Eikonal approximation

The treatment presented so far is quite general and formally exact. It combines the eikonal representation for nuclear motions and the time-dependent density matrix in an approach which could be named as the Eik/TDDM approach. The following section reviews how the formalism can be implemented in the eikonal approximation of short wavelengths for the nuclear motions, and for specific choices of electronic states leading to the TDHF equations for the one-electron density matrix, and to extensions of TDHF. 

The time-dependent eikonal approximation for the density operator can be introduced by analogy to the treatment for stationary states. Here the steps leading to the working equations are first outlined for a general system. For nuclear motions with short de Broglie wavelengths A = ft/P, it is assumed that the xi vary little over a wavelength so that... 

Solving the Eqs. (C.6-C.8,C.12,C.13) comprise what is known as the Ehrenfest dynamics method. This method has appeared under a number of names and derivations in the literature such as the classical path method, eikonal approximation, and hemiquantal dynamics. It has also been put to a number of different applications, often using an analytic PES for the electronic degrees of freedom, but splitting the nuclear degrees of freedom into quantum and classical parts. 

To consider next the partitioning approach for many-atom systems, we start again with the full density operator f(t), describing both electrons and nuclei. We describe a procedure useful in a first principles dynamics where the nuclear motions are described in an eikonal approximation with Q = Q(t), so that the electronic part of the problem involves the calculation of an electronic density operator r[Q(f), Q t), f] = pei t). This satisfies an electronic L-vN equation... 

J. M. Cohen and D. A. Micha. Electronically diabatic atom-atom collisions A self-consistent eikonal approximation. J. Chem. Phys., 97 1038, 1992. 

Dissipative structures can sustain long-range correlations. The temperature and chemical potential are well defined with the assumption of local equilibrium, and the stationary probability distribution is obtained in the eikonal approximation so the fluctuation-dissipation relation for a chemical system with one variable is... 

Due to the short-wave character of projectile states, the straightforward use of DWBA approach needs enormous calculation efforts. Even with an additional, eikonal, approximation [24,25], the complexity of calculations restricts significantly the possibilities of application of this approach. One can notice, however, that the promising feature of the considered problem is that the projectile motion could be described in the quasiclassical approach. This goal can be achieved in the following way [26]. 

Recently we developed the non-linear-complex-eikonal approximation. Our goal was to set a standard mathematical treatment for the analysis of the propagation of light waves through non-linear media. The linear eikonal approximation is a cornerstone for optical systems design and it is hoped that the non-linear eikonal method will gain some of the universality of its linear counterpart. 

Further applications of the non-linear eikonal approximation are discussed here. We are interested in absorptive non-linearity whose sources are molecular photoexcitatons. Propagation effects of an intense light field in organic molecular media are discussed and their connection to molecular parameters is emphasized. 

M. Orenstein, J. Katriel and S. Speiser "The nonlinear complex eikonal approximation. I Optical bistability in absorbing media", (to be published). 

The notation is close to that in Chap. 3. The concentrations in the N boxes are denoted by vectors with arrows over the symbols. The magnitudes of the changes in X and Y are given by r (see text after (3.1)). We use again the eikonal approximation according to which the stationary probability of... 

Thus the stationary probability distribution of the master equation in the eikonal approximation is a Lyapunov function, which gives necessary and sufficient conditions of the existence and stability of non-equilibrium stationary states and provides a measure of relative stability on the basis of inhomogeneous fluctuations, (6.17). 

Let the imposed flux be J = k X. The stationary solution of the lowest order eikonal approximation of the master equation for the system (11.15) with the imposed flux is... 

In the eikonal approximation for ordered systems without environmental fluctuations aU cumulants (([AT/ ] )) are proportional to the volume 12 of the system in the limit 12 —oo [10],... 

Two most impressive examples of the proposed combination of different approaches are the unitarization of the eikonal eimplitude, that made possible not only to increase the range of validity of the eikonal approximation, but also to incorporate resonance states into the formalism, and the adiabatic scaling relations, that were obtained by combining the scaling relations from the sudden approximation with adiabatic formulae for the excitation of the ground state. 

In most practical applications, discussed in this book, one dynamical approximation is used - so called generalized eikonal approach. It is very refined and effective combination of the uniform quasiclassical representation of the scattering amplitude and the eikonal approximation for propagator, that seems to be very natural compromise between the complexity of the problem and our ability to understand its main features aird tendencies. 

For small deviations of angles from the specular direction a simplified expression for the scattei ing indicatrix has been obtained by Sergeev (Sergeev 1978). This distribution is very close to Gaussian one. The comparison of the results of calculations for Ar-Ag(lll) system on the base of this simplified formula with experimental results, presented in Fig. 35, seems satisfactory. The simplicity of the eikonal approximation makes it very convenient for applied aerodynamical investigations. 

A measure of the great practical importance of the charge transfer process is the very considerable experimental and theoretical effort devoted to that process. The areas in which significant progress has very recently been recorded include atom capture as well as electron capture, the eikonal approximation, and versions of the Glauber approximation. Unfortunately, space permits only one topic, an important step in our understanding of asymmetric charge transfer. 

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