think of heating the system (by increasing its kinetic energy), followed by a coo ling back to its initial thermo- dynamic state. Although this process is straightforward, t he dynamics becomes biased and non physical during the heating and cooling processes. More over, it is difficult to control accurately the heating and cooling rates that should be chos en for any system. This latter aspect is particularly critical for the 38-atom Lennard-Jo nes cluster where the narrow and deepest funnel is hard to reach even at high temperatures. Because of the inherent difficulties of molecular dynamics, M C approaches can be es- pecially useful for dealing with the problem of crossing hig h energy barriers. Monte Carlo methods have been developed in previous work10,17in the microcanonical ensemble. In these approaches the microcanonical sampling is at fixed ene rgy without any additional constraints. Such methods can be contrasted with isoenerge tic molecular dynamics where the total, linear and angular momenta are also conserved. Th ese additional constraints must be considered even at zero angular momentum.18–20To differentiate microcanonical simula- tions, where only the energy is fixed, from molecular dynamic s simulations, where additional constraints are imposed, we identify the former to be simula tions in the microcanonical en- semble and identify the latter simulations to be in the molec ular dynamics ensemble. The differences in the two ensembles can be particularly importa nt when the angular momentum is large enough to induce structural (centrifugal) distort ions.20Because dynamical properties are calculated using molecular dynamics methods, in this wo rk we find that a combination of Monte Carlo and molecular dynamics methods are most conve nient for developing ergodic approaches to dynamics. In this paper, we adapt the parallel tempering method to both the microcanonical and molecular dynamics ensembles. The application of parallel tempering in the molecular dy- namics ensemble requires the incorporation of the conserva tion of the total linear and angular momenta into the probabilities. In order to extract ergodic dynamical properties, we com- bine Monte Carlo methods with molecular dynamics to develop a hybrid ergodic MC/MDalgorithm. The efficiency of the simulation tools developed i n this work is demonstrated by applications to the 38-atom Lennard-Jones cluster, which e xhibits a solid-solid transition prior to melting.13,16This transition to an equilibrium phase between truncated o ctahedral and icosahedral geometries makes this cluster an ideal cand idate for investigating how the ergodic hypothesis can influence the dynamical behavior of a complex system. The contents of the remainder of this paper are as follows. In the next section, we re- call the basic principle of Monte Carlo sampling in the micro canonical ensemble, and we present the simple modifications needed to include parallel tempering. We test microcanon- ical parallel tempering methods on the 38-atom Lennard-Jon es cluster, and compare the microcanonical results with those found in I using the canon ical ensemble. We focus on equilibrium properties, including the caloric curve and th e isomers distributions. In section III we review the characteristics of the molecular dynamics ensemble at fixed total linear and angular momenta and fixed total energy. We extend the para llel tempering Monte Carlo method to the MD ensemble, and we combine microcanonic al parallel tempering with molecular dynamics to produce an ergodic MD method. We also a pply these methods to several dynamical properties of LJ 38; in particular the diffusion constant and the largest Lyapunov exponent. We summarize our findings and discuss our results in section IV. II. PARALLEL TEMPERING MONTE CARLO IN THE MICROCANONICAL ENSEMBLE The fundamental quantity in the microcanonical ensemble is the density of states Ω. For a system having Nidentical particles, volume Vand total energy E, Ω is defined by Ω(N,V,E ) =1 N!h3N/integraldisplay δ[H(r,p)−E]d3Nr d3Np (1) wherehis Planck’s constant and where H(r,p) denotes the classical Hamiltonian function of the coordinates rand momenta pof theNparticles. Knowing the microcanonical density of states Ω, one can calculate the canonical partition funct ionQ(N,V,T ) by a Laplacetransformation.10The kinetic part of the Hamiltonian His quadratic in the momenta, and Eq. (1) can be partly integrated to give10,21 Ω(N,V,E ) =/parenleftbigg2πm h2/parenrightbigg3N/21 N!Γ(3N/2)/integraldisplay Θ[E−U(r)][E−U(r)]3N/2−1d3Nr. (2) In Eq.(2), Γ is the Gamma function, mis the mass of each particle, U(r) =H−Kis the potential energy and Θ is the Heaviside step function: Θ( x) = 1 ifx≥0, 0 otherwise. Microcanonical averages of a coordinate-dependent variab leA(r) can be expressed /an}bracketle{tA/an}bracketri}ht(N,V,E ) =/integraldisplay Θ[E−U(r)][E−U(r)]3N/2−1A(r)d3Nr /integraldisplay Θ[E−U(r)][E−U(r)]3N/2−1d3Nr. (3) The microcanonical entropy Scan be defined by S(N,V,E ) =kBln Ω(N,V,E ) withkBthe Boltzmann constant. The thermodynamic temperature T(N,V,E ) is given by the thermo- dynamic relation ( ∂S/∂E )N,V= 1/T, and can be obtained from a microcanonical average21 T(N,V,E ) =2 3N−21 /an}bracketle{tK−1/an}bracketri}ht. (4) This expression is slightly different from the kinetic tempe rature 2 /an}bracketle{tK/an}bracketri}ht/3N, which is a con- sequence of our choice in the definition of the entropy. As dis cussed by Pearson and co- workers,21it is also possible to define the entropy using the phase space volume Φ(N,V,E ) =/integraldisplayE 0Ω(N,V,E′)dE′. (5) Definitions of the temperature based on Ω differ from the tempe rature based on Φ to order 1/N, and the two definitions agree only in the thermodynamic limi t. Monte Carlo simulations can be used to explore the microcano nical ensemble by per- forming a random walk in configuration space. In the standard Metropolis scheme, a trial move from configuration roto configuration rnis accepted with the probability22 acc(ro→rn) = min/parenleftBigg 1,ρE(rn)T(rn→ro) ρE(ro)T(ro→rn)/parenrightBigg , (6) whereT(ro→rn) is a trial probability. The acceptance probability expres sed in Eq.(6) insures detailed balance so that the random walk visits poin ts in configuration space pro- portional to the equilibrium distribution ρE(r) defined byρE(r) =ζ−1Θ[E−U(r)][E−U(r)]3N/2−1(7) whereζis the normalization. In practice, T(ro→rn) is a uniform distribution of points of width ∆ centered about ro, and ∆ is adjusted as a function of the energy so that not too many trial moves are either accepted or rejected. Implementation of microcanonical Monte Carlo is as easy as i ts canonical version. Be- cause Monte Carlo methods are based on random walks in configu ration space, in principle the system can cross energy barriers higher than the availab le energy. However, in difficult cases like LJ 38, large atomic displacements having poor acceptance ratios are needed to reach ergodicity. Parallel tempering23–26has proved to be an important approach to insure ergodicity in canonical Monte Carlo simulations, and parallel temperi ng can be easily adapted to the microcanonical ensemble by replacing the Boltzmann factor s in the acceptance probability by the microcanonical weight ρE(r). In the parallel tempering scheme, several microcanonica l MC simulations are performed simultaneously at different to tal energies {Ei}. With some predetermined probability, two simulations at energies EiandEjattempt to exchange their current configurations, respectively riandrj, and this exchange is accepted with probability min/parenleftBigg 1,ρEi(rj)ρEj(ri) ρEi(ri)ρEj(rj)/parenrightBigg . The acceptance ratio is analogous to the canonical expressi on given in I. In microcanonical simulations the potential energies must be smaller than min (Ei,Ej); otherwise the move is rejected. Parallel tempering microcanonical MC works in the same way as in standard canonical MC. As with canonical parallel tempering MC, the g aps between adjacent total energies must be chosen to be small enough so that exchanges b etween the corresponding trajectories are accepted with a reasonable probability. By using a histogram reweighting technique,27it is possible to extract from the MC simulations the density of states Ω, and then all the thermod ynamic quantities in both the microcanonical and the canonical ensembles. The procedure is similar to that described in Ref. 28, and relies on the calculation of the distribution P(U,E) of potential energyUat the total energy E.Pis fitted to the microcanonical form P(U,E) = ΩC(U)(E− U)3N/2−1/Ω(E), where Ω Cstands for the configurational density of states, and Ω( E) is extracted by convolution of Ω C(U) and (E−U)3N/2−1. We have tested the parallel tempering Monte Carlo algorithm in the microcanonical ensemble on the 38-atom Lennard-Jones cluster previously i nvestigated. Forty different to- tal energies ranging from −172.4737εto−124εhave been used, and the same simulation conditions have been chosen as in I. In addition to a constrai ning sphere of radius 2 .25σto prevent evaporation, exchanges have been attempted every 1 0 passes, with the same method for choosing exchanging trajectories as described in the pr evious article. The simulations are begun with random configurations of the cluster geometry, an d consist of 1 .3×1010points accumulated following equilibration moves consisting of 9 5×106Metropolis points(no ex- changes) followed by 190 ×106points using parallel tempering. The microcanonical heat capacity calculated in this fashion and shown in Fig. 1, is qu alitatively the same as the canonical heat capacity [see I]. The melting peak in the micr ocanonical heat capacity occurs at the same calcuated temperature as the temperature of the m elting peak in the canon- ical heat capacity, and there are slope change regions at tem peratures that correspond to equilibrium between the icosahedral basin and the truncate d octahedral global minimum structure. The present simulations are also used to obtain s tructural insight about the clus- ter as a function of total energy. We have calculated the orde r parameter Q4as defined in I as a function of temperature, and compared the classifica tion into the three categories of isomers (truncated octahedral, icosahedral or liquid-l ike) using the energy criterion also outlined in I. In Fig. 2 we show the caloric curve T(E) determined from our parallel tempering mi- crocanonical MC simulations. We also present the canonical curve for comparison. The melting transition near T∼0.166ε/kBis reflected in the change in slope of the temperature as a function of the energy. The microcanonical curve does no t display a van der Waals loop, and remains very close to the canonical curve. The aver age value of the order param- eter/an}bracketle{tQ4/an}bracketri}htis displayed in the lower panel of Fig. 2 as a function of the to tal energy. Ashas been discussed in I for the canonical simulation, the ord er parameter begins to drop at energies where there is the onset of isomerization transiti ons to the icosahedral basin (near E=−160ε), and the order parameter reaches its lowest value at the mel ting transition. The isomer distributions have been evaluated either using t he parameter Q4or using the energy criterion (see the discussion in paper I). The result s have been plotted in Fig. 3 as a function of the total energy. The behavior of the isomer d istributions as a function of energy is similar to the canonical distributions as a func tion of temperature, and the cluster exhibits equilibrium between truncated octahedra l and icosahedral geometries in the energy range −160ε<∼E<∼−150ε, prior to the solid-like to liquid-like phase change. As in the canonical case, the icosahedral distribution is a symme tric function of the energy when the energy criterion is used rather than the definition based onQ4. This difference reflects the differences between the two definitions of icosahedral an d liquid basins. The oscillatory structure observed at the peak of PQ4for the icosahedral distribution in the upper panel of Fig. 3 is smaller than the calculated errors (two standard de viations of the mean are shown). Whether the observed structure would persist for a longer si mulation is not known to us. Because the definition that assigns configurations to the ico sahedral basin is arbitrary, we have chosen not to investigate this structure further. It is useful to contrast the current results with previous co nstant energy studies of LJ 38. Previous simulations have used molecular dynamics methods where no attempt has been made to insure ergodicity. To contrast these past studies wi th the molecular dynamics tech- nique discussed in the next section of this paper, we define standard molecular dynamics to represent the usual molecular dynamics method where no spec ial procedure is introduced to insure ergodicity. Simulations of LJ 38using standard molecular dynamics invariably lead to a caloric curve with a clear van der Waals loop and a melting temperature higher than that inferred from Fig. 2.29From the results of Ref. 29, the cluster is trapped in the octa - hedral basin, and the system does reflect the true dynamical c oexistence state between the truncated octahedron and the icosahedral basin. This is the common situation encountered in MD simulations of the LJ 38system; the cluster chooses either to remain trapped in theoctahedral basin, or to escape and coexist between the icosa hedral solid-like and liquid-like forms. Because the system is unable to return from the octahe dral basin, the microcanonical temperature decreases. In the usual case, van der Waals loop s arise when there are large energy gaps between the lowest-energy isomers.30In the specific case of LJ 38, it appears that the presence of extra (icosahedral) isomers only slightly h igher in energy than the octahedral structure eliminates this loop in the ergodic microcanonic al caloric curve. In order to extract dynamical quantities, the Monte Carlo me thod we have presented must be modified to sample the MD ensemble. The modification is the subject of the next section. III. ERGODIC MOLECULAR DYNAMICS The molecular dynamics ensemble differs from the microcanon ical ensemble in that two quantities are conserved in addition to the total energy E, volumeVand number of particles N. These two quantities are the total linear momentum Pand total angular momentum L. If their values are prescribed, the density of states remai ns the fundamental quantity of interest, and is now defined by Ω(N,V,E, P,L) = 1 N!h3N/integraldisplay δ[H(r,p)−E]δ/parenleftBigg P−N/summationdisplay i=1pi/parenrightBigg δ/parenleftBigg L−N/summationdisplay i=1ri×pi/parenrightBigg d3Nrd3Np. (8) As is the case in the microcanonical ensemble [see Eq.(2)], f or atomic systems the momentum integrations in Eq.(8) can be evaluated explicitly.18–20Because the thermodynamic proper- ties are not affected by the translational motion of the cente r of mass, we can assume that P= 0. We then obtain20 Ω(N,V,E, P= 0,L) = /parenleftbigg2πm h2/parenrightbigg3N/2−31 N!Γ(3N/2−3)/integraldisplay Θ[E−UL(r)][E−UL(r)]3N/2−4d3Nr√ detI, (9) where Iis the inertia matrix and UL(r) =U(r) +L†I−1L/2 is the effective rovibrational energy. This effective potential energy includes the kineti c energy contribution of the ro-tating cluster considered as a rigid body.31,32The landscape of rotating clusters has been investigated by Miller and Wales in order to study cluster ev aporation.33Averages in the MD ensemble are now expressed as /an}bracketle{tA/an}bracketri}ht=/integraldisplay Θ[E−UL(r)][E−UL(r)]3N/2−4A(r)d3Nr√ detI/integraldisplay Θ[E−UL(r)][E−UL(r)]3N/2−4d3Nr√ detI. (10) As in the microcanonical ensemble, we define the entropy in th e molecular dynamics ensem- ble byS=kBln Ω. The differences between the microcanonical and molecul ar dynamics ensembles are the exponent 3 N/2 which is reduced by 3 owing to the geometrical con- straints, the potential energy which now includes the contr ibution of the centrifugal energy, and the weight 1 /√ detIwhich is a consequence of the conservation of the angular mom en- tum. Monte Carlo simulations can sample the MD ensemble by pe rforming a random walk in configuration space. The acceptance probability from con figuration roto configuration rnis acc(ro→rn) = min/parenleftBigg 1,ρE,L(rn)T(rn→ro) ρE,L(ro)T(ro→rn)/parenrightBigg (11) in the Metropolis scheme. The microcanonical weight ρE(r) is now replaced by the MD weightρE,Lgiven by ρE,L(r) =ζ−11√ detIΘ[E−UL(r)][E−UL(r)]3N/2−4, (12) whereζis a normalization. The expression for the acceptance proba bility is similar to Eq. (6), and a practical implementation of Monte Carlo in the MD e nsemble is made in the same way as in the true microcanonical ensemble, given the vector L. Parallel tempering can be also easily combined with the MC simulations. The acceptanc e probability of exchanging the configurations riandrjinitially at the total energies EiandEjrespectively is then min 1,/parenleftBigg[Ei−UL(rj)][Ej−UL(ri)] [Ei−UL(ri)][Ej−UL(rj)]/parenrightBigg3N/2−4  provided that all quantities inside brackets are positive ( otherwise the move is rejected). It is remarkable that the geometrical weights have canceled in this expression.The Monte Carlo method we have just described allows samplin g of configuration space rigorously equivalent to the sampling we would obtain using molecular dynamics, but with the additional possibility of crossing the energy barriers higher than the available energy. The method can be used in its present form to extract equilibr ium properties only dependent on the energy or geometry, as has been illustrated in the prev ious section. To compute dy- namical quantities, the method can also provide a database o f configurations representative of a given total energy. Instead of performing a few very long MD simulations that are in principle unable to reach other parts of the energy surface s eparated by barriers higher than the available energy, we choose to perform a statistical num ber of short simulations starting from configurations obtained by parallel tempering Monte Ca rlo in the MD ensemble with same total energy and angular momentum. By construction, if the MC method is correctly ergodic, then the hybrid MD method we have suggested can be ex pected to yield ergodic dynamical observables. We now illustrate this ergodic molecular dynamics method on the LJ 38problem. Two essentially dynamical parameters have been calculated. Th e first is the self diffusion constant D, obtained from the derivative of the average mean square ato mic displacement D=1 6d dt/an}bracketle{t[r(t)−r(0)]2/an}bracketri}ht, (13) where the average is taken over all particles of the system an d over all short MD simulations. The other parameter is the largest Lyapunov exponent λ, that measures the exponential rate of divergence of the distance between two initially close tr ajectories in the phase space. If we write the equation describing the Hamiltonian dynamics i n condensed form as ˙ψ(t) = F(ψ) whereFis a nonlinear function and ψ={r,p}the phase space point, then a small perturbation δψevolves according to the simple equation dδψ/dt = (∂F/∂ψ )δψ. The largest Lyapunov exponent λis obtained from the time evolution of the vector δψ: λ= lim t→∞lim δψ(0)→01 tln/bardblδψ(t)/bardbl /bardblδψ(0)/bardbl. (14) In Eq.(14), /bardbl·/bardblis a metric on the phase space. In principle, any metric can be used, and we choose the Euclidian metric including both the momenta and t he coordinates. The numericalprocedure34involves a periodic renormalization of the vector δψto prevent its exponential divergence. The successive lengths are accumulated and con tribute to the average value of λ. In I, the clusters have been defined using a hard sphere constr aining potential. Because the angular momentum is not conserved after reflection from s uch hard wall boundaries, in the molecular dynamics simulations we have chosen a soft rep ulsive spherical wall Ucdefined with respect to the center of mass of the cluster for each part icle by Uc(r) =  0, r<R c κ(r−Rc)4/4, r≥Rc.(15) In this equation, the atomic distances rare measured with respect to the cluster center of mass. The simulations have been performed setting the ang ular momentum to zero for simplicity. We stress that even in this case (with L= 0), the weight 1 /√ detImust be included in the Monte Carlo probabilities so that we effectiv ely sample the MD ensemble. The actual thermodynamic behavior in the MD ensemble at zero angular momentum is nevertheless nearly identical to the microcanonical behav ior. The application to the LJ 38cluster has been made by performing 1010MC steps following 107equilibration steps in a parallel tempering simulation in t he MD ensemble. The same 40 total energies have been chosen as in the previous section , and 105configurations have been stored every 105steps for each simulation. Short molecular dynamics runs of 104time steps following 103equilibration steps have been performed for each of these co nfigurations, with the same total energy as the corresponding MC trajector y of origin, and with zero total linear and angular momenta as well. The parameters use d for the constraining wall are respectively Rc= 2.25σandκ= 100ε, for both the MC and MD runs. A simple Verlet algorithm has been used to propagate the MD trajectory with t he time step δt= 0.01 reduced LJ units. The propagation of the tangent trajectory to calculate the Lyapunov exponent has been determined with a fourth order Runge-Kutt a scheme. The final values of Dandλare an average over the 105MD simulations. The variations of Dandλwith total energy are depicted in Fig. 4. In both cases, two curves have b een plotted, calculated eitherfrom standard molecular dynamics (with 108time steps following 107equilibration steps, and starting initially from the lowest-energy structure), or from our hybrid ergodic molecular dynamics method. For both quantities, the two MD schemes cle arly yield distinct values in the energy range where equilibrium between truncated octah edral and icosahedral geometries occurs. The thermodynamic temperature, not plotted here, h as the same variations as the caloric curve of Fig. 2 when calculated with ergodic MD. Stan dard molecular dynamics predicts a van der Waals loop centered at T∼0.18ε/kB. For standard MD, the cluster is trapped in the icosahedral basin and is, in practice, unable to reach the octahedral basin. Only the equilibrium between the icosahedral basin and liqu id-like structures occurs. As can be seen from the upper panel of Fig. 4, this change in curva ture of the temperature is also present for the diffusion constant, which exhibits stro ng variations at the energy where the octahedral structure vanishes when standard MD is used. In contrast, the variations in ergodic MD are smooth. The melting temperature implied by the largest Lyapunov exp onent is also higher in standard MD than in ergodic MD, even though the variations of the Lyapunov exponent are continuous in both MD schemes.29Indeed, using ergodic molecular dynamics we observe a shift of the curve obtained by standard MD toward the lower en ergies. As shown by Hinde, Berry, and Wales,35–37the Lyapunov exponent and the Kolmogorov entropy are quanti ties essentially dependent on the local properties of the energy landscapes. One contribution comes from the negative curvature of the landscape, and anot her contribution is the fluctu- ation of positive curvature.38Both contributions are affected by the cluster being trapped either inside the truncated octahedral basin or inside the i cosahedral basin. In this latter case in particular, the different isomers belonging to the ic osahedral basin are connected through regions of negative curvature, while only one isome r defines the octahedral funnel. Because ergodic molecular dynamics allows the cluster to be found in both basins prior to melting, the dynamical behavior is likely to be very differ ent (and more chaotic) with respect to the dynamical behavior of the cluster confined to t he octahedral funnel. This difference is precisely what we observe on the lower panel of F ig. 4.IV. CONCLUSION In this paper, we have explored the parallel tempering metho d in simulations in the microcanonical ensemble. The implementation of the parall el tempering algorithm in this ensemble is straightforward, the Boltzmann factor exp( −βU) being replaced by the micro- canonical weight ( E−U)3N/2−1. Application to the LJ 38cluster has shown the thermody- namic behavior in the microcanonical ensemble to be similar to the behavior in the canonical ensemble. The solid-liquid phase change is preceded by a sol id-solid phase change where the cluster is in equilibrium between truncated octahedral and icosahedral geometries. This phase equilibrium is well reproduced in the simulations owi ng to the power of parallel tem- pering. The calculated microcanonical caloric curve, whic h does not display a van der Waals loop, is consistent with the single peaked heat capacity obs erved in I.16 We have extended the parallel tempering microcanonical Mon te Carlo algorithm to sam- ple the molecular dynamics ensemble at constant total energ y, linear momentum and angular momentum. Combined with standard molecular dynamics, this method circumvents the lack of connectivity between regions of the potential energy sur face. The method can ensure er- godicity in microcanonical simulations, which is much more difficult to achieve than in the canonical ensemble. Ironically, this ergodic MD method can be viewed as the counterpart of the techniques developed by Chekmarev and Krivov to study the dynamics of systems confined to only one catchment basin in the energy surface.39 We have performed a statistical number of short molecular dy namics runs starting from configurations stored periodically in parallel tempering M onte Carlo simulations. These sim- ulations sample the MD ensemble at the same total energies, l inear and angular momenta as the standard molecular dynamics runs. In fact, the length of the MD runs is mainly dictated by the large number of starting configurations. One may think of reducing drasti- cally this number, to allow for the calculation of parameter s varying on longer time scales. Unfortunately, if the energy landscape is not known in advan ce, then it is hard to guess how important are the contributions of the basins not selected a s starting configurations. In thecase of LJ 38having only 3 main regions on the energy surface, one possibi lity is to compute a dynamical property as the average value over 3 different sim ulations starting either from the truncated octahedral geometry, one icosahedral geomet ry or a low-lying liquid geometry, all carried out at the same total energy. However, as we have s een in Fig. 3, it is not obvious how to choose properly the weights of each basin in this avera ge because of the difficulty in distinguishing between icosahedral and liquid structur es in many cases. For this reason, we believe that the first parallel tempering MC step of the hyb rid ergodic method is essen- tial in the vicinity of phase changes to capture many startin g configurations that are used subsequently in standard molecular dynamics. The enhanced sampling offered by parallel tempering can also act as a statistical representation of th e energy surface at a given total energy, and the long time dynamics may be further investigat ed by using master equations after searching the saddle points.15,40 We have calculated two dynamical quantities with the presen t hybrid MD/MC method, the diffusion constant and the largest Lyapunov exponent in t he 38-atom Lennard-Jones cluster. The variations of both quantities with the total en ergy are significantly different when evaluated with standard (non-ergodic) molecular dyna mics or with our hybrid ergodic MD method. These results emphasize the different contributi ons of the two funnels of the energy landscape to the average value of the parameters esti mated. The algorithms developed in this investigation allow the ca lculation of thermodynamic, structural, or dynamical properties of systems such as LJ 38that can be expressed as phase space or time averages. Parallel tempering works using a cri terion based on the potential energy but not on the geometry. Consequently permutational isomers can be introduced in the course of the simulation. Quantities such as fluctuati ons of configuration-dependent properties are much more difficult to extract than actual aver ages. For instance, the Lin- demann index δ, which measures the root mean square bond length fluctuation , is often considered to be a reliable parameter for detecting melting in atomic and molecular sys- tems. This quantity cannot be properly estimated with the er godic MD scheme, and the same difficulty persists for other methods based on the use of d ifferent trajectories.Although the idea of combining Monte Carlo sampling with sta ndard molecular dynamics can be applied to other techniques such as jump-walking, we b elieve that parallel tempering is the key to the success in the case of LJ 38. As in the canonical version, the equilibrium phase between truncated octahedral and icosahedral struct ures is correctly reproduced in an energy range preceding the melting region, because in thi s range configurations may be accessed either from higher energy trajectories containin g mainly icosahedral geometries, or from lower energy trajectories acting as a reservoir for t he octahedral geometry. As noticed by Falcioni and Deem,25the parallel tempering algorithm is especially useful at lo w temperatures, or in our case, at low energy. The long relaxat ion times inherent in systems like clusters, proteins, critical or glassy liquids, are a s erious difficulty for standard simulation methods. We expect the present ergodic method to be particul arly useful to deal with the dynamics of such systems. The method we have presented works at constant total energy. It is possible to improve ergodicity in constant-temperature MD either by using cano nical parallel tempering as in the work of Sugita and Okamoto41, or by coupling parallel tempering canonical Monte Carlo to short Nos´ e-Hoover trajectories. In the Nos´ e-Hoover ap proach such molecular dynamics simulations do conserve a zero angular momentum, so a rigoro us MC sampling should in- clude the geometrical weight 1 /√ detIin the probabilities also in this case. The present microcanonical scheme can be easily used for rotating bodie s, which makes the method suit- able for investigating the strong influence of centrifugal e ffects on phase changes in atomic clusters. ACKNOWLEDGMENTS Some of this work has been motivated by the attendance of two o f us (DLF and FC) at a recent CECAM meeting on ‘Overcoming broken ergodicity i n simulations of condensed matter systems.’ We would like to thank CECAM, J.E. Straub an d B. Smit who orga- nized the meeting, and those who attended the workshop for st imulating discussions. Thiswork has been supported in part by the National Science Found ation under grant numbers CHE-9714970 and CDA-9724347. This research has been suppor ted in part by the Phillips Laboratory, Air Force Material Command, USAF, through the u se of the MHPCC under cooperative agreement number F29601-93-0001. The views an d conclusions contained in this document are those of the authors and should not be interpret ed as necessarily representing the official policies or endorsements, either expressed or im plied, of Phillips Laboratory or the US Government.REFERENCES 1B. A. Berg and T. Neuhaus, Phys. Rev. Lett. 68, 9 (1991). 2I. Andricioaei and J. E. Straub, J. Chem. Phys. 107, 9117 (1997). 3U. E. Hansmann and Y. Okamoto, Phys. Rev. E 56, 2228 (1997). 4E. Marinari and G. Parisi, Europhys. Lett. 19, 451 (1992). 5J. I. Siepmann and D. Frenkel, Mol. Phys. 75, 59 (1992). 6C. Pangali, M. Rao, and B. J. Berne, Chem. Phys. Lett. 55, 413 (1978). 7D. D. Frantz, D. L. Freeman, and J. D. Doll, J. Chem. Phys 93, 2769 (1990). 8D. D. Frantz, D. L. Freeman, and J. D. Doll, J. Chem. Phys 97, 5713 (1992). 9A. Matro, D. L. Freeman, and R. Q. Topper, J. Chem. Phys. 104, 8690 (1996). 10E. Curotto, D. L. Freeman, and J. D. Doll, J. Chem. Phys. 109, 1643 (1998). 11H. Xu and B. J. Berne, J. Chem. Phys. 110, 10 299 (1999). 12J. P. K. Doye and D. J. Wales, Phys. Rev. 80, 1357 (1998). 13J. P. K. Doye, D. J. Wales, and M. A. Miller, J. Chem. Phys. 109, 8143 (1998). 14J. P. K. Doye, M. A. Miller, and D. J. Wales, J. Chem. Phys. 110, 6896 (1999). 15M. A. Miller, J. P. K. Doye, and D. J. Wales, Phys. Rev. E 60, 3701 (1999). 16J. P. 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Phys. 78, 151 (1993). 31J. Jellinek and D. H. Li, Phys. Rev. Lett. 62, 241 (1989). 32D. H. Li and J. Jellinek, Z. Phys. D 12, 177 (1989). 33M. A. Miller and D. J. Wales, Mol. Phys. 89, 533 (1996). 34G. Benettin, L. Galgani, and J.-M. Strelcyn, Phys. Rev. A 14, 2338 (1974). 35R. J. Hinde, R. S. Berry, and D. J. Wales, J. Chem. Phys. 96, 1376 (1992). 36R. J. Hinde and R. S. Berry, J. Chem. Phys. 99, 2942 (1993). 37D. J. Wales and R. S. Berry, J. Phys. B 23, L351 (1991). 38V. Mehra and R. Ramaswamy, Phys. Rev. E 56, 2508 (1997). 39S. F. Chekmarev and S. V. Krivov, Phys. Rev. E 57, 2445 (1998).40K. D. Ball and R. S. Berry, J. Chem. Phys. 111, 2060 (1999). 41Y. Sugita and Y. Okamoto, Chem. Phys. Lett. 314, 141 (1999).FIGURES FIG. 1. The heat capacity as a function of energy calculated i n the microcanonical ensemble. The melting peak occurs at the same calculated temperature i n the microcanonical ensemble as found in the canonical ensemble, but the height of the microc anonical peak is significantly higher than the canonical peak [compare with Fig. 1 in I]. Both the mi crocanonical and canonical heat capacities display a region of change in slope at the transit ion between the truncated octahedron and the icosahedral basin. The error bars represent two stan dard deviations of the mean. FIG. 2. Upper panel: the microcanonical caloric curve for LJ 38obtained from parallel temper- ing Monte Carlo simulations. The temperature is plotted as a function of the total energy, both expressed in reduced LJ units. The circles are the direct res ults of microcanonical simulations. The solid line is a fit obtained by the histogram reweighting tech nique. Also plotted as a dotted line is the caloric curve in the canonical ensemble. Lower panel: av erage value of the order parameter Q4 as a funciton of the total energy. For both panels, the error b ars are smaller than the size of the symbols. FIG. 3. Upper panel: the probability distribution of the ord er parameter Q4as a function of the total energy. Lower panel: the probability distributio n of the energy of the quenched structure as a function of the total energy. For both quantities, FCC la bels the truncated octahedron, IC labels structures from the icosahedral basin and LIQ labels structures from the liquid region. In the lower panel, the error bars are smaller than the size of th e symbols. In the upper panel, the error bars represent two standard deviations of the mean. FIG. 4. Two dynamical parameters calculated for LJ 38using either standard molecular dy- namics starting from the lowest-energy structure (empty sy mbols) or the hybrid ergodic MD/MC method (full symbols), as a function of the total energy. The results are expressed in Lennard-Jones time units t0. Upper panel: diffusion constant D; lower panel: largest Lyapunov exponent λ. For both panels, the error bars are smaller than the size of the sy mbols.−180 −170 −160 −150 −140 −130 −120 E/ε345678〈CV〉/NkB00.10.20.30.4εT/kB −180 −170 −160 −150 −140 −130 −120 E/ε00.050.10.150.2〈Q4〉00.20.40.60.81PQ4FCC IC LIQ −180 −170 −160 −150 −140 −130 −120 E/ε00.20.40.60.81P (Uquench) FCC IC LIQ10−810−610−410−2100D/t0Standard Ergodic −180 −170 −160 −150 −140 −130 −120 E/ε00.511.522.5λt0Standard Ergodic

arXiv:physics/0003073v1 [physics.acc-ph] 24 Mar 2000Temporary Acceleration of Electrons While Inside an Intens e Electromagnetic Pulse Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princet on, NJ 08544 Konstantine Shmakov GlobalStar, Inc., San Jose, CA 95461 (Nov. 20, 1997) A free electron can temporarily gain a very significant amount of energy if it is overrun by an intense electromag- netic wave. In principle, this process would permit large en - hancements in the center-of-mass energy of electron-elect ron, electron-positron and electron-photon interactions if th ese take place in the presence of an intense laser beam. Prac- tical considerations severely limit the utility of this con cept for contemporary lasers incident on relativistic electron s. A more accessible laboratory phenomenon is electron-positr on production via an intense laser beam incident on a gas. In- tense electromagnetic pulses of astrophysical origin can l ead to very energetic photons via bremsstrahlung of temporaril y accelerated electrons. PACS numbers: 03.65.Sq, 12.15.-y 41.75.Fr, 52.40.Mj, 97.3 0.- b The prospect of acceleration of charged particles by intense plane electromagnetic waves has excited interest since the suggestion by Menzel and Salisbury [1] that this mechanism might provide an explanation for the origin of cosmic rays. However, it has generally been recognized that if a wave overtakes a free electron, the latter gains energy from the wave only so long as the electron is still in the wave, and reverts to its initial energy once the wave has past [2–5]. There is some controversy as to the case of a “short” pulse of radiation, for which modest net energy transfer between a wave and electron appears possible [6–10]. Acceleration via radiation pressure is negligible [11]. It has been remarked that even in the case of a “long” pulse, some of the energy transferred from the wave to the electron can be extracted if the electron undergoes a scattering process while still inside the wave [3,5]. This paper is an elaboration of that idea. We do not discuss here the observed phenomenon that an electron ionized from an atom in a strong wave can emerge from the wave with significant energy [12]. We consider a plane electromagnetic wave (often called the background wave) with dimensionless, invariant field strength η=e/radicalbig /angbracketleftAµAµ/angbracketright mc2=eErms mω0c=eErmsλ0 mc2. (1) Here the wave has laboratory frequency ω0, reduced wavelength λ0, root-mean-square electric field Erms, and four-vector potential Aµ;eandmare the charge and mass of the electron, and cis the speed of light. A practical realization of such a wave is a laser been. Laser beams with parameter ηclose to one have beenused in recent plasma-physics experiments [9] and in high-energy-physics experiments [13,14]. When such a wave overtakes a free electron, the lat- ter undergoes transverse oscillation (quiver motion), wit h relativistic velocities for η>∼1 [2–5,15,16]. The v×B force then couples the transverse oscillation to a longi- tudinal drift in the direction of the wave. In the nonrel- ativistic limit, this effect is often said to be due to the “ponderomotive potential” associated with the envelope of the electromagnetic pulse [3]. The resulting temporary energy transfer to the longitudinal motion of the electron can in principle be arbitrarily large. A semiclassical description of this process exists as well. A quantum-mechanical electron inside a classical plane wave can be described by the Volkov solutions to the Dirac equation [17,18]. Such electrons are sometimes described as “dressed”, and they can be characterized by a quasimomentum q=p+ǫk0, (2) where the invariant ǫis given by ǫ=m2η2 2(p·k0), (3) with ( p·k0) being the 4-vector product of the 4-momenta pof the electron and k0of a photon of the background wave. The factor ǫneed not be an integer, and can be thought of as an effective number of wave photons “dragged” along with the electron as a result of a small difference between the large rates of absorption and emis- sion (back into the wave) of wave photons by the electron. (Strictly speaking, the wave used in the Volkov solution is classical and, hence, contains no photons.) As a result, the electron inside the wave has an effective mass, m, that is greater than its free mass m[3]: m2=q2=m2(1 +η2). (4) From a classical view, the quasimomentum qis the result of averaging over the transverse oscillations (quiv er motion) of the electron in the background wave. When discussing conservation of energy and momentum in the classical view, both transverse and longitudinal motion of the electron must be considered; but in a quantum analysis, quasimomentum is conserved and no mention is made of the classical transverse oscillations. 1Throughout this paper the background wave propa- gates in the + zdirection, and the 4-momentum of a pho- ton of this wave is written k0= (ω0,0,0, ω0). (5) From now on, we use units in which cand ¯hequal one. We first consider a relativistic electron moving along the + zaxis with 4-momentum p= (E,0,0, P) =γm(1,0,0, β), (6) where EandPare the energy and the momentum of the electron prior to the arrival of the wave, β≈1 is the electron’s velocity and γ= 1//radicalbig 1−β2≫1. Then (p·k0) =ω0(E−P) =m2ω0 E+P, (7) so ǫ=η2(E+P) 2ω0≈γmη2 ω0, (8) where the approximation holds for a relativistic electron. For a wave of optical frequencies (such as a laser), ǫ≫1. The quasienergy, q0, is then large: q0=E(1 +η2). (9) The electron has been accelerated from energy Eoutside the wave to energy E(1 +η2) inside the wave. Since η can in principle be large compared to 1, this acceleration can be very significant. Can we take advantage of this acceleration in a high- energy-physics experiment? The example of Compton scattering of an electron by one laser beam while in a second laser beam has recently been reported elsewhere [19]. Here, we consider examples of possibly enhanced production of electroweak gauge bosons in high-energy eeandeγcollisions in the presence of an intense laser. Suppose the electron pcollides head-on with a positron p′, all inside the background wave. The positron 4- momentum is then p′= (E′,0,0,−P′), (10) where E′≫min the relativistic case. Then (p′·k0) =ω0(E′+P′)≈2E′ω0. (11) The corresponding quasimomentum is q′=p′+ǫ′k0, (12) where ǫ′=m2η2 2(p′·k0)≈m2η2 2E′ω0. (13) The factor ǫ′is not large in general, and the energy of a relativistic positron (or electron) moving against an optical wave is almost unchanged.However, the center-of-mass (cm) energy of the e+e− system is increased when the collision occurs inside the background wave. The cm-energy squared is s= (q+q′)2≈4EE′(1 +η2), (14) which is enhanced by a factor 1 + η2compared to the case of no background wave. For example, the Z0boson could be produced in e+e− collisions with 33- rather than 46.6-GeV beams, if the collision took place inside a background wave of strength η= 1. Of course, the background wave Compton scatters off the positron beam at a high rate if η>∼1, which results in substantial smearing of the energy of that beam. In practice, the cm-energy enhancement by a background wave would not be very useful in e+e−oreecollisions. Note, however, that Compton scattering is insignifi- cant when the background wave and electron move in the same direction, unless the wave is extraordinarily strong. By an application of the Larmor formula in the (average) rest frame of the electron, we find that the fraction of the electron’s (laboratory) energy radiated in one cycle of its motion in the wave is of order αη2(ω0/E), where αis the fine-structure constant. Suppose instead that the electron collides head-on with a high-energy photon of frequency ωand 4-momentum p′=k= (ω,0,0,−ω). (15) Then eq. (14) holds on substituting ωforE′; the cm- energy squared is again enhanced by the factor 1 + η2. The background wave can, of course, interact directly with the high-energy photon to produce e+e−pairs, but if 4ωω0< m2(1 +η2), the pair-production rate is much suppressed [14]. Thus, there is a regime in which e+ photon collisions in a strong background wave are cleaner thane+e−oreecollisions in the wave. In practice, we could get the high-energy photon from Compton scattering of the background wave off an elec- tron beam. One might not want to backscatter the wave off a positron beam because of “backgrounds” from e+e−→Z0. A physics topic of interest would be the reaction k+e−→W−+ν, (16) which proceeds via the triple-gauge-boson coupling γWW , and whose angular distribution is sensitive to the magnetic moment of the Wboson [20,21]. The “back- ground” process k+e−→Z0+e−(17) could be suppressed by suitable choice of polarization of the electron and background wave. For electron beams of 46.6 GeV as at the Stanford Linear Accelerator Center, green laser light backscatters into photons of energies up to about 30 GeV. Thus if the 2laser had η= 1, the cm energy would extend up to 106 GeV, well above the threshold for reactions (16-17). However, the enhancement factors 1+ η2in the electron energy, eq. (9), and in the cm-energy squared, eq. (14), of eeor electron-photon collisions are very much dependent on the idealization that the background wave is highly collinear with the electron. We reconsider the preceding, but now suppose that the electron makes angle θ≪1 to the zaxis, The 4- momentum of the electron is p= (E, Psinθ,0, Pcosθ), (18) and (p·k0) =Eω0(1−βcosθ)≈mω0 2γ(1 +γ2θ2).(19) As a consequence, the (quasi)energy of the electron inside the wave is now q0=p0+m2η2ω0 2(p·k0)≈E/parenleftbigg 1 +η2 1 +γ2θ2/parenrightbigg , (20) which reduces to eq. (9) as θgoes to zero. However, if θ > η/γ , then the electron is hardly accelerated by the background wave. Electrons of present interest in high-energy physics typically have energies in the range 1-1000 GeV, corre- sponding to γ≈103-106. This places very severe require- ments on the alignment of the background wave with the electron beam. Indeed, the angular divergence of an elec- tron beam is often larger than 1 /γ, so that no alignment of the background wave could impart large energy en- hancements to the entire beam. Furthermore, optical waves with η≈1 can only be obtained at present in focused laser beams for which the characteristic angular spread is ∆ θ>∼0.1. So even if the central angle of the beam could be aligned to better than 1/γ, only a very small fraction of the beam power would lie within a cone of that angle. We also note that for the quasimomentum qto be meaningful, the electron must have resided inside the strong background field for at least one cycle. A rela- tivistic electron moves distance 2 γ2(1 +η2)λ0while the background wave advances one wavelength relative to the electron [22]. However, the strong-field region of a fo- cused laser is characterized by its Rayleigh range, which is typically a few hundred wavelengths when η≈1. Fur- ther, the transverse extent of the (classical) trajectory is of order γηλ0. Hence, in present laser systems, the strong-field region is not extensive enough that the en- ergy transfer (9) could be realized for γ>∼10. While physical consequences of the temporary acceler- ation of relativistic electrons inside an intense laser bea m may be difficult to demonstrate, there is also interest in the case where the electron is initially at rest, or nearly so, such as electrons ionized from gas atoms by the pas- sage of the background laser pulse [12].An interesting process is so-called trident production, e+A→e′+A′+e+e−, (21) of an electron-positron pair in the interaction of an ion- ization electron with a nucleus Aof a gas atom. For a very heavy nucleus A, its final state A′has a different mo- mentum but the same energy. Then the initial electron must provide the energy to create the e+e−pair as well as that for the final electron. The least energy required is when all three final-state electrons and positrons are at “rest” ( i.e., they have zero net longitudinal momentum; they must always have quiver motion when they are in the wave). Thus, the minimum total quasienergy of the final-state electrons and positrons is 3 m. We conclude that the quasienergy q0of the initial elec- tron must be at least 3 mfor reaction (21) to occur. If the electron is at rest prior to the arrival of the back- ground wave its 4-momentum is p= (m,0,0,0). (22) As the electron is overtaken by a wave of strength ηand 4-momentum given by (5), it takes on quasimomentum q= (m(1 +η2/2),0,0, mη2/2)≡(mγ,0,0,mγβ z).(23) Thus, the net longitudinal velocity of the electron inside the wave is βz=qz/q0= (η2/2)/(1 +η2/2). As ex- pected, inside a very strong wave the electron can take on relativistic longitudinal motion. We could have trident production while the electron is still in the wave if the quasienergy q0=m(1 +η2/2) exceeds 3 m. For an electron initially at rest, this requires η≥/radicalbig 16 + 12√ 2 = 5.74. The trident process is still possible within a wave with η <5.74 provided the electron has quasienergy q0≥3m. This might arise, for example, because of acceleration of the electron by the plasma-wakefield effect [23]. It is conceivable that the electron creates the pair in a linearly polarized wave at a phase when its (classical) kinetic energy is high, but the final electron and the pair all appear with a lower kinetic energy corresponding to some other phase of the wave. This can’t happen if the interaction takes place at a well-defined point, since the phase of the wave is a unique function space and time. It might occur if the final particles “tunnel” to another space-time point before appearing, and the instantaneous kinetic energy is lower at that point. However, we will find shortly that such tunneling is not consistent with energy conservation. To be as definite as possible, we consider ordinary energy along the classical trajectories, rather than quasimomentum. The latter is taken into account in the sense that the electron and positron are not created at rest, but with the transverse velocities appropriate to phase of the background wave at the spacetime point at which the pair appears. It is sufficient to consider only those trajectories with zero average momentum ( i.e., zero quasi-3-momentum). 3For circular polarization of the background wave, the electron trajectory is a circle in the plane perpendicu- lar to the zaxis, with radius a/ω0, velocity β=aand Lorentz factor γcirc=1√ 1−a2=/radicalbig 1 +η2, (24) where parameter ais given by a2=η2 1 +η2,0≤a2≤1. (25) For a background wave that is linearly polarized in the x-direction, the trajectory can be parametrized as [4,15] x=√ 2a ω0sinδ, z =a2 4ω0sin 2δ, (26) where δ=ω0τ/radicalbig 1 +η2=ω0τ/√ 1−a2, and τis the proper time. Expression (26) describes the well-known figure-8 trajectory. Now dx/dτ = (dx/dt)(dt/dτ) =γβx, soγ2= 1 + γ2β2= 1 + ( dx/dτ )2+ (dz/dτ )2. We find that γlin=1 +1 2[a2−(ω0x)2]√ 1−a2. (27) From expression (26) for the x-trajectory we see that 0≤(ω0x)2≤2a2, so γmin=1 +η2/2/radicalbig 1 +η2,and γmax=1 + 3η2/2/radicalbig 1 +η2.(28) These values surround the result that γcirc=/radicalbig 1 +η2 always for circular polarization. For small η,γmin≈1 + η4/8,γmax≈1 +η2, and γcirc≈1 +η2/2; for large η, γmin≈η/2,γmax≈3η/2, and γcirc≈η. Suppose an electron interacts with a nucleus at the place where its Lorentz factor is γmaxand reappears along with an electron-positron pair at a location where γmin holds at that moment. The nucleus absorbs the excess momentum of the initial electron. Conservation of (ordi- nary) energy requires that γmax= 3γmin. But this is not satisfied for any value of ηaccording to (28). That is, the hypothetical tunneling process is not possible under any circumstances. In sum, even when in a background wave an electron can produce positrons off nuclei only if the electron has sufficient longitudinal momentum that the corresponding (quasi)energy is three times the (effective) electron mass. We close by returning to the astrophysical context that began the historical debate on acceleration by in- tense electromagnetic waves. Gunn and Ostriker [24] have given an extensive discussion the possibility of elec- tron acceleration in the rotating dipole field of a millisec- ond pulsar, where the field strength ηcan be of order 1010. Their argument does not primarily address free electrons overtaken by a wave, but rather electrons “in- jected” or “dropped at rest’ into the wave. Neutron decayis a candidate process for injection. In very strong fields (η≫1) this decay takes place together with the absorp- tion by the electron (and proton) of a very large num- ber of wave photons, so that the electron is created with (quasi)energy ≈mη2/2 (compare eq. (23)) [25]. Because the fields of the pulsar fall off as 1 /rwhere (coinciden- tally) rpulsar≈λ0, the wavelength of the rotating dipole radiation, the field region is “short”, and the electron may emerge with some fraction of the large energy it had at the moment of its creation. An example closer to the theme of the present paper would be an electron that is overtaken by the intense electromagnetic pulse of a supernova (or other transient astrophysical occurrence, perhaps including gamma-ray bursters), and thereby temporarily accelerated to en- ergymη2/2. Such pulses could have significant fields at optical frequencies, where the transverse scale, ηλ0, of the motion of accelerated electrons is less than the Chandrasekhar radius for η < 1010. In general, the electron has low energy before and after the passage of the pulse. However, high-energy photons can arise via bremsstrahlung of the electron when it interacts with a plasma nucleus while still in the pulse. In this view, the primary astrophysical evidence of temporarily accel- erated electrons would be high-energy photons which, of course, could transfer some of their energy to protons and other charged particles in subsequent interactions. This work was supported in part by DoE grants DE- FG02-91ER40671 and DE-FG05-91ER40627. [1] D.H. Menzel and W.W. Salisbury, Nucleonics, 2, No. 4, 67 (1948). [2] G. Toraldo di Francia, Nuovo Cim. 37, 1553 (1965). [3] T.W.B. Kibble, Phys. Rev. 138, B740, (1965); 150, 1060 (1966); Phys. Rev. Lett. 16, 1054 (1966); Carg` ese Lec- tures in Physics , Vol. 2, ed. by M. L´ evy (Gordon and Breach, New York, 1968), p. 299. [4] E.S. Sarachik and G.T. Schappert, Phys. Rev. D 1, 2738 (1970). [5] P.E. Kaw and R.M. Kulsrud, Phys. Fluids 16, 321 (1973). [6] H.M. Lai, Phys. Fluids 23, 2373 (1980). [7] P.H. Bucksbaum et al., Phys. Rev. Lett. 58, 349 (1987). [8] L. J¨ onsson, J. Opt. Soc. Am. B 4, 1422, (1987). [9] G. Malka et al., Phys. Rev. Lett. 78, 3314 (1997). [10] J.X. Wang et al., Phys. Rev. E 58, 6575 (1998). [11] K. Hagenbuch, Am. J. Phys. 45, 693 (1979). [12] P.H. Bucksbaum, in Atoms in Strong Fields , C. Nicholaides et al. eds. (Plenum Press, 1990), p. 381. [13] C. Bula et al., Phys. Rev. Lett. 76, 3116 (1996). [14] D.L. Burke et al., Phys. Rev. Lett. 79, 1626 (1997). [15] L. Landau and E.M. Lifshitz, The Classical Theory of Fields , 4th ed. (Pergamon Press, Oxford, 1975), prob. 2, §47; p. 112 of the 1941 Russian edition. [16] E.M. McMillan, Phys. Rev. 79, 498 (1950). 4[17] D.M. Volkov, Zeits. f. Phys. 94, 250 (1935). [18] See also Secs. 40 and 101 of V.R. Berestetskii, E.M. Lif- shitz and L.P. Pitaevskii, Quantum Electrodynamics , 2nd ed. (Pergamon Press, Oxford, 1982). [19] F.W. Hartemann, Phys. Plasmas 5, 2037 (1998). [20] K.O. Mikaelian, Phys. Rev. D 17, 750 (1978); 30, 1115 (1984). [21] I.F. Ginzburg et al., Nucl. Phys. B228 , 285 (1983). [22] Y.W. Chan, Phys. Lett. 35A, 305 (1971). [23] D. Umstadter et al., Science, 273, 472 (1996). [24] J.P. Ostriker and J.E. Gunn, Ap. J. 157, 1395 (1969); 165, 523 (1971). [25] A.I. Nikishov and V.I. Ritus, Sov. Phys. JETP 19, 1191 (1964); 58, 14 (1983); V.I. Ritus, ibid.,29, 532 (1969). 5

arXiv:physics/0003074v1 [physics.gen-ph] 25 Mar 2000HIERARCHIC THEORY OF CONDENSEDMATTER : New state equation &Interrelation between mesoscopic and macroscopic properties Alex Kaivarainen JBL, University of Turku, FIN-20520, Turku, Finland URL: http://www.karelia.ru/˜alexk H2o@karelia.ru Materials, presented in this original article are based on: [1]. Book by A. Kaivarainen: Hierarchic Concept of Matter and Field. Water, biosystems and elementary par- ticles. New York, 1995 and two articles: [2]. New Hierarchic Theory of Matter General for Liq- uids and Solids: dynamics, thermodynamics and mesoscopic structure of water and ice (see: http://arXiv.org/abs/physics/0003044 and http:// www.karelia.ru/˜alexk [New articles]); [3]. Hierarchic Concept of Condensed Matter and its Interaction with Light: New Theories of Light Refraction, Brillouin Scattering and M¨ ossbauer effect (see URL: http:/ /www.karelia.ru/˜alexk [New articles]). Computerized verification of described here new theo- ries are presented on examples of WATER and ICE, using special computer program (copyright, 1997, A. Kaivarainen ). CONTENTS Summary of Hierarchic Theory of Matter and Field 1 The state equation for real gas 2 New state equation for condensed matter 3 Vapor pressure 4 Surface tension 5 Mesoscopic theory of thermal conductivity 6 Mesoscopic theory of viscosity for liquids and solids 7 Brownian diffusion 8 Self-diffusion in liquids and solids 9 Mesoscopic approach to proton conductivity in water, ice and other systems containing hydrogen bonds 10 Regulation of pH and shining of water by electromag- netic and acoustic fields 1Summary of Hierarchic Theory of Condensed Matter (http://arXiv.org/abs/physics/0003044) A basically new hierarchic quantitative theory, general fo r solids and liquids, has been developed. It is assumed, that unharmonic oscillations of particles in any con- densed matter lead to emergence of three-dimensional (3D) s uperpo- sition of standing de Broglie waves of molecules, electroma gnetic and acoustic waves. Consequently, any condensed matter could b e con- sidered as a gas of 3D standing waves of corresponding nature . Our approach unifies and develops strongly the Einstein’s and De bye’s models. Collective excitations, like 3D standing de Broglie waves o f molecules, representing at certain conditions the mesoscopic molecul ar Bose con- densate, were analyzed, as a background of hierarchic model of con- densed matter. The most probable de Broglie wave (wave B) length is deter- mined by the ratio of Plank constant to the most probable impu lse of molecules, or by ratio of its most probable phase velocity to fre- quency. The waves B are related to molecular translations (t r) and librations (lb). As far the quantum dynamics of condensed matter does not foll ow in general case the classical Maxwell-Boltzmann distribution, the re al most probable de Broglie wave length can exceed the classical thermal de Brog lie wave length and the distance between centers of molecules many times. This makes possible the atomic and molecular Bose condensat ion in solids and liquids at temperatures, below boiling point. It is one o f the most important results of new theory, confirmed by computer simulations on e xamples of water and ice. Four strongly interrelated new types of quasiparticles (collective excita- tions) were introduced in our hierarchic model: 1.Effectons (tr and lb) , existing in ”acoustic” (a) and ”optic” (b) states represent the coherent clusters in general case ; 2.Convertons , corresponding to interconversions between trandlbtypes of the effectons (flickering clusters); 3.Transitons are the intermediate [ a⇋b] transition states of the trandlb effectons; 4.Deformons are the 3D superposition of IR electromagnetic or acoustic waves, activated by transitons andconvertons. Primary effectons (tr and lb) are formed by 3D superposition of the most probable standing de Broglie waves of the oscillating ions, atoms or molecules. The volume of effectons (tr and lb) may contain fro m less than one, to tens and even thousands of molecules. The first condition m eans validity ofclassical approximation in description of the subsystems of the effect ons. 2The second one points to quantum properties of coherent clusters due to molecular Bose condensation . The liquids are semiclassical systems because their primar y (tr) effectons contain less than one molecule and primary (lb) effectons - mo re than one molecule. The solids are quantum systems totally because both kind of t heir primary effectons (tr and lb) are molecular Bose condensates .These conse- quences of our theory are confirmed by computer calculations . The 1st order [ gas→liquid ] transition is accompanied by strong decreas- ing of rotational (librational) degrees of freedom due to em ergence of primary (lb) effectons and [ liquid→solid] transition - by decreasing of translational de- grees of freedom due to mesoscopic Bose-condensation in for m of primary (tr) effectons. In the general case the effecton can be approximated by par- allelepiped with edges corresponding to de Broglie waves le ngth in three selected directions (1, 2, 3), related to the symmetry of the molecular dynamics. In the case of isotropic molecular moti on the effectons’ shape may be approximated by cube. The edge-lengt h of primary effectons (tr and lb) can be considered as the ”parame ter of order”. The in-phase oscillations of molecules in the effectons corr espond to the effecton’s (a) - acoustic state and the counterphase oscillations correspond to their (b) - optic state. States (a) and (b) of the effectons differ in potential energy only, however, their kinetic energies, impulses and spatial dimensions - are the same. The b-state of the effectons has a common feature with Fr¨ olich’s polar mode. The(a→b)or(b→a)transition states of the primary effectons (tr and lb), defined as primary transitons, are accompanied b y a change in molecule polarizability and dipole moment withou t density fluctuations. At this case they lead to absorption or radiati on of IR photons, respectively. Superposition (interception) of t hree internal standing IR photons of different directions (1,2,3) - forms p rimary electromagnetic deformons (tr and lb). On the other hand, the [lb ⇋tr]convertons andsecondary transitons are accompanied by the density fluctuations, leading to absorption or radiation of phonons . Superposition, resulting from interception of standing phonons in three di- rections (1,2,3) is termed: secondary acoustic deformons (tr and lb). Correlated collective excitations of primary and secondary effectons and deformons (tr and lb) ,localized in the volume of primary trandlb electromag- netic deformons ,lead to origination of macroeffectons, macrotransitons andmacrodeformons (tr and lb respectively) . Correlated simultaneous excitations of tr and lb macroeffec tonsin the vol- ume of superimposed trandlbelectromagnetic deformons lead to origination ofsupereffectons. In turn, the coherent excitation of both: tr andlb macrodeformons and 3macroconvertons in the same volume means creation of superdeformons. Su- perdeformons are the biggest (cavitational) fluctuations, leading to microbub- bles in liquids and to local defects in solids. Total number of quasiparticles of condensed matter equal to 4!=24, reflects all of possible combinations of the four basic ones [ 1-4], intro- duced above. This set of collective excitations in the form o f ”gas” of 3D standing waves of three types: de Broglie, acoustic and el ectro- magnetic - is shown to be able to explain virtually all the pro perties of all condensed matter. The important positive feature of our hierarchic model of ma tter is that it does not need the semi-empiric intermolecular potentials f or calculations, which are unavoidable in existing theories of many body systems. T he potential energy of intermolecular interaction is involved indirectly in di mensions and stability of quasiparticles, introduced in our model. The main formulae of theory are the same for liquids and solid s and include following experimental parameters, which take into ac- count their different properties: [1]- Positions of (tr) and (lb) bands in oscillatory spectra; [2]- Sound velocity; [3]- Density; [4]- Refraction index (extrapolated to the infinitive wave leng th of photon ). The knowledge of these four basic parameters at the same temp erature and pressure makes it possible using our computer program, to ev aluate more than 300 important characteristics of any condensed matter. Amo ng them are such as: total internal energy, kinetic and potential energies, heat capacity and ther- mal conductivity, surface tension, vapor pressure, viscos ity, coefficient of self- diffusion, osmotic pressure, solvent activity, etc. Most of calculated parameters are hidden, i.e. inaccessible to direct experimental measu rement. The new interpretation and evaluation of Brillouin light sc attering and M¨ ossbauer effect parameters may also be done on the basis of h ierarchic the- ory. Mesoscopic scenarios of turbulence, superconductivi ty and superfluity are elaborated. Some original aspects of water in organization and large-sc ale dynamics of biosystems - such as proteins, DNA, microtubules, membrane s and regulative role of water in cytoplasm, cancer development, quantum neu rodynamics, etc. have been analyzed in the framework of Hierarchic theory. Computerized verification of our Hierarchic theory of matte r on examples of water and ice is performed, using special comput er pro- gram: Comprehensive Analyzer of Matter Properties (CAMP, c opy- right, 1997, Kaivarainen). The new optoacoustic device, ba sed on this program, with possibilities much wider, than that of IR, Ram an and Brillouin spectrometers, has been proposed (see URL: http: //www.karelia.ru/˜alexk [CAMP]). 4This is the first theory able to predict all known experimenta l temperature anomalies for water and ice. The conformity bet ween theory and experiment is very good even without any adjustab le pa- rameters. The hierarchic concept creates a bridge between micro- and m acro- phenomena, dynamics and thermodynamics, liquids and solid s in terms of quantum physics. *************************************************** ***************** 1. The state equation for real gas The Clapeyrone-Mendeleyev equation sets the relationship between pressure (P), volume ( V) and temperature ( T) values for the ideal gas containing N0 molecules (one mole): PV=N0kT=RT (1) In the real gases interactions between the molecules and the ir sizes should be taken into account. It can be achieved by entering the corres ponding amend- ments into the left part, to the right or to the both parts of eq . (1). It was Van der Waals who choused the first way more than a hundre d years ago and derived the equation: /parenleftBig P+a V2/parenrightBig/parenleftbigV−b/parenrightbig =RT (2) where the attraction forces are accounted for by the amendin g term (a/V2), while the repulsion forces and the effects of the excluded vol ume accounted for the term (b). Equation (2) correctly describes changes in P,V and T relate d to liquid-gas transitions on the qualitative level. However, the quantit ative analysis by means of (2) is approximate and needs the fitting parameters. The pa rameters (a) and (b) are not constant for the given substance and depend on tem perature. Hence, the Van der Waals equation is only some approximation descri bing the state of a real gas. We propose a way to modify the right part of eq.(1), substitut ing it for the part of the kinetic energy (T) of 1 mole of the substance (eq.4 .31 in [1, 2]) in real gas phase formed only by secondary effectons and deformo ns with nonzero impulse, affecting the pressure: PV=2 3¯Tkin=2 3V01 Z·/summationdisplay tr,lb/bracketleftBigg ¯nef/summationtext3 1/parenleftbig¯Ea 1,2,3/parenrightbig2 2m/parenleftbig va ph/parenrightbig2/parenleftbig¯Pa ef+¯Pb ef/parenrightbig + 5+ ¯nd/summationtext3 1/parenleftBig ¯E1,2,3 d/parenrightBig2 2m(vs)2¯Pd  tr,lb(3) The contribution to pressure caused by primary quasipartic les as Bose-condensate with the zero resulting impulse is equal to zero also. It is assumed when using such approach that for real gases the model of a system of weakly interacted oscillator pairs is va lid. The validity of such an approach for water is confirmed by availab le ex- perimental data indicating the presence of dimers, trimers and larger H2Oclusters in the water vapor (Eisenberg and Kauzmann, 1975). Water vapor has an intensive band in oscillatory spectra at ˜ν= 200cm−1. Possibly, it is this band that characterizes the frequenci es of quantum beats between ”acoustic” (a) and ”optic” (b) transl ational oscillations in pairs of molecules and small clusters. The f requencies of librational collective modes in vapor are absent. The energies of primary gas quasiparticles (hνaandhνb)can be calculated on the basis of the formulae used for a liquid (Cha pter 4 of [1] or [2]). However, to calculate the energies of secondary quasiparti cles in (¯ a) and ( ¯b) states the Bose-Einstein distribution must be used for the case when the temperature is higher than the Bose-conden sation temperature (T >T 0)and the chemical potential is not equal to zero (µ<0). According to this distribution: /braceleftbigg ¯Ea=h¯νa=hνa exp(hνa−µ kT)−1/bracerightbigg tr,lb /braceleftBigg ¯Eb=h¯νb=hνb exp/parenleftBig hνb−µ kT/parenrightBig −1/bracerightBigg tr,lb(4) The kinetic energies of effectons (¯a)tr,lband(¯b)tr,lbstates are equal, only the potential energies differ as in the case of condensed matter. All other parameters in basic equation (3) can be calculated as previously described [1, 2]. 2. New state equation for condensed matter Using our eq.(4.3 from [1,2]) for the total internal energy o f condensed matter (Utot), we can present state equation in a more general form than (3 ). For this end we introduce the notions of internal pressure (Pin), including all type of interactions between particles of matter and excluded molar volume (Vexc): 6Vexc=4 3πα∗N0=V0/parenleftbiggn2−1 n2/parenrightbigg (5) whereα∗is the acting polarizability of molecules in condensed matt er (see Part 1 of [3]);N0is Avogadro number, and V0is molar volume. The general state equation can be expressed in the following form: PtotVfr= (Pext+Pin)(V0−Vexc) =Uef (6) where:Uef=Utot(1+V/Tt kin) =U2 tot/Tkinis the effective internal energy and: (1 +V/Tkin) =Utot/Tkin=S−1 is the reciprocal value of the total structural factor ( eq.2.46aof[1]);Ptot= Pext+Pinis total pressure, PextandPinare external and internal pressures; Vfr=V0−Vexc=V0/n2(see eq.5) is a free molar volume; Utot=V+Tkinis the total internal energy, V and Tkinare total potential and kinetic energies of one mole of matter. For the limit case of ideal gas, when Pin= 0;Vexc= 0; and the potential energyV= 0, we get from (6) the Clapeyrone - Mendeleyev equation (see 1): PextV0=Tkin=RT One can use equation of state (6) for estimation of sum of all types of internal matter interactions , which determines the internal pressure Pin: Pin=Uef Vfr−Pext=n2U2 tot V0Tkin−Pext (7) where: the molar free volume: Vfr=V0−Vexc=V0/n2; and the effective total energy: Uef=U2 tot/Tkin=Utot/S. For solids and most of liquids with a good approximation: Pin≫[Pext∼1 atm. = 105Pa]. Then from (7) we have: Pin∼=n2Utot V0S=n2 V0·Utot/parenleftbigg 1 +V Tkin/parenrightbigg (8) whereS=Tkin/Utotis a total structural factor; Tkinand V are total kinetic and potential energies, respectively. For example for 1 mole of water under standard conditions we o btain: Vexc= 8.4cm3;Vfr= 9.6cm3;V0=Vexc+Vfr= 18cm3; Pin∼=380000 atm. = 3 .8·1010Pa(1 atm. =105Pa). 7The parameters such as sound velocity, molar volume, and the positions of translational and librational bands in oscillatory spectr a that determine Uef(4.3) depend on external pressure and temperature. The results of computer calculations of Pin(eq.7) for ice and water are presented on Fig. 1 a,b. Polarizability and, consequently, free volume ( Vfr) andPinin (6) depend on energy of external electromagnetic fields (see Part 1 of [3]. Fig. 1. (a) Theoretical temperature dependence of internal pres- sure (Pin) in ice including the point of [ice ⇔water] phase transition; (b) Theoretical temperature dependence of internal pressu re (Pin) in water. Computer calculations were performed using eq. (7). The minima of Pin(T) for ice at −1400and−500Cin accordance with eq.(9) correspond to the most stable structure of this matter, rela ted to temperature transition. In water some kind of transition appears at 350C, near physiological temperature. There may exist conditions when the derivatives of internal pressure P inare equal to zero: (a) :/parenleftbigg∂Pin ∂Pext/parenrightbigg T= 0 and ( b) :/parenleftbigg∂Pin ∂T/parenrightbigg Pext= 0 (9) This condition corresponds to the minima of potential energy, i.e. to the most stable structure of given matter. In a general case there may be a few metastable states when conditions (9) are fulfilled. Equation of state (7) may be useful for the study of mechanica l properties of condensed matter and their change under different influenc es. Differentiation of (6) by external pressure gives us at T = const: 8Vfr+∂Pfr ∂Pext(Pex+Pin) +Vfr∂Pin ∂Pext=∂Pef ∂Pext(10) Dividing the left and right part of (10) by free volume Vfrwe obtain: /parenleftbigg∂Pin ∂Pext/parenrightbigg T=/parenleftbigg∂Pef ∂Pext/parenrightbigg T−/bracketleftbig1 +βT(Pext+Pin)/bracketrightbig T(11) where:βT=−(∂Vfr/∂Pext)/Vfris isothermal compressibility. From (9) and (11) we derive condition for the maximum stability of matter structure: /parenleftbigg∂Pef ∂Pext/parenrightbigg T= 1 +β0 TPopt tot (12) where:Popt tot=Pext+Popt inis the ”optimum” total pressure. The derivative of (6) by temperature gives us at Pext=const: Ptot/parenleftbigg∂Vfr ∂T/parenrightbigg Pext+Vfr/parenleftbigg∂Pin ∂T/parenrightbigg Pext=/parenleftbigg∂Uef ∂T/parenrightbigg Pext=CV (13) where /parenleftbigg∂Vfr ∂T/parenrightbigg Pext=/parenleftbigg∂V0 ∂T/parenrightbigg Pext−4 3πN0/parenleftbigg∂α∗ ∂T/parenrightbigg Pext(14) and/parenleftbigg∂Vtot ∂T/parenrightbigg Pext=∂Pin ∂T(14a) From our mesoscopic theory of refraction index (Part 1 of [3] ) the acting polar- izabilityα∗is: α∗=/parenleftBig n2−1 n2/parenrightBig 4 3πN0 V0(15) When condition (9b) is fulfilled, we obtain for optimum inter nal pressure ( Popt in) from (13): Popt in=CV//parenleftbigg∂Vfr ∂T/parenrightbigg Pext−Pext (16) or 9Popt in=C Vfrγ−Pext, (17) where γ= (∂Vfr/∂T)/Vfr (18) is the thermal expansion coefficient; Vfris the total free volume in 1 mole of condensed matter: Vfr=V0−Vexc=V0/n2(19) It is taken into account in (13) and (19) that (∂Vexc/∂T)∼=0 (20) because, as has been shown earlier (Fig.25a of [1] and Part 1 o f [3]), ∂α∗/∂T∼=0 Dividing the left and right parts of (13) by PtotVfr=Uef, we obtain for the heat expansion coefficient: γ=CV Uef−1 Ptot/parenleftbigg∂Pin ∂T/parenrightbigg Pext(21) Under metastable states, when condition (9 b) is fulfilled, γ0=CV/Uef (22) Putting (8) into (12), we obtain for isothermal compressibi lity of metastable states corresponding to (9a) following formula: β0 T=V0Tkin n2U2 tot/parenleftbigg∂Uef ∂Pext−1/parenrightbigg (23) It seems that our equation of state (7) may be used to study diff erent types of external influences (pressure, temperature, electromag netic radiation, defor- mation, etc.) on the thermodynamic and mechanic properties of solids and liquids. 103. Vapor pressure When a liquid is incubated long enough in a closed vessel at co nstant tem- perature, then an equilibrium between the liquid and vapor i s attained. At this moment, the number of molecules evaporated and conde nsed back to liquid is equal. The same is true of the process of sublimat ion. There is still no satisfactory quantitative theory for vapor pressure calcula- tion. We can suggest such a theory using our notion of superdeformons , represent- ing the biggest thermal fluctuations (see Table 1 and Introdu ction). The basic idea is that the external equilibrium vapor pressure is rela ted to internal one (PS in) with coefficient determined by the probability of cavitatio nal fluctuations (superdeformons) in the surface layer of liquids or solids. In other words due to excitation of superdeformons with prob ability (PS D), the internal pressure ( PS in) in surface layers, determined by the total contribu- tions of all intramolecular interactions turns to external one - vapor pressure (PV). It is something like a compressed spring energy realizati on due to trigger switching off. For taking into account the difference between the surface an d bulk internal pressure (Pin) we introduce the semiempirical surface pressure factor ( qS) as: PS in=qSPin−Pext=qS·n2Utot V0S−Pext (24) where: P incorresponds to eq.(7);S=Tkin/Utotis a total structure factor. The value of surface factor ( qS) for liquid and solid states is not the same: qS liq<qS sol (25) Fig. 2. a) Theoretical ( −) and experimental ( ··) temperature de- pendences of vapor pressure ( Pvap) for ice (a) and water (b) includ- ing phase transition region. Computer calculations were pe rformed using eq. (26). 11Multiplying (24) to probability of superdeformons excitat ion we obtain for vapor pressure, resulting from evaporation or sublimation , the following formu- lae: Pvap=PS in·PS D=/parenleftbigg qSn2U2 tot V0Tkin−Pext/parenrightbigg ·exp/parenleftbigg −ES D kT/parenrightbigg (26) where: PS D= exp/parenleftbigg −ES D kT/parenrightbigg (27) is a probability of superdeformons excitation (see eqs. 3.3 7, 3.32 and 3.33). We can assume, that the difference in the surface and bulk inte rnal pressure is determined mainly by difference in total internal energy ( Utot) but not in kinetic one ( Tk). Then a pressure surface factor could be presented as: qS=γ2= (Uin/Utot)2 where:γ=US tot/Utotis the surface energy factor , reflecting the ratio of surface and bulk total energy. Theoretical calculated temperature dependences of vapor p ressure, described by (26) coincide very well with experimental ones for water a tqS liq= 3.1 (γl= 1.76) and for ice at qS sol= 18 (γs= 4.24) (Fig. 2). The almost five-times difference between qS solandqS liqmeans that the surface properties of ice differ from bulkones much more than for liquid water. The surface factors qS liqandqS solshould be considered as a fit pa- rameters. The qS=γ2is the only one fit parameter that was used in our hierarchic mesoscopic theory. Its calculation from t he known vapor pressure or surface tension can give an important info rmation itself. 4. Surface tension The resulting surface tension is introduced in our mesoscop ic model as a sum: σ= (σtr+σlb) (28) where:σtrandσlbare translational and librational contributions to surfac e tension. Each of these components can be expressed using our mesoscopic state equation (6, 7), taking into account the difference between s urface and bulk total energies ( qS), introduced in previous section: 12σtr,lb=1 1 π(Vef)2/3 tr,lb/bracketleftbiggqSPtot(PefVef)tr,lb−Ptot(PefVef)tr,lb (Pef+Pt)tr+ (Pef+Pt)lb+ (Pcon+PcMt)/bracketrightbigg (29) where (Vef)tr,lbare volumes of primary tr and lib effectons, related to their concentration ( nef)tr,lbas: (Vef)tr,lb= (1/nef)tr,lb; rtr,lb=1 π(Vef)2/3 tr,lb is an effective radius of the primary translational and libra tional effectons, lo- calized on the surface of condensed matter; qSis the surface factor, equal to that used in eq.(24-26); [ Ptot=Pin+Pext] is a total pressure, corresponding to eq.(6); (Pef)tr,lbis a total probability of primary effecton excitations in the (a) and (b) states: (Pef)tr= (Pa ef+Pb ef)tr (Pef)lb= (Pa ef+Pb ef)lb (Pt)trand (Pt)lbin (29) are the probabilities of corresponding transiton ex cita- tion; Pcon=Pac+Pbcis the sum of probabilities of [ a] and [ b]convertons; PcMt= Pac·Pbcis the probability of Macroconverton (see Introduction and Chapter 4). The eq. (29) contains the ratio: (Vef/V2/3 ef)tr,lb=ltr,lb (30) where:ltr= (1/nef)1/3 trandllb= (1/nef)1/3 libare the length of the ribs of the primary translational and librational effectons, approxim ated by cube. Using (30) and (29) the resulting surface tension (28) can be presented as: σ=σtr+σlb=πPtot(qS−1)·/bracketleftbig(Pef)trltr+ (Pef)llb/bracketrightbig (Pef+Pt)tr+ (Pef+Pt)lb+ (Pcon+PcMt)(31) where translational component of surface tension is: σtr=πPtot(qs−1)(Pef)trltr (Pef+Pt)tr+ (Pef+Pt)lb+ (Pcon+PcMt)(32) 13and librational component of σis: σlb=πPtot(qS−1)(Pef)lbllb (Pef+Pt)lb+ (Pef+Pt)lb+ (Pcon+PcMt)(33) Under the boiling condition when qS→1 as a result of ( US tot→Utot), then σtr, σlbandσtends to zero. The maximum depth of the surface layer, which determines the σlbis equal to the length of edge of cube ( llb), that approximates the shape of primary librational effectons. It decreases from about 20 ˚A at 00Ctill about 2.5 ˚A at 1000C(see Fig. 7b of [1] or Fig. 4b of [2]). Mono- tonic decrease of ( llb)with temperature could be accompanied by nonmonotonic change of probabilities of [lb/tr] convertons and macrocon vertons excitations (see comments to Fig. 7a of [1] or to Fig 4a of [2]). Consequent ly, the temper- ature dependence of surface tension on temperature can disp lay anomalies at definite temperatures. This consequence of our theory is con firmed experimen- tally (Adamson, 1982; Drost-Hansen and Lin Singleton, 1992 ). The thickness of layer ( ltr), responsible for contribution of translational effectons in surface tension ( σtr) has the dimension of one molecule in all tem- perature interval for liquid water. The results of computer calculations of σ(eq.31) for water and experimental data are presented at Fig.3. Fig. 3. Experimental ( ) and theoretical (- - -) temperature dependences of the surface tension for water, calculated fr om eq.(31). It is obvious, that the correspondence between theory and ex per- iment is very good, confirming in such a way the correctness of our model and Hierarchic concept in general. 5. Mesoscopic theory of thermal conductivity 14Thermal conductivity may be related to phonons, photons, fr ee electrons, holes and [electron-hole] pairs movement. We will discuss here only the main type of thermal conductivi ty in condensed matter, related to phonons. The analogy with the known formula for thermal conductivity (κ) in the framework of the kinetic theory for gas is used [4]: κ=1 3CvvsΛ (34) where C vis the heat capacity of condensed matter, vsis sound velocity, charac- terizing the speed of phonon propagation in matter, and Λ is t he average length of free run of phonons. The value of Λ depends on the scattering and dissipation of ph onons at other phonons and different types of defects. Usually decrea sing temperature increases Λ. Different factors influencing a thermal equilibrium in the sy stem of phonons are discussed. Among them are the so called U- and N- processe s describing the types of phonon-phonon interaction. However, the tradi tional theories are unable to calculate Λ directly. Mesoscopic theory introduce two contributions to thermal c onductivity: re- lated to phonons, irradiated by secondary effectons and form ingsecondary translational and librational deformons ( κsd)tr,lband to phonons, irradiated by aandbconvertons [ tr/lb], forming the convertons-induced deformons ( κcd)ac.bc: κ= (κsd)tr,lb+ (κcd)ac.bc=1 3Cvvs[(Λsd)tr,lb+ (Λ cd)ac,bc] (35) where: free runs of secondary phonons (tr and lb) are represented as: 1/(Λsd)tr,lb= 1/(Λtr) + 1/(Λlb) = (νd)tr/vs+ (νd)lb/vs consequently: 1/(Λsd)tr,lb=vs (νd)tr+ (νd)lb(36) and free runs of convertons-induced phonons: 1/(Λcd)ac,bc= 1/(Λac) + 1/(Λbc) = (νac)/vs+ (νbc)/vs consequently: (Λ sd)tr,lb=vs (νd)tr+ (νd)lb(37) The heat capacity: CV=∂Utot/∂Tcan be calculated also from our theory (see Chapter 4 and 5). 15Fig. 4. Temperature dependences of total thermal conductiv- ity for water and contributions, related to acoustic deform ons and [lb/tr]convertons. The dependences were calculated, using eq. (3 7). Quantitative calculations show that formula (35), based on our mesoscopic model, works well for water (Fig. 4). It could be used for any other con- densed matter also if positions of translational and librat ional bands, sound velocity and molar volume for this matter at the same te mper- ature interval are known. The small difference between experimental and theoretical d ata can reflect the contributions of non-phonon process in therm al con- ductivity, related to macrodeformons, superdeformons and macro- convertons, i.e. big fluctuations. 6. Mesoscopic theory of viscosity for liquids and solids The viscosity is determined by the energy dissipation as a result of medium (liquid or solid) structure deformation. Viscosity corres ponding to the shift deformation is named shear viscosity . So- called bulk viscosity is related to deformation of volume parameters and corresponding dissip ation. These types of viscosity have not the same values and nature. The statistical theory of irreversible process leads to the following expression for shear viscosity (Prokhorov, 1988): η=nkTτ p+ (µ∞−nkT)τq (38) where n is the concentration of particles, µ∞is the modulus of instant shift characterizing the instant elastic reaction of medium, τpandτqare the relaxation times of impulses and coordinates, respectively. 16However, eq.(38) is inconvenient for practical purposes du e to difficulties in determination of τp,τqandµ∞. Sometimes in a narrow temperature interval the empiric Ondr ade equation is working: η=A(T)·exp(β/T) (39) A(T) is a function poorly dependent on temperature. A good results in study the microviscosity problem were obta ined by combin- ing the model of molecular rotational relaxation [5] and the Kramers equation (˚Akesson et al., 1991). However, the using of the fit parameter s was necessarily in this case also. We present here our mesoscopic theory of viscosity. To this end the dissipation processes, related to ( A⇋B)tr.lbcycles of translational and li- brational macroeffectons and (a,b)- convertons excitations were used. The same approach was employed for elaboration of mesoscopic theory of diffusion in con- densed matter (see next section). In contrast to liquid state, the viscosity of solids is determined by the biggest fluctuations: supereffectons andsuperdeformons , resulting from simultane- ous excitations of translational and librational macroeffe ctons and macrodefor- mons in the same volume. The dissipation phenomena and ability of particles or molec ules to diffusion are related to the local fluctuations of the free volume (∆ vf)tr,lb. According to mesoscopic theory, the fluctuations of free volume and that o f density occur in the almost macroscopic volumes of translational and librat ional macrodeformons and in mesoscopic volumes of macroconvertons , equal to volume of primary librational effecton at the given conditions. Translationa l and librational types of macroeffectons determine two types of viscosity, i.e. tra nslational ( ηtr) and librational ( ηlb) ones. They can be attributed to the bulk viscosity. The con- tribution to viscosity, determined by (a and b)- convertons is much more local and may be responsible for microviscosity and mesoviscosit y. Let us start from calculation of the additional free volumes (∆vf) originat- ing from fluctuations of density, accompanied the translati onal and librational macrodeformons (macrotransitons). For 1 mole of condensed matter the following ratio between fr ee volume and concentration fluctuations is true: /parenleftbigg∆vf vf/parenrightbigg tr,lb=/parenleftbigg∆N0 N0/parenrightbigg tr,lb(40) whereN0is the average number of molecules in 1 mole of matter and (∆N0)tr,lb=N0/parenleftbiggPM D Z/parenrightbigg tr,lb(41) 17is the number of molecules changing their concentration as a result of transla- tional and librational macrodeformons excitation. The probability of translational and librational macroeffe ctons excitation (see eqs. 3.23; 3.24): /parenleftbiggPM D Z/parenrightbigg tr,lb=1 Zexp/parenleftbigg −ǫM D kT/parenrightbigg tr,lb(42) where Z is the total partition function of the system (Chapte r 4 of [1, 2]). Putting (41) to (40) and dividing to Avogadro number ( N0), we obtain the fluctuating free volume, reduced to 1 molecule of matter: ∆v0 f=∆vf N0=/bracketleftbiggvf N0/parenleftbiggPM D Z/parenrightbigg/bracketrightbigg tr,lb(43) It has been shown above (eq.19) that the average value of free volume in 1 mole of matter is: vf=V0/n2 Consequently, for reduced fluctuating (additional) volume we have: (∆v0 f)tr,lb=V0 N0n21 Zexp/parenleftbigg −ǫM D kT/parenrightbigg tr,lb(44) Taking into account the dimensions of viscosity and its phys ical sense, it should be proportional to the work (activation energy) of flu ctuation-dissipation, necessary for creating the unit of additional free volume: ( EM D/∆v0 f), and the period of (A⇋B)tr.lbcycles of translational and librational macroeffectons τA⇋B,determined by the life-times of all intermediate states (eq .46). In turn, the energy of dissipation should be strongly depend ent on the structural factor (S): the ratio of kinetic energy of m atter to its total internal energy. We assume here that this dependen ce for viscosity calculation is cubical: (Tk/Utot)3=S3. Consequently, the contributions of translational and libr ational macrodefor- mons to resulting viscosity we present in the following way: ηM tr,lb=/bracketleftBigg EM D ∆v0 f·τM/parenleftbiggTk Utot/parenrightbigg3/bracketrightBigg tr,lb(45) where: reduced fluctuating volume (∆ v0 f) corresponds to (44); the energy of macrodeformons: [ EM D=−kT·(lnPM D)]tr,lb. 18The cycle-periods of the trandlibmacroeffectons has been introduced as: /bracketleftbig τM=τA+τB+τD/bracketrightbig tr,lb(46) where: characteristic life-times of macroeffectons in A, B- states and that of transition state in the volume of primary electromagnetic d eformons can be presented, correspondingly, as follows: /bracketleftBig τA= (τa·τa)1/2/bracketrightBig tr,lband/bracketleftBig τA= (τa·τa)1/2/bracketrightBig tr,lb(47) /bracketleftBig τD=|(1/τA)−(1/τB)|−1/bracketrightBig tr,lb Using (47, 46 and 44) it is possible to calculate the contribu tions of (A⇋B) cycles of translational and librational macroeffectons to v iscosity separately, using (45). The averaged contribution of macroexcitations (tr and lb)i n viscosity is: ηM=/bracketleftbig(η)M tr·(η)M lb/bracketrightbig1/2(48) The contribution of aandb convertons to viscosity of liquids could be pre- sented in a similar to (44-48) manner after substituting the parameters of tr and lb macroeffectons with parameters of a and b convertons: ηac,bc=/bracketleftBigg Ec ∆v0 fτc/parenleftbiggTk Utot/parenrightbigg3/bracketrightBigg ac,bc(49) where: reduced fluctuating volume of ( aandb) convertons (∆ v0 f)ac,bccor- responds to: (∆v0 f)ac,bc=V0 N0n21 ZPac,bc (50) where:PacandPbcare the relative probabilities of tr/libinterconversions between aandbstates of translational and librational primary effectons ( see Introduction and Chapter 4); EacandEbcare the excitation energies of ( aand b) convertons correspondingly (see Chapter 4 of [1] and [2]); Characteristic life-times for ac-convertons and bc-convertons [ tr/lb] in the volume of primary librational effectons (”flickering cluste rs”) could be presented as: 19τac= (τa)tr+ (τa)lb= (1/νa)tr+ (1/νa)lb τbc= (τb)tr+ (τb)lb= (1/νb)tr+ (1/νb)lb(51) The averaged contribution of the both types of convertons in viscosity is: ηc= (ηac·ηbc)1/2(52) This contribution could be responsible for microviscosity or better term: meso- viscosity , related to volumes, equal to that of primary librational eff ectons. The resulting viscosity (Fig.5) is a sum of the averaged cont ributions of macrodeformons and convertons: η=ηM+ηc (53) Fig. 5. Theoretical and experimental temperature dependences of viscosities for water. Computer calculations were perfo rmed using eqs. (44 - 53) and (4.3; 4.36). The best correlation between theoretical and experimental data was achieved after assuming that only ( π/2 = 2π/4) part of the period of above described 20fluctuation cycles is important for dissipation and viscosi ty. Introducing this factor to equations for viscosity calculations gives up ver y good correspondence between theory and experiment in all temperature interval ( 0-1000C) for water (Fig.5). As will be shown below the same factor, introducing the effect ive time of fluctuations [τ π/2], leads to best results for self-diffusion coefficient calcul ation. In the classical hydrodynamic theory the sound absorption c oefficient (α) obtained by Stokes includes share ( η) and bulk ( ηb) averaged macroviscosities: α=Ω 2ρv3s/parenleftbigg4 3η+ηb/parenrightbigg , (54) where Ω is the angular frequency of sound waves; ρis the density of liquid. Bulk viscosity ( ηb) is usually calculated from the experimental ηandα. It is known that for water: (ηb/η)∼3. The viscosity of solids In accordance with our model, the biggest fluctuations: supereffectons and superdeformons (see Introduction) are responsible for viscosity and diffus ion phenomena in solid state. Superdeformons are accompanied b y the emergency of cavitational fluctuations in liquids and the defects in so lids. The presentation of viscosity formula in solids ( ηs) is similar to that for liquids: ηS=ES (∆v0 f)S·τS/bracketleftbiggTk Utot/bracketrightbigg3 (55) where: reduced fluctuating volume, related to superdeformo ns excitation (∆v0 f)sis: (∆v0 f)S=V0 N0n21 ZPS (56) where:Ps= (PM D)tr·(PM D)lbis the relative probability of superdeformons, equal to product of probabilities of tr and lb macrodeformons excitation (see 42);Es=−kT·lnPsis the energy of superdeformons (see Chapter 4); Characteristic cycle-period of ( A∗⇋B∗) transition of supereffectons is re- lated to its life-times in A∗,B∗and transition D∗states (see eq.46) as was shown in section 4.3: 21τS=τA∗+τB∗+τD∗ (56a) The viscosity of ice, calculated from eq.(55) is bigger than that of water (eq.53) to about 105times. This result is in accordance with available experime ntal data. 7. Brownian diffusion The important formula obtained by Einstein in his theory of B rownian mo- tion is for translational motion of particle: r2= 6Dt=kT πηat (57) and that for rotational Brownian motion: ϕ2=kT 4πηa3t (58) where: a- radius of spherical particle, much larger than dimension o f molecules of liquid. The coefficient of diffusion D for Brownian motion is equal to: D=kT 6πηa(59) If we take the angle ¯ ϕ2= 1/3 in (59), then the corresponding rotational corre- lation time comes to the form of the known Stokes- Einstein eq uation: τ=4 3πa31 k/parenleftBigη T/parenrightBig (60) All these formulas (57 - 60) include macroscopic share visco sity (η) correspond- ing to our (53). 8. Self-diffusion in liquids and solids Molecular theory of self-diffusion, as well as general conce pt oftransfer phe- nomena in condensed matter is extremely important, but still unres olved prob- lem. Simple semiempirical approach developed by Frenkel leads t o following ex- pression for diffusion coefficient in liquid and solid: 22D=a2 τ0exp(−W/kT ) (61) where [a] is the distance of fluctuation jump; τ0∼(10−12÷10−13)sis the average period of molecule oscillations between jumps; W - a ctivation energy of jump. The parameters: a, τ0and W should be considered as a fit parameters. In accordance with mesoscopic theory , the process of self-diffusion in liquids,like that of viscosity , described above, is determined by two contribu- tions: a) the collective, nonlocal contribution , related to translational and librational macrodeformons ( Dtr,lb); b) the local contribution, related to coherent clusters flickering: [dissocia- tion/association] of primary librational effectons ( aandb)- convertons ( Dac,bc). Each component of the resulting coefficient of self-diffusion (D) in liquid could be presented as the ratio of fluctuation volume cross-s ection surface: [∆v0 f]2/3to the period of macrofluctuation ( τ). The first contribution to co- efficient D,produced by translational and librational macrodeformons is: Dtr,lb=/bracketleftbigg/parenleftbig ∆v0 f/parenrightbig2/3·1 τM/bracketrightbigg tr,lb(62) where: the surface cross-sections of reduced fluctuating fr ee volumes (see eq.43) fluctuations in composition of macrodeformons ( tr and lb) are: (∆v0 f)2/3 tr,lb=/bracketleftBigg V0 N0n21 Zexp/parenleftbigg −ǫM D kT/parenrightbigg tr,lb/bracketrightBigg2/3 (63) (τM)tr,lbare the characteristic ( A⇔B) cycle-periods of translational and li- brational macroeffectons (see eqs. 46 and 47). The averaged component of self-diffusion coefficient, which t akes into ac- count both types of nonlocal fluctuations, related to transl ational and librational macroeffectons and macrodeformons, can be find as: DM= [(D)M tr·(D)M lb]1/2(64) The formulae for the second, local contribution to self-diff usion in liquids, related to ( aandb) convertons ( Dac,bc) are symmetrical by form to that, presented above for nonlocal processes: Dac,bc=/bracketleftbigg (∆v0 f)2/3·1 τS/bracketrightbigg ac,bc(65) 23where: reduced fluctuating free volume of ( aandb) convertons (∆ v0 f)ac,bc is the same as was used above in mesoscopic theory of viscosit y (eq.50): (∆v0 f)ac,bc=V0 N0n21 ZPac,bc (66) where:PacandPbcare the relative probabilities of tr/libinterconversions between aandbstates of translational and librational primary effectons ( see Introduction and Chapter 4) The averaged local component of self-diffusion coefficient, w hich takes into account both types of convertons (ac and bc) is: DC= [(D)ac·(D)bc]1/2(67) In similar way we should take into account the contribution o f macroconver- tons (DMc): DMc=/parenleftbiggV0 N0n21 ZPMc/parenrightbigg2/3 ·1 τMc(67a) where:PMc=Pac·Pbcis a probability of macroconverton excitation; the life-time of macroconverton is: τMc= (τac·τbc)1/2(67b) The cycle-period of ( ac) and (bc) convertons are determined by the sum of life-times of intermediate states of primary translationa l and librational effec- tons: τac= (τa)tr+ (τa)lb; andτbc= (τb)tr+ (τb)lb (67c) The life-times of primary and secondary effectons (lb and tr) ina- and b- states are the reciprocal values of corresponding state fre quencies: [τa= 1/νa;τa= 1/νa; andτb= 1/νb;τb= 1/νb]tr,lb (67d) [νaandνb]tr,lbcorrespond to eqs. 4.8 and 4.9; [ νaandνb]tr,lbcould be calculated using eqs.2.54 and 2.55. The resulting coefficient of self-diffusion in liquids (D) is a sum of nonlocal (DM) and local ( Dc, DMc) effects contributions (see eqs.64 and 67): D=DM+Dc+DMc (68) The effective fluctuation-times were taken the same as in prev ious section for viscosity calculation, using the correction factor [( π/2)·τ]. 24Fig. 6. Theoretical and experimental temperature dependences of self-diffusion coefficients in water. Theoretical coefficie nt was cal- culated using eq. 68. Like in the cases of thermal conductivity, viscosity and vap or pressure, the results of theoretical calculations of self-diffusion coeffi cient coincide well with experimental data for water (Fig. 6) in temperature interva l (0−1000C). The self-diffusion in solids In solid state only the biggest fluctuations: superdeformons, representing simultaneous excitation of translational and librational macrodeformons in the same volumes of matter are responsible for diffusion and the v iscosity phenom- ena. They are related to origination and migration of the def ects in solids. The formal presentation of superdeformons contribution to sel f-diffusion in solids (Ds) is similar to that of macrodeformons for liquids: DS= (∆v0 f)2/3 S·1 τS(69) where: reduced fluctuating free volume in composition of sup erdeformons (∆v0 f)Sis the same as was used above in mesoscopic theory of viscosit y (eq.56): (∆v0 f)S=V0 N0n21 ZPS (70) where:PS= (PM D)tr·(PM D)lbis the relative probability of superdeformons, equal to product of probabilities of tr and lb macrodeformons excitation (see 42). 25Characteristic cycle-period of supereffectons is related t o that of tr and lb macroeffectons like it was presented in eq.(56a): τs=τA∗+τB∗+τD∗ (71) The self-diffusion coefficient for ice, calculated from eq.69 is less than that of water (eq.53) to about 105times. This result is in accordance with available experimental data. Strong decreasing of D in a course of phase transition: [wate r→ice] pre- dicted by our mesoscopic theory also is in accordance with ex periment (Fig. 7). Fig. 7. Theoretical temperature dependences of self-diffus ion coefficients in ice. All these results allow to consider our mesoscopic theory of trans- fer phenomena as a quantitatively confirmed one. They point t hat the ”mesoscopic bridge” between Micro- and Macro Worlds is w ide and reliable indeed. It gives a new possibilities for unders tanding and detailed description of very different phenomena in solids a nd liquids. One of the important consequences of our theory of viscosity and diffusion is the possibility of explaining numerous nonmono tonic tem- perature changes, registered by a number of physicochemica l methods in various aqueous systems during the study of temperature d epen- dences ([6], [7], [8], [9]; [10]; [11], [12]). Most of them are related to diffusion or viscosity processes a nd may be explained by nonmonotonic changes of the refraction index, included in our equations: 44, 45, 50 for viscosity and eqs. 69, 70 for self-d iffusion. For water 26these temperature anomalies of refraction index were revea led experimentally, using few wave lengths in the temperature interval 3 −950[13]. They are close to Drost-Hansen temperatures. The explanation of the se effects, related to periodic variation of primary librational effectons stab ility with monotonic temperature change was presented as comments to Fig. 7a of [1 ] or Fig.4a of [2]. Another consequence of our theory is the elucidation of a big dif- ference between librational ηlb(48), translational ηtr(45)viscosities and mesoviscosity, determined by [lb/tr]convertons (49 and 52). The effect of mesoviscosity can be checked as long as the volum e of a Brownian particle does not exceed much the volume of prim ary librational effectons (eq. 15). If we take a Brownian particl e, much bigger than the librational primary effecton, then its motio n will re- flect only averaged share viscosity (eq.53). The third consequence of the mesoscopic theory of viscosity is the prediction of nonmonotonic temperature behavior of the sound absorption coefficient α(51).Its temperature dependence must have anomalies in the same regions, where the refraction index ha s. The experimentally revealed temperature anomalies of (n) a lso fol- low from our theory as a result of nonmonotonic (a⇔b)lbequilibrium behavior, stability of primary lb effectons and probability of [lb/tr] convertons excitation (see Discussion to Fig.7a of [1] or to Fig.4a of [2]). Our model predicts also that in the course of transition from the laminar type of flow to the turbulent one the share viscosity ( η) will increases due to increasing of structural factor (Tk/Utot)in eq. 45. The superfluidity (η→0)in the liquid helium could be a result of inability of this liquid at the very low temperature for tran slational and librational macroeffectons excitations, i.e. τM→0. In turn, it is a consequence of tending to zero the life-times of secondary effectons and deformons in eqs.(45), responsible for dissi- pation processes, due to their Bose-condensation and trans formation to primary ones (see Chapter 12 of [1]). The polyeffectons, st abi- lized by Josephson’s junctions between primary effectons fo rm the superfluid component of liquid helium. 9. Mesoscopic approach to proton conductivity in water, ice and other systems, containing hydrogen bonds The numerous models of proton transitions in water and ice ar e usually related to migration of two types of defects in the id eal Bernal- Fouler structure [14]: 271. Ionic defects originated as a result of 2H2Odissociation to hy- droxonium and hydroxyl ions: 2H2O⇔H3O++ OH− 2. Orientational Bjerrum defects are subdivided to D (doppl et) and L (leer) ones. D-defect (positive) corresponds to situation, when 2 proto ns are placed between two oxygen atoms, instead of the normal struc ture of hydrogen bond: O...H −Ocontaining 1 proton. L-defect (negative) corresponds to opposite anomaly, when even 1 proton between two oxygens is absent. Reorientation of dip ole moment of H2Oin the case of D- and L-defects leads to origination of charges: qB=q+ D=|q− L|= 0.38e (72) The interrelation between the charge of electron (e), Bjerr um charge (qB)and ionic charge (eI) (Onsager, Dupius, 1962) is: e=eI+qB (73) The general approach to problem of proton transition takes i nto account both types of defects: ionic and orientational. It w as assumed that orientational defects originate and annihilate in the process of continuous migration of ions H+and OH−through the water medium. Krjachko (1987) considers DL-pairs as a cooperative water c luster with linear dimensions of about 15˚Aand with ”kink”. The Bjerrum’s DL-pair is a limit case of such model. The protons conductivity in water must decrease with temper ature increas- ing due to decreasing and disordering of water clusters and c hains. Thekink-soliton model of orientational defects migration along the H2O chain was developed by Sergienko (1986). Mobility of ionic d efects exceeds the orientational ones about 10 times. But it is important to point out that the strong experimental evidence con- firming the existence of just Bjerrum type orientational def ects are still absent. Our mesoscopic model of proton diffusion in ice, water and oth er hydrogen bonds containing systems includes following stag es: 1. Ionization of water molecules in composition of superdef ormons and ionic defects origination; 2. Bordering by H+ 3OandHO−the opposite surface-sides of pri- mary librational effectons; 3. Tunneling of proton through the volume of primary effecton as a coherent water cluster (Bose-particle); 284. Diffusion of ions H+ 3OandHO−in the less ordered medium between primary effectons can be realized in accordance with fluctu- ation mechanism described above in Section 8. The velocity o f this stage is less than tunneling. Transitions of protons and hydroxyl groups can occur also du e to exchange processes (Antonchenko, 1991) like: H+ 3O+H2O⇔H2O+H+ 3O (74) H2O+HO−⇔HO−+H2O (75) The rate of ions transferring due to exchange is about 10 time s more, than diffusion velocity, but slower than that, determined by tunneling jumps. 5. The orientational defects can originate as a result of H2Omolecules rearrangements and conversions between translational and librational effectons in composition of superdeformons. Activation ene rgy of superdeformons and macroconvertons in water is 10.2 kcal/M and about 12kcal/M in ice (see 6.12; 6.13). The additional activation energy about 2-3 kcal/M is necessary for subsequent reorien tation of surrounding molecules (Bjerrum, 1951). Like the ionic defects, positive (D) and negative (L) defect s can form a separated pairs on the opposite sides of primary effect ons, approximated by parallelepiped. Such pairs means the effect ons po- larization. Probability of H+orHO−tunneling through the coherent cluster - primary effecton in the (a)-state is higher than that in the (b)-state as far (see 1.30-1.32 of [1]): [Ea=Ta kin+Va]<[Eb=Tb kin+Vb] (76) where:Ta kin=Tb kinare the kinetic energies of (a) and (b) states; Eb−Ea= Vb−Va. is the difference between total and potential energies of th ese states. In accordance with known theory of tunneling, the probabili ty of passing the particle with mass (m)through the barrier with wideness (a) and height (ǫ)has a following dependence on these parameters: |ψa|∼exp/parenleftBig −a b/parenrightBig = exp/parenleftbigg −a(2mǫ)1/2 /planckover2pi1/parenrightbigg (77) where: 29b=/planckover2pi1/(2mǫ)1/2(78) is the effective wave function fading length. Parameter ( b) is similar to wave B most probable amplitude (AB) with total energy EB=ǫ(see eq. 2.22 of [1]): b=AB=/planckover2pi1/(2mEB)1/2(79) With temperature decreasing the (a⇔b)tr,lbequilibrium of primary effectons shifts to the left: Ka⇔b= (Pa/Pb)→ ∞ (80) wherePa→1andPb→0are the thermoaccessibilities of (a) and (b) states of primary effectons (see eqs. 4.10-4.12). The linear dimensions of primary effectons of ice also tend to infinity at T →0. In water the tunneling stage of proton conductivity can be re lated to primary librational effectons only and their role increas e with tem- perature decreasing. Dimensions of translational effecton s in water does not exceed that of one molecule as it leads from our compu ter calculations. Increasing of protons conductivity in ice with respect to wa ter, in accordance with our model, is a consequence of participat ion of translational primary effectons in tunneling of [H+]besides librational ones, as well as significant elevation of primary librationa l effectons dimensions. Increasing of the total contribution of tunnel ing process in protons migration in ice rise up their resulting transfer ring velocity comparing to water. The external electric field induce: a) redistribution of positive and negative charges on the su rface of primary effectons determined by ionic defects and corresp onding orientational defects; b) orientation of polarized primary effectons in field, makin g quasi- continuous polyeffectons chains and that of the effectons orc hestrated superclusters. These effects create the conditions for relay mechanism of [H+] andH+ 3Otransmitting in the direction of electric field and [HO−]in the opposite one. In accordance with our hierarchic model, t he[H+] transition mechanism includes the alternation of tunnelin g, exchange and usual diffusion processes. 10. Regulation of pH and shining of water by electromagnetic and acoustic fields 30In accordance with our model, water dissociation reaction: H2O⇔H++HO− leading to increase of protons concentration is dependent on probability of [A∗ S→B∗ S] transitions in supereffectons. This means that stimulatio n of [A∗ S→B∗ S] transitions (superdeformons) by ultrasound with resonant frequencies, corre- sponding to frequency of these transitions, should lead to d ecreasing of pH, i.e. to increasing the concentration of protons [ H+]. The[AS→BS]transitions of supereffectons can be accompanied by origination of cavitational fluctuations (cavitational mi crobubbles). The opposite [BS→AS]transitions are related to the collapse of these microbubbles. As a result of this adiabatic process, water v apor in the bubbles is heated up to 4000−60000K. The usual energy of su- perdeformons in water (Section 6.3): ǫS D= 10.2 kcal/M≃RT∗(81) correspond to local temperature T∗≃50000K. For the other hand it is known, that even 20000Kis enough already for partial dissociation of water molecules inside bubbles (about 0.01% of total amou nt of bubble water). The variable pressure (P), generated by ultrasound in liqui d is dependent on its intensity ( I, wt/cm2) like: P= (ρvsI)1/2·4.6·10−3(atm) (82) whereρis density of liquid; vs- sound velocity ( m/s). [A∗ S→B∗ S] transitions and cavitational bubbles origination can be s timu- lated also by IR radiation with frequency, corresponding to the activation energy of corresponding big fluctuations, described in mesoscopic theory by superde- formons andmacroconvertons. In such a way, using IR radiation and ultrasound it is possibl e to regulate a lot of different processes in aqueous systems, depending on pH and water activity. The increasing of ultrasound intensity leads to increased c avita- tional bubble concentration. The dependence of the resonan ce cavity radius (Rres)on ultrasound frequency (f) can be approximately ex- pressed as: Rres= 3000/f (83) At certain conditions the water placed in the ultrasound fiel d, begins to shine in the region: 300−600nm[15]. This shining (sonolumi- nescense) is a consequence of electronic excitation of wate r ions and molecules in the volume of cavitational bubbles. 31When the conditions of ultrasound standing wave exist, the n um- ber of bubbles and intensity of sonoluminescense is maximal . The intensity of shining is nonmonotonicly dependent on tem perature with maxima around 15 ,30,45 and650[6]. This temperature corresponds to ex- tremes of stability of primary librational effectons, relat ed to the number of H2Oper effecton’s edge ( κ) (see comments to Fig. 7a of [1] or to Fig 4a of [2]). An increase of inorganic ion concentration, destabil izing (a)-state of these effectons, elevate the probability of superdeformons and co nsequently, shining intensity. The most probable reason of photon radiation is recombinati on of water molecules, turning it into exited state: −OH+H+⇋H2O∗→H2O+hνp (84) Very different chemical reactions can be stimulated in the vo lume of cavitational fluctuation by the external fields. The optimal resonant para meters of these fields could be calculated using hierarchic theory. We propose here that the reaction of water molecules recombi na- tion (84) could be responsible for coherent ”biophotons” ra diation by cell’s and microbes cultures and living organisms in visibl e and ultra- violet (UV) range. The advances in biophoton research are de scribed by Popp et al., 1992 [16]. In accordance to our model, the cell’s body filaments - micro- tubules (MTs) ”catastrophe” (cooperative reversible disa ssembly of MTs) is a result of the internal water cavitational fluctuati ons due to superdeformons excitation. Such collective process sho uld be ac- companied by dissociation and recombination (84) of part of water molecules, localized in the hollow core of microtubules, le ading to high-frequency electromagnetic radiation (see: http://a rXiv.org/abs/physics/0003045). The coherent biophotons in the infrared (IR) range are a cons e- quence of (a⇋b)tr,lbtransitions of the water primary effectons in microtubules. We can see that lot of well working new theoretical models for different physical phenomena, based on our Hierarchic theor y of condensed matter, can be elaborated. It means that this theo ry may serve as new convenient scientific language. =================================================== =========== REFERENCES 32[1]. Kaivarainen A. Hierarchic Concept of Matter and Field. Water, biosystems and elementary particles. New York, NY,1995, pp. 485. [2]. Kaivarainen A. New Hierarchic Theory of Matter General for Liquids and Solids: dynamics, thermodynamics and mesoscopic structure of water and ice (see URL: http://www.karelia.ru/˜alexk) and: [3]. Kaivarainen A. Hierarchic Concept of Condensed Matter and its Interaction with Light: New Theories of Light Refraction, Brillouin Scattering and M¨ ossbauer ef- fect (see URL: http://www.karelia.ru/˜alexk). [4]. Blakemore J.S. Solid state physics. Cambridge Uni- versity Press, Cambridge, N.Y. e.a, 1985. [5]. Dote J.L., Kivelson D., Schwartz H. J.Phys.Chem. 1981, 85, 2169. [6]. Drost-Hansen W. In: Colloid and Interface Science. Ed. Kerker M. Academic Press, New York, 1976, p.267. [7]. Drost-Hansen W., Singleton J. Lin. Our aqueous heritage: evidence for vicinal water in cells. In: Funda- mentals of Medical Cell Biology, v.3A, Chemistry of the living cell, JAI Press Inc.,1992, p.157-180. [8]. Johri G.K., Roberts J.A. Study of the dielectric response of water using a resonant microwave cavity as a probe. J.Phys.Chem. 1990,94,7386. [9]. Aksnes G., Asaad A.N. Influence of the water struc- ture on chemical reactions in water. A study of proton- catalyzed acetal hydrolysis. Acta Chem. Scand. 1989,43,726− 734. [10]. Aksnes G., Libnau O. Temperature dependence of esther hydrolysis in water. Acta Chem.Scand. 1991,45,463− 467. [11]. K¨ aiv¨ ar¨ ainen A.I. Solvent-dependent flexibility o f proteins and principles of their function. D.Reidel Publ.C o., Dordrecht, Boston, Lancaster, 1985,pp.290. [12]. K¨ aiv¨ ar¨ ainen A., Fradkova L., Korpela T. Sepa- rate contributions of large- and small-scale dynamics to th e heat capacity of proteins. A new viscosity approach. Acta Chem.Scand. 1993,47,456−460. [13]. Frontas’ev V.P., Schreiber L.S. J. Struct. Chem. (USSR )6(1966)512 . [14]. Antonchenko V.Ya. Physics of water. Naukova dumka, Kiev, 1986. [15]. Guravlev A.I. and Akopjan V.B. Ultrasound shin- ing. Nauka, Moscow, 1977. 33[16]. Popp F.A., Li K.H. and Gu Q. Recent advances in biophoton research. Singapore: World Scientific, 1992. 34

arXiv:physics/0003075v1 [physics.atom-ph] 26 Mar 2000Very long storage times and evaporative cooling of cesium at oms in a quasi-electrostatic dipole trap H. Engler, T. Weber, M. Mudrich, R. Grimm, and M. Weidem¨ ulle r∗ Max-Planck-Institut f¨ ur Kernphysik†, 69029 Heidelberg, Germany (January 1, 2014) We have trapped cesium atoms over many minutes in the focus of a CO 2-laser beam employing an extremely simple laser system. Collisional properties of the unpolarized at oms in their electronic ground state are investigated. Inelast ic binary collisions changing the hyperfine state lead to trap l oss which is quantitatively analyzed. Elastic collisions resu lt in evaporative cooling of the trapped gas from 25 µK to 10 µK over a time scale of about 150 s. The focus of a CO 2-laser beam constitutes an almost perfect realization of a conservative trapping potential for neutral atoms [1,2]. Atoms are confined in all three spatial dimensions by the optical dipole force pointing to- wards the maximum of the intensity [3]. The CO 2-laser wavelength of 10 .6µm is far below any optical transitions from the ground state which has important consequences. As one consequence, the optical potential becomes quasi- electrostatic, i.e. the static polarizability of the parti cle determines the depth of the trap ( quasi-electrostatic trap , QUEST). Therefore, different atomic species [4] and even molecules [5] can be confined in the same trap. All sub- states of the electronic ground state experience the same trapping potential in contrast to magnetic traps. Atoms can thus be trapped in their absolute energetic ground state which excludes loss through inelastic binary colli- sions. Another consequence of the large laser detuning from resonance is the negligibly small photon scattering rate so that heating by the photon momentum recoil does not occur. In this Rapid Communication we show that storage times of many minutes can be achieved in a focused- beam dipole trap with a low-cost, easy-to-use CO 2laser usually employed for cutting and engraving of materi- als. The laser posseses neither frequency stabilization nor longitudinal mode selection. Despite the simplicity of the laser system, laser-noise induced heating rates, as first identified by Savard et al. [6], are found to be below 100nK/s. Comparable storage times in a QUEST have recently been realized by O’Hara et al., who utilized an ultrastable, custom-made CO 2laser [7]. The simplicity ∗Email: m.weidemueller@mpi-hd.mpg.de †Website: http://www.mpi-hd.mpg.de/ato/lasercoolof our trap setup in combination with the long storage times are ideal prerequisites for experiments on interest- ing collisional properties of the trapped gas. As one ap- plication of the trap, we have studied hyperfine-changing collisions of unpolarized cesium atoms. As another im- portant application, evaporative cooling of the trapped gas is demonstrated, which has so far only once been observed in an optical dipole trap [8]. The trapping potential of a CO 2-laser beam with a spatial intensity distribution I(r) is given by U(r) = αstatI(r)/2ε0cwhere αstatdenotes the static polarizabil- ity of the atoms [1,3]. For a focused beam of power Pand waist wone gets a trap potential with a depth U0=αstatI0/ε0cwithI0= 2P/πw2. The CO 2laser (Synrad 48-2WS) provides 25W of power in a nearly TEM 00transversal mode characterized by M2= 1.2. The laser beam is first expanded to a waist of 11mm by a telescope, and then focused into the vacuum chamber by a lens of 254mm focal length. The focus has a waist of 110 µm and a Rayleigh range zRof 2.4mm, yielding a trap depth 120 µK (in units of the Boltzmann constant kB). Gravity lowers the potential height along the ver- tical direction to 92 µK. The axial and radial oscillation frequencies in the harmonic approximation are given by ωz= (2U0/mz2 R)1/2andωr= (4U0/mw2)1/2withmde- noting the cesium mass. For our experimental values one gets an axial oscillation frequency ωz/2π= 8.1Hz and a radial frequency ωr/2π= 254Hz. Atoms are transferred into the dipole trap from a magneto-optical trap (MOT) containing about 106 atoms. The MOT is loaded from an atomic beam which is Zeeman-slowed in the fringe fields of the MOT mag- netic quadrupole field. The main vacuum chamber at a background pressure of about 2 ×10−11mbar is con- nected to the oven chamber by a tube which is di- vided into differentially-pumped sections to assure a suf- ficiently large pressure gradient. To optimize transfer from the MOT into the dipole trap, the atoms are fur- ther cooled and compressed by decreasing the detuning of the MOT trapping laser from initially −2 Γ (natural linewidth Γ /2π= 5.3MHz) to −20Γ with respect to the 62S1/2(F= 4) - 62P3/2(F= 5) transition of the cesium D2 line. After 40ms of compression, the distribution of atoms in the MOT has a rms radius of 120 µm corre- sponding to an mean density of 1010atoms/cm3. The temperature is 25 µK as measured by ballistic expansion of the cloud after release from the MOT. After the laser beams and the magnetic field of the MOT were turned off, the atoms are trapped in the focus of the CO 2laser 1beam which was present during the whole loading phase. Atoms are prepared in either the F= 3 or the F= 4 cesium hyperfine ground state by shuttering the MOT trapping laser 1 ms after or before the MOT repumping laser has been shuttered, respectively. The number and spatial distribution of atoms trapped in the optical dipole trap are measured by taking an ab- sorption image of the trapped atoms. A weak, resonant probe beam of 1 µW/cm2intensity is pulsed for 100 µs and absorption of the atomic cloud is imaged onto a CCD camera. Fig. 1(a) shows a typical absorption image of atoms trapped in the QUEST. The image is taken 5 s after transfer from the MOT. The transmitted intensity Itof the probe laser through the atomic sample is de- scribed by It(x, y) =I0exp (−Aη(x, y)) where I0denotes the laser intensity and Athe absorption cross section for the resonant transition. The column density ηis given by the integral of the density distribution n(x, y, z ) along the direction of the laser beam. By fitting a thermal equilibrium distribution [9] to the data, we derive the mean density ¯ n, the total number of trapped atoms Nand the temperature T. The max- imum absorption of the trapped cloud is typically 17% in the center of the distribution yielding a mean density of 4×109atoms/cm3in the dipole trap. The atoms have axially expanded into an rms extension of 750 µm while the radial rms extension is 30 µm. The temperature is the same as in the MOT before transfer indicating that the atoms are cooled into the dipole trap by the MOT. Typically 105atoms are transferred into the dipole trap. Assuming sufficient ergodicity, the number of atoms transferred from the MOT into the dipole trap can be determined from the phase-space integral/integraltext fMOT(x,p)θ(U0−ǫ)dxdpwhich describes the projec- tion of the phase-space distribution fMOT(x,p) in the MOT onto the trapping region of the dipole trap. The Heavyside step function θ(U0−ǫ) equals 1 when the total energy ǫ=U(r) +p2/2mof atoms in the dipole trap is smaller than the trap depth, and 0 elsewhere. Taking our experimental parameters, one expects 2 ×105atoms to be transferred into the dipole trap in reasonable agreement with the actual value. To measure the radial oscillation frequency of atoms in the dipole trap, we take advantage of the fast switching capability of the CO 2laser. The laser can be turned off within about 200 µs by simply switching the RF power supply driving the gas discharge. By turning the laser off for a short time interval (around 1ms), the trapped ensemble moves ballistically until the laser is turned on again ( release-recapture ). Part of the atoms will have escaped from the trap, while the recaptured atoms con- stitute a non-equilibrium distribution which oscillates a t twice the oscillation frequency. When the first release- recapture cycle is followed by a second one, the number of finally recaptured atoms depends on the phase of the oscillation and thus on the delay time between the two cycles. In Fig. 1(b), the number of recaptured atoms is plot-ted as a function of delay time between the two release- recapture cycles. One observes some cycles of coherent oscillation of the ensemble until the oscillation dephases mainly due to the anharmonicity of the potential. The oscillation period of 1.8ms gives a radial oscillation fre- quency of 270Hz in good agreement with the value ex- pected for the trap parameters. The solid line in Fig. 1(b) shows the result of a Monte-Carlo simulation of the clas- sical atom trajectories for the sequence of two release- recapture cycles with variable time delay between them. The lifetime of atoms in the dipole trap is determined by recapturing the atoms back into the MOT after a variable storage time. For each loading and trapping cycle, three camera pictures are taken. The first pic- ture gives the flourescence of atoms in the MOT shortly before transfer into the dipole trap. The second one shows the fluorescence distribution after recapture into the MOT. The last picture is taken after the atoms have been released from the recapture-MOT to provide the background which is then subtracted from the first two images. The number of atoms is determined from the in- tegral over the fluorescence image. The particle number determined by this method is estimated to be accurate within a factor of two. By normalizing the number of recaptured atoms to the number of atoms initially in the MOT, fluctuations due to variations in the initial number of atoms can be cancelled out. Variations of the parti- cle number in the MOT, however, are found to be below 10%, so that normalization was not used in the data pre- sented here. The decay of the number of stored atoms is shown in Fig. 2 for the two hyperfine ground states. Even after 10min storage time, a few hundred atoms can be de- tected when prepared in the F= 3 state. For atoms in the energetically higher F= 4 hyperfine ground state, inelastic binary collisions between trapped atoms lead to additional trap loss [10]. The energy of h×9.2GHz released in the collisions is much larger than the trap depth, therefore both collision partners are ejected from the trap. The decay curve for atoms in the F= 4 state is fit- ted by the solution to the differential equation dN/dt = −γN−β¯nNwhere γis the rate for trap loss through collisions with background gas and βis the rate coeffi- cient for inelastic binary collisions. The fit yields a decay constant γ−1= 165(25)s. This value is consistent with the expected loss rates for collisions with background gas atoms at a pressure of 10−11mbar. No indication for laser-noise induced trap loss is found. From the fit one finds a decay coefficient β= 2(1) ×10−11cm3/s with the error being mainly due to the uncertainty in the absloute particle number. A previous measurement of this quan- tity in an opto-electric trap [11] gave a similar result. Atoms in the energetic ground state F= 3 can not undergo inelastic collisions. One would therefore expect a purely exponential decay. However, the decay curve in Fig. 2 shows a faster loss of particles at higher particle numbers. It was checked that more than 95% of the 2particles are initially prepared in the absolute ground state so that inelastic collisions between trapped particl es can be excluded. At particle numbers below 104, the decay curve approaches a pure exponential with a decay constant of γ−1= 140(20)s in agreement with the value found for atoms in the F= 4 state. The faster initial trap loss can be attributed to evap- oration of high-energetic atoms from the trap leading to cooling [12]. At the initial temperature, the trap depth U0is only about 5 kBT. Therefore there is a certain prob- ability that atoms leave the trap after an elastic col- lision. The rate for elastic collisions is given by ¯ nσ¯v with the cross section σand the mean relative veloc- ity ¯v= 4(kBT/πm )1/2. For evaporative cooling to be effective, the ratio between the elastic collision rate (pro - viding thermalization and evaporation) and the rate for inelastic collisions (causing losses and heating) has to be large. Up to now, only one experiment is reported on evap- orative cooling in an optical dipole trap [8]. In order to achieve sufficiently high densities, Adams et al. stored sodium atoms in a crossed-beam dipole trap which pro- vides tight confinement in all three dimensions. In the simple focused-beam geometry used in our experiment it is rather surprising to find evaporative cooling since the achievable densities are rather low. The reason that evap- oration actually takes place lies in the long storage times of our trap on the one hand, and the anomalously large elastic cross section of cesium at low temperatures [13] on the other hand. Although the density of cesium atoms in the dipole trap is only of the order of 109atoms/cm3, one expects a thermalization time of only a few seconds due to the existence of a zero-energy resonance [14]. This time scale is indeed more than an order of magnitude smaller than the storage time. In Fig. 3, the temperature of the trapped atoms in the F= 3 state is plotted versus storage time. Temperatures derived from the density distribution shown as the dots are in good agreement with measurements of ballistic ex- pansion after release from the trap, which are shown as the three additional points in Fig. 3. One clearly observes cooling of the gas caused by evaporation of atoms from the trap at constant trap depth ( plain evaporation ). The final temperature of about 10 µK corresponds to roughly 1/10 of the potential depth. Although the temperature is reduced by roughly a factor of 2 after 150s, the phase- space density remains almost constant since the particle number diminishes by a factor of 10 at the same time. The temperature evolution for atoms in the F= 4 state shows a similar behaviour, but with a slower decrease of the temperature. This is to be expected due to the faster initial density decrease through inelastic collisions and the according decrease in the rate of elastic collisions. We apply a model developed by Luiten et al. [9] to simulate the temperature and particle number evolution during evaporation. Given the shape of the potential and the corresponding density of states, the model pro- vides two coupled differential equations for the evolutionof temperature and particle number. For the true po- tential function of a focused Gaussian beam, the den- sity of states diverges as the energy approaches the es- cape energy of the trap. We therefore approximate the trap potential by a three-dimensional Gaussian U(r)≃ U0exp(−2(x/w)2−2(y/w)2)−(z/zR)2) for which the density of states remains finite. The potential is fully determined by the trap param- eters, therefore the only adjustable parameter in the model is the the rate γfor inelastic collisions with back- ground gas and the cross section σfor elastic collisions. In the temperature range considered here, mainly s-wave collisions contribute to the cross section. The large scat- tering length of cesium atoms in the F= 3 ground state [13] results in a temperature-dependent effective cross section σ(T) =π2¯h2/mkBT(unitarity limit ) [14]. Note, that the cross section in the unitarity limit is com- pletely determined by the temperature. Since the model explicitly assumes a constant cross section for elastic col - lisions [9], we have fixed the effective cross section to the value σ= (930 a0)2(a0= Bohr radius) corresponding to a temperature of 15 µK. The result of the simulation for γ−1= 130s is shown by the solid line in Fig. 3. The model prediction for the time scale of temperature decrease is in reasonable agreement with the experimental data. As can be seen from Fig. 3, however, the model slightly overestimates the rate of evaporation. This discrepancy can be ex- plained by the influence of gravity on the one hand, and the temperature dependence of the collision cross sec- tion on the other hand. Both effects are not contained in the model, adn are difficult to be included. Due to gravity, evaporation predominantly takes place in only one spatial dimension which slows down the cooling pro- cess [15]. The temperature-dependent cross section gives rise to an increased thermalization time [14] so that the rate of evaporative cooling is decreased. Nevertheless, the final temperature of the evaporation process is cor- rectly reproduced by the model. The model also shows that the initial faster loss of particles in the F= 3-state (see curve in Fig. 2) mainly stems from the evaporation process. Employing a low-cost, off-the-shelf CO 2laser without any longitudinal mode selection, our major concern was possible heating by laser-noise induced fluctuations of the trap potential. From the observation of evaporative cool- ing with a temperature decrease of about 10 µK over a time scale of 100s, one can infer that heating rates by laser noise are much lower than 100nK/s. From the stor- age times of several minutes one can infer a similar upper bound for the laser-noise induced heating rates. Due to its simple ingredients, our setup represents a minimalistic version of an optical dipole trap providing very long storage times. The universality of the QUEST opens way to experiments aiming at fundamental ques- tions. Atomic spin states can be prepared in the QUEST with very long coherence times which is of importance for applications in quantum computing and for the search of 3a permanent electric dipole moment of atoms. Another intriguing prospect is the formation and the storage of cold molecules. In the same dipole trap, we have re- cently also trapped lithium atoms achieving similar stor- age times as for cesium [4]. Currently, we are studying the properties of a simultaneously stored mixture of both species in view of sympathetic cooling and photoassocia- tion of cold heteronuclear dimers. We gratefully acknowledge contributions of M. Nill in the early stage of the experiment and stimulating discus- sions with A. Mosk. We are indebted to D. Schwalm for encouragement and support. [1] T. Takekoshi and R.J. Knize, Opt. Lett. 21, 77 (1996). [2] S. Friebel et al., Phys. Rev. A 57, R20 (1998). [3] A review on optical dipole traps for neutral atoms is give n by R. Grimm, M. Weidem¨ uller, and Yu.B. Ovchinnikov, Adv. At. Mol. Opt. Phys. 42, 95 (2000). [4] M. Weidem¨ uller et al., in:Laser Spectroscopy XIV , edited by R. Blatt et al. (World Scientific, Singapore, 1999), p. 336. [5] T. Takekoshi, B.M. Patterson, and R.J. Knize, Phys. Rev. Lett. 81, 5105 (1998). [6] T.A. Savard, K.M. O’Hara, and J.E. Thomas, Phys. Rev. A56, R1095 (1997). [7] K.M. O’Hara et al., Phys. Rev. Lett. 82, 4204 (1999). [8] C.S. Adams et al., Phys. Rev. Lett. 74, 3577 (1995). [9] O.J. Luiten, M.W. Reynolds, and J.T.M. Walraven, Phys. Rev. A, 53, 381 (1996). [10] J. Weiner et al., Rev. Mod. Phys. 71, 1 (1999). [11] P. Lemonde et al., Europhys. Lett. 32, 555 (1995). [12] W. Ketterle and N.J. van Druten, Adv. At. Mol. Opt. Phys.37, 181 (1996). [13] S.A. Hopkins et al., Phys. Rev. A 61, 032707 (2000). [14] M. Arndt et al., Phys. Rev. Lett. 79, 625 (1997). [15] P.W.H.Pinkse et al., Phys. Rev. A 57, 4747 (1998). (a) (b)1 mm FIG. 1. (a) Distribution of cesium atoms trapped in the focus of a CO 2laser beam. The image shows the absorption of a weak resonant probe beam. (b) Measurement of the radial oscillation frequency of cesium in the dipole trap. T wo cycles of release-recapture are seperated by a variable del ay time. The free expansion time in each cycle is 0.7 ms. The number of recaptured atoms Nrecapafter the second cycle is normalized to the number of atoms after the first cycle N0. The number of recaptured atoms oscillates at twice the radia l oscillation frequency. The result of a Monte-Carlo simulat ion is depicted by the solid line. FIG. 2. Evolution of the number of trapped cesium atoms as a function of storage time. The atoms are initially prepar ed in either the F= 3 or F= 4 hyperfine ground state. 4FIG. 3. Evaporative cooling of the trapped cesium gas. The dots give the temperature derived from the axial exten- sion of the trapped atom cloud while the diamonds show the result of a ballistic expansion measurement. The solid line depicts a simulation of plain evaporative cooling in the tra p. 5

arXiv:physics/0003076v1 [physics.flu-dyn] 26 Mar 2000Nonlinear Modes of Liquid Drops as Solitary Waves A. Ludu and J. P. Draayer Department of Physics and Astronomy, Louisiana State Unive rsity, Baton Rouge, LA 70803-4001 (December 16, 2013) Abstract The nolinear hydrodynamic equations of the surface of a liqu id drop are shown to be directly connected to Korteweg de Vries (KdV, MKd V) systems, giving traveling solutions that are cnoidal waves. They gen erate multiscale patterns ranging from small harmonic oscillations (linear ized model), to non- linear oscillations, up through solitary waves. These non- axis-symmetric lo- calized shapes are also described by a KdV Hamiltonian syste m. Recently such “rotons” were observed experimentally when the shape o scillations of a droplet became nonlinear. The results apply to drop-like sy stems from cluster formation to stellar models, including hyperdeformed nucl ei and fission. 47.55.Dz, 24.10.Nz, 36.40.-c, 97.60.Jd Typeset using REVT EX 1A fundamental understanding of non-linear oscillations of a liquid drop (NLD), which reveals new phenomena and flows more complicated than linear theory suggests, is needed in diverse areas of science and technology. Besides their dire ct use in rheological and surfactant theory [1–7], such models apply to cluster physics [8], supe r- and hyper-deformed nuclei [1], nuclear break-up and fission [2,3,8], thin films [9], radar [4 ] and even stellar masses and supernova [1,10]. Theoretical approaches are usually base d on numerical calculations within different NLD models, [2–4] and explain/predict axis-symme tric, non-linear oscillations that are in very good agreement with experiment [1,5–7]. However , there are experimental results which show non-axis-symmetric modes; for example, traveli ng rotational shapes [5,6] that can lead to fission, cluster emission, or fusion [5–7]. In this letter the existence of analytic solutions of NLD mod els that give rise to traveling solutions which are solitary waves is proven. Higher order n on-linear terms in the devi- ation of the shape from a sphere produce surface oscillation s that are cnoidal waves [11]. By increasing the amplitude of these oscillations, the non- linear contribution grows and the drop’s surface, under special conditions (non-zero ang ular momentum), can transform from a cnodial wave form into a solitary wave. This same evolu tion can occur if there is a non-linear coupling between the normal modes. Thus this ap proach leads to a unifying dynamical picture of such modes; specifically, the cnoidal s olution simulates harmonic os- cillations developing into anharmonic ones, and under spec ial circumstances these cnoidal wave forms develop into solitary waves. Of course, in the lin ear limit the theory reproduces the normal modes of oscillation of a surface. Two approaches are used: Euler equations [2,3], and Hamilto nian equations, which de- scribe the total energy of the system [2]. We investigate fini te amplitude waves, for which the relative amplitude is smaller than the angular half-wid th. These excitations are also “long” waves, important in the cases of externally driven sy stems, where the excited wave- length depends by the driving frequency. The first original o bservations of travelling waves on liquid drops are described in [5]. Similar travelling or r unning waves are also discussed or quoted in [2,6]. These results suggest that higher amplitud e non-linear oscillations can lead 2to a traveling wave that originates on the drop’s surface and developes towards the interior. This is shown to be related in a simply way to special solitary wave solutions, called “rotons” in the present analysis. Recent experiments and numerical t ests [8,12] suggest the existence of stable traveling waves for a non-linear dynamics in a circ ular geometry, re-enforcing the theory. A new NLD model for describing an ideal, incompressible fluid drop exercising irrota- tional flow with surface tension, is employed in the analysis . Series expansion in terms of spherical harmonics are replaced by localized, nonlinear s hapes shown to be analytic solu- tions of the system. The flow is potential and therefore gover ned by Laplace’s equation for the potential flow, △Φ = 0, while the dynamics is described by Euler’s equation, ρ(∂t/vector v+ (/vector v· ∇)/vector v) =−∇P+/vectorf, (1) where Pis pressure. If the density of the external force field is also potential, /vectorf=−∇Ψ where Ψ is proportional to the potential (gravitational, el ectrostatic, etc.), then Eq. (1) reduces to Bernoulli’s scalar equation. The boundary condi tions (BC) on the external free surface of the drop, Σ1, and on the inner surface Σ2, [2,3,11] , are ˙ r|Σ1= (rt+rθ˙θ+ rφ˙φ)|Σ1and ˙r|Σ2= 0, respectively. Φ r= ˙ris the radial velocity, Φ θ=r2˙θ, Φφ=r2sinθ˙φ are the tangential velocities. The second BC occurs only in t he case of fluid shells or bubbles. A convenient geometry places the origin at the cent er-of-mass of the distributon r(θ, φ, t) =R0[1 +g(θ)η(φ−V t)] and introduces for the dimensionless shape function gηa variable denoted ξ. Here R0is the radius of the undeformed spherical drop and Vis the tangential velocity of the traveling solution ξmoving in the φdirection and having a constant transversal profile gin the θdirection. The linearized form of the first BC, ˙ r|Σ1=rt|Σ1, allows only radial vibrations and no tangential motion of th e fluid on Σ1, [2,3,11]. The second BC restricts the radial flow to a spherical layer of depth h(θ) by requiring Φ r|r=R0−h= 0. This condition stratifies the flow in the surface layer, R0−h≤r≤R0(1 +ξ), and the liquid bulk r≤R0−h. In what follows the flow in the bulk will be considered neglig ible compared to the flow in the surface layer. This condition does not restrict the generality of 3the argument because hcan always be taken to be R0. Nonetheless, keeping h < R 0opens possibilities for the investigation of more complex fluids, e.g. superfluids, flow over a rigid core, multilayer systems [2,7] or multiphases, etc. Instea d of an expansion of Φ in term of spherical harmonics, consider the following form Φ =∞/summationdisplay n=0(r/R0−1)nfn(θ, φ, t). (2) The convergence of the series is controlled by the value of th e small quantity ǫ=max|r−R0 R0|, [11]. The condition max|h/R 0| ≃ǫis also assumed to hold in the following development. Laplace’s equation introduces a system of recursion relati ons for the functions fn,f2= −f1−△ Ωf0/2, etc., where △Ωis the ( θ, φ) part of the Laplacean. Hence the set of unknown fn’s reduces to f0andf1. The second BC, plus the condition ξφ=−V ξt, for traveling waves, yields to second order in ǫ, f0,φ=V R3 0sin2θξ(1 + 2 ξ)/h+O3(ξ), (3) i.e., a connection between the flow potential and the shape, w hich is typical of nonlinear systems. Eq.(3) together with the relations f1≃R2 0ξt≃2h R0f2≃ −h△Ωf0 R0+2h, which follow from the BC and recursion, characterize the flow as a function of th e surface geometry. The balance of the dynamic and capillary pressure across the sur face Σ1 follows by expanding up to third order in ξthe square root of the surface energy of the drop [2,3,11], US=σR2 0/integraldisplay Σ1(1 +ξ)/radicalBig (1 +ξ)2+ξ2 θ+ξ2 φ/sin2θdΣ, (4) and by equating its first variation with the local mean curvat ure of Σ1 under the restriction of the volume conservation. The surface pressure, in third o rder, reads P|Σ1=σ R0(−2ξ−4ξ2− △ Ωξ+ 3ξξ2 θctgθ), (5) where σis the surface pressure coefficient and the terms ξφ,θ, ξφ,φandξθ,θare neglected because the relative amplitude of the deformation ǫis smaller than the angular half-width L,ξ=ξφφ≃ǫ2/L2≪1, as most of the experiments [6,7,9,12] concerning traveli ng surface patterns show. Eq.(5) plus the BC yield, to second order in ǫ, 4Φt|Σ1+V2R4 0sin2θ 2h2ξ2 =σ ρR0(2ξ+ 4ξ2+△Ωξ−3ξ2ξθctgθ). (6) The linearized version of Eq. (6) together with the lineariz ed BC, Φ r|Σ1=R0ξt, yield a limiting case of the model, namely, the normal modes of oscil lation of a liquid drop with spherical harmonic solutions [2,3]. Differentiation of Eq. (6) with respect to φtogether with Eqs.(3,5) yields the dynamical equation for the evolution o f the shape function η(φ−V t): Aηt+Bηφ+Cgηη φ+Dηφφφ= 0, (7) which is the Korteweg-de Vries (KdV) equation [11] with coeffi cients depending parametri- cally on θ A=VR2 0(R0+ 2h) sin2θ h, B=−σ ρR0(2g+△Ωg) g, C= 8/parenleftbiggV2R4 0sin4θ 8h2−σ ρR0/parenrightbigg , D =−σ ρR0sin2θ. (8) In the case of a two-dimensional liquid drop, the coefficients in Eq. (8) are all constant. Eq. (7) has traveling wave solutions in the φdirection if Cg/(B−AV) and D/(B−AV) do not depend on θ. These two conditions introduce two differential equations forg(θ) andh(θ) which can be solved with the boundary conditions g=h= 0 for θ= 0, π. For example, h1=R0sin2θandg1=P2 2(θ) is a particular solution which is valid for h≪R0. It represents a soliton with a quadrupole transvere profile, be ing in good agreement with [2,6]. The next higher order term in Eq. (6), −3ξ2ξθctgθ, introduces a η2ηφnonlinear term into the dynamics and transforms the KdV equation into the modifie d KdV equation [11]. The traveling wave solutions of Eq. (7) are then described by the Jacobi elliptic function [11] η=α3+ (α2−α3)sn2/parenleftbigg/radicalBigg C(α3−α2) 12D(φ−V t);m/parenrightbigg , (9) where the αiare the constants of integration introduced through Eq. (7) and are related through the velocity V=C(α1+α2+α3)/3A+B/Aandm2=α3−α2 α3−α1.m∈[0,1] is the free parameter of the elliptic snfunction. This result for Eq. (9) is known as a cnoidal wave 5solution with angular period T=K[m]/radicalBig C(α3−α1)/3Dwhere K(m) is the Jacobi elliptic integral. If α2→α1→0, then m→1, T→ ∞ and a one-parameter ( η0) family of traveling pulses (solitons or anti-solitons) is obtained, ηsol=η0sech2[(φ−V t)/L], (10) with velocity V=η0C/3A+B/Aand angular half-width L=/radicalBig 12D/Cη 0. Taking for the coefficients AtoDthe values given in Eq.(8) for θ=π/2 (the equatorial cross section) andh1,g1from above, one can calculate numerical values of the parame ters of any roton excitation function of η0only. The soliton, among other wave patterns, has a special shape- kinematic dependence η0≃ V≃1/L; a higher soliton is narrower and travels faster. This relat ion can be used to experimentally distinguish solitons from other modes or tu rbulence. When a layer thins (h→0) the coefficient Cin eq.(8) approaches zero on average, producing a break in th e traveling wave solution ( Lbecomes singular) because of the change of sign under the squ are root, eq.(9). Such wave turbulence from capillary waves on t hin shells was first observed in [9]. For the water shells described there, eq.(8) gives h(µm)≤20ν/k, that is h=15- 25µm at V=2.1-2.5m s−1for the onset of wave turbulence, in good agreement with the abrupt transition experimentaly noticed. The cnoidal solu tions provide the nonlinear wave interaction and the transition from competing linear wave m odes ( C≤0) to turbulence (C≃0). In the KdV eq.(7), the nonlinear interaction balances or even dominates the linear damping and the cnoidal (roton) mode occurs as a bend mode ( hsmall and coherent traveling profile) in agreement with [9]. The condition for the existen ce of a positive amplitude soliton isgCD≥0 which, for g≤0, limits the velocity from below to the value V≥hω2/R0where ω2is the Lamb frequency for the λ= 2 linear mode [2,3]. This inequality can be related to the “independent running wave” described in [6], which li es close to the λ= 2 mode. Moreover, since the angular group velocity of the ( λ, µ) normal mode, Vλ,µ=ωλ/µ, has practically the same value for λ= 2 (µ= 0,±1, tesseral harmonics) and for λ=µ, any λ(sectorial harmonics) this inequality seems to be essentia l for any combination of rank 2 6tesseral or sectorial harmonics, in good agreement with the conclusions in [2]. The periodic limit of the cnoidal wave is reached for m≃0, that is, α2−α3≃0, and the shape is characterized by harmonic oscillations ( sn→sin in Eq. (9)) which realize the quadrupole mode of a linear theory Yµ 2limit [2,3] or the oscillations of tesseral harmonics [2], F ig. 1. The NLD model introduced in this paper yields a smooth transi tion from linear oscilla- tions to solitary traveling solutions (“rotons”) as a funct ion of the parameters αi; namely, a transition from periodic to non-periodic shape oscillati ons. In between these limits the surface is described by nonlinear cnoidal waves. In Fig.1 th e transition from a periodic limit to a solitary wave is shown, in comparison with the correspon ding normal modes which can initiate such cnoidal nonlinear behavior. This situation i s similar to the transformation of the flow field from periodic modes at small amplitude to travel ing waves at larger amplitude [2,6]. The solution goes into a final form if the volume conser vation restriction is enforced: /integraltext Σ(1 +g(θ)η(φ, t))3dΩ = 4 πand requires η(φ, t) to be periodic. The periodicity condition, nK[(α3−α2)/(α3−α1)] =π√α3−α1for any positive integer n, is only fulfilled for a finite number of nvalues, and hence a finite number of coresponding cnoidal mod es. In the roton limit the periodicity condition becomes a quasi-periodic o ne because the amplitude decays rapidly. This approach could be extended to describe elasti c modes of surface as well as their nonlinear coupling to capillary waves. The double-pe riodic structure of the elliptic solutions [11] could describe the new family of normal wave m odes predicted in [4]. The development up to this point was based on Euler’s equatio n. The same result will now be shown to emerge from a Hamiltonian analysis of the NLD system. Recently, Natarajan and Brown [2] showed that the NLD is a Lagrangian sy stem with the volume con- servation condition being a Lagrange multiplier. In the thi rd order deviation from spherical, the NLD becomes a KdV infinite-dimensional Hamiltonian syst em described by a nonlin- ear Hamiltonian function H=/integraltext2π 0Hdφ. In the linear approximation, the NLD is a linear wave Hamiltonian system [2,3]. If terms depending on θare absorbed into definite integrals (becoming parameters) the total energy is a function of ηonly. Taking the kinetic energy from [2,3], Φ from Eq. (2) and using the BC, the dependence of t he kinetic energy on the 7tangential velocity along θdirection, Φ θ, becomes negligible and the kinetic energy can be expressed as a T[η] functional. For traveling wave solutions ∂t=−V ∂φ, to third order in ǫ, after a tedious but feasible calculus, the total energy is: E=/integraldisplay2π 0(C1η+C2η2+C3η3+C4η2 φ)dφ, (11) where C1= 2σR2 0S1,0 1,0,C2=σR2 0(S1,0 1,0+S1,0 0,1/2) +R6 0ρV2C3,−1 2,−1/2,C3=σR2 0S1,0 1,2/2 + R6 0ρV2(2S3,−1 −1,2R0+S5,−2 −2,3+R0S6,−2 −2,3)/2,C4=σR2 0S−1,0 2,0/2, with Sk,l i,j=R−l 0/integraltextπ 0hlgigj θsinkθdθ. Terms proportional to ηη2 φcan be neglected since they introduce a factor η3 0/L2which is small compared to η3 0, i.e. it is in the third order in ǫ. If Eq. (11) is taken to be a Hamilto- nian,E→H[η], then the Hamilton equation for the dynamical variable η, taking the usual form of the Poisson bracket, gives /integraldisplay2π 0ηtdφ=/integraldisplay2π 0(2C2ηφ+ 6C3ηηφ−2C4ηφφφ)dφ. (12) Since for the function η(φ−V t) the LHS of Eq.(12) is zero, the integrand in the RHS gives the KdV equation. Hence, the energy of the NLD model, in the th ird order, is interpreted as a Hamiltonian of the KdV equation [7,11]. This is in full agre ement with the result finalized by Eq. (7) for an appropriate choice of the parameters and the Cauchy conditions for g, h. The dependence of E(α1, α2)|V ol=constant , Eq.(11), shows an energy minimum in which the solitary waves are stable, [12]. The nonlinear coupling of modes in the cnoidal solution coul d explain the occurence of many resonances for the l= 2 mode of rotating liquid drops, at a given (higher) angular velocity, [13]. The rotating quadrupole shape is close to th e soliton limit of the cnoidal wave. On one hand, the existence of many resonances is a consequenc e of by the multi-valley profile of the effective potential energy for the KdV, (MKdV) equatio n:η2 x=aη+bη2+cη3+(dη4), [11]. The frequency shift predicted by Busse in [13] can be re produced in the present theory by choosing the solution h1=R0sinθ/2. It results the same additional pressure drop in the form of V2ρR2 0sin2θ/2 like in [13], and hence a similar result. For a roton emerged from a l= 2 mode, by calculating the half-width ( L2) and amplitude ( ηmax,2) which fitt the quadrupole 8shape it results a law for the frequency shift: ∆ ω2/ω2= (1±4L2(α3−α2)/3R0)−1V/ω 2, showing a good agreement with the observations of Annamalai et al in [13], i.e. many resonances and nonlinear dependence of the shift on Ω = V. The special damping of the l= 2 mode for rotating drops could also be a consequence of the e xistence of the cnoidal solution. An increasing in the velocity Vproduces a modification of the balance of the coefficients C/Dwhich is equivalent with an increasing in dispersion. The model introduced in this article proves that traveling a nalytic solutions exist as cnoidal waves on the surface of a liquid drop. These travelin g deformations (“rotons”) can range from small oscillations (normal modes), to cnoidal os cillations, and on out to solitary waves. The same approach can be applied to bubbles as well, ex cept that the boundary condition on Σ 2is replaced by a far-field condition [2,3] (recently importa nt in the context of single bubble sonoluminiscence). Nonlinear phenomena c an not be fully investigated with normal linear tools, e.g. spherical harmonics. Using analy tic non-linear solutions sacrifices the linearity of the space but replaces it with multiscale dy namical behavior, typical for non- linear systems (solitons, wavelets, compactons [12]). The y can be applied to phenomena like cluster formation in nuclei, fragmentation or cold fission, the dynamics of the pellet surface in inertial fusion, stellar models, and so forth. Supported by the U.S. National Science Foundation through a regular grant, No. 9603006, and a Cooperative Agreement, No. EPS-9550481, tha t includes matching from the Louisiana Board of Regents Support Fund. 9REFERENCES [1] R. E. Apfel et al,Phys. Rev. Lett. 78, 1912 (1997). [2] R. Natarajan and R. A. Brown, J. Fluid Mech. 183, 95 (1987); Phys. Fluids 29, 2788 (1986); J. A. Tsamopoulos and R. A. Brown, J. Fluid Mech. 127, 519 (1983). [3] H-L. Lu and R. E. Apfel, J. Fluid Mech. ,222, 351 (1991); T. Shi and R. E. Apfel, Phys. Fluids 7, 1545 (1995). [4] Y. Tian, R. G. Holt and R. E. Apfel, Phys. Fluids 7, 2938 (1995). [5] E. H. Trinh, P. L. Marston and J. L. Robey, J. Colloid Interface Sci. 124, 95 (1988). [6] E. Trinh and T. G. Wang, J. Fluid Mech. 122, 315 (1982). [7] E. H. Trinh, R. G. Holt, and D. B. Thiessen, Phys. Fluids 8, 43 (1995); P. L. Marston and S. G. Goosby, Phys. Fluids 28, 1233 (1985). [8] A. Ludu et al,Int. J. Modern Phys. E 2, 855 (1993); J. Phys. G: Nucl. Part. Phys. 21, 1715 (1995). [9] R. G. Holt and E. H. Trinh, Phys. Rev. Lett. 77, 1274 (1996). [10] R. H. Durisen et al Astrophys. J. 305, 281 (1996). [11] G. L. Lamb, Elements of Soliton Theory , (John Wiley & Sons, New York, 1980). [12] Y. G. Kevrekidis; A. Ustinov; A. Ludu and J. P. Draayer, T heCNLS 17thAnn. Conf. , May 12-16, 1997, Los Alamos; P. Rosenau and J. M. Hyman, Phys. Rev. Let. 70, 564 (1993); S. Dramanyan, et al,Phys. Rev. E 55, 7662 (1997). [13] F. H. Busse, J. Fluid. Mech. 142, 1 (1984); P. Annamalai, E. Trinh and T. G. Wang, J. Fluid. Mech. 158, 317 (1985). 10FIGURES FIG. 1. The cnoidal solution for θ= 0. The soliton limit and a 3- and 4-mode solution is shown. The closest spherical harmonics to each of the cnoida l wave profiles (labelled Cn and Sol, respectively) is given for comparison. The labels λ,µand the parameters α1,2,3of the coresponding cnoidal solution are given. 11

arXiv:physics/0003077v1 [physics.flu-dyn] 26 Mar 2000Patterns on liquid surfaces: cnoidal waves, compactons and scaling A. Ludu and J. P. Draayer Department of Physics and Astronomy, Louisiana State Unive rsity, Baton Rouge, LA 70803-4001, U.S.A. February 2, 2008 Abstract Localized patterns and nonlinear oscillation formation on the bounded free surface of an ideal incompressible liquid are analytically investigated . Cnoidal modes, solitons and compactons, as traveling non-axially s ymmetric shapes are discused. A finite-difference differential generalized Kort eweg-de Vries equation is shown to describe the three-dimensional motion of the fluid s urface and the limit of long and shallow channels one reobtains the well known KdV equation. A tenta- tive expansion formula for the representation of the genera l solution of a nonlinear equation, for given initial condition is introduced on a gra phical-algebraic basis. The model is useful in multilayer fluid dynamics, cluster for mation, and nuclear physics since, up to an overall scale, these systems display liquid free surface behavior. PACS : 47.55.Dz, 68.10.Cr, 47.20.Ky, 47.20.Dr, 97.60.j, 83 .70, 36.40.-c . 11 Introduction Liquid oscillations on bounded surfaces have been studied i ntensively, both theoretically [1-3] and experimentally [4-6]. The small-amplitude oscil lations of incompressible drops maintained by surface tension are usually characterized by their fundamental linear modes of motion in terms of spherical harmonics [1-3]. Nonli near oscillations of a liquid drop introduce new phenomena and more complicated patterns (higher resonances, solitons, compactons, breakup and fragmentation, fractal structures, superdeformed shapes) than can be described by a linear theory. Nonlineari ties in the description of an ideal drop demonstrating irrotational flow arise from B ernoulli’s equation for the pressure field and from the kinematic surface boundary co nditions [7]. Computer simulations have been carried for non-linear axial oscilla tions and they are in very good agreement with experiments [4-6]. The majority of experiments show a rich variety of complicat ed shapes, many re- lated to the spinning, breaking, fission and fusion of liquid drops. There are experiments [6] and numerical simulations [2] where special rotational patterns of circulation emerge: a running wave originates on the surface of the drop and then p ropagates inward. Re- cent results (superconductors [8], catalytic patterns [9] , quasi-molecular spectra [10], numerical tests on higher order non-linear equations [11] a nd analytical calculations on the non-compact real axis [12-13]) show shape-stable tra veling waves for nonlinear systems with compact geometry. Recent studies showed that a similar one-dimensional analysis for the process of cluster emission from heavy nucl ei and quasi-molecular spec- tra of nuclear molecules yields good agreement with experim ent [10]. Such solutions are stable and express to a good extent the formation and stabili ty of patterns, clusters, droplets, etc. However, even localised, they have nor compa ct support neither period- icity (excepting some intermediate steps of the cnoidal sol utions, [10,13]), creating thus difficulties when analysing on compact surfaces. In the present paper we comment on the cnoidal-towards-soli tons solution inves- tigated in [9,12], especially from the energy point of view. We introduce here a new nonlinear 3-dimensional dynamical model of the surface, in compact geometry (pools, droplets, bubbles, shells), inspired by [12], and we invest igate the possibilities to obtain 2compacton-like solutions for this model. We also study the s cale symmetries of such solutions. The model in [9,12] consider the nolinear hydrodynamic equa tions of the surface of a liquid drop and show their direct connection to KdV or MKdV s ystems. Traveling so- lutions that are cnoidal waves are obtained [10,13] and they generate multiscale patterns ranging from small harmonic oscillations (linearized mode l), to nonlinear oscillations, up to solitary waves. These non-axis-symmetric localized s hapes are described by a KdV Hamiltonian system, too, which results as the second ord er approximation of the general Hamiltonian, next corrextion from the linear harmo nic shape oscillations. Such rotons were observed experimentally when the shape oscilla tions of a droplet became nonlinear [4,6,8,13]. 2 Liquid drop cnoidal and soliton solutions from Hamiltonian approach The dynamics governing one-dimensional surface oscillati ons of a perfect ( ρ=const.), irrotational fluid drop (or bubble, shell) can be described b y the velocity field Φ and a corresponding Hamiltonian [1-3,7,10,13]. By expanding th e Hamiltonian and dynamical equations in terms of a small parameter, i.e. the amplitude o f the perturbation ηover the radius of drop R0, the usual linear theory is recovered in the first order. High er order non-linear terms introduce deviations and produce la rge surface oscillations like cnoidal waves [7]. These oscillations, under conditions of a rigid core of radius R0−h and non-zero angular momentum, transform into solitary wav es. In the following, by using the calculation developed in [10], we present the Hami ltonian approach for the liquid drops nonlinear oscillations. However, this approa ch is different from the nuclear liquid drop model point of view in [10], since we do not use her e the nuclear interaction (shell corrections) responsible for the formation of differ ent potential valleys. The total hydrodynamic energy Econsists of the sum of the kinetic Tand po- tential Uenergies of the liquid drop. The shape function is assumed to factorize, r(θ, φ, t) =R0(1 +g(θ)η(φ, t)). All terms that depend on θare absorbed in the coeffi- cients of some integrals and the energy reduces to a function al ofηonly. The potential 3energy is given by the surface energy US=σ(Aη− A 0)|V0, where σis the surface pres- sure coefficient, Aηis the area of the deformed drop, and A0the area of the spherical drop, of constant volume V0. The kinetic energy T=ρ/contintegraltext ΣΦ∇Φ·d/vectorS/2, [1-3,10,13], the kinematic free surface boundary condition Φ r=∂tr+ (∂θr)Φθ/r2+ (∂φr)Φφ/r2sinθ, and the boundary condition for the radial velocity on the inn er surface ∂rΦ|r=R0−h= 0, [7], result in the expression [2,3,10] T=R2 0ρ 2/integraldisplayπ 0/integraldisplay2π 0R0Φηtsinθ+1 R0gηφΦΦφ(1−sinθ) /radicalBig 1 +g2 θη2+g2η2 φdθdφ. (1) If the total energy, written in the second order in η, is taken to be a Hamiltonian H[η], the time derivative of any quantity F[η] is given by Ft= [F, H]. Defining F= /integraltext2π 0η(φ−V t)dφit results ([10], last reference) dF dt=/integraldisplay2π 0ηtdφ=/integraldisplay2π 0(2C2ηφ+ 6C3ηηφ−2C4ηφφφ)dφ= 0, (2) which leads to the KdV equation. Here C2=σR2 0(S1,0 1,0+S1,0 0,1/2) +R6 0ρV2C3,−1 2,−1/2, C3=σR2 0S1,0 1,2/2 +R6 0ρV2(2S3,−1 −1,2R0+S5,−2 −2,3+R0S6,−2 −2,3)/2,C4=σR2 0S−1,0 2,0/2, with Sk,l i,j= R−l 0/integraltextπ 0hlgigj θsinkθdθ. Terms proportional to ηη2 φcan be neglected since they introduce a factor η3 0/L2which is small compared to η3 0, i.e. it is in the third order. In order to verify the correctness of the above approximations, we present, fo r a typical soliton solution η(φ, t), some terms occuring in the expresion of E, Fig. 1. All details of calculation are given in [10,13]. Therefore, the energy of the non-linea r liquid drop model can be interpreted as the Hamiltonian of the one-dimensional KdV e quation. The coefficients in eq.(2) depend on two stationary functions of θ(the depth h(θ) and the transversal profile g(θ)), hence, under the integration, they involve only a parame tric dependence. The KdV equation has the following cnoidal wave (Jacobi elli ptic function) as exact solution η=α3+ (α2−α3)sn2/parenleftbigg/radicalBigg C3(α3−α2) 12C4(φ−V t)/vextendsingle/vextendsingle/vextendsingle/vextendsinglem/parenrightbigg , (3) where α1, α2, α3are constants of integration, m2= (α3−α2)/(α3−α1). This solution oscillates between α2andα3, with a period T= 2K(m)/radicalBig (α3−α2)C3 3C4, where K(m) is the period of a Jacobi elliptic function sn(x|m). The parameter Vis the velocity of the 4cnoidal waves and α1+α2+α3=3(V−C2) 2C. In the limit α1=α2= 0 the solution eq.(3) approaches η=η0sech2/bracketleftbigg/radicalBigg η0C3 12C4(φ−V t)/bracketrightbigg , (4) which is the soliton solution of amplitude η0. Small oscillation occur when α3→α2and m→0, T→π/2. Consequently, the system has two limiting solutions, a pe riodic and a localized traveling profile, which deform one into the othe r, by the initial conditions and the velocity parameter V. A figure showing the deformation from the l= 5 cnoidal mode towards a soliton is shown in Figs. 2. The cnoidal solution eq.(3) depends on the parameters αisubjected to the volume conservation and the periodicity condition of the solution (for the final soliton state this condition should be taken as a quasi-periodicity reali sed by the rapidly decreasing profile. This a problem of the basic model, [10]). The periodi city restriction reads K/parenleftbigg/radicalBigg α3−α2 α3−α1/parenrightbigg =π n√α3−α1, n= 1,2, . . .,2√α3−α1. (5) Hence, a single free parameter remains, which can be taken ei ther one out of the three α’s,Vorη0. Equatorial cross-sections of the drop are shown in Fig. 2b f or the cnoidal solution at several values of the parameter η0. All explicite calculations are presented in detail in [10]. In Fig. 3 we present the total energy plotted versus the param etersα1, α2for constant volume. From the small oscillation limit ( α2≃3 in the figure) towards the solitary wave limit ( α2= 1 in the figure) the energy increases and has a valley for α1≃0.1 and α2∈(1.2,1.75) (close to the l= 2 mode). In order to introduce more realistic results, the total hydrodynamic energy is plotte d versus α1, α2for constant volume, too but we marked those special solutions fulfilling the periodicity condition. In Fig. 4 we present the total energy valley, from the small os cillations limit towards the solitary wave limit. We notice that the energy constantl y increases but around α2∈(1.2,1.75) (close to the linear l= 2 mode) it has a valley providing some stability for solitary solution (also called roton [13]). 53 The three-dimensional nonlinear model In the following we introduce a sort of generalized KdV equat ion for fluids. We consider the three-dimensional irrotational flow of an ideal incompr essible fluid layer in a semi- finte rectangular channel subjected to uniform vertical gra vitation ( ginzdirection) and to surface pressure [12]. The depth of the layer, when the fluid is at rest is z= h. Boundary conditions at the finite spaced walls consist in an nilation of the normal velocity component, i.e. on the bottom of the layer ( z= 0) and on the walls x=x0±L/2 of the channel of width L. The following results remain valid if the walls expand arbitrary, e.g. L→ ∞ , and the flow is free. We choose for the potential of the velocities the form Φ =/summationdisplay k≥0αk(t) coskπ(x−x0) Lcosh√ 2kπ(y−y0) Lcoskπz L, (6) where αk(t) are arbitrary functions of time and Lis a free parameter. Eq.(7) fulfils △Φ = 0 and the above boundary conditions at the walls. However t here is another boundary condition at the free surface of the fluid [7] (Φz−ηt−ηxΦx)z=h+η= 0, (7) where η(x, t) describes the shape of the free surface. By introducing the function f(x, t) =∞/summationdisplay k=0αk(t)kπ L/parenleftbigg sinkπ(x−x0) Lcosh√ 2kπ(y−y0) L/parenrightbigg , (8) the velocity field on the free surface can be written Φx|z=h+η=−cosh(z∂x)f, −Φz|z=h+η=−sinh(z∂x)f. (9) Eqs (10) do not depend on Land the case L→ ∞ of unbounded channels and free travelling profiles remains equaly valid. Since the unique f orce field in the problem is potential, the dynamics is described by the Bernoulli equ ation, which, at the free surface, reads Φxt+ Φ xΦxx+ Φ zΦxz+gηx+1 ρPx= 0. (10) 6HerePis the surface pressure obtained by equating P’s first variation with the local mean curvature of the surface, under the restriction of the v olume conservation P/vextendsingle/vextendsingle/vextendsingle/vextendsingle z=h+η=σηxx (1 +η2 x)3/2, (11) andσis the surface pressure coefficient. The pressure in eq.(12) a pproaches −σηxx, for small enough relative amplitude of the deformation η/h. In order to solve the system of the two partial differential equations (8,11) with respec t to the unknown functions f(x, t) and η(x, t), we consider the approximation of small perturbations of t he surface compared to the depth, a=max|η(k)(x, t)|<< h , where k= 0, ...,3 are orders of differentiation. Inspired by [12] and using a sort of perturb ation technique in a/h, we obtain from eqs.(6-11) the generalised KdV equation ηt+c0 hsin(h∂)η+c0 h(ηxcosh(h∂)η+ηcosh(h∂)ηx)0. (12) If we approximate sin( h∂)≃h∂−1 6(h∂)3, cosh( h∂)≃1−1 2(h∂)2, we obtain, from eq.(9), the polynomial differential equation: a˜ηt+ 2c0ǫ2h˜η˜ηx+c0ǫh˜ηx−c0ǫh3 6˜ηxxx−c0ǫ2h3 2/parenleftbigg ˜ηx˜ηxx+ ˜η˜ηxxx/parenrightbigg = 0,(13) where ǫ=a h. The first four terms in eq.(20) correspond to the zero order a pproximation terms in f, obtained from the boundary condition at the free surface, i .e. the traditional way of obtaining the KdV equation in shallow channels. In order to find an exact solution for eq.(12) we can write it in the form: Ahu X(X) +u(X+h)−u(X−h) 2i+uX(X)u(X+h) +u(X−h) 2 +u(X)uX(X+h) +uX(X−h) 2= 0, (14) where X=x+Ac0tandAis an arbitrary real constant. We want to stress here that eq.(14) is a finite-difference differential equation, which i s rather the exception than the rule fir such systems. Hence, it may contain among its symm etries, the scaling symmetry. Actualy, the first derivative of u(X) is shown to be alinear combination of translated versions of the original function. In this way, t he theory of such equations 7can be related with the wavelet, or other self-similarity sy stems, theory, [13]. In the following we study the solutions with a rapid decreasing at i nfinity and make a change of variable: v=eBXforx∈(−∞,0) and v=e−BXforx∈(0,∞), with Ban arbitrary constant. Writing u(X) =−hA+f(v), and choosing for the solution the form of a power series in v: f(v) =∞/summationdisplay n=0anvn, (15) we obtain a nonlinear recurrsion relation for the coefficient san: /parenleftbigg Ahk+sin(Bhk) B/parenrightbigg ak =−k−1/summationdisplay n=1n/parenleftbigg cosh (Bh(k−n)) + cosh( Bh(k−1))/parenrightbigg anak−n. (16) With the coefficients given in eq.(16) the general solution ηcan be written analyticaly. In order to verify the consistency of this solution we study a limiting case of the relation, by replacing sin and cosh expressions with their lowest nonv anishing terms in their power expansions Thus, eq.(16) reduces to αk=6 B2h3k(k2−1)k−1/summationdisplay n=1nαnαk−n, (17) and αk=/parenleftbigg1 2B2h3/parenrightbiggk−1 k (18) is the solution of the above recurrence relation. In this app roximation, the solution of eq.(12) reads η(X) = 2 B2h3∞/summationdisplay k=1k/parenleftbigg −e−B|X|/parenrightbiggk =B2h3 21 (cosh( BX/2))2, (19) which is just the single-soliton solution of the KdV equatio n and it was indeed obtained by assuming hsmall in the recurrence relation (16). Hence, we have shown t hat the KdV equation describing the shallow liquids can be generalised for any depths and lengths. This result may be the starting point to search for more inter esting symmetries. It would be interesting to interpret the generalized-KdV eq.( 12) as the Casimir element of a certain algebra. 84 Compacton and self-similar solutions Eq.(12) has a special character, namely contains both infini tesimal and finite difference operators. This particularity relates it to another field of nonlinear systems, that is scaling functions and wavelet basis, functions or distribu tions with compact support and self-similarity properties. In the following we investiga te a particular case of eq.(12), that is when h≪η,h≪δ, where δis the half-width of the solution, if this has bounded or compact support. In this approximations, from eq.(12) we keep only the terms 1 c0ηt+ηx+1 hηηx−h 2ηxηxx+1 hηηx−h 2ηηxxx+O3≃0. (20) This equation is related to another intergable system, name ly the K(2,2) equation, investigated in [11] ηt+ (η2)x+ (η2)xxx= 0. (21) The main property of the K(2,2) equation is the equal occuren ce of non-linearity, dis- persion and the existence of a Lagrangian and Hamiltonian sy stem associated with it. The special solutions of this equation are the compactons ηc=4η0 3cos2/parenleftbiggx−η0t 4/parenrightbigg ,|x−η0t| ≥2π, (22) andηc= 0 otherwise. This special solutions have compact support a nd special proper- ties concerning the scattering between different such solut ions. As the authors comment in [11], the robustness of these solutions makes it clear tha t a new mechanism is un- derlying this system. In this respect, we would like to add th at, taking into account eq.(12), this new mechanism might be related to selfsimilar ity and multiscale properties of nonlinear systems. 5 Conclusions In the present paper we introduced a non-linear hydrodynami c model describing new modes of motion of the free surface of a liquid. The total ener gy of this nonlinear liquid 9drop model, subject to non-linear boundary conditions at th e free surface and the inner surface of the fluid layer, gives the Hamiltonian of the Korte weg de Vries equation. We have studied the stability of the cnoidal wave and solitary w ave solutions, from the point of view of minima of this Hamiltonian. The non-linear terms yield rotating steady-state solution s that are cnoidal waves on the surface of the drop, covering continuously the range f rom small harmonic oscilla- tions, to anharmonic oscillations, and up to solitary waves . The initial one-dimensional model [10] was extend to a three-dimensional model. A kind of new generalized KdV equation, together with some of its analytical solutions ha ve been presented. We also found a connection between the obtained generalized KdV equ ation, and another one (i.e. K(2,2)), in a certain approximation. In this case, com pacton solutions have been found and new symmetries (e.g. self-similarity) were put in to evidence. The analytic solutions of the non-linear model presented in this paper, make pos- sible the study of clusterization as well as to explain or pre dict the existence of new strongly deformed shapes, or new patterns having compact su pport or finite wavelength. The model applies not only in fluid and rheology theories, but may provide insight into similar processes occurring in other fields and at other scal es, such as the behavior of superdeformed nuclei, supernova, preformation of clust er in hydrodynamic models (metallic, molecular, nuclear), the fission of liquid drops (nuclear physics), inertial fu- sion, etc. Supported by the U.S. National Science Foundation through a regular grant, No. 9603006, and a Cooperative Agreement, No. EPS-9550481, tha t includes a matching component from the Louisiana Board of Regents Support Fund. One of the authors (A.L.) would like to thank Peter Herczeg from the T5 Division at Los Alamos National Laboratory, and at the Center for Nonlinear Studies at Los Al amos for hospitality. References 10[1] H-L. Lu and R. E. Apfel, J. Fluid Mech. 222351 (1991); T. Shi and R. E. Apfel, Phys. Fluids 71545 (1995); Y. Tian, R. G. Holt and R. E. Apfel, Phys. Fluids 7 2938 (1995); W. T. Shi, R. E. Apfel and R. G. Holt, Phys. Fluids 72601 (1995). [2] R. Natarajan, R. A. Brown, J. Fluid Mech. 18395 (1987); Phys. Fluids 292788 (1986). [3] J. A. Tsamopoulos and R. A. Brown, J. Fluid Mech. 127519 (1983). [4] E. H. Trinh, R. G. Holt and D. B. Thiessen, Phys. Fluids 843 (1995); P. L. Marston and S. G. Goosby, Phys. Fluids 281233 (1985); E. Trinh and T. G. Wang, J. Fluid Mech. 122315 (1982). [5] R. E. Apfel et al,Phys. Rev. Lett. 781912 (1997) [6] E. Trinh and T. G. Wang, J. Fluid Mech. 122(1982) 315 [7] G. L. Lamb, Elements of Soliton Theory (John Wiley & Sons, New York, 1980); C. Rebbi and G. Soliani, Solitons and Particles (World Scientific, Singapore, 1984); S. Novikov, S. V. Manakov, Theory of Solitons: The Inverse Scattering Method (Consultants Bureau, New York, 1984); R. K. Bullough and P. J . Caudrey, Eds., Solitons (Topics in Current Physics, Springer-Verlag, Berlin, 1980 ). [8] A. Ustinov, Solitons in Josephson Junctions ,non-linear Waves and Solitons in Physical Systems (May 12-16, 1997, CNLS, Los Alamos, NM) to appear in Physica D. [9] Y. G. Kevrekidis, Catalytic Pattern on Microdesigned Domains ,non-linear Waves and Solitons in Physical Systems (May 12-16, 1997, CNLS, Los Alamos, NM) to appear in Physica D . [10] A. Ludu, A. Sandulescu and W. Greiner, Int. J. Modern Phys. E 1169 (1992) ; 24 (1993) 855; J. Phys. G: Nucl. Part. Phys. 211715 (1995); J. Phys. G: Nucl. Part. Phys. 23343 (1997). 11[11] P. Rosenau and J. M. Hyman, Phys. Rev. Let. 70564 (1993); F. Cooper, J. M. Hyman and A. Khane, to be published. [12] A. Ludu and W. Greiner, Found. Phys. 26665 (1996). [13] A. Ludu and J. P. Draayer, in preparation; Internal semi nar CNLS and T8, Los Alamos, May, 1996. [14] R. Abraham and J. E. Marsden, Foundations of Mechanics (The Ben- jamin/Cummings Publishing Company, Inc., Reading, Massac husetts, 1978). 12FIGURE CAPTIONS Fig. 1 The order of smallness of four typical terms depending on φand occuring in the Hamiltonian, eqs.(1,2). Order zero holds for η2, order 1 for η2 φ, order 2 for η3,and order 3 forηη2 φ. Figs. 2 2a. The transition of the cnoidal solution, from a l= 5 mode to the soliton limit: shape of the cross-section for θ=π/2 function as a function of α2withα1,3fixed by the volume conservation and periodicity conditions. 2b. Cnoidal solutions (cross-sections of Σ1 for θ=π/2) subject to the volume con- servation constraint. Results for the l= 6 mode to the l= 2 mode and a soliton are shown. The corresponding linear modes, i.e. spherical harm onics, are superimposed on the non-linear solutions. 2c. Pictorial view of a soliton deformation of a drop, on the top o f the original unde- formed sphere. The supporting sphere for the soliton has sma ller radius because of the volume conservation. Fig. 3 The energy plotted versus α1, α2for constant volume. From the small oscillation limit ( α2≃3) towards the solitary wave limit ( α2= 1) the energy increases and has a valley for α1≃0.1 and α2∈(1.2,1.75) (close to the l= 2 mode). 13Fig. 4 The total energy plotted versus α1, α2for constant volume (small circles). Larger circles indicate the patterns fulfilling the periodicity co ndition. From the small oscilla- tions limit ( α2≃3) towards the solitary wave limit ( α2= 1) the energy increases but forα2∈(1.2,1.75) (close to l= 2 mode) it has a valley.. 14-2 -1 0 1 2 3 x00.020.040.060.08 0 312l=3 l=2 Solitonl=6l=5 l=40 0.5 1 1.5 2a2 0.50.7511.251.5 a10102030 E 0 0.5 1 1.5 2a21 1.5 2a2 1 1.2 1.4 1.6a10.20.40.60.81 0.20.40.60.8

arXiv:physics/0003078v1 [physics.comp-ph] 27 Mar 2000A fast algorithm for generating a uniform distribution inside a high-dimensional polytope Andr´ e van Hameren∗and Ronald Kleiss† University of Nijmegen, Nijmegen, the Netherlands February 18, 2014 Abstract We describe a uniformly fast algorithm for generating point s/vector x uniformly in a hypercube with the restriction that the differ ence be- tween each pair of coordinates is bounded. We discuss the qua lity of the algorithm in the sense of its usage of pseudo-random sour ce num- bers, and present an interesting result on the correlation b etween the coordinates. ∗andrevh@sci.kun.nl †kleiss@sci.kun.nl1 Introduction In this paper we shall discuss the problem of generating sets of points /vector x= (x1, x2, . . ., x m) inside an m-dimensional hypercube with an additional re- striction. The points /vector xare required to satisfy the conditions |xk|<1,|xk−xl|<1 for all k, l . (1) These conditions define a m-dimensional convex polytope P. The reason for tackling this problem is the following. In a recently develo ped Monte Carlo algorithm, SARGE [1], we address the problem of generating configurations of four-momenta pµ i,i= 1,2, . . ., n ofnmassless partons at high energy, with a distribution that has, as much as possible, the form of a so- called QCD antenna:1 s12s23s34···sn−1,nsn1, s kl= (pk+pl)2, where sklis the invariant mass squared of partons kandl, with the addi- tional requirement that the total invariant mass squared of all the partons is fixed to s, andevery skl(also those not occurring explicitly in the antenna) exceeds some lower bound s0: in this way the singularities of the QCD matrix elements are avoided. The SARGE algorithm has a structure that is, in part, similar to the RAMBO algorithm [2], where generated momenta are scaled so as to attain the correct overall invariant mass. Obviously, inSARGE this is more problematic because of the s0cut, but one should like to implement this cut as far as possible. Note that out of the n(n−1)/2 different skl,n occur in the antenna, and each of these must of course be bound ed by s0 from below and some sM< sfrom above. The scale-invariant ratios of two of these masses are therefore bounded by s0 sM≤sij skl≤sM s0, (2) The structure of the SARGE algorithm is such [1] that there are m= 2n−4 of these ratios to be generated. By going over to variables x(···)= log( sij/skl)/log(sM/s0), and inspecting all ratios that can be formed from the chosen mones, we arrive at the condition of Eq.(1). Note that, inside SARGE , a lot of internal rejection is going on, and events satisfying Eq.(1) maystill be discarded: however, if Eq.(1) is not satisfied, the event is certainly discarded, and it therefore pays to include this condition from the start. 12 The algorithm The most straightforward way of implementing is of course th e following: generate xk,k= 1, . . ., m byxk←2ρ−1, and reject if the conditions are not met. Here and in the following, each occurrence of ρstands for a call to a source of iid uniform pseudo-random numbers betwee n in [0 ,1). The drawback of this approach is that the efficiency, i.e.the probability of success per try, is given by 2−mVm(P) (where Vm(P) is the volume of the polytope P) and becomes very small for large m, as we shall see. To compute the volume Vm(P) we first realize that the condition |xk− xl|<1 is only relevant when xkandxlhave opposite sign. Therefore, we can divide the xvariables in m−kpositive and knegative ones, so that Vm,k(P) =1/integraldisplay 0dy1dy2···dykdxk+1dxk+2···dxmθ/parenleftbigg 1−max ixi−max jyj/parenrightbigg , Vm(P) =m/summationdisplay k=0m! k!(m−k)!Vk(P), (3) where we have written yk=−xk. By symmetry we can always relabel the indices such that xm= max ixiandy1= max jyj. The integrals over the other x’s and y’s can then easily be done, and we find Vm,k(P) = k(m−k)1/integraldisplay 0dy1yk−1 11−y1/integraldisplay 0dxmxm−k−1 m =k1/integraldisplay 0dy1yk−1 1(1−y1)m−k=k!(m−k)! m!, (4) and hence Vm(P) =m+ 1. (5) The efficiency of the straightforward algorithm is therefore equal to ( m+ 1)/2m, which is less than 3% for nlarger than 6. We have given the above derivation explicitly since it allow s us, by work- ing backwards, to find a rejection-free algorithm with unit e fficiency. The algorithm is as follows: 1. Choose a value for k. Since each kis exactly equally probably we simply have k←⌊(m+ 1)ρ⌋. 22. For k= 0 we can simply put xi←ρ , i= 1, . . ., m , while for k=mwe put xi←−ρ , i= 1, . . ., m . 3. For 0 < k < m ,y1has the unnormalized density yk−1 1(1−y1)m−k between 0 and 1. An efficient algorithm to do this is Cheng’s rej ection algorithm BA for beta random variates (cf. [3])1, but the following also works: v1←−log/parenleftBiggk/productdisplay i=1ρ/parenrightBigg , v2←−log m−k+1/productdisplay j=1ρ , y1←v1 v1+v2. The variable xmhas unnormalized density xm−k−1 m between 0 and 1−y1 so that it is generated by xm←(1−y1)ρ1/(m−k). The other x’s are now trivial: x1←−y1, x i←x1ρ, i= 2,3, . . ., k , xi←xmρ, i=k+ 1, k+ 2, . . ., m−1. Finally, perform a random permutation of the whole set ( x1, x2, . . ., x m). 3 Computational complexity The number usage S, that is, the expected number of calls to the random number source ρper event can be derived easily. In the first place, 1 number is used to get kfor every event. In a fraction 2 /(m+ 1) of the cases, only m calls are made. In the remaining cases, there are k+ (m−k+ 1) = m+ 1 calls to get y1, and 1 call for all the other xvalues. Finally, the simplest 1There is an error on page 438 of [3], where “ V←λ−1U1(1−U1)−1” should be replaced by “V←λ−1log[U1(1−U1)−1]”. 3permutation algorithm calls m−1 times [4]. The expected number of calls is therefore S= 1 +2m m+ 1+m−1 m+ 1(m+ 1 + ( m−1) + (m−1)) =3m2−m+ 2 m+ 1.(6) For large mthis comes to about 3 m−1 calls per event. Using a more sophisticated permutation algorithm would use at least 1 ca ll, giving S= 1 +2m m+ 1+m−1 m+ 1(m+ 1 + ( m−1) + (1)) = 2 m . (7) We observed that Cheng’s rejection algorithm to obtain y1uses about 2 calls per event. Denoting this number by Cthe expected number of calls becomes S=2m2+ (C−1)m−C+ 3 m+ 1∼2m+C−1 (8) for the simple permutation algorithm, while the more sophis ticated one would yield S=m2+ (C+ 2)m−C+ 1 m+ 1∼m+C+ 2. (9) We see that in all these cases the algorithm is uniformly effici ent in the sense that the needed number of calls is simply proportional to the problem’s complexity m, asmbecomes large. An ideal algorithm would of course still need mcalls, while the straightforward rejection algorithm rath er has S=m2m/(m+ 1)∼2mexpected calls per event. In the testing of algorithms such as this one, it is useful to s tudy expec- tation values of, and correlations between, the various xi. Inserting either xi orxixjin the integral expression for V(P), we found after some algebra the following expectation values: E(xi) = 0 ,E(x2 i) =m+ 3 6(m+ 1),E(xixj) =m+ 3 12(m+ 1)(i/negationslash=j),(10) so that the correlation coefficient between two different x’s is precisely 1/2 in all dimensions! This somewhat surprising fact allows for a simple but powerful check on the correctness of the algorithm’s implem entation. 44 Extension Let us, finally, comment on one possible extension of this alg orithm. Suppose that the points /vector xare distributed on the polytope P, but with an additional (unnormalized) density given by F(/vector x) =m/productdisplay i=1cos/parenleftbiggπxi 2/parenrightbigg , (11) so that the density is suppressed near the edges. It is then st ill possible to compute Vm,k(P) for this new density: Vk,m(P) = k(m−k)1/integraldisplay 0dy1cos/parenleftbiggπy1 2/parenrightbigg1−y1/integraldisplay 0dxmcos/parenleftbiggπxm 2/parenrightbigg  y1/integraldisplay 0dycos/parenleftbiggπy 2/parenrightbigg k−1 xm/integraldisplay 0dxcos/parenleftbiggπx 2/parenrightbigg m−k−1 =k(m−k)/parenleftbigg2 π/parenrightbiggm1/integraldisplay 0dsin/parenleftbiggπy1 2/parenrightbigg/parenleftbigg sin/parenleftbiggπy1 2/parenrightbigg/parenrightbiggk−1 cos(πy1 2)/integraldisplay 0dsin/parenleftbiggπxm 2/parenrightbigg/parenleftbigg sin/parenleftbiggπxm 2/parenrightbigg/parenrightbiggm−k−1 =2m−1k πm1/integraldisplay 0ds sk/2−1(1−s)(m−k)/2 =/parenleftbigg2 π/parenrightbiggmΓ(1 + k/2)Γ(1 + ( m−k)/2) Γ(1 + m/2), (12) where we used s=/parenleftBig sin/parenleftBigπy1 2/parenrightBig/parenrightBig2. Therefore, a uniformly efficient algorithm can be constructed in this case as well, along the following l ines. Using the Vk,m, the relative weights for each kcan be determined. Then sis generated as aβdistribution. The generation of the other x’s involves only manipula- tions with sine and arcsine functions. Note that, for large m, the weighted volume of the polytope Pis V(P) =m/summationdisplay k=0/parenleftbigg2 π/parenrightbiggm/parenleftBig k 2/parenrightBig !/parenleftBig m−k 2/parenrightBig ! /parenleftBig m 2/parenrightBig !m! k!(m−k)! 5∼m/radicalbiggπ 8/parenleftbigg8 π2/parenrightbiggm/2 , (13) so that a straightforward rejection algorithm would have nu mber usage S∼/radicalBigg 8 π/parenleftBiggπ2 2/parenrightBiggm/2 , (14) and a correspondingly decreasing efficiency. References [1] P. Draggiotis, A. van Hameren and R. Kleiss, in preparati on. [2] S.D. Ellis, R. Kleiss and W.J. Stirling, A new Monte Carlo treatment of multiparticle phase space at high energy , Comp. Phys. Comm. 40 (1986) 359. [3] L. Devroye, Non-Uniform Random Variate Generation , (Springer, 1986). [4] D.E. Knuth, The Art of Computer Programming, Vol.2. 2d ed. (Prince- ton, 1991). 6

arXiv:physics/0003079v1 [physics.flu-dyn] 27 Mar 2000Emergence of intense jets and Jupiter Great Red Spot as maximum entropy structures. F. BOUCHET1and J. SOMMERIA2 1UMR 5582, Institut Fourier, BP 74, 38402 Saint Martin d’H` er es Cedex, France 2CNRS, LEGI/Coriolis, 21 av. des Martyrs, 38 000 Grenoble, Fr ance July 22, 2013 Abstract We explain the emergence and robustness of intense jets in hi ghly turbulent plan- etary atmospheres, like on Jupiter, by a general approach of statistical mechanics of potential vorticity patches. The idea is that potential vor ticity mixing leads to the formation of a steady organized coarse grained flow, corresp onding to the statistical equilibrium state. Our starting point is the quasi-geostro phic 1-1/2 layer model, and we consider the relevant limit of a small Rossby radius of def ormation. Then narrow jets are obtained, scaling like the Rossby radius of deforma tion. These jets can be either zonal, or closed into a ring bounding a vortex. Taking into account the effect of the beta effect and a sublayer deep shear flow, we predict an org anization of the turbu- lent atmospheric layer into an oval-shaped vortex amidst a b ackground shear. Such an isolated vortex is centered over an extremum of the equivale nt topography (determined by the deep shear flow and beta-effect). This prediction is in a greement with analysis of wind data in major Jovian vortices (Great Red Spot and Oval BC). 1 Introduction Atmospheric and oceanic flows are often organized into narro w jets. They can zonally flow around the planet like the jet streams in the earth stratosph ere, or the eastward jet at 24 in the northern hemisphere of Jupiter ( Maxworthy 1984). Jet s can alternatively organize into rings, forming vortices, like the rings shed by the mean dering of the Gulf-Stream in the western Atlantic Ocean. The flow field in Jupiter most famo us feature, the Great Red Spot, is an oval-shaped jet, rotating in the anticycloni c direction and surrounding an interior area with a weak mean flow ( Dowling and Ingersoll 198 9), see figure 1(a). Robust cyclonic vortices have a similar jet structure ( Hatzes et al 1981 ), see figure 1(b). Such jets and vortices are in a turbulent surrounding, and th e persistence of their strength and concentration in the presence of eddy mixing is intriguing. The explanation proposed in this paper is based on a statistical mechanical a pproach: the narrow jet or vortex appears as the most probable state of the flow after a tu rbulent mixing of potential vorticity, taking into account constraints due to the quant ities conserved by the dynamics, 1especially energy. Such a statistical theory has been first p roposed for the two-dimensional Euler equations by Kuz’min (1982), Robert (1990) , Robert an d Sommeria (1991), Miller (1990). See Brandt et al (1999) for a recent review and discus sion. This theory predicts an organization of two-dimensional turbulence into a steady fl ow (with fine scale, ’microscopic’ vorticity fluctuations). Complete vorticity mixing is prev ented by the conservation of the energy, which can be expressed as a constraint in the accessi ble vorticity fields. A similar, but quantitatively different, organization had been previo usly obtained with statistical me- chanics of singular point vortices with the mean field approx imation, instead of continuous vorticity fields (Onsager 1949, Joyce and Montgomery 1973). Extension to the quasi-geostrophic (QG) model has been disc ussed by Sommeria et al (1991), Michel & Robert (1994), Kazantsev Sommeria and Ve rron (1998). This model describes a shallow water system with a weak vorticity in com parison with the planetary vorticity (small Rossby number), such that the pressure is i n geostrophic balance, and the corresponding free surface deformation is supposed small i n comparison with the layer thick- ness. For Jupiter the free surface would be rather at the bott om of the active atmospheric layer, floating on a denser fluid, as discussed by Dowling and I ngersoll (1989), see Dowling (1995) for a review. The gradient of planetary vorticity is a ccounted by a beta-effect. An additional beta-effect, depending on the latitude coordina tey, is introduced to represent the influence on the active atmospheric layer of a steady zona l flow in the deep interior, as discussed by Dowling and Ingersoll (1989). The free surface deformability, representing the strength of the density stratification, is controlled by the Rossby radius of deformation R∗. The two-dimensional Euler equation is recovered in the limit of very strong stratification for whic hR∗→ ∞. We shall consider in this paper the opposite limit of weak stratification for wh ichR∗is much smaller than the scale of the system L. This limit is appropriate for large scale oceanic currents , as the radius of deformation is typically 10-100 km. For Jupiter, R∗is estimated to be in the range 500-2500 km, while the Great Red Spot extends over 20,0 00 km in longitude, and 10,000 km in latitude, so the limit R∗/L→0 seems relevant. We show that in this limit the statistical equilibrium is made of quiescent zones with well mixed uniform potential vorticity, bounded by jets with thickness of order R∗. This provides therefore a general justification of jet persistence. Some of the ideas used have been already sketched in Sommeria et al (1991), but we here provide a systematic deriv ation and thorough analysis. In principle, the Quasi Geostrophic approximation breaks d own for scales much larger than the radius of deformation, so that the limit R∗/L→0 seems inconsistent with the QG approximation. However the relevant scale is the jet width, which remains of order R∗, so that the Quasi Geostrophic approximation remains valid in t his limit. This point has been discussed by Marcus (1993) for the Great Red Spot, which he su pposes to be a uniform Potential Vorticity ( PV ) spot surrounded by a uniform Poten tial Vorticity background ( we here justify this structure as the result of Potential Vor ticity mixing with constraints on the conserved quantities ). Analyzing wind data in the Great Red Spot, Dowling & Ingersoll (1989) concluded that the QG approximation is good up within typically 30% error, which is reasonable to a first approximation. Statistical mechanics of the more general shallow water system (to be published), predicts a similar jet structure. The present Quasi Geostrophic results therefore provide a good description as a first appro ximation. We first consider the case without beta-effect in section 2. We furthermore assume 2periodic boundary conditions (along both coordinates) in t his section to avoid consideration of boundary effects. Starting from some initial condition wi th patches of uniform PV, we find that these patches mix with uniform density (probabilit y) in two sub-domains, with strong density gradient at the interface, corresponding to a free jet. The coexistence of the two sub-domains can be interpreted as an equilibrium betwee n two thermodynamic phases. We find that the interface has a free energy per unit of length, and its minimization leads to a minimum length at equilibrium. This results in a constant r adius of curvature, in analogy with surface tension effects in thermodynamics, leading to s pherical bubbles or droplets. The range of the vortex interaction is of the order R∗, therefore becoming very small in the limit of small radius of deformation, so the statistical equilibrium indeed behaves like in usual thermodynamics with short range molecular interac tions. This contrasts with the case of Euler equation, with long range vortex interacti ons, analogous to gravitational effects (Chavanis Sommeria and Robert 1996, Chavanis 1998). Figure 7 summarizes the calculated equilibrium states, dep ending on the total energy and a parameter Brepresenting an asymmetry between the initial Potential Vo rticity patch areas, before the mixing process. We obtain straight jets fo r a weak asymmetry and circular jets for higher asymmetry. Such circular a jet reduces to an a xisymmetric vortex, with radius of order R∗, in the limit of low energy. We discuss the influence of the beta-effect or the deep zonal flo w in section 3. The channel geometry, representing a zonal band periodic in the longitudexis appropriate for that study. With the usual beta-effect βy, linear in the transverse coordinate y, statistical equilibrium is, depending on the initial parameters, a zona l flow, or a meandering eastward jet, or a uniform velocity vm=R2βwhose induced free surface slope cancels the beta-effect ( uniformization of Potential Vorticity ) on which circular vortices can coexist. For more general beta-effects, due to the deep zonal flow, we fin d that the jet curvature depends on latitude y. In particular a quadratic beta effect ay2leads to oval-shape jets, similar to the Great Red Spot. Using the determination of the sublayer flow from Voyager data by Dowling and Ingersoll (1989), we show in section 4, th at such a quadratic effective beta-effect is indeed a realistic model for Jupiter atmosphe re in the latitude range of the Great Red Spot and the White Ovals, the other major coherent v ortices on Jupiter. Using these data on beta-effect, as well as the shear in the zonal flow at the latitude of the Great Red Spot, the jet width and its maximum velocity, we deduce al l the parameters of our model. 2 The case with periodic boundary conditions 2.1 The dynamical system We start from the barotropic Quasi Geostrophic (QG) equatio n : ∂q ∂t+v· ∇q= 0 (1) q=−∆ψ+ψ R2−h(y) (2) 3v=−ˆz∧ ∇ψ (3) whereqis the potential vorticity ( PV ), advected by the non-diverg ent velocity v,ψis the stream function, Ris the internal Rossby deformation radius between the layer of fluid under consideration and a deep layer unaffected by the dynami cs.xandyare respectively the zonal and meridional coordinates ( xis directed eastward and ypoleward ). The termh(y) represents the combined effect of the planetary vorticity g radient and of a given stationary zonal flow in the deep layer, with stream function ψd(y):h(y) =−βy+ψd/R2. This deep flow induces a constant deformation of the free surf ace, acting like a topography on the active layer. We shall therefore call h(y) the ’topography’, and study its influence in section 3. Let us assume h(y) = 0 in this section. We define the QG equations (1,2) in the non-dimensional square D= [−1 2,1 2]2.Ris then the ratio of the internal Rossby deformation radius R∗to the physical scale of the domain L. Let/angb∇acketleftf/angb∇acket∇ight ≡/integraltext Dfd2rbe the average of fonDfor any function f. Physically, as the stream function ψis related to the geostrophic pressure, /angb∇acketleftψ/angb∇acket∇ightis proportional to the mean height at the interface between the fluid layer and the bottom layer, and due to the mass conservation it must be constant (Pedlosky 1987). We make th e choice /angb∇acketleftψ/angb∇acket∇ight= 0 (4) without loss of generality. The total circulation is /angb∇acketleftq/angb∇acket∇ight=/angb∇acketleft−∆ψ+ψ/R2/angb∇acket∇ight=/angb∇acketleftψ/R2/angb∇acket∇ightdue to the periodic boundary conditions. Therefore /angb∇acketleftq/angb∇acket∇ight= 0 (5) We note that the Dirichlet problem (2) on Dwith periodic boundary conditions has a unique solution ψfor a given PV field. Due to the periodic conditions for ψ, the linear momentum is also equal to 0, /angb∇acketleftv/angb∇acket∇ight= 0 (6) The energy E=1 2/integraldisplay Dqψd2r=1 2/integraldisplay D[ (∇ψ)2+ψ2 R2]d2r (7) is conserved ( we note that the first term in the right hand side of (7) is the kinetic energy whereas the second one is the gravitational available poten tial energy ). The integrals Cf(q) =/integraldisplay Df(q)d2r (8) for any continuous function fare also conserved, in particular the different moments of th e PV. In the case of an initial condition made of a finite number o f PV levels, the areas initially occupied by each of these levels is conserved, and this is equ ivalent to the conservation of all the constants of motion (8). 42.2 The statistical mechanics on a two PV levels configuratio n. 2.2.1 The macroscopic description. The QG equations (1) (2) are known to develop very complex vor ticity filaments. Because of the rapidly increasing amount of information it would req uire, as time goes on, a de- terministic description of the flow for long time is both unre alistic and meaningless. The statistical theory adopts a probabilistic description for the vorticity field. The statistical equilibrium depends on the energy and of the global probabil ity distribution of PV levels. Various previous studies (Sommeria & al 1991), (Kazantsev & al 1998) indicate that a model with only two PV levels provides a good approximation i n many cases. The determi- nation of the statistical equilibrium is then simplified as i t depends only on the energy, on the two PV levels, denoted q=a1andq=a−1and on their respective areas Aand (1 −A) inD. The number of free parameters can be further reduced by appr opriate scaling. Indeed a change in the time unit permits to define the PV levels up to a m ultiplicative constant, and we choose for the sake of simplicity : a1−a−1 2= 1 (9) and define the non-dimensional parameter Bas : B≡a1+a−1 2(10) The condition (5) of zero mean PV imposes that a1A+a−1(1−A) = 0. This means that a1 anda−1must be of opposite sign and, using (9) and (10), A= (1−B)/2. The distribution of PV levels is therefore fully characterized by the single a symmetry parameter B, which takes values between -1 and +1. The symmetric case of two PV pa tches with equal area A= 1/2 corresponds to B= 0, while the case of a patch with small area (but high PV, such that /angb∇acketleftq/angb∇acket∇ight= 0) corresponds to B→1. Note that we can restrict the discussion to B≥1 as the QG system is symmetric by a change of sign of the PV. The two PV levels mix due to turbulent effects, and the resulti ng state is locally de- scribed by the local probability (local area proportion) p(r) to find the first level at the location r. The probability to find the complementary PV level a−1is 1−p, and the locally averaged PV at each point is then q(r) =a1p(r) +a−1(1−p(r)) = 2/parenleftbigg p−1 2/parenrightbigg +B (11) where the second relation is obtained by using (9) and (10). As we consider the evolution of two PV patches, the conservat ion of all invariants (8) is equivalent to the conservation of the area Aof the patch with PV value a1( the area of the other PV level a−1being 1 −A). The integral of pover the domain must be therefore equal to the initial area A( the patch with PV level a1is mixed but globally conserved ), A≡1−B 2=/integraldisplay Dp(r)d2r (12) 5As the effect of local PV fluctuations is filtered out by integra tion, the stream function and the velocity field are fully determined by the locally ave raged PVqas the solution of q=−∆ψ+ψ R2;ψ periodic (13) andv=−ˆz∧ ∇ψ Therefore the energy is also expressed in terms of the field q: E=1 2/integraldisplay D/bracketleftbigg (∇ψ)2+ψ2 R2/bracketrightbigg d2r=1 2/integraldisplay Dqψd2r (14) Here the energy of the ’microscopic’ PV fluctuations has been neglected (replacing qby q), as justified in the case of Euler equation by Robert and Somm eria (1991). Indeed, considering a ’cutoff’ for the microscopic fluctuations much smaller than R, the small scale dynamics coincides with the Euler case. The central result of the statistical mechanics of the QG equ ations (1,2) is that, under an ergodic hypothesis, we expect the long time dynamics to co nverge towards the Gibbs states defined by maximizing the mixing entropy S=−/integraldisplay D[p(r)lnp(r) + (1 −p(r))ln(1 −p(r)) ]d2r (15) under the constraints of the global PV distribution (12) and energy (14). It can be shown that the microscopic states satisfying the constraints giv en by the conservation laws are overwhelmly concentrated near the Gibbs state, which is the refore likely to be reached after a complex flow evolution. A good justification of this st atement is obtained by the construction of converging sequences of approximations of the QG equation (1,2), in finite dimensional vector spaces, for which a Liouville theorem ho lds. This is a straightforward translation of the work of Robert (1999) for 2D Euler equatio ns. The sequence of such Liouville measures has then the desired concentration prop erties as (1,2) enters in the context considered in Michel & Robert (1994). 2.2.2 The Gibbs states Following Robert & Sommeria (1991), we seek maxima of the ent ropy (15) under the constraints (12) and (14). To account for these constraints , we introduce two corresponding Lagrange multipliers, which we denote 2 αand−C/R2for convenience in future calculations. Then the first variation of the functionals satisfies : δS−2αδA+C R2δE= 0 for all variations δpof the probability field p. After straightforward differentiation we obtain: δS=−/integraldisplay D[ lnp−ln(1−p) ]δpd2r, δA =/integraldisplay Dδpd2r δE=/integraldisplay Dψδqd2r=/integraldisplay D2ψδpd2r (16) 6where the expression of δEhas been obtained by integrating by part and expressing qby (11). Then we can write the first variation under the form/integraltext D[−lnp+ ln(1 −p)−2α+ 2Cψ/R2]δp d2rwhich must vanish for any small variation δp. This implies that the integrand must vanish, and yields the equation for the optim um state: p=1−tanh(α−Cψ R2) 2, (17) and using (11) and (13), the partial differential equation q=−∆ψ+ψ R2=B−tanh/parenleftbigg α−Cψ R2/parenrightbigg (18) determining the Gibbs states (statistical equilibrium). F rom now on we forget the qover- line for the locally averaged PV and refer to it as the PV. Therefore, we have shown that for any solution of the variati onal problem, two constants αandCexist such that ψsatisfies (18). Conversely it can be proved that for any such two constants, a solution to equation (18), in general not un ique, always exists. Then p associated with one of these solutions by (17) is a critical p oint of the ’free energy’ −S(p)+ 2αA(p)−C R2E(p) (i.e. its first variation vanishes). Then the Lagrange mult ipliers are not given but have to be calculated by prescribing the constrain ts (12) and (14) corresponding to the two parameters BandErespectively, given by the initial condition. Furthermore , among the states of given energy Eand asymmetry parameter B, we shall select the actual maxima. Finally, let us find a lower bound for the parameter Cof the Gibbs states with non-zero energy (i.e. ψis not constant on D). Multiplying (18) by −∆ψ, integrating by part and definingf(Cψ)≡B−tanh(α−Cψ R2), we obtain : C=/integraltext D/parenleftbig (∆ψ)2+1 R2(∇ψ)2/parenrightbig d2r/integraltext D−f′(Cψ)(∇ψ)2d2r From which, using 0 <−f′(Cψ)≤1 R2it follows that when ψis not constant : C >1 (19) 2.3 The limit of small Rossby deformation radius As suggested by oceanographic or Jovian parameters, we seek solutions for the Gibbs states equation in the limit of a small ratio between the Rossby defo rmation radius and the length scale of the domain : R<< 1 with our non-dimensional coordinates1 1Modica (1987) considered the minimization of the functiona lEǫ(u) =/integraltext Ω[ǫ(∇u)2+W0(u)]dxwith the constraint/integraltext Ωu(x)dx=min the limit ǫ→0+where W0is a real function with two relative minima. He proved, in a mathematical framework, working with bounded v ariations functions, that if ( uǫ) are solutions of this variational problem, for any subsequence of ( uǫ) converging in L1(Ω) as ǫ→0, this subsequence converge to a function u0which takes only the values where W0reaches its minima ; with the interface between the corresponding subdomains having minimal area ( See Modica (1987) for a precise statement ). We note that the Euler equation of this variational problem m ay be the same as the Gibbs States equation (18) for a convenient choice of W0. However as the variational problem itself is different this beautiful result cannot be used in our context. 72.3.1 The uniform subdomains Then we expect that the Laplacian in the Gibbs states equatio n (18) can be neglected with respect toψ/R2, except possibly in transition regions of small area. This t ransforms (18) into the algebraic equation : q=ψ R2=B−tanh/parenleftbigg α−Cψ R2/parenrightbigg (20) Depending on the parameters, this equation has either one, t wo or three solutions, denotedψ−1,ψ0andψ1in increasing order (see figure 2 ). The case with a single solu tion would correspond to a uniform ψ, which should be equal to 0 due to the condition /angb∇acketleftψ/angb∇acket∇ight= 0. This is only possible for E= 0. Otherwise, we have therefore two or three solutions, wit h different solutions occurring in subdomains. This conditio n of multiple solutions requires that the maximum slope for the right hand side of (20) must be g reater than 1 /R2; this is always realized due to the inequality (19). Furthermore αmust be in an interval centered inCB(α=CBin the symmetric case of figure 2 ). At the interface between two constant stream function subdo mains, a strong gradient of ψnecessarily occurs, corresponding to a jet along the interf ace. These jets give first order contributions to the entropy and energy, but let us first desc ribe the zero order problem. Suppose that ψtakes the value ψ1( respψ−1) in subdomains of total area A1( resp A−1). The reason why we do not select the value ψ0will soon become clear. Using (11) we conclude that the probability ptakes two constants values p±1in their respective subdomains. The two areas A±1( measured from the middle of the jet ) are complementary such that : A1+A−1= 1 (21) Furthermore the constraint (4) of zero domain average for ψimplies at zero order, ψ1A1+ψ−1A−1= 0, (22) or equivalently, using q±1=ψ±1/R2, (11) and (21) : 2A1/parenleftbigg p1−1 2/parenrightbigg + 2(1 −A1)/parenleftbigg p−1−1 2/parenrightbigg =−B (23) This can be obtained as well from the constraint on the (micro scopic) PV patch area (12). The energy inside the subdomains reduces to the potential te rmψ2/2R2, since velocity vanishes. This area energy EAcan be computed in terms of p±1usingq±1=ψ±1/R2and (11) : EA≡ψ2 1A1+ψ2 −1A−1 2R2=A1e(p1) + (1 −A1)e(p−1) withe(p)≡R2/parenleftBigg 2/parenleftbigg p−1 2/parenrightbigg2 + 2B/parenleftbigg p−1 2/parenrightbigg +B2 2/parenrightBigg(24) 8There is also an energy in the jet at the interface of subdomai ns, but it is small with respect toEA. Indeed the velocity in the jet, of width R, is of order ( ψ1−ψ−1)/R∼ψ/R, and the corresponding integrated kinetic energy is of order ψ2/R. This is small in comparison with the area energy EA(mostly potential) of order ψ2/R2. A precise calculation will confirm this estimate in next sub-section. We need to determine three unknown, the area A1and the probabilities p±1of the PV levela1in each subdomain, while the constraints (23) and (24) provi de two relations. An additional relation will be given by entropy maximization. As we neglect the jet area, the entropy reduces at order zero to the area entropy : SA≡A1s(p1) + (1 −A1)s(p−1) withs(p)≡ −plogp−(1−p)log(1 −p)(25) Thus the zero order problem corresponds to the maximization of the area entropy (25) with respect to the 3 parameters p±1andA1, under the 2 constraints (23) and (24). A necessary condition for a solution of this variational prob lem is the existence of two Lagrange parameters α0andC0, associated respectively with the circulation constraint (23) and with the energy constraint (24), such that the first variations of the total free energy, FA≡ −SA−C0 R2EA+α0/angb∇acketleftψ/angb∇acket∇ight R2, (26) vanish. Let us calculate FAusing (23) and (24): FA=A1f(p1) + (1 −A1)f(p−1), withf(p)≡ −s(p)−2C0/parenleftbigg p−1 2/parenrightbigg2 −2(C0B−α0)/parenleftbigg p−1 2/parenrightbigg −C0B2 2−α0B.(27) The vanishing of the variations with respect to p1andp−1gives that f(p±1) are local minima of the free energy f(p). It is easily proven that the function fhas two local minima and one local maximum ( for C0>1 and (C0B−α0) small enough ) ( see figure 3). The local maximum is achieved for p0corresponding to the value ψ0. It is the reason why it has not been taken into account in this analysis. In addition, the va nishing of the first variations with respect to the area A1imposes the free energies f(p±1) in the two subdomains to be equal. This is like the condition of thermodynamic equili brium for a chemical species shared by two coexisting phases. In the expression (27) of f(p), the entropy term s(p) is symmetric with respect to p=1 2, as well as the quadratic term. Therefore if the linear term i n/parenleftbig p−1 2/parenrightbig vanishes the two maxima are equal, with p±1symmetric with respect to1 2. The addition of a linear term obviously breaks this condition of two equal maxima, so the coefficient of the linear term must vanish, thus : α0=C0B. (28) Sincep±1are symmetric with respect to1 2, we introduce the parameter uby : p±1=1 2(1±u). (29) 9Using (11),(23) we deduce: ψ±1=R2(B±u) (30) From (22) we state that the two constant stream function (30) have to be of opposite sign, so thatu>|B|. Introducing (29) in the circulation constraint (23), and u sing (21), we get : A±1=1 2/parenleftbigg 1∓B u/parenrightbigg . (31) Using these results, the energy (24) becomes E≃EA=R2 2(u2−B2) (32) This relates the parameter uto the given energy Eand asymmetry parameter B. Finally the condition that f(p±1) are maxima of fleads to : u= tanh(C0u), (33) which determines the ’temperature’ parameter C0, as represented in figure 4. Therefore all the quantities are determined from the asymmetry parameter Band from the parameter u, related to the energy by (32). In the limit of low energy, u→ |B|, when for instance B >0, thenA1goes to zero, so thatψ−1tends to occupy the whole domain. This state is the most mixed one com- patible with the constraint of a given value of B(or equivalently a given initial patch area A= (1−B)/2). In the opposite limit u→1, we see from (30) that in the two subdomains q=ψ/R2tends to the two initial PV levels a1= 1+Banda−1=−1+B. Thus, this state is an unmixed state. It achieves the maximum possible energy E=R2 2(1−B2) under the constraint of a given patch area. We conclude that the parame teru, or the related ’tem- perature’C0, linked with the difference between the energy and the maximu m accessible energy for the two given initial PV levels, characterizes th e mixing of these two PV levels. We shall call uthe segregation parameter, as it quantifies the segregation of the PV level a1( or its complementary a−1) between the two phases. Let us now study the interface between the subdomains. 2.3.2 Interior Jets At the interface between two constant stream function subdo mains, a strong gradient of ψ necessarily occurs, corresponding to a jet along the interf ace. To study these jets, we come back to the Gibbs state equation (18). We expect the Lagrange parameters αandCto be close to the zero order parameters α0andC0, computed in the previous sub-section, so we useα=α0andC=C0to calculate the jet structure. In such a jet, we cannot negle ct the Laplacian term in (18), but a boundary layer type approximat ion can be used: we neglect the tangential derivative with respect to the derivative al ong the coordinate normal to the 10interface,ζ. Accordingly, we neglect the inverse of the curvature radiu s of the jet with respect to 1 /R. Thus, from the Gibbs states equation (18), using (28), we ded uce the jet equation : −d2ψ dζ2+ψ R2=B−tanh/parenleftbigg C0/parenleftbigg B−ψ R2/parenrightbigg/parenrightbigg (34) As the stream function depends only on the normal coordinate ζ, the velocity is tangent to the interface, forming a jet with a typical width scaling lik eR. We thus make the change of variables defined by : τ≡ζ R;φ≡ −B+ψ R2, (35) leading to the rescaled jet equation: d2φ dτ2=−tanh(C0φ) +φ (36) The jet equation (36) is similar to a one dimensional equatio n of motion for a particle ( with the position φdepending on a time τ) under the action of a force −dU/dφ deriving from the potential, U(φ) =ln(cosh(C0φ)) C0−φ2 2, (37) represented in figure 2(b). In its trajectory the particle en ergy is conserved : 1 2/parenleftbiggdφ dτ/parenrightbigg2 +U(φ) =Cst (38) Letφi≡ψi/R2−B,i=−1,0,1, corresponding to the solutions ψiof the algebraic equation (20). From (30), we have φ±1=±u. Note that the stationary limit of (36), which must be reached for lim τ→±∞, yields again (33). Moreover, the particle energy conserva tion (38) imposes the integrability condition, U(φ−1) =U(φ1), (39) which is indeed satisfied due to the symmetry of the potential U. We note that the Lagrange parameter determination (28) and the symmetry of the probab ilities (29) with respect to 1 2could have been deduced from this integrability condition ( 39) instead of minimizing the free energy (27) (we shall proceed in this way in section 3 to take into account the beta-effect). The jet equation (36) has been numerically integrated. Figu re 5 shows a typical stream function and velocity profile in the jets. Figure 6(a) shows h ow the jet width depends on the segregation parameter u. We note that the width of the jet is an increasing function of the mixing and therefore a decreasing function of the ener gy. Figure 6(b) shows the dependence in uof the total non-dimensional energy e(u) =1 2/integraltext+∞ −∞(dφ/dτ )2dτand of the 11maximum non dimensional jet velocity ( dφ/dτ )max. As the jet structure (36) does not depend on the coordinate ta ngent to the jet, we can define the jet entropy ( respectively energy, free energy ) pe r unit length SJet( respectively EJet,FJet). Multiplied by the jet length, these quantities are the firs t order corrections to the entropy ( respectively energy, free energy ). Using th e change of variables (35), we calculate the jet entropy per unit length : SJet=R/integraldisplay+∞ −∞[s(p(τ))−s(p±1)]dτ wheresis defined in (25), and p±1are defined in (29). Using the probability equation (17) and (35) we obtain : SJet=R/integraldisplay+∞ −∞[˜s(φ))−˜s(φ±1))]dτ (40) involving the function ˜ s(φ)≡ln(cosh(C0φ))−C0φtanh(C0φ). Similarly we straightfor- wardly calculate the potential and kinetic energy per unit l ength for the jet : EP Jet=R3 2/integraldisplay+∞ −∞(φ2−φ2 1)dτ EK Jet=R3 2/integraldisplay+∞ −∞/parenleftbiggdφ dτ/parenrightbigg2 dτ We use the integral (38) to calculate dφ/dτ and conclude : EJet=R3/integraldisplay+∞ −∞[˜g(φ)−˜g(φ±1)]dτ, (41) with ˜g(φ) =−ln(cosh(C0φ) C0+φ2. Due to the symmetry of the jets, the jets provide no perturbation to the zero-order circulation, so there is no c irculation term in the jet free energy expression : FJet=−SJet−C0/R2EJet. Then FJet=C0R/integraldisplay+∞ −∞[˜h(φ)−˜h(φ±1)]dτ (42) with˜h(φ) =−φ(φ−tanh(C0φ)). Let us study the sign of FJet. Asφ1verifiesφ1= tanh(C0φ1) and asφ(τ) is an increasing function of τwith limτ→+∞φ(τ) =φ1we conclude that ˜h(φ)−˜h(φ1)>0 for anyτ >0. ThusFJet>0. Using the analogy with usual thermodynamics, the ‘surfac e tension’ is positive. This favors large ‘bubbles’ which minimize the in terfacial length and therefore the corresponding free energy (27). Our initial hypothesis of well separated domains with uniformψis thus supported, as discussed more precisely in next subse ction. 2.3.3 Selection of the sub-domain shape The above analysis has permitted us to determine the areas of subdomains on which the stream function ψtakes the constant values ψ±1, but the subdomains shape is still to be selected. There is an analogy with two phases coexisting in t hermodynamic equilibrium, for 12instance a gas bubble in a liquid medium, for which a classica l thermodynamic argument explains the spherical shape of the bubble by minimizing its free energy, proportional to the bubble area. Our system is isolated rather than in a therm al bath, but the jet energy is small (of order R) with respect to the total energy. Therefore the subdomain i nterior behaves like a thermal bath with respect to the jet, so the usu al argument on free energy minimization applies. We shall now show this more precisely by directly maximizing the total entropy with constraints, taking into account the jet contribution. The jet with length Lhas an entropy SjetLand energy EjetL. Since the total energy E=EA(C) +LEJetis given, the jet has also an indirect influence in the area ene rgy EA. This small energy change δEAresults in a corresponding change in the area entropy δSA=−(C0/R2)δEA, from the condition (2.2.2) of zero first variations. Note th at there is no area change δAsince the jet is symmetric and has therefore no influence in th e condition (12) of a given integral of p(the difference in pwith the case of two uniform patches with boundary at the jet center has zero integral). Therefore, ad ding the direct and indirect contribution of the jet entropy leads to the total entropy S=SA(C0) +/parenleftbiggC0 R2EJet+SJet/parenrightbigg L=SA(C0)−FJetL (43) whereSA(C0) is the zero order, area entropy, obtained in the limit of van ishing jet width2 We deduce from (43) that the maximization of the entropy is ac hieved by minimizing the total free energy FJetL, which we have proved to be positive at the end of previous sub-section. Thus we conclude that the maximum entropy stat e minimizes the jet length, with a given area of the subdomains (31). The subdomains shap e will therefore be a circle or a stripe. More precisely if A1<1/πthe jet forms a circle enclosing the positive constant stream function domain ( the jet bounds a cyclonic vortex ), i f 1/π < A 1<1−1/πtwo straight lines jets form a stripe and if A1>1−1/πthe jet form a circle enclosing the negative constant stream function domain ( the jet bounds an ti-cyclonic vortex ). The different types of solutions can be summarized in a (E,B) d iagram : figure 7. The outer parabola is the maximum energy achievable for a fixed B : E=R2(1−B2)/2. The frontier lines between the straight jets and the circular je ts corresponds to A1= 1/πor A−1= 1/π. It has been calculated using (31) and (32) : E=R2B2(2π−2)/(π−2)2. Note that the maximum accessible energy is in R2, but it has been scaled by the normalization condition (9) on PV levels, so the real energy is not bounded. All this analysis assumes that the vortex size is much larger than the jet width λ, given by figure 6. In other words, the area A1orA−1(31) must be larger than (2 λ)2. This is satisfied on the right side of the dashed line represented in fi gure 7. The dashed line itself corresponds to the equality, and the condition of large vort ex is clearly not satisfied on its left side, for low energy. The position of the dashed line dep ends on the numerical value of R(it has been here numerically computed for R= 0.03), and it gets closer to the origin as R→0. We shall now determine the statistical equilibrium in thi s case of low energy. 2This reasoning to obtain the first order entropy can be precis ed by evaluating explicitly the first order modification of the Lagrange parameter C( let say C1≡C−C0) due to the jet energy, and the first order modification of the Lagrange parameter α( let say α1≡α−α0) due to the jet curvature and computing the first order entropy from its definition (15). We have calcu lated the first order entropy in this way to actually obtain (43). 132.3.4 Axisymmetric vortices We have noted in subsection (2.3.1) that, when for instance B >0, in the limit of small energy (E→0 or equivalently u→ |B|, for fixedBandR), the areaA−1occupied by ψ−1 tends to 1, the whole domain. Therefore, in this limit, the co mplementary area A1tends to 0 and the vortex becomes smaller than the deformation radius , so we can no more neglect the curvature radius of the jet. In this limit u→ |B|, as the vortex has a small area with respect to the total domai n, it is not affected by the boundary conditions, so it can be supp osed axisymmetric. From the general Gibbs states equation (18), we deduce the axisym metric vortex equation : −d2ψ dζ2−1 ζdψ dζ+ψ R2=B−tanh/parenleftbigg α−Cψ R2/parenrightbigg (44) AsRwill be a typical scale of the vortex, we make the change of var iable, ζ=Rr;φ=−α C+ψ R2, (45) leading to the rescaled axisymmetric vortex equation : −d2φ dr2−1 rdφ dr+φ+α C=B+ tanh(Cφ)) (46) From now on, we shall consider the case B >0 (the case B <0 is just the symmetric case of a negative vortex). For this equation to describe a localized vortex, we impose limr→∞φ(r) =φ−1≡ −α/C+ψ−1/R2, whereψ−1is the positive solution of the algebraic equation (20). Sin ce nearly the whole fluid domain is covered by the asymptotic str eam function ψ−1outside the vortex, the condition of zero total circulation /angb∇acketleftq/angb∇acket∇ight=/angb∇acketleftψ/angb∇acket∇ight/R2= 0 imposes that ψ−1≃0 (it is of order R), so thatφ−1=−α/C, and the algebraic equation (20) then leads to: α= arg tanh( B) (47) We can thus eliminate αin (46), leading to an equation depending on two parameters, BandC, d2φ dr2=−1 rdφ dr−tanh(Cφ) +φ−B+arg tanhB C dφ dr(r= 0) = 0andlim r→∞φ(r) =−arg tanhB C(48) where the regularity condition at r= 0 has been included. Let us consider, as in section (2.3.2), the analogy of equation (48) with a one particle mot ion with ’position’ φand ’time’ r. The last four terms on the right-hand side of (46) can be writt en as the derivative −dU/dφ of the potential, U(φ) =ln(cosh(Cφ)) C−φ2 2+/parenleftbigg B−arg tanhB C/parenrightbigg φ, (49) 14(represented in figure 8), while the first term can be interpre ted as a friction effect. Indeed, an integration of (48) leads to: U(φ−1)−U(φ(r= 0)) = −/integraldisplay+∞ 01 r/parenleftbiggdφ dr/parenrightbigg2 dr<0 (50) Thus, in figure 8(a), the hatched area on the right side must be greater than the one on the left (since ( U(φ1)−U(φ−1))>U(φ(r= 0)−U(φ−1)>0). It is clear from figure 8 that this is possible only if φ0<0 andα/C < B , or, using (47), C > α/B = arg tanh B/B. The valueC=α/Bcorresponds to the integrability condition (39) when the eff ect of jet curvature is neglected. This effect is now taken into account by the departure of Cfrom this value, which we shall denote ∆ C≡C−arg tanhB/B. Then ∆C >0 and we expect to recover the results of section 2.3.2 in the limit ∆ C→0. Moreover, we must reach a uniform stream function at large distance, solution of the algebrai c equation(20), so it must have three solutions. We see in figure 8 that the corresponding ∆ Cmust not exceed a maximal value, denoted ∆ Cmax. We can prove that for any B >0 and arg tanh B/B < C < arg tanhB/B+ ∆Cmax, equation (48) has a unique solution. Such solutions have bee n numerically obtained for B= 0.75 and 0<∆C <∆Cmax. Corresponding stream function profiles are shown in figure 9. As ∆Cis decreased from ∆ Cmaxto zero, two stages can be seen in figure 9. First the maximum value for the stream function is increased while the mean width of the vortex remains of the order of R. In a second stage, when ∆ Cgoes to zero, as we are closer to the integrability condition for big vortices (39), φremains longer in the vicinity of φ1so the vortex size increases. Note that the energy monotically increases as ∆ Cis decreased, first by an increase in the vortex maximum stream function and then by an increase in size. Finally the case of a jet with negligible curvature studied i n subsection 2.3.2 is reached when ∆C→0. In conclusion, we have shown that in the limit of small energy , with fixed BandR, the Gibbs states are approximated by axisymmetric vortices, wh ose radial structure depends on the parameter ∆ C, which monotonically decreases from ∆ Cmaxto 0 as energy is increased. 2.3.5 The linear approximation for the Gibbs states The previous discussion of axisymmetric vortices was conce rned with the limit of small energy with fixed Band (small) fixed R. We consider now the limit of small EandB, i.e. the neighborhood of the origin in the phase diagram of fig ure 7. Then from (32), u→ |B|<<1. Figure 6 shows that for |u|<<1, the jet width diverges and therefore the jet tends to develop on the scale of the whole domain, so the ap proximation of a localized jet, or an isolated axisymmetric vortex, falls down. In this limit of small EandB, we can however linearize the Gibbs states equation (18), following the work of Chavanis and Sommeria (1996) for Euler equation. After linearization, solutions are expressed in terms of the eigenmodes of the Lap lacian, and only the first eigenmodes can be entropy maxima. These results are unchanged by the linear deformation term ψ/R2, so the work of Cha- vanis and Sommeria (1996) directly applies here. With the pe riodic boundary conditions, 15the first eigenmode of the Laplacian, a sine function of one of the coordinates, for instance y, is thus selected. This corresponds to the low energy limit o f the two jet configuration shown in figure 7. The next eigenmode, in sin( πx)sin(πy), has the topology of the vortex states. A competition between these two modes is expected in the neighborhood of the origin for small EandB. Note that the range of validity of the linear approximation is limited to a smaller range of parameters than in the Euler cas e, and this range of validity gets smaller and smaller as R→0. The dominant solution with uniform subdomains and interfacial jets relies by contrast on the tanh like relatio n between PV and stream function, and it is genuinely non-linear. 3 The channel case We now consider the channel geometry, which represents a zon al band around a given latitude. It is then natural to introduce a beta-effect, or a m ean sublayer zonal flow ( topography ), with the term h(y) in (1). We shall study the two cases of a linear h(y) ( beta effect and/or uniform velocity for the sublayer flow ) or a quadratic h(y). We shall follow the presentation for the periodic boundary con ditions, stressing only the new features. 3.1 The dynamical system Let us consider the barotropic QG equations (1, 2 and 3) in a ch annelD= [−1 2,1 2]2with the velocity vtangent to the boundary for y=±1 2and 1-periodicity in the zonal direction. Thus we choose for the boundary conditions a constant ψ, denotedψb, the same on the two boundaries y=±1 2. We note that, due to these conditions, the physical momentu m (6) is equal to zero. It is always possible to satisfy this conditio n by a change of reference frame with a zonal velocity Vsuch that it moves with the center of mass of the fluid layer, an d a corresponding change of the deep flow, resulting in an additi onal beta effect h→h+Vy R2. As in section (2), we need to specify the gage constant in the s tream function ψ, and we generalize the integral condition (4) as, /angb∇acketleftψ/angb∇acket∇ight R2− /angb∇acketlefth(y)/angb∇acket∇ight= 0. (51) The total mass /angb∇acketleftψ/angb∇acket∇ightis then constant in time (but not the boundary value ψbin general). With these conditions, the Dirichlet problem (2) has a uniqu e solutionψfor a given PV fieldq. We note that the scale unit is chosen such that the area of Dis equal to 1. The integral of any functions of the potential vorticity (8) is still conserved. Let in particular Γ be the global PV, or circulation : Γ≡ /angb∇acketleftq/angb∇acket∇ight=/integraldisplay D−∆ψd2r=/integraldisplay ∂Dv.dl (52) By contrast with the doubly periodic boundary conditions, t he circulation Γ is not neces- sarily equal to zero. The expression of the energy in terms of the PV (see equation (7)) is 16therefore modified (due to the boundary term in the integrati on by parts) : E=1 2/integraldisplay D/bracketleftbigg (∇ψ)2+ψ2 R2/bracketrightbigg d2r=1 2/integraldisplay D(q+h(y))ψd2r−1 2Γψb (53) Due to the invariance under zonal translation of the system, another conserved quantity exists : M=/integraldisplay Dyqd2r (54) This constant moment fixes the ‘center of mass’ latitude for t he PV field. 3.2 General form of the Gibbs states et us consider the statistical mechanics on a two PV level con figuration : the initial states is made of patches with two levels of potential vorticity, q=a1andq=a−1, occupying respectively the areas Aand (1 −A) inD. We keep the normalization (9) and definition (10) forB. Now, since the circulation Γ is non-zero, the area Ais related to BbyA= (1−B)/2 + Γ/2. The boundary term in the expression of the energy (53) lead s to an obvious change in the energy variation (16). Let γbe the Lagrange multiplier associated with the conservation (54) of the momentum M. Adapting the periodic case computations, we then calculate the probability equation and the Gibbs state equation : p=1−tanh/parenleftBig α′−Cψ R2+γy/parenrightBig 2(55) q=−∆ψ+ψ R2−h(y) =B−tanh/parenleftbigg α′−Cψ R2+γy/parenrightbigg (56) withα′≡α+Cψb/R2. These results generalize (17) and (18) of section 2. In the case of a Gibbs state depending on x, the Lagrange parameter γis related to a zonal propagation of the equilibrium structure. The statis tical theory only predicts a set of equilibria shifted in x, but introducing the result back in the dynamical equation y ields the propagation. Indeed the Gibbs state equation (56) is of t he formq=f(ψ,y), which can be inverted (as it is monotonous in ψ, see Robert and Sommeria (1991)) to yield: ψ=g(q) +R2γy C(57) wheregis a function of the potential vorticity. From this relation we calculate the velocity using (3) : v=R2γ Cˆx−g′(q)ˆz∧ ∇q. As PV is advected ( equation (1) ) we obtain : ∂q ∂t+R2γ C∂q ∂x= 0 (58) Thus the PV field is invariant in a frame propagating with the z onal speed Vsr=R2γ C. 173.3 The limit of small Rossby deformation radius. In this sub-section we propose to analyze the Gibbs state equ ation (56) in the limit of small deformation radius ( R << 1 ). The main difference with the periodic case resides in the latudinally depending topography h(y), resulting in two effects. Firstly the subdomains of uniform PV are no more strictly uniform, and contain a weak zo nal flow. Secondly the jet curvature is no longer constant in general, but depends o n the local topography. Using the same boundary layer approximation as in the periodic cas e, the Laplacian term in the Gibbs state equation (56) will be neglected, except possibl y in an interior jet and in the vicinity of the boundaries y=±1 2( boundary jets ). Outside such jets, (56) reduces to an algebraic equation, li ke previously : ψ R2−h(y) =B−tanh/parenleftbigg α′−Cψ R2+γy/parenrightbigg . (59) This is like (20), replacing the constants BbyB+h(y) andαbyα′+γy. The three solutions can be still visualized by figure 8, but the positio n of the straight line with respect to the tanh curve now depends on y, due to the terms h(y) andγy. We assume that this dependence is linear in yor varies on scales much larger than Rso that the Laplacian term remains indeed negligible. The zero order Lagrange paramet ersα′,C,γ, involved in this expression, can be obtained by directly maximizing the entr opy by the same method as in section (2.3.1). A relation between the jet curvature and to pography is then obtained at first order. This approach is developed in appendix (A). However it is more simple to proceed differently : we start fro m the jet equation and show that its integrability condition provides the relatio n between the jet curvature and topography. To catch this effect, we take into account the rad ius of curvature of the jet, denotedr, like in section 2.3.4, but state that ris constant across the jet, assumed thin. From the Gibbs state equation (56), using the boundary layer approximation, we thus obtain the jet equation in term of the transverse coordinate ζ: −d2ψ dζ2−ǫ1 rdψ dζ+ψ R2−h(y) =B−tanh/parenleftbigg α′−Cψ R2+γy/parenrightbigg . (60) We have introduced ǫ=±1 to account for the direction of curvature (keeping r >0). We defineǫ= 1 (respectively -1) if the curvature of the jet is such that φ1( respectively φ−1) is in the inner part of the jet. Note that, in the case of a vorte x, as in our notations ψis proportional to the opposite of the pressure the case ǫ= 1 ( respǫ=−1 ) corresponds to a cyclone ( resp an anticyclone ). The algebraic equation (59) depends on three Lagrange param eters, instead of two for the periodic case of previous section, but we have three addi tional constraints, the condition thatψhas the same value at the two boundaries, the circulation con straint (52) and the momentum constraint (54). This will be achieved in general b y boundary jets. Let us first study the interior jets. To study the interior jet, we make the change of variables : τ≡ζ R;φ≡ −α′+γy C+ψ R2(61) 18We assume the variations of yin the jet width are negligible ( R << scale of variation of h(y) ), so that yis treated as a constant. Then we obtain the jet equation : −d2φ dτ2−ǫR rdφ dτ+φ+α′+γy C=B+h(y) + tanh(Cφ) (62) withφ→φ±1forτ→ ±∞ , where again φ±1corresponds to the solutions of the algebraic equation (59), rescaled as φ+α′ C−B+γy C−h(y) = tanh(Cφ). (63) Let us consider, as in section (2.3.4), the analogy of the equ ation (62) with the motion equation of a particle in the potential : U(φ) =ln cosh(Cφ) C−φ2 2+/parenleftbigg B+h(y)−α′+γy C/parenrightbigg φ. (64) Like in section 2.3.4., integration of (62) from −∞to +∞imposes the integrability condition : U(φ1)−U(φ−1) =ǫR r/integraldisplay+∞ −∞/parenleftbiggdφ dτ/parenrightbigg2 dτ (65) The second term of the l.h.s. of equation (62) can be interpre ted as a friction term: if ǫ= 1, the ’particle’ starting from rest at φ1can reach a state of rest at φ−1only if the difference of ’potential’ corresponds to the energy loss (65) by fricti on (ifǫ=−1 the same is true in the reversed direction). As in the periodic case we have made the thin jet assumption R << r , so that the friction term ( rhs of (65)) is a correction of order R/r:U(φ1)−U(φ−1) =O(R/r). We first neglect it to get the zero order results, so we write U(φ) =U(φ−1). Therefore the two hatched areas in figure 8 must be equal, like in figure 2. Due to the symmetry of the tanh function, this clearly implies that the central sol ution of the rescaled algebraic equation (63) must be φ0= 0, so that α′ 0/C0−B+γy/C 0−h(y) = 0 (denoting the zero order Lagrange parameters by the index 0). This is possible a t different latitudes yonly if γy/C 0−h(y) = 0, or is of order R(so that it can be neglected at zero order). Then the integrability condition becomes α′ 0=C0B. (66) Furthermore φ±1are symmetric with respect to 0, of the form φ±1=±u, determined by equation (33), like in section 2.3. This parameter uis again related to the energy by (32). Finally, the terms γy/C 0−h(y) = 0 and the curvature term disappears in the jet equation (62), which therefore reduces to (36), discussed in section 2.3.2. The first order solution outside the jet is obtained as a small correctionδφ±1to the zero order solutions ±u, with also a small correction α1andC1to the parameters α′ 0/C0 andC0, φ±1=±u+δφ±1(y),α′ C=α′ 0 C0+α1, C=C0+C1 (67) 19From (65), we deduce that U(φ1)−U(φ−1) has the sign of ǫ. As may be seen in figure 2, whenα1is positive, the line φ+α′ Cmoves upward, so that U(φ1)< U(φ−1). Thusα1 has the sign opposite to the sign of U(φ1)−U(φ−1) ; and we conclude that −ǫα1is always positive. Introducing this expansion (67) in the algebraic equation (63), using the zero order results (33) and (66), we obtain : C0δφ±1(y) =−α1C0+C1[B±C0u(1−u2)]−γy+C0h(y) 1−C0(1−u2)(68) Coming back to the stream function ψ, using (61), we deduce the corresponding velocity vby differentiation with respect to y. This velocity outside the jet is zonal, along the unit vector ˆx, and verifies: v=R2 dh(y) dy−γ(1−u2) 1−C0(1−u2) ˆx (69) It is therefore a constant plus a term proportional to the loc al beta-effectdh dy. Notice that the corresponding sheardv dyis stronger than the deep shear dψ d/dy2=R2d2h/dy2by the factor [1 −C0(1−u2)]−1>1. The integrability condition (65) now provides the curvat ure of the jet. We can approximate the r.h.s. of this relation, of orderR, by the zero order jet profile (36), denoting : e(u)≡1 2/integraldisplay+∞ −∞/parenleftbiggdφ dτ/parenrightbigg2 dτ. (70) ( see figure 6(b) ). The l.h.s. can be expanded, using (67). We fi rst expand the ex- pression (64) of the potential, U(φ) =U0(φ) +C1/C0[φtanh(C0φ)−log cosh(C0φ)/C0] + [h(y)−γy/C 0−α1]φ. We can approximate φ≃ ±uin the correction terms, and ex- pandU0(φ) =U0(±u) +dU0/dφ(±u). The zero order equilibrium condition imposes that dU0/dφ(±u) = 0, so that (65) becomes ǫu/parenleftbigg h(y)−γy C0−α1/parenrightbigg =e(u)R r(71) This equation (71), expresses the dependence in latitude yof the curvature radius rof the curve on which the jet is centered, thus defining the shape of t he sub-domain interface as a function of the topography. Without topography and for γ= 0, we get a constant jet curvature. The same result was obtained in sub-section 2.3.3 by a different argument of f ree energy minimization. The parameteru, related to the energy by (32) and to C0by (33), quantifies the strength of the jet. By contrast, the vortex area is determined by the con straint on PV patch area (parameter B), but it is also related to the jet curvature, proportional b y (71) to the small shiftα1in chemical potential and temperature. Likewise the equili brium temperature at a liquid-gas interface slightly depends on the bubble curvat ure, due to capillary effects. As explained in the end of section (3.2) the parameter γis linked to the zonal propaga- tion speed of the structure. The term γyin (71), combined with a usual beta effect (linear 20topography term h(y)), leads to an oscillation with latitude yof the jet curvature 1 /r, i.e. a meandering jet. Another possibility is an exact compensat ion of the beta-effect by the γy term, leading to a propagating circular vortex, and the sele ction between these two alter- natives is discussed in next sub-section. An oval shaped, zo nally elongated vortex, such as on Jupiter, is obtained when this compensation occurs, but w ith an additional quadratic topography h(y). Indeed, to get a zonally elongated vortex, supposed latit udinally centered in zero, the radius of curvature of the jet must decrease for y>0 and increase for y<0. As a consequence, we deduce from (71) that the topography must b e extremal at the latitude on which the vortex is centered (it actually admits a maximum in the cyclonic case and a minimum in the anticyclonic case ). Moreover, we deduce from (72) that the surrounding flow must have a zero velocity at the latitude on which the vort ex is centered and that the shear is cyclonic when the vortex is a cyclone and anticyc lonic when the vortex is an anticyclone. More generally, the curvature can be related t o the zonal velocity outside the jet, eliminating the topography between (72) and (71), v=R2/parenleftbiggγ C0+ǫe(u) u(1−C0(1−u2))d dy/parenleftbiggR r/parenrightbigg/parenrightbigg ˆx (72) Let us recall our approximations. Writing down the jet equat ion (60), when making the boundary layer approximation, we have assumed R << r . We also assumed that in the jet width the topography can be considered as a constant. If 1/√ais a typical length scale for the topography variation this gives aR2<<1. Moreover we have assumed that the effective topography effect h(y)−γy/C 0remains small along the jet. If LVdenotes the jet extension ( for example a vortex latitudinal size ) this a pproximation is valid as long as aL2 V<<1. 3.3.1 Beta effect or linear topography. Leth(y)≡ −βyin the following of this section. βmay mimic the beta-effect or a uniform velocity in the sublayer ( but we will refer it as the beta-effe ct ). A first class of equilibrium states corresponds to a single so lutionψ(y) of the algebraic equation (59). This determines a smooth zonal flow, with poss ibly intense jets at the boundaries y=±1 2. The solution depends on the unknown parameters C,α′andγ, which are indirectly determined by the energy E, momentum M, and the condition /angb∇acketleftψ/angb∇acket∇ight= 0. The limit of small energy corresponds to C→ ∞, for which we can neglect the term ψ/R2on the left hand side of (59), which then reduces to ψ=R2/C[arg tanh(βy−B) +γy+α′]. This corresponds indeed to arbitrarily small values of ψ(small energy) as C→ ∞. When the particular energy value E=R2β2/24 is reached, a uniform PV is possible, withψ/R2=−βy. Then PV mixing is complete, which clearly maximizes the mix ing entropy. In this case, γ=−Cβ, so thatγycancels the term Cψ/R2in (59). Physically, the uniform westward zonal velocity, vm=−R2β (73) tilts the free surface with uniform slope by the geostrophic balance, and the corresponding topographic beta-effect exactly balances the imposed beta- effect. 21For a still higher energy, a first possibility is that again γ=−Cβ (74) so that the beta effect exactly balanced by the γyterm in the jet equation of previous subsection. This cancellation is directly obtained in the g eneral Gibbs state equations (55) and (56). Indeed the modified stream function ψ′=ψ+R2βysatisfies the same equations as in the doubly-periodic case. Therefore in the limit of sma llR, the Gibbs states are made of subdomains with uniform ψ′(uniform PV), separated by straight zonal jets or circular vortices. However jets or a vortex persist in this sea of uniform PV due t o the constraint of energy conservation. The vortex moves westward at the same velocit yvm, according to (58) , so they are just entrained by the background flow, without relat ive propagation (this can be physically understood by the cancellation of the beta-effec t). The selection of the subdomain areas and PV values is given li ke in the periodic case of section 2, just replacing ψbyψ′=ψ−βR2y. Therefore we get again probabilities p±1= (1±u) in the two subdomains with respective areas A±1given by (31), and stream function, ψ±1=R2(B±u)−R2βy (75) From this relation, we can calculate the energy E=1 2(ψ2 −1A−1+ψ2 1A1)/R2, so the energy condition (32) then becomes E=R2 2(u2−B2) +R2β2 24(76) Therefore these solutions with canceled beta effect can be ob tained only beyond a minimum energyR2β2/24, corresponding to the potential energy of the surface til ting associated with the drift velocity vm. Then the excess energy will control the organization in two uniform PV areas. The shape of these subdomains can be obtained again by minimi zation of the jet free energy. However, unlike in the periodic case, jets occur at t he boundaries y=±1 2as well as at subdomain interfaces. Indeed, such boundary jets are i n general necessary to satisfy /angb∇acketleftv/angb∇acket∇ight=0, or equivalently that the stream function ψbmust be equal at the two boundaries y=±1 2. In particular, the solutions (75) necessarily involve a st ream function difference (or mass flux) −R2βassociated with the drift velocity vm. This stream function difference must be compensated by boundary eastward jets with opposite total mass flux. We show in Appendix B that for two PV levels with similar initial areas a single eastward jet, separating two regions of uniform PV and weak westward drift, is the sele cted state (instead of two opposite jets in the periodic case). In the case of a strong PV level with a small initial area, the system organizes in a circular vortex like in the periodi c case. In the limit u→ |B|, as one of the areas A±1goes to zero, the jet approximation falls down. The correspo nding analysis of axisymmetric vortices and of the linear approxi mation for the Gibbs states, as performed in section 2, is still valid here. Up to now we have ignored the constraint of the momentum M(54). This constraint imposes the latitude y0of the equilibrium structure ( a circular patch or a zonal ban d with 22uniform PV ). For instance in the case B>0, for which A1>A−1(as seen from (31)), we definey0≡/integraltext A1yd2r/A1. Then M≡/integraldisplay Dyqd2r=/integraldisplay A1y(B+u)d2r+/integraldisplay A−1y(B−u)d2r= 2uy0A1 (77) We thus deduce the latitudinal position of the equilibrium s tructure : y0=M 2uA1(78) In the case of a single eastward jet, the subdomain position h as been already fixed by the area (y0=A1/2). Then the only possibility to satisfy a moment Mdifferent from uA2 1 is that the jet oscillates in latitude with some amplitude Λ ( thenM−uA2 1≃uΛ2). This is possible if γ/negationslash=−βCaccording to (71), which becomes 1 r=b(y−y0) (79) whereb≡ −(u(γ+βC))/(2Ce(u)R)<0 andy0≡α1/b. This equation clearly leads to a jet oscillating around the mean latitude y0( as the curvature ris positive for y <y 0and negative for y > y 0; recall that the curvature is by definition positive when pos itive PV is in the inner part of the jet ). Note that this oscillation pr opagates eastward at speed R2γ C( given by (58) ). Since b <0,γ C>−β, this speed is eastward with respect to the background drift vm(73). 3.3.2 Quadratic sublayer topography As explained in section (3.3), in the limit of small Rossby de formation radius R, the Gibbs state equation has solutions consisting of a vortex bounded by a strong jet on the scale of R. This corresponds to the case of an initial patch with strong PV and small area ( the asymmetry parameter Bis sufficiently large ) with an energy sufficiently strong to get a structure of closed jet ( see figure 7). In the presence of a mod erate topography h(y), this internal jet is no more circular but its radius of curvature r<<R depends on yaccording to (71). We have seen in previous subsection that a linear top ographyh(y) =−βyleads to jets oscillating ( or to circular jets when γ=−Cβ). We shall discuss here how a quadratic term inh(y) modifies the shape of closed jets. We therefore assume a topo graphyh(y) of the form : h(y)≡ay2+by (80) This corresponds to a uniform deep zonal shear, with velocit yvd=R2d(h−βy)/dy= 2aR2y+b−β. We focus our attention on vortex solutions, seeking close c urves solutions of equation (71). The vortices will be typically oval shaped as the ones seen on Jupiter. We then study how this shape ( for instance the ratio of the great axis of the oval to the small one ) depends on the topography ( sublayer flow ) and on the jet p arameters. Application of these results to Great Red Spot observations will be discu ssed in next section. To make equation (71) more explicit, let sbe a curvilinear parameterization of our curve, T(s) the tangent unit vector to the curve and θ(s) the angular function of the curve 23defined by T(s) = (cosθ(s),sinθ(s)) for anys. Then the radius of curvature rof the curve is linked to θ(s) by 1/r=dθ/ds and (71) yields the differential equations : /braceleftBigg dθ dS=−dY2+ 1 dY dS= sinθ(S)(81) anddX dS= cosθ(S) (82) withc′X=x,c′Y=y−y0,c′S=s; whered= (e2(u)R2a)/(ǫα3 1u2),y0≡e(u)R(C0b− γ)/(ǫα1C0u). The space coordinates X,YandSare here non dimensional and have been obtained by dividing the real coordinates by the scale c′≡e(u)R/(ǫα1u). Note that as explained in section (3.3) ǫα1>0, so thatc′>0. We further assume that a >0, so that d>0. We first note that the two variables θandYare independent of X. We will therefore consider the system formed by the two first differential equat ions (81). It is easily verified that this system is Hamiltonian, with θandYthe two conjugate variables and H≡cosθ−dY3 3+Y (83) the Hamiltonian. Thus His constant on the solution curves. We look for vortex soluti ons of our problem (81 and 82). Thus we require θto be a monotonic function of S. Moreover the curves must close, that is XandYmust be periodic. For symmetry reasons, it is easily verified that the solutions of (82,81) with initial conditio nsθ(0) =π 2,Y(0) = 0 (H= 0 ) and someX(0) are periodic. We prove in appendix (C) that these initial conditions are the only ones leading to closed curves. We also prove that the sol utions of (81 and 82) when d>dmax≡4 9does not define θas a monotonic function of S. They contain double points and thus are not possible solution for our problem. Once give n these initial conditions, we can easily prove that the structure has both a zonal symmetry axis and a latitudinal one passing through y0. To study the shape of the jets, we numerically solve equation s (81,82) with initial conditions : θ(0) =π 2, y(0) = 0. We obtain closed curves with oval shapes, as shown in figure 10. In figure 11 we have represented the width, the lengt h and the aspect ratio of these vortices versus the parameter d. Whendtends to4 9, the vortex width tends to a maximum value : wmax=3 2whereas the length diverges. In this limit, the vortices are thus very elongated. 4 Application to the Jupiter’s Great Red Spot and Oval BC In previous sections, we have found maximum entropy states w ith the following properties : •The fluid domain is partitioned in two subdomains with weak ve locity, separated by jets whose width scales as the Rossby deformation radius. A s trong initial PV level occupying a small area mixes in a subdomain with the form of a v ortex bounded by an annular jet. 24•In the presence of a parabolic topography h(y) (due to a sublayer zonal flow with uniform shear), outside the jet, exists a zonal flow (69) with uniform shear. The velocity at the latitude of the vortex center vanishes in the reference frame of the vortex. •The curvature of the jet is linked to the topography by (71). F or the parabolic topography, solutions are oval shaped vortices, symmetric in latitude and longitude. These properties of our solutions are the main qualitative p roperties of the Jovian vortices. Moreover, would this description be correct, it w ould predict that the topography has an extremum at the center of the vortex. Dowling and Ingersoll (1989) derive the bottom topography, using the GRS and Oval BC velocity fields obtained from cloud motion. They analyze t he results in the frame of a 1-1/2 shallow water model (SW), with an active shallow layer floating on a much deeper layer. This deep layer is in steady zonal motion which acts li ke a topography h2. The SW topography h2is defined by fvd=−1/Rl(λ)∂(gh2)/∂λwherevdis the deep layer flow andRl(λ) is the latitudinal radius of curvature of Jupiter. Dowling and Ingersoll (1989) have deduced this SW topography h2by assuming the conservation of the shallow water potential vorticity ( ω+f)/h1(h1is the upper layer thickness ) along the streamlines of the observed steady vortex flow. The vorticity ωis deduced from the measured velocity field, and the planetary vorticity fis known, so that the variation of halong each streamline is deduced. The pressure field is then obtained from the Bernoul li relation and the hydrostatic balance, leading to the field h2. The result depends on the radius of deformation of Rossby ( R∗=gh0/f0, whereh0is the mean upper layer height and f0the mean Coriolis parameter ), a free parameter in this analysis. Three test values have bee n chosen,R∗ 1= 1700,R∗ 2= 2200 andR∗ 3= 2600 km for the GRS and R∗ 1= 1100,R∗ 2= 1600 and R∗ 3= 2000 km for the Oval BC ( we denote by a star superscript the physical parameters, to distinguish them from the non-dimensional quantities used earlier ). The height h2under each vortex has been found to depend only on latitude, and has been fitted as a quartic the planetographic latitude λ: gh2=A0+A1λ+A2λ2+A3λ3+A4λ4. (84) The values obtained for the coefficients Aiin the vortex reference frame, for each of the vortices and for each of the values R∗ 1,R∗ 2,R∗ 3are reported in table 1 of Dowling and Ingersoll (1989). Our model is the QG limit of this shallow water system. Starti ng from (SW) equations, we can derive the QG equations (1) by assuming the geostrophi c balance and weak free surface deformation in comparison with the mean layer thick nessh0. The validity of this QG approximation has been discussed by Dowling and Ingersol l (1989) and was found reasonably good as a first approach, although not accurate. W e furthermore use the beta- plane approximation, linearizing the planetary vorticity around a reference latitude λ0( λ0is taken to be −230for the GRS and −33.50for the Oval BC). Therefore we write f=f0+βy, withf0= 2Ωsinλ 0andβ≡2Ω cosλ0/rz(λ0) ( Ω is the planetary angular speed of rotation : 2 π/Ω = 9 h 55 mn 29.7 s and rz(λ0) is the zonal planetary radius, which slightly depends on the latitude λ0, due to the ellipsoidal planetary shape, see formula (4) of Dowling and Ingersoll (1989) ) . We then obtain the QG poten tial vorticity (2) with the 25QG topography h∗(y∗) linked to the SW topography (84) by: h∗(y∗) =gh2 f0R∗2+βy∗(85) We have computed the QG topography (85) using results of Dowl ing and Ingersoll (1989) for the SW topography (84) for the three values of the R ossby deformation radius R∗ 1,R∗ 2andR∗ 3, for the GRS and for the Oval BC. The result in figure 12 shows th at, for both the GRS and the Oval BC, the QG topography has an extremum at a latitude which is nearly the center of the vortex. As far as we know, this fact has not been noticed in the literature. This result is in agreement with the prediction s of our model. We note moreover that the two extrema of the topography are minima, thus our mo del predicts anticyclonic shears around the GRS and the Oval BC, as observed. Figure 85 s hows a comparison of the QG topography derived from Dowling results with a quadra tic approximation, in the caseR∗=R∗ 2= 2200 km. This shows that the quadratic approximation h∗(y∗) =a∗y∗2is a good approximation on the latitudinal extension of the GRS . This also provides values of the parameter a∗(80) :a∗= 9.2 10−13,a∗= 7.2 10−13anda∗= 6.4 10−13km−2s−1for R∗=R∗ 1,R∗ 2andR∗ 3respectively. Let us deduce the corresponding non-dimensional parameter s. First, the PV levels were normalized by (9), so our time unit T⋆will depend on the real PV level difference : (a⋆ 1−a⋆ −1)/2≡1 T⋆. The other parameters are the Rossby deformation radius R⋆, the segregation parameter uand the topography coefficient a⋆(80). We will consider R⋆as a free parameter and use the following data from GRS observ ation : •The jet width l⋆.Let us define the width of the jet l⋆as the width on which the jet velocity is greater than one half of the maximum velocity . We use velocity mea- surement within the GRS of Mitchell et al (1981). They have us ed small clouds as tracers to measure velocities, and have observed that the ve locity is nearly tangential to ellipses. Using a grid of concentric ellipses of constant eccentricity, the velocities have been plotted with respect to the semi-major axe Aof the ellipse on which the measurement point lies. Results have been fitted by a quartic inAand may be seen in figure 13. Using these results, assuming that the velocity profile in the jet is sym- metric with respect to its maximum, we choose as jet width l⋆= 5.6 103km. In our model, the normalized jet width l=l⋆/R⋆can be computed from the jet equation (34) as shown in figure 6(a). This determines the parameter ufrom the parameter R⋆. The corresponding theoretical jet velocity can be compare d to observations in figure 13, for R⋆= 2500 km (the shape is not very sensitive to this parameter). The computed results for uversusR⋆is shown in figure 15. •The maximum jet velocity v⋆ max.We will use the value v⋆ max= 110 ms−1( Mitchell et al 1981). Using (35) and giving real dimension gives : v⋆ max=R⋆ T⋆dφ dτ|max(u). (86) 26dφ/dτ |max(u) has been obtained by solving the non-dimensional jet equat ion (34), and is shown in figure 6(b). This now determine T⋆fromR⋆. •The velocity shear surrounding the vortex. The ambient zonal shear measured at the latitude of the GRS from Limaye et al. (1986) is σ⋆= 1.5e−5 s−1. Using (69) in its dimensional form, for a quadratic topography gives : a⋆=σ⋆ 2R⋆/parenleftbigg 1−C0 cosh2(C0u)/parenrightbigg (87) This permits to compute a⋆as a function of R⋆(sinceuhas been determined, as well asC0, related to uby (33) ). The computed results for a⋆andT⋆versusR⋆are shown in figure 14. This shows that our determination of the topography is in agreement with the QG topography deduced from the shallow water model of Dowling and Ingersoll (1989) within a factor of two. The corresponding PV level difference a⋆ 1−a⋆ −1is comparable to the planetary vorticity f0at the latitude of the center of the GRS when R≃2400 km. For this value of R∗, figure 15 shows that uis very close to 1. Furthermore, as the GRS area is very small c ompared to the global area of a latitudinal band centered around the GRS , the non dimensional area occupied by the positive PV is very close to 1. Using this area expression (31) we conclude thatBis very close to 1. Using the definition of B(10), we conclude that a⋆ 1≃0 and a⋆ −1≃ −f0. As discussed below a forcing mechanism by convective plume s incoming from the sublayer is expected to yield this result. The shape of the jet depends on the parameter d=e2(u) ǫα3 1u2a⋆R⋆2 a⋆ 1−a⋆ −1and on the length scale c′∗≡e(u)R∗ ǫα1u. We can determine these two parameters from the previously d etermined values ofa∗,T∗andu, and from the observed half width of the Great Red Spot : y⋆ max= 4900km. This permits to calculate c′∗,α1anddversusR∗. Figure 16 show dversusR⋆. The dot line represents the critical value d=4 9beside which a vortex solution exists. The ratio of the length to the width of the GRS is approximatively 2, which wou ld correspond to d= 0.441 ( figure 11 ) ; this is very close to the critical value4 9. From figure 16, our model predicts that the Rossby deformation radius is R⋆= 1800 km. Figure 10 shows the actual shape of the vortex for d= 0.441. However, in the jet shape analyze, to obtain (81,82) we have s upposeda⋆L⋆2 V<<(a⋆ 1− a⋆ −1)/2 whereL⋆ Vis the maximal latitudinal extension of the vortex ( the topo graphy part of the PV remains negligible with respect to the PV ). For the v alue ofR⋆calculated above, we finda⋆y⋆ max2= 3.910−5s−1whereasa∗ 1−a∗ −1= 1.310−4s−1. We are thus at the limit of validity of our assumption. Now we can reverse the procedure and propose a predictive mod el of the Great Red Spot. Assume a steady deep zonal flow with uniform shear, vd= 2a∗R∗2(y∗−y∗ I), vanishing at an origin yIdepending on our reference frame (which we shall choose in or der to cancel the vortex drift). This flow is presumably generated by deep t hermal convection but we are concerned here only with the dynamics of the upper layer, assumed stably stratified (due to cooling by radiative effects). We model this stratifie d upper layer as a shallow layer 27with radius of deformation R⋆≃2400 km. This layer is submitted to a total beta effect or ’topography’ h(y) =ay2−(β+ 2ayI)y. Assume that PV spots with value −f0, occupying small area proportion are randomly generated in this layer. This would be the result of intense i ncoming thermal plumes, as recently discussed by Ingersoll et al (2000), : conservatio n of the absolute angular momen- tum during the radial expansion leads to strong decrease of t he local absolute vorticity, which comes close to zero. This means that in the planetary re ference frame, a local vor- ticity patch with value −f0is created. The opposite vorticity is globally created by th e subducting flow, but it is close to 0 due the much larger area. T his gives our time unit (2f0)−1andB= 1−2A. The outcome of random PV mixing with the constraint of the con servation laws is then a zonal velocity (69) in the observed upper layer and a vortex with area (31) with velocity profile shown by the dot curve in fig 14. The vortex moves with th e upper velocity at y= 0, it drifts with respect to the deep layer at velocity (73) so th at the beta effect is suppressed. The shape is an oval symmetric in xandy, with aspect ratio computed from the shape parameterd( figure 11 ). Note that a slightly larger area , or stronger energy could le ad to very elongated vortices. Then argument of free energy minimization show that this wou ld lead to a single eastward zonal jet. This may explain the jet observed in the Northern h emisphere of Jupiter at the same latitude as the GRS. For smaller Jovian vortices such as the Oval BC, the size of th e vortices is comparable with the Rossby deformation radius. Thus such vortices may b e described as done for axisymmetric vortices ( section 2.3.4 ). This explain why su ch vortices do not have a quiescent core as the GRS. Let us describe dark brown cyclonic spots ( ’Barges’ ) at 14 N o n Jupiter. Their first interest is to stress that cyclonic vortices embedded in cyc lonic shear exist on Jupiter. Let us go further. The greater of these barges is studied from Voy ager observations by Hatzes at al (1981). The meridional velocities measured at the lati tude of the center of the barge ( Hatzes et al (1981), figure 7 ) show a boundary jet organizati on around the perimeter of the barge ( vmax= 25 ms−1), see figure 1(b). The surrounding shear is such that the shea r velocity at the maximum latitude of the barge is the same as th e maximum jet velocity. Thus our approximation aL2 V<<1 is not good. We can however explain the elongated shape, similar to figure 10(b) obtained for dvery close to dmax. We conclude that the Gibbs state equation (56) derived from m aximization of entropy of the QG model (1) is in the limit of small Rossby deformation ra dius a model that explains the main qualitative features of Jovian vortices. The stati stical mechanics itself explains the organization of a turbulent flow in coherent structures. 5 Conclusion Our first result is to provide a general explanation for the em ergence and robustness of intense jets in atmospheric or oceanic turbulent flows. In th e absence of topography or beta-effect turbulence mixes potential vorticity in subdom ains, and such jets occur at the 28interface of these subdomains, with a width of the order of th e deformation radius. From a thermodynamic point of view, this is like coexistence of tw o phases. Indeed the vortex interaction becomes short ranged in the limit of small defor mation radius, and statistical mechanics leads to a thermodynamic equilibrium between two ’phases’, with different con- centrations of the 2 Potential Vorticity levels. Another ap proach leading to the same result is to consider the general partial differential equation (18 ) characterizing the equilibrium states. This equation reduces to the algebraic equation (20 ) in the limit of small deforma- tion radius. The two uniform subdomains correspond to two so lutionsψ−1andψ1of this equation. At the interface of these subdomains, the general pde reduces to the equation (36), whose solution determines the jet profile. In addition , a solvability condition of this equation confirms the relation of equilibrium between the tw o ’phases’, which was obtained in the thermodynamic approach. All our results have been obtained for a 2 potential vorticit y level case, but cases with more levels would lead to qualitatively similar results, al though the quantitative analysis would be more involved due to the additional parameters. In the presence of beta effect or topography, for low energy, p urely zonal flows with gentle variation in latitude are obtained. A critical energ y is the energy of the state where the zonal flow just compensates the beta-effect. For this stat e the PV is strictly uniform in the whole domain. For greater energy, two well mixed domain s eparated by jets appears, as in the without topography case. However the PV is no more st rictly uniform in the well mixed subdomain : a zonal flow exists. Also the jet curvature d epends on latitude. With ordinary beta effect this yields an intense eastward jet, pur ely zonal or wavy depending on the constraint on the momentum M. With the quadratic beta-effect generated by a deep shear, this can produce an oval shaped vortex. This vortex th en drift so as to compensate the beta-effect. In other words, in the vortex reference fram e the equivalent topography h(y) admits an extremum, and this is in agreement with the data of Dowling and Ingersoll (1989). Our quasi analytical approach therefore explains most of th e basic features of the Great Red Spot and other Jovian vortices. It can be developed into a more accurate predictive model along the following lines. First the approximation R<<r of thin jet is convenient for a qualitative understanding but is only marginally sati sfied. However this limitation can be overcomed by numerical determination of the equilibr ium state equation (18) by methods like used by Turkington and Whitaker (1995) or using relaxation equations toward the maximum entropy state as described by Robert et Sommeria (1992). Furthermore, extension to the more general shallow water model is desirab le, as the Rossby number ( ≃0.36 where it a maximal ) is not very small. This can be formally a chieved ( in preparation ). Finally the results rely on an assumption of ergodicity, or c omplete potential vorticity mixing consistent with the constraints on the conservation laws. Various numerical and laboratory experiments in the case of Euler equations ( see e .g. Brands et al. 1989 ) indicate that mixing may not be global but be restricted to ac tive regions. Organization into local vortices, rather that at the scale of the whole dom ain, is more likely with a small radius of deformation, as vortex interactions leadin g to coalescence are then screened. This is observed for instance in the numerical computations of Kukharkin Orszag and Yakhot (1995). By contrast, the zonal shear here promote vor tex encounters ( as observed 29in Voyager data ) and we expect a much better relaxation towar d the global statistical equilibrium, which involves always a single vortex in a give n shear zone ( as it minimizes the interfacial free energy ). 6 Acknowledgments The authors thank R. Robert for collaboration on statistica l mechanics approach and for useful comments on the present work. A Determination of the Gibbs state by direct entropy max- imization, in the presence of a topography ( beta-effect ) . In section (3.3), we studied the limit of small Rossby deform ation radius in the Gibbs state equation (56) by considering the jet equation (60) and its in tegrability condition (65). We deduced that the Gibbs states are composed of subdomains in w hichψverifies the algebraic equation (59) separated by an interfacial jet whose curvatu re verifies (71). The aim of this annex is to prove that these results can be obtained by direct ly maximizing the entropy, adapting the method used in section (2.3.1). Let us make the following assumptions : 1. In the limit of small Rossby deformation radius, the proba bilitypof finding the PV levela1takes two values p±1(y), depending only on y. We are looking for vortex solutions. The vortex shape is described by the length l(y) on which the probability ptakes the value p−1(y) ( See figure 17 ). 2. The two subdomains where ptake the two values p±1(y) are separated by a jet. The probabilities p±1(y) are supposed to be close to the their values without topogra phy p±1=±u, such that the free energy per unit length of the jet is well ap proximated by the one calculated without topography (42). If LVdenotes the vortex size, 1 /√a a typical length on which topography varies, we will show thi s approximation to be valid as soon as aL2 V<<1. 3. The boundary conditions can be relaxed ; that is no boundar y term appears in the variation of the free energy at the order considered here. ( S ee discussion concerning boundary jets in section (3.3). Given these hypothesis, the Gibbs states is described by the 3 functions p±1(y) and l(y). We will determine them by maximizing the entropy S(15) under the 3 constraints : energy (53), mass conservation (51) and momentum (54). A ne cessary condition for a solution to this variational problem is the existence of 3 La grange parameters C,αandγ such that the first variations of the free energy : F≡ −S−C0 R2E+α0/angb∇acketleftψ/angb∇acket∇ight R2+γM (88) 30vanish. Using (15),(53),(51) and (54) ; (2) and (11) and the a bove hypothesis ( see figure 17 ), a direct calculation shows that the free energy (88) is u p to a constant : F=/integraldisplayymax ymin[f(p1(y),y)(1−l(y)) +f(p−1(y),y)l(y)]dy+/integraldisplayymin −1 2f(p1(y),y)dy+ /integraldisplay1 2 ymaxf(p1(y),y)dy+LFJet(u) withf(p,y)≡(plogp+ (1−p)log(1 −p))−2C0/parenleftbigg p−1 2/parenrightbigg2 −2(C0B−α0)/parenleftbigg p−1 2/parenrightbigg − 2(C0h(y)−γy)/parenleftbigg p−1 2/parenrightbigg (89) and whereLis the jet length, FJet(u) is the jet free energy per unit length (42), calculated without topography. Considering first variations of the free energy (89) under va riations ofp1(y) ( respp−1(y) ) proves that ∂f/∂p (p1(y),y) = 0 and that ∂f/∂p (p−1(y),y) = 0. A direct calculation shows that : 2/parenleftbigg p±1−1 2/parenrightbigg = tanh/parenleftbigg 2C0/parenleftbigg p±1−1 2/parenrightbigg +C0h(y)−γy+C0B−α/parenrightbigg (90) Using (11) and (2) ; recalling that we neglect the Laplacian t erm, a straight calculation shows that (90) is equivalent to the algebraic equation (59) . Let us consider now first variations of the free energy (89) un der small variations δl(y) ofl(y). Using that the length of the jet is given by L= 2/integraltextymax ymin/radicalBig 1 +1 4(dl/dy)2dy, a straightforward calculation shows that δL=−/integraltextymax yminδl(y)/rdywhereris the radius of curvature of the jet. We thus deduce from first variations of t he free energy (89) : FJet(u) r=f(p1(y),y)−f(p−1(y),y) (91) Hypothesis (2) : aL2 V<<1 permitted us to consider FJet(u) as independent of y. In accordance with this hypothesis we evaluate f(p1(y),y)−f(p−1(y),y) at order zero, with p±1=1 2(1±u) at this order. We obtain : f(p1(y),y)−f(p−1(y),y) = 2u(α1+C0h(y)−γy). Moreover using the free energy per unit length expression (4 2), (37) and (38), one can show thatFJet(u) = 2e(u)C0Rwheree(u) is defined by (70). These two last results show that (91) is equivalent to (71) the expression for the radius of curvat urerfound by the integrability condition for the jet. B Boundary jets in the channel case. Beta-effect or linear sublayer topography. Let us derive the boundary jet properties in the case of a beta -effect (or linear sublayer topography). For the sake of simplicity, we will treat the ca se of zero circulation Γ = 0. 31Letφ+ bandφ− bbe the values of φon they=1 2andy=−1 2boundary respectively. The boundary jet satisfies the jet equation (36), but with bounda ry conditions φ(τ= 0) =φ± b andφ(τ→+∞) =φ1orφ−1. Thus, using (38), we deduce that : 1 2/parenleftbiggdφ dτ/parenrightbigg2/parenleftbigg y=±1 2/parenrightbigg +U(φ± b) =U(φ1) =U(φ−1) (92) (where the last equality comes from the integrability condi tion for the interfacial jet). We can relate dφ/dτ to the derivative dψ/dζ normal to the boundary ( denoting the normal coordinate ζ=±1 2∓y), using (61), dφ dτ/parenleftbigg y=±1 2/parenrightbigg =1 Rdψ dζ/parenleftbigg y=±1 2/parenrightbigg ∓Rγ C(93) Furthermore the condition Γ = 0 imposes Γ =−/integraldisplay y=1 2dψ dydx+/integraldisplay y=−1 2dψ dydx=dψ dζ/parenleftbigg y=1 2/parenrightbigg +dψ dζ/parenleftbigg y=−1 2/parenrightbigg = 0, so thatdψ dζ(y=1 2) =−dψ dζ(y=−1 2). Then (92) becomes : U(φ± b) =U(φ1)−1 2/parenleftbigg1 Rdψ dζ/parenleftbigg y=1 2/parenrightbigg −Rγ C/parenrightbigg2 (94) so that U(φ+ b) =U(φ− b) (95) Note that the two values φ± bcannot be equal. They must indeed satisfy, from (61) and the condition of zero mass flux ( ψ+=ψ−), φ+ b−φ− b=−γ/C (96) To solve these two equations (95) and (96), let us have a look a t the potential Uin figure 19. We have to compare φ+ b−φ− bandφ1−φ−1. We thus distinguish two cases. Using (96) and φ±1=±u, we calculate : ( φ+ b−φ− b)/(φ1−φ−1) =−γ/(2Cu). We recall thatuis of order unity. •Low beta-effect case : −γ/C < 2u. For an x-independent statistical equilibrium, one possibi lity is a zonal band with φ−1 inside the band and φ1near both boundaries, with φ+ b1andφ− b1at the boundaries. The symmetric solution, with φ+ b−1andφ− b−1at the boundaries and φ−1near the boundaries, has the same free energy ( due to the symmetry of t he potential U). However the solution which maximizes the total free energy o f the jets corresponds toφ−1in the lower part of the domain, and φ1in the upper part, with φ+ b0at the upper boundary ( y=1 2) andφ− b0at the lower boundary ( y=−1 2), and a single eastward interior jet. 32•High beta-effect case : β>2u In this case, we note that φ+ b−φ− b>φ1−φ−1. Then the two equations (96) and (95) determine a unique solution for ( φ+ b,φ− b) ( see figure 19 ). We are then necessarily in the case with φ=φ1near the boundaries y=1 2andφ=φ−1near the other one. It again involves a single eastward internal jet. This jet can o scillate in latitude due to the momentum constraint ( according to (79) ). Onceφ± bare fixed, we use equation (61) on the two boundaries y=±1 2to conclude thatα=−1 2(φ+ b+φ− b). Then using the integrability condition (66) we calculate : ψb=R2/parenleftbigg B+φ+ b+φ− b 2/parenrightbigg (97) This closes the determination of our parameters. C Boundary condition for the equation (81,82) of the oval- shaped vortices boundary We want to see whether equations (81,82) with d >0, defining the curve formed by the jet, admit periodic solutions both in xandycorresponding to vortices, or not. As stressed in section (3.3.2), equations (81) derive from the Hamilton ian (83) ;yandθare the two conjugated variables. Let us study the phase portrait of H. Forθin [0,2π[, there are 4 critical points : P1= (0,1/d), P2= (0,−1/d), P3= (π,1/d), P4= (π,−1/d). By linearization around these fixed points, one easily prove th atP1andP3are stable fixed points whereas P2andP4are hyperbolic fixed points. This permits to draw the phase portrait : figure 18. Using H(83), we obtain that the unstable manifolds are given by 1−2/(3√ d) =Hand−1+2/(3√ d) =Hrespectively. The parameter dgoverns a transition of the phase space structure. This transition occurs when th e two unstable manifolds merge ; this permits to compute the transition value for d:d=4 9. We are looking for vortices solutions of (81,82). We recall that θis the angle with the xaxe. We thus impose θto be a monotonic function of son a trajectory. Thus areas cof the phase portrait (figure 18 ) are forbidden ; they would correspond to θvarying in a finite interval. We won’t address their analyze there, but oscillating jet solutions can be found in these areas. Area b2 : it can be shown that, as on such trajectories, as θis not strictly increasing, double points exists, thus forbidding area b2. Conversely areas aandb1 are admissible trajectories, giving yas periodic functions of θ.We have to had the condition that x(82) must also be a periodic function of θfor the curve to close. Let us denote ∆ xthexvariation when θruns in [0,2π]. We thus impose the condition ∆ x= 0. Using (82) and (81) we calculate : ∆x=/integraldisplayL 0cosθds=/integraldisplay2π 0cosθdθ −dy2(θ) + 1=/integraldisplayπ 2 −π 2cosθd[y2(θ)−y2(θ−π)] (−dy2(θ) + 1)( −dy2(θ−π) + 1)dθ= 0 (98) ( The last expression is obtained rewriting the integral as a sum on/bracketleftbig −π 2,π 2/bracketrightbig plus a sum on/bracketleftbigπ 2,3π 2/bracketrightbig ; and performing a variable change ). Let us study the sign of y2(θ)−y2(θ−π). 33Using (83) we deduce that −dy3(θ) 3+y(θ) =H−cosθand−dy3(θ−π) 3+y(θ−π) =H+cosθ. From these two relations we conclude that y(θ−π) =y(θ) implies cos( θ) = 0 and that y(θ−π) =−y(θ) impliesH= 0. Thus if H/negationslash= 0 ;y2(θ)−y2(θ−π) does not change sign on/bracketleftbig −π 2,π 2/bracketrightbig . Moreover, on the areas a) and b1) ( −dy2(θ) + 1) does not change sign. Thus ifH/negationslash= 0, the argument of the last integral of (98) does not change s ign and ∆xcan not be zero. We thus conclude that the only solution where xis a periodic function of θis the solution corresponding to H= 0. This solution is the one obtained from (81,82) with initial conditions y(0) = 0 and θ(0) =π 2. As we have previously excluded the area b2 ( when d>4 9) we conclude that no vortex solution exists when d>4 9. We conclude that equations (81,82) with d>0, defining the curve formed by the jet, admit one periodic sol ution both inxandycorresponding to vortices, only when d<dmax=4 9. This solution corresponds toH= 0 (83). The vortex then admits a latitudinal and a zonal axes of symmetry. References [1] BRANDS, H. CHAVANIS, P-H. PASMANTER, R. and SOMMERIA, J. 1999 Maxi- mum Entropy versus Minimum Enstrophy Vortices. [2] CHAVANIS, P.H. & SOMMERIA, J. 1996a Classification of sel f-organized vortices in two-dimensional turbulence: the case of a bounded domain. J. Fluid Mech. 134, 267. [3] CHAVANIS, P.H. SOMMERIA, J. & ROBERT, R. 1996b Statistic al mechanics of two-dimensional vortices and collisionless stellar syste ms.Astr. Jour. 471, 385-399. [4] CHAVANIS, P.H. 1998 From Jupiter’s great red spot to the s tructure of galaxies: statistical mechanics of two-dimensional vortices and ste llar systems Annals of the New York Academy of Sciences. 867, 120-141. [5] DOWLING, T.E 1995 Dynamics of Jovian Atmospheres, Annu. Rev. Fluid Mech ,27, 293-334. [6] DOWLING, T.E. & INGERSOLL, A.P. 1989 Potential vorticit y and layer thick- ness variations in the flow around Jupiter’s Great Red Spot an d White Oval BC. J.Atmos.Sci. 45, 1380-1396 . [7] HATZES, A. WENKERT, D.D. INGERSOLL, A.P. DANIELSON, G.E 1981 Oscilla- tions and Velocity Structure of a Long-Lived Cyclonic Spot. J.Geop.Res. 86, (A10), 8745-8749. [8] INGERSOLL, A.P. GIERASCH, P.J BANFIEL, D. VASAVADA, A.R . and the Galileo Imaging Team. 2000 Moist convection as an energy source for t he large-scale motions in Jupiter’s atmosphere. Nature 403, 630-632 [9] JOYCE, G. & MONTGOMERY, D. 1973 Negative temperature sta tes for the two- dimensional guiding center plasma. J. Plasma Phys 10, 107-121 [10] KAZANTZEV, E. SOMMERIA, J. & VERRON, J. 1998 Subgridsca le Eddy Param- eterization by Statistical Mechanics in a Barotropic Ocean Model. J.Phys.Ocean. 28 (6) 1017-1042. 34[11] KUKHARKIN, N. ORSZAG, S.A. and YAKHOT, V. 1995 Quasicry stallisation of Vor- tices in Drift-Wave Turbulence. Phys. Rev. Lett. 7513 2486-2489. [12] KUZ’MIN, G.A. 1982 Statistical mechanics of the organi zation into two-dimensional coherent structures. in “Structural Turbulence” (ed M. A. Golgshtik), pp 103-114. Acad. Naouk CCCP Novosibirsk, Institute of Thermophysics. [13] LIMAYE, S.S. 1986. Jupiter : new estimates of the mean zo nal flow at the cloud level Icarus 65335-52. [14] MARCUS, P.S. 1993. Jupiter’s Great Red Spot and other vo rtices. Annu. Rev. Astron. Astrophys. 31523-73. [15] MAXWORTHY, T. 1984. The dunamics of a high-speed Jovian jet.Planet. Space Sci. 328. 1053-1058. [16] MICHEL, J. & ROBERT, R. 1994a Statistical mechanical th eory of the great red spot of Jupiter. J. Phys. Stat. 77(3/4), 645-666. [17] MICHEL, J. & ROBERT, R. 1994b Large Deviations for Young measures and sta- tistical mechanics of infinite dimensional dynamical syste ms with conservation law. Commun. Math. Phys. 159, 195-215. [18] MILLER, 1990 Statistical mechanics of Euler’s equatio n in two dimensions. Phys. Rev. Lett.65(17), 2137. [19] MITCHELL, J.L. BEEBE, R.F. INGERSOLL, A.P. & GARNEAU, G .W. 1981 Flow fields within Jupiter’s Great Red Spot and White Oval BC. J. Geophys.Res. 86, 8751-8757. [20] MODICA, L. 1987 Gradient theory of phase transitions an d minimal interface criteria. Arch. Rational. Mech. Anal. 98123-142. [21] ONSAGER, L. 1949 Statistical hydrodynamics. Nuovo Cimento Suppl. 6(3), 279. [22] PEDLOSKY, J. 1987 Geophysical Fluid Dynamics, Second E dition. Springer Verlag [23] ROBERT, R. 1990 Etat d’´ equilibre statistique pour l’´ ecoulement bidimensionnel d’un fluide parfait. C. R. Acad. Sci. Paris 311(I), 575-578. [24] ROBERT, R. 2000 On the statistical mechanics of 2D Euler and 3D Vlasov Poisson equations, Comm. Math. Phys. in press. [25] ROBERT, R. & SOMMERIA, J. 1991 Statistical equilibrium states for two- dimensional flows, J. Fluid. Mech. 229, 291-310 [26] ROBERT, R. & SOMMERIA, J. 1992 Relaxation towards a stat istical equilibrium state in two-dimensional perfect fluid dynamics, Phys. Rev. Lett. 69, 2776-2779 [27] ROBERT, R. & ROSIER, C. 1997 The modeling of small scales in 2D turbulent flows: a statistical mechanics approach. J. Stat. Phys. 86, 481-515 35[28] SOMMERIA, J. NORE, C. DUMONT,T & ROBERT, R. 1991 Th´ eori e statistique de la Tache Rouge de Jupiter. C.R. Acad. Sci t. 312 , Srie II p 999-1005. [29] SOMMERIA, J. STAQUET, C. and ROBERT, R. 1991 Final equil ibrium state of a two dimensional shear layer J.Fluid Mech. 233, 661. [30] TURKINGTON, B. and WHITAKER, N. 1996 Statistical equil ibrium computations of coherent structures in turbulent shear layers SIAM J. Sci. Comput. 17(16), 1414. 36Figure Captions. Figure 1 : Annular jets observed in the atmosphere of Jupiter . a) Velocity field in the Great Red Spot of Jupiter (200South), from Dowling and Ingersoll (1989). b) Velocity field in the cyclonic Barge of Jupiter (140North) from Hatzes et al (1981) Figure 2 : (a) Graphical representation of the algebraic equ ation (20), with the rescaled variableφ≡ −α/C+ψ/R2. The three solutions are at the intersection of the curve (le ft- hand side) and straight line (right-hand side). Here the int egrability condition α=C0B for the differential equation (36) Figure 3 : The free energy density f(p) (27) versus the probability p. ForC0>1 and (C0B−α0) small enough f(p) has two local minima and one local maximum, allowing to obtain two values p±1in the maximization of entropy under constraints. Figure 4 : The parameter uversus the Lagrange parameter C0, as the solution of (33). Figure 5 : Typical stream function profile in a jet ( u= 0.75 ) versus the transverse coordinate τ=ζ/R( Top ) and corresponding velocity profile ( Bottom ). Figure 6 : Jet properties versus the segregation parameter u. a) Jet width, defined as the width of the region with velocity greater than half the maximum jet velocity. b) Maximum velocity ( dφ/dτ )maxand jet kinetic energy e(u) ( dotted line). Figure 7 : Phase diagram of the Gibbs states versus the energy Eand the asymmetry pa- rameter B. The outer line is the maximum energy achievable fo r a fixed B : E=R2 2(1−B2). The frontiers line between the straight jets and the circula r jets corresponds to A1= 1/π orA−1= 1/π. it as been calculated using (31) and (32) : E=R2B2(2π−2)/(π−2)2. The dot line represents the frontiers between axisymmetric vor tices and the circular jets. We define it as the energy value for which the circular vortex are aA1orA−1(31) is equal to (2l)2, wherelis the typical jets width ( figure 6 ). Such a line depends on the numerical value of R the ratio of the Rossby deformation radius to the do main scale. It has been here numerically calculated for R = 0.03. Figure 8 : (a) Graphical representation of the algebraic equ ation (20), with the rescaled variableφ≡ −α/C+ψ/R2, like in figure 2, but in the case of a Gibbs state with an axisymmetric vortex (∆ C >0). Then the rhs hatched area is greater than the lhs one. (b) The corresponding potential U(φ), given by (49), is asymmetric to compensate for the friction term in equation (50). Figure 9 : Various axisymmetric stream-function profiles fo r decreasing ∆ C( ∆C= [0.90.60.30.10.050.030.01] andB= 0.75. Figure 10 : a ) Typical sub-domain shape with a topography h(y) =ay2. The parame- terdhas been chosen such that the ratio of the length on the width b e 2 ; as on Jupiter’s 37GRS. b ) Typical sub-domain shape with a topography h(y) =ay2when the parameter did very close to its maximum value d=4 9. The shape is then very elongated, with latitudinal boundaries quasi parallel, as for instance the Jovian cyclo nic vortices ( ’Barges’ ) described by ( Hatzes et al 1981 ). Figure 11 : a ) Sub-domain non dimensional length and width ve rsus the parameter d ( topography h(y) =ay2). b ) Sub-domain aspect ratio versus the parameter d. Figure 12 : QG topography ( units s−1) versus latitude computed from data of Dowling and Ingersoll (1989) : a) under the GRS ; b) under the Oval BC. Figure 13 : Velocity profile within the GRS from ( Mitchell et a l 1981 ). They have observed that the velocity is nearly tangential to ellipses . Using a grid of concentric ellipses of constant eccentricity, the velocities have been plotted with respect to the semi-major axisAof the ellipse on which the measurement point lies. Results h ave been fitted by a quartic in A.l⋆is the jet width, defined as the width on which the jet velocity is greater than half the maximum velocity. Figure 14 : A) Coefficient afor a quadratic topography h(y) =ay2versusR⋆com- puted from our QG model (87). The three cross show the coefficie ntacomputed from QG topography deduced from Dowling SW observed results. B) D ifference of PV lev- elsa1−a−1versusR⋆computed from our QG model (86). The dot line represents the planetary vorticity f0at the latitude of the center of the GRS. This show that one of t he PV levels may be interpreted as vorticity generated by conve ction-plumes from the sublayer. Figure 15 : The segregation parameter uversus the Rossby deformation Radius R∗for the GRS.uhas been computed using the actual jet maximum velocity and w idth ( see section (4) ). Figure 16 : The non dimensional parameter giving the shape of the curve : d( see (81) ) with respect to R⋆in our model of the GRS. The dot line represents the critical v alue d=4 9below which a vortex solution exists. The ratio of the length to the width of the GRS is approximately 2. From figure 11 we conclude that this co rrespond to dvery close to the critical value4 9. From this figure, our model predicts that the Rossby deforma tion radius isR∗= 1800 km ( see section (4) for comments ). Figure 17 : Definition of l(y). Figure 18 : Phase portraits of the Hamiltonian H (83) for y0= 0, governing the jet shape via differential equations (81) ( two periods in θ). For vortices, we are looking for periodic solutions in y. Thus only trajectories of areas a) and b) are under interest . Conversely trajectories of area c) could correspond to osci llating jets. The parameters d governs a transition between two type of phase portraits. A) Ford <4 9( hered= 0.075 ), trajectories of area a) can define yas a function of θcorresponding to convex vortices. B ) Ford>4 9( hered= 0.075 ), for trajectories of area a), the curve y(θ) admits double 38points. Thus they can not define vortex boundaries. Figure 19 : Resolution of the equations (94 and 96). The long- doted and the doted lines represent the two cases discussed in appendix (B). 39Figure 1: Annular jets observed in the atmosphere of Jupiter . a) Velocity field in the Great Red Spot of Jupiter (200South), from Dowling and Ingersoll (1989). b) Velocity field in the cyclonic Barge of Jupiter (140North) from Hatzes et al (1981) . 40Figure 2: (a) Graphical representation of the algebraic equ ation (20), with the rescaled variableφ≡ −α/C+ψ/R2. The three solutions are at the intersection of the curve (le ft- hand side) and straight line (right-hand side). Here the int egrability condition α=C0B for the differential equation (36) is verified, so the two hatc hed areas are equal. b) The corresponding potential U(φ), given by (37), integral from 0 to φof the difference between the two curves (hatched area in (a)). 0-C - B0 ( C0 ) b )a ) CBαα/φ+ φtanh -2.5 -1.5 -0.5 0.5 1.5 2.5 0.6 0.3 0.0 -2.5 -1.5 -0.5 0.5 1.5 2.5φ φ −1φ1 φφ)U(012 -1 -2 41Figure 3: The free energy density f(p) (27) versus the probability p. ForC0>1 and (C0B−α0) small enough f(p) has two local minima and one local maximum, allowing to obtain two values p±1in the maximization of entropy under constraints. p p1 -1 0.0 0.5 1.0f(p) p Figure 4: The parameter uversus the Lagrange parameter C0, as the solution of (33). C01.0 0.8 0.6 0.4 0.2 0.0 1.0 1.5 2.0 2.5 3.0 3.5u 42Figure 5: Typical stream function profile in a jet ( u= 0.75 ) versus the transverse coordinate τ=ζ/R( Top ) and corresponding velocity profile ( Bottom ) τ =τ =φ = − Β + ζ/Rζ/Rψ/R2 1.1 0.5 -0.5 -1.10 -9 -6 -3 0 3 6 9 -9-6 -3 0 3 6 90.40 0.30 0.20 0.10 0.00 43Figure 6: Jet properties versus the segregation parameter u. a) Jet width, defined as the width of the region with velocity greater than half the maxim um jet velocity. b) Maximum velocity (dφ/dτ )maxand jet kinetic energy e(u) ( dotted line). φd__ τda ) b ) |max ; e(u) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.20.40.60.81.0 00.0 0.5 1.00510152025l u u 44Figure 7: Phase diagram of the Gibbs states versus the energy Eand the asymmetry parameter B. The outer line is the maximum energy achievable for a fixed B : E=R2 2(1− B2). The frontiers line between the straight jets and the circu lar jets corresponds to A1= 1/πorA−1= 1/π. it as been calculated using (31) and (32) : E=R2B2(2π−2)/(π−2)2. The dot line represents the frontiers between axisymmetric vortices and the circular jets. We define it as the energy value for which the circular vortex a reaA1orA−1(31) is equal to (2l)2, wherelis the typical jets width ( figure 6 ). Such a line depends on the numerical value of R the ratio of the Rossby deformation radius to the do main scale. It has been here numerically calculated for R = 0.03. -101 -101 EB 45Figure 8: (a) Graphical representation of the algebraic equ ation (20), with the rescaled variableφ≡ −α/C+ψ/R2, like in figure 2, but in the case of a Gibbs state with an axisymmetric vortex (∆ C >0). Then the rhs hatched area is greater than the lhs one. (b) The corresponding potential U(φ), given by (49), is asymmetric to compensate for the friction term in equation (50). φU( )0Cφ+ α/ - B 0Ctanh( )φ φ φ φ φ(0)0 1 −1 φφ -2.0 -1.2 -0.4 0.4 2.0012 -1 -2 -2.0 -1.2 -0.4 0.4 1.2 2.000.10.20.30.40.50.60.7 46Figure 9: Various axisymmetric stream-function profiles fo r decreasing ∆ C( ∆C= [0.90.60.30.10.050.030.01] andB= 0.75. rφ(r) ∆∆ C = 0.9C = 0.01 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15-0.973-0.538-0.1040.3310.7651.200 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15-0.973-0.538-0.1040.3310.7651.200 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15-0.973-0.538-0.1040.3310.7651.200 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15-0.973-0.538-0.1040.3310.7651.200 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15-0.973-0.538-0.1040.3310.7651.200 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15-0.973-0.538-0.1040.3310.7651.200 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15-0.973-0.538-0.1040.3310.7651.200 47Figure 10: a ) Typical sub-domain shape with a topography h(y) =ay2. The parameter d has been chosen such that the ratio of the length on the width b e 2 ; as on Jupiter’s GRS. b ) Typical sub-domain shape with a topography h(y) =ay2when the parameter did very close to its maximum value d=4 9. The shape is then very elongated, with latitudinal boundaries quasi parallel, as for instance the Jovian cyclo nic vortices ( ’Barges’ ) described by ( Hatzes et al 1981 ). a ) b )-3 0 3-202y x -6 0 6-404y x 48Figure 11: a ) Sub-domain non dimensional length and width ve rsus the parameter d( topography h(y) =ay2). b ) Sub-domain aspect ratio versus the parameter d. 4 _ 9a ) b ) 0.0 0.1 0.2 0.3 0.4 0.50.91.31.72.12.52.9 0.0 0.1 0.2 0.3 0.4 0.50.91.31.72.12.52.9 0.0 0.1 0.2 0.3 0.4 0.50.91.31.72.12.52.9 Half width0.0 0.1 0.2 0.3 0.4 0.50.91.31.72.12.52.9 0.0 0.1 0.2 0.3 0.4 0.50.91.31.72.12.52.9 Half length 49Figure 12: QG topography ( units s−1) versus latitude computed from data of Dowling and Ingersoll (1989) : a) under the GRS ; b) under the Oval BC. h( λ) ( 10 s )-4 -1h( λ) ( 10 s )-4 -1 λ λGRS ( degree )( degree ) Oval BC-30 -26 -22 -18 -14-0.5-0.20.10.40.71.0 -30 -26 -22 -18 -14-0.5-0.20.10.40.71.0 -30 -26 -22 -18 -14-0.5-0.20.10.40.71.0 -30 -26 -22 -18 -14-0.5-0.20.10.40.71.0 R = 1700km R = 2200km R = 2600km -38 -36 -34 -32 -301.21.41.61.82.02.2 -38 -36 -34 -32 -301.21.41.61.82.02.2 -38 -36 -34 -32 -301.21.41.61.82.02.2 R = 1100km R = 1600km R = 2000kmR = 1700km a = 9.257E-13 km^-2s^-1 50Figure 13: Velocity profile within the GRS from ( Mitchell et a l 1981 ). They have observed that the velocity is nearly tangential to ellipses. Using a g rid of concentric ellipses of constant eccentricity, the velocities have been plotted wi th respect to the semi-major axis Aof the ellipse on which the measurement point lies. Results h ave been fitted by a quartic inA.l⋆is the jet width, defined as the width on which the jet velocity is greater than half the maximum velocity. *v ( m.s )-1 l_ 2* A ( 10 km )3 0 2 4 6 8 10 12-101030507090110 0 2 4 6 8 10 12-101030507090110 51Figure 14: A) Coefficient afor a quadratic topography h(y) =ay2versusR⋆computed from our QG model (87). The three cross show the coefficient acomputed from QG topography deduced from Dowling SW observed results. B) Diff erence of PV levels a1−a−1 versusR⋆computed from our QG model (86). The dot line represents the p lanetary vorticityf0at the latitude of the center of the GRS. This show that one of t he PV levels may be interpreted as vorticity generated by convection-pl umes from the sublayer. R ( 10 km)*3R ( 10 km)*3*-6 -2 -1a ( 10 m s ) f0 1a* -a* ( 10 s ) -1-1-5A ) 1.5 2.0 2.5 3.0 3.50.00.51.01.52.0 1.5 2.0 2.5 3.0 3.50.00.51.01.52.0 1.5 2.0 2.5 3.0 3.50.00.51.01.52.0 1.5 2.0 2.5 3.0 3.50.00.51.01.52.0 1.5 2.0 2.5 3.0 3.55101520253035 52Figure 15: The segregation parameter uversus the Rossby deformation Radius R∗for the GRS.uhas been computed using the actual jet maximum velocity and w idth ( see section (4) ). u R ( 10 km )* 3 1.5 2.0 2.5 3.0 3.50.80.91.01.1 Figure 16: The non dimensional parameter giving the shape of the curve : d( see (81) ) with respect to R⋆in our model of the GRS. The dot line represents the critical v alued=4 9 below which a vortex solution exists. The ratio of the length to the width of the GRS is approximately 2. From figure 11 we conclude that this corresp ond todvery close to the critical value4 9. From this figure, our model predicts that the Rossby deforma tion radius isR∗= 1800 km ( see section (4) for comments ). * ( 10 km )3 R 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.10.10.20.30.40.50.60.7d 53Figure 17: Definition of l(y). l(y)yp (y)p (y) -1 1 54Figure 18: Phase portraits of the Hamiltonian H (83) for y0= 0, governing the jet shape via differential equations (81) ( two periods in θ). For vortices, we are looking for periodic solutions in y. Thus only trajectories of areas a) and b) are under interest . Conversely trajectories of area c) could correspond to oscillating jet s. The parameters dgoverns a transition between two type of phase portraits. A) For d<4 9( hered= 0.075 ), trajectories of area a) can define yas a function of θcorresponding to convex vortices. B ) For d>4 9 ( hered= 0.075 ), for trajectories of area a), the curve y(θ) admits double points. Thus they can not define vortex boundaries. /21 d /21 d /21 d /21 dP1P2 PP3 0π2π 3π 4π0d = 0.075 d = 2 0 π 2π 3π 4π 0a ac cc cca ac c ccA ) B )θ θyy b1 b2 --c 4 -8.00.08.0 -2.50.02.5 55Figure 19: Resolution of the equations (94 and 96). The long- doted and the doted lines represent the two cases discussed in appendix (B). U(φ) φφ φ φ- - +- φ+φ φ+ b-1b-1 b 0b 0 b 1b 1φ φ1 -1 -4 -2 0 2 4-1.4-1.0-0.6-0.20.20.61.0 56