Eikonal expansion

An eikonal expansion of the potential scattering T matrix is evaluated, without approximation, through third order in the inverse momentum. Based on the results, their correspondence with the WKB approximation and a new statement of the unitarity constraint, we propose a sequence of four approximations to the exact impact parameter (Fourier-Bessel) representation of the scattering matrix. The sequence consists of the Glauber approximation and three systematic corrections to the Glauber approximation. The corrections are analytic functions of the impact parameter for Yukawa and Gaussian potentials; they vanish for a Coulomb potential.The sequence of eikonal amplitudes is convergent at high energy and is clearly established for small momentum transfer. Validity for all momentum transfer is conjectured based on systematic cancellation, explicitly verified through third order in the expansion, of momentum transfer dependence in the eikonal impact parameter representation. Such cancellation is shown to occur in the explicit construction of the eikonal expansion of the second Born amplitude for a Yukawa potential.Numerical tests of the sequence of eikonal amplitudes show systematic increase of the angular range of validity by comparison with partial wave results for continuous potentials; the theory is not convergent for discontinuous potentials.The WKB phase shift formula is shown to produce a systematic connection with eikonal expansion results. From this we deduce a generating function for the eikonal phase corrections of arbitrary order and also conjecture a sum of the eikonal expansion valid in the limit of high energy and arbitrary potential strength.     

Eikonal expansion in electron scattering: I. Elastic scattering

A systematic eikonal expansion for the scattering of high-energy electrons from nuclei is derived which starts from the iterated Dirac equation. The resulting scattering amplitude is written in an impact parameter representation depending on eikonal phases which are proportional to inverse powers of the energy. The first two correction terms to the leading Glauber-Baker amplitude are calculated. For a Coulomb potential they agree with a sinθ-expansion of the relativistic Coulomb scattering amplitude. In the case of scattering from an extended charge distribution at sufficiently high energies numerical partial wave calculations are accurately reproduced.     

Atom-surface interaction in inelastic scattering: He/Ag(111)

The amplitude of inelastic He scattering from Rayleigh waves on a Ag(111) surface is calculated in the framework of the distorted wave Born approximation (DWBA) for a two-body exponential potential. Using the potential parameters recently obtained by Bortolani et al. from a quantum mechanical calculation, agreement is obtained with the experimental data. Simple explicit formulas are presented both in the DWBA and in the eikonal approximation. The essential factor in both cases describes an approximate exponential decay of the intensity for increasing parallel momentum transfer.     

An impact parameter representation for the off-energy-shell scattering amplitude

The unitarized Born approximation for the scattering amplitude, suggested previously, is considered in the short wavelength limit. It is found that its on-energy-shell component is equivalent to the well-known impact parameter representation developed by Glauber. An approximate expression is then derived for the off-energy-shell scattering amplitude on the basis of the unitarized Born approximation which is expected to be as accurate as the Glauber approximation.     

Long-range Coulomb forces in DIS: missed radiative corrections?

The Born approximation, one photon exchange, used for DIS (deep inelastic scattering) is subject to virtual radiative corrections which are related to the long-range Coulomb forces. They may be sizeable for heavy nuclei since Zα is not a small parameter. So far, these corrections are known only for two processes, elastic scattering and bremsstrahlung on the Coulomb field of a point-like target. While the former amplitude acquires only a phase, in the latter case also the cross-section is modified. Although the problem of Coulomb corrections for DIS on nuclei is extremely difficult, it should be challenged rather than “swept under the carpet”. The importance of these radiative corrections is questioned in the present paper. We show that, in the simplest case of a constant hadronic current, the Coulomb corrections provide a phase to the Born amplitude, therefore the cross-section remains the same. Inclusion of more realistic hadronic dynamics changes this conclusion. The example of coherent production of vector mesons off nuclei reveals large effects. So far a little progress has been made deriving lepton wave functions in the Coulomb field of an extended target. Employing available results based on the first-order approximation in Zα, we conclude that the Coulomb corrections are still important for heavy nuclei. We also consider an alternative approach for extended nuclear targets, the eikonal approximation, which we demonstrate to reproduce the known exact results for Coulomb corrections. Calculating electroproduction of vector mesons, we again arrive at a large deviation from the Born approximation. We conclude that one should accept with caution the experimental results for nuclear effects in DIS based on analyses done in the Born approximation. Received: 16 May 2001 / Accepted: 4 July 2001     

Transport equations of amplitudes in the eikonal approximation of dynamical diffraction equations

An approach of the eikonal approximation of the dynamical diffraction equations of X-rays in deformed crystals, based on the second-order differential equations for the transmitted and diffracted waves, is presented. By analogy with usual optics, this approach allows one not only to obtain the eikonal equation and to study the behavior of the amplitude in zero-order approximation, which usually is performed in the eikonal dynamical diffraction theory, but also to establish for all orders of the amplitude asymptotic expansion the corresponding transport equations and to present their solutions as integrals over the amplitude propagation trajectory. Summarizing the transport equations, an equation for the total amplitude, analogous with the parabolic diffraction equation in optics, is obtained.     

Improving the Born approximation for the scattering of ultrasound in elastic solids

It has recently been demonstrated that the Born approximation for predicting the scattering response of flaws can be improved through the use of simple modifications called the "doubly distorted Born approximation". In this paper the doubly distorted Born approximation itself is modified with phase and amplitude corrections that further improve the Born scattering results for isotropic elastic media. The reliability of this new modification of the Born approximation has been evaluated by comparison with the exact solution for spherical inclusions obtained with the method of separation of variables. Unlike the ordinary Born approximation which works well only for very weak scattering inclusions, our modification of the doubly distorted Born approximation gives improved scattering results for both weak and strong scattering inclusions.     

Quasiclassical green function in an external field and small-angle scattering processes

A representation is obtained for the quasiclassical Green functions of the Dirac and Klein-Gordon equations allowing for the first nonvanishing correction in an arbitrary localized potential which generally possesses no spherical symmetry. This is used to obtain a solution of these equations in an approximation similar to the Furry-Sommerfeld-Maue approximation. It is shown that the quasiclassical Green function does not reduce to the Green function obtained in the eikonal approximation and has a wider range of validity. This is illustrated by calculating the amplitude of small-angle scattering of a charged particle and the amplitude of Delbrück forward scattering. A correction proportional to the scattering angle was obtained for the amplitude of charged particle scattering in a potential possessing no spherical symmetry. The real part of the Delbrück forward scattering amplitude was calculated in a screened Coulomb potential.     

Electron capture processes in ion-atom collisions

The development of the theory of electron capture processes in ion-atom collisions is reviewed in the present work. The formal theory of scattering is used to obtain the Born and the distorted wave Born series for the rearrangement scattering matrix. On the basis of these series, the applications of the first and second Born approximation, the OBK approximation, the DWBA and the Padé approximants to electron capture processes are discussed. The impulse approximation, the Faddeev method and the second order potential method are also explained. The results of the investigations of the high energy behaviour of the charge transfer cross section based on the Faddeev-Watson multiple scattering expansion are analysed.The methods of molecular and atomic eigenfunction expansions are presented and their relative merits discussed. The semi-classical treatment for determining the differential electron capture probability is reviewed. Results obtained in works utilising expansions in Sturmian, Gaussian and pseudo-state wave functions are also presented. An integral equation approach to the close coupling approximation is described. The field theoretic approach to investigate the electron capture process is discussed in the last section.The results obtained by different theoretical methods are compared with available experimental results.     

High energy limit in rearrangement collisions

The cross section for particle exchange from ground state to ground state in a three-particle collision is investigated. An asymptotic expansion for high incident energies can be given if the Fourier transforms of the interaction potentials allow an expansion in powers of 1/k for largek, which includes Coulomb interaction. It is shown that in general the first and second Born approximations provide the asymptotically leading terms. For special mass ratios, scattering into special (critical) angles becomes predominant. These angles, which are independent of energy, can also be determined employing classical arguments. Near the critical angles single terms of either the first or second Born approximation become dominant. If the exchanged mass is small, the interaction between the two heavy masses cancels near forward direction, implying the validity of the impact parameter theory. The results are discussed for Coulomb interaction as the simplest example (electron exchange). Here, the contributions by critical scattering are small unless the energies become exceedingly large.     

Eikonal theory of electron- and positron-atom collisions

We review recent developments in the application of eikonal methods to the field of electron and positron collisions with atoms. The foundations of the eikonal approximation are first analyzed within the framework of potential scattering, with particular attention to those aspects of the theory which can be generalized to atomic collision processes. We next discuss various many-body applications of the eikonal method, namely: the Glauber approximation, the eikonal-Born series method, optical model theories, the eikonal distorted wave method and the multistate eikonal approximation. We also analyze eikonal exchange amplitudes. Applications of these methods are then considered, first for the case of elastic scattering and then for various inelastic processes.     

On Born approximation in black hole scattering

A massless field propagating on spherically symmetric black hole metrics such as the Schwarzschild, Reissner–Nordström and Reissner–Nordström–de Sitter backgrounds is considered. In particular, explicit formulae in terms of transcendental functions for the scattering of massless scalar particles off black holes are derived within a Born approximation. It is shown that the conditions on the existence of the Born integral forbid a straightforward extraction of the quasi normal modes using the Born approximation for the scattering amplitude. Such a method has been used in literature. We suggest a novel, well defined method, to extract the large imaginary part of quasinormal modes via the Coulomb-like phase shift. Furthermore, we compare the numerically evaluated exact scattering amplitude with the Born one to find that the approximation is not very useful for the scattering of massless scalar, electromagnetic as well as gravitational waves from black holes.     

Wigner transform and the eikonal approximation for nonlocal potentials

The same approximation scheme which leads to the Thomas-Fermi theory in bound state problems is applied to scattering by a nonlocal potential. Based on the Wigner transform of the evolution operator, it gives in a very simple way the eikonal approximation for the scattering amplitude. As an application a generalized Fermi gas model for inclusive scattering from nuclei is derived.     

Scattering by a potential well in an intense radiation field

Scattering by a spherically symmetric potential well in a laser radiation field is examined in the eikonal approximation. The total scattering cross section is found for radiation of an arbitrary number of photons. The range of values of the Born parameter is obtained for which the influence of the radiation field on the scattering cross sections is significant.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 73–76, April, 1989.     

Exact Path-Integral Representations for the T-Matrix in Nonrelativistic Potential Scattering

Several path integral representations for the T-matrix in nonrelativistic potential scattering are given which produce the complete Born series when expanded to all orders and the eikonal approximation if the quantum fluctuations are suppressed. They are obtained with the help of ??phantom?? degrees of freedom which take away explicit phases that diverge for asymptotic times. Energy conservation is enforced by imposing a Faddeev?CPopov-like constraint in the velocity path integral. An attempt is made to evaluate stochastically the real-time path integral for potential scattering and generalizations to relativistic scattering are discussed.     

Correlation effects in small-angle multiple neutron scattering

The dependence of the spectra of small-angle multiple neutron scattering on the volume fraction occupied by scattering grains is considered. The concentration expansion is used to develop scattering theory in the eikonal approximation. The leading term of the expansion reproduces the standard low-concentration theory (Mollier). Some properties of the first correction term are analyzed, and it is shown that the angular distribution narrows with an increase in concentration, in qualitative agreement with the experimental data.     

An efficient method for computing backscattering from Born objects of arbitrary shape

A method is presented for efficiently computing the propagating pressure field backscattered by an arbitrarily shaped, weakly scattering, three-dimensional object. This is accomplished by drawing upon a previously reported relationship between the boundary condition on a two-dimensional radiating aperture and the pressure propagating along an axis normal to the aperture, and the fundamental theorem of diffraction tomography, which relates the Fourier transform of an object function to its scattered pressure field. Together, these two results are used to derive an integral formula that expresses the pressure field backscattered from an object as a one-dimensional Fourier transform of its scattering amplitude. This formula is then utilized to compute the backscattered pressure field from a uniform fluid sphere in the first Born approximation; the results of which are compared to the rigorous partial wave expansion.     

Coulomb corrections for coherent electroproduction of vector mesons: Eikonal approximation

Virtual radiative corrections due to the long-range Coulomb forces of heavy nuclei with charge Z may lead to sizeable corrections to the Born cross-section usually used for lepton-nucleus scattering processes. An introduction and presentation of the most important issues of the eikonal approximation is given. We present calculations for forward electroproduction of rho-mesons in a framework suggested by the VDM (vector dominance model), using the eikonal approximation. It turns out that Coulomb corrections may become relatively large. Some minor errors in the literature are corrected.Received: 3 October 2003, Revised: 2 December 2003, Published online: 6 July 2004PACS: 25.30.-c Lepton-induced reactions - 25.30.Rw Electroproduction reactions - 13.40.-f Electromagnetic processes and properties - 25.30.Bf Elastic electron scattering     

Hamilton-Jacobi expansion of the scattering amplitude

The calculation of the scattering amplitude is reduced to the problem of solving a set of classical Hamilton-Jacobi equations. This allows one to incorporate classical intuition into approximations at a fundamental level. The result is actually an iterative expansion for the scattering amplitude which is expected to be convergent in the high-energy limit. The first term in the expansion is shown to be the Glauber approximation, which is an approximation used extensively in nuclear as well as atomic and particle physics.     

Exchange amplitudes and dispersion relations for electron-atom scattering

The analytic structure of the second Born approximation to the forward elastic exchange amplitude is analyzed for e^?+H scattering, and the importance of the interplay between bound and continuum intermediate state contributions is emphasized.