Magnus expansion and eikonal approximation in potential scattering

An expansion procedure due to Magnus is applied to the case of potential scattering. This technique, which is superior to the usual Born-Dyson expansion in that it preserves unitarity at each state, leads in first approximation to an expression for the scattering amplitude which contains as special cases the eikonal approximation and also the second Born approximation. In particular the first Magnus approximation provides a basis for the ad hoc addition of the real part of the second Born contribution to the eikonal scattering amplitude as was considered by Byron et al.     

A reciprocal phase-perturbation theory for rough-surface scattering

We study the scattering of a scalar plane wave from a two-dimensional, randomly rough surface, on which the Dirichlet boundary condition is satisfied. The scattering amplitude is obtained in the form of the Fourier transform of an exponential, in which the exponent is written as an expansion in powers of the surface profile function. It is shown that the latter expansion can be written in such a way that the corresponding scattering matrix is manifestly reciprocal. Numerical results for the reflectivity, and for the contribution to the mean differential reflection coefficient from the incoherent component of the scattered field, are obtained and compared with the predictions of small-amplitude perturbation theory and the Kirchhoff approximation. As the wavelength of the incident wave is varied continuously the results of the phase-perturbation theory change continuously from those of the small-amplitude perturbation theory to those of the Kirchhoff approximation.     

On the Fermi approximation in slow neutron scattering

In this paper a general connection between the scattering from a bound and a free nucleus is derived in terms of formal scattering theory. The basic idea is to eliminate the interaction Hamiltonian between neutron and nucleus. Then theT-operator for a bound nucleus can be expressed by that of the free nucleus and its binding potential. From this equation an expansion is given as a power series in the binding potential for the nucleus. For slow neutron scattering the first term of the series leads to Fermi's approximation. The second term is the first correction to Fermi's approximation which contains no divergences for point scatterers contrarily to theories of other authors. In particular the correction to Fermi's approximation of the scattering amplitude is calculated for an elastically bound proton in the limit of zero-energy neutron.     

Eikonal description of quantal scattering

Elastic and inelastic quantal scattering is described by a theory in which the contribution of a range of impact parameters to the scattering amplitude is determined by a phase integral (“eikonal”) which is integrated along a real curved “quantal” trajectory. This amplitude reduces to the Glauber expression in the high-energy, forward-angle limit, and to the usual semiclassical amplitude in the classical limit. The formulation can be applied to the study of heavy-ion scattering. The quantal trajectories are investigated analytically for the case of Coulomb scattering. A numerical analysis of elastic ¹⁶O¹⁶O scattering is carried out. The results show appreciable improvement as compared with the Glauber approximation.     

A reciprocal phase-perturbation theory for rough-surface scattering

Abstract

We study the scattering of a scalar plane wave from a two-dimensional, randomly rough surface, on which the Dirichlet boundary condition is satisfied. The scattering amplitude is obtained in the form of the Fourier transform of an exponential, in which the exponent is written as an expansion in powers of the surface profile function. It is shown that the latter expansion can be written in such a way that the corresponding scattering matrix is manifestly reciprocal. Numerical results for the reflectivity, and for the contribution to the mean differential reflection coefficient from the incoherent component of the scattered field, are obtained and compared with the predictions of small-amplitude perturbation theory and the Kirchhoff approximation. As the wavelength of the incident wave is varied continuously the results of the phase-perturbation theory change continuously from those of the small-amplitude perturbation theory to those of the Kirchhoff approximation.     

Glauber amplitude for ionization of helium by electron impact

The high-energy Glauber approximation is used to derive the formula for scattering amplitude for ionization of helium atom by electron impact. The scattering amplitude is expressed as a one dimensional integral involving MeijerG-functions. This may further be expressed in a closed form as simple sums of Meijer functions by writing down the series expansion for Bessel and Meijer functions. Further, the asymptotic behaviour of these amplitudes is examined for both large and small momentum transfers.     

Virtual Compton Scattering off Spin-Zero Particles at Low Energies

We discuss the low-energy behaviour of the virtual Compton scattering amplitude off a spin-zero target. We first compare various methods of obtaining a low-energy expression based either on the soft-photon approximation or the use of Ward-Takahashi identities. We point out that structure-dependent terms are defined with respect to a low-energy approximation of the pole terms, which commonly is separated from the full amplitude. We derive a general expression for the structure-dependent terms in an expansion in terms of the momenta and of the initial and final virtual photon, respectively, up to and including terms of order . At order two terms appear that are related to the usual electric and magnetic polarizabilities of real Compton scattering. At order we find nine new structures of which five can only be determined using virtual photons. Received October 23, 1997; accepted for publication December 29, 1997     

Semiclassical scattering theory with complex trajectories. I. Elastic waves

The WKB approximation to the one-particle Schrödinger equation is used to obtain the wave function at a given point as a sum of semiclassical terms, each of them corresponding to a different classical trajectory ending up at the same point. Besides the usual, real trajectories, also possible complex solutions of the classical equations of motion are considered. The simplicity of the method makes its use easy in practical cases and allows realistic calculations. The general solution of the one-dimensional WKB equations for an arbitrary number of complex turning points is given, and the solution is applied to calculate the position of the Regge poles of the scattering amplitude. The solution of the WKB equations in three dimensions for a central analytical potential is also obtained in a way that can be easily generalized to N-dimensions, provided the problem is separable. A multiple reflection series is derived, leading to a separation of the scattering amplitude into a smooth “background” term (single reflection approximation) that can be treated using classical but complex trajectories and a second resonating term that can be treated using the Sommerfeld-Watson transformation. The physical interpretation of the complex solutions of the classical equations of motion is given: they describe diffractive effects such as Fresnel, Fraunhofer diffraction, or the penetration of the quantal wave into shadow regions of caustics. They arise also in the scattering by a complex potential in an absorptive medium. The comparison with exact quantal calculations shows an astonishingly good agreement, and establishes the complex semiclassical approximation as a quantitative tool even in cases where the potential varies rapidly within a fraction of a wavelength. An approximate property of classical paths is discussed. The general pattern of the trajectories depends only on the product ? = EΘ, and not on energy and angle separately. This property is confirmed by experiments and besides the signature it gives for the semiclassical behavior, it simplifies considerably the search for all trajectories scattering through the same angle. Finally, a general classification of the different types of elastic heavy ion cross sections is given.     

Funktionalmittelwerte in der statistischen Theorie von quantenmechanischen und klassischen Systemen vieler Teilchen

The grand-canonical partition function of an interacting many-particle-system is represented as a functional integral with Gaussian random variables. The representation can be regarded as a Gaussian average over the partition function of free particles in an external fluctuating potential. The latter partition function is studied by means of diagrammatical techniques. The set of diagrams of a particularly simple structure is summed up by introducing the full scattering amplitude for the scattering in the external potential. The thermodynamicalGibbs' potential proves to be stationary with respect to the true particle density. It is shown that a variational procedure leads directly to an approximation which may be regarded as the renormalized form of the well-known Random-Phase-Approximation (RPA). The main feature of the approximation is thatGibbs' potential is stationary with respect to the two-particle-density correlation function. The classical limit of the renormalized RPA yields the results of the Debye-Hückel theory. In case of an hard-core potential the approximation applies only to the long-range part of the potential. The results are similar to recent developments in the theory of the Ising model and of real gases.     

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.     

An extension of the small-slope approximation for rough surface scattering

Solutions are derived for scattering from a rough one-dimensional pressure-release surface in the form of a functional series in the surface slope. These solutions are obtained by solving an integral equation of the first kind for the surface potential to obtain a representation for the scattering amplitude. It is shown that the subsequent expansion of terms occurring in the scattering amplitude to obtain a functional series in the slope does not yield a unique result. The result obtained contains a free parameter that may be arbitrarily selected. Thus, this result is an extension or generalization of the small-slope approximation of Voronovich (1985 Sov. Phys.-JETP 62 65-70) that differs at second order in the slope from his result. It is also shown that the free parameter can be selected such that each term of the functional series is reciprocal and exhibits a limiting grazing angle dependence consistent with the requirements of flux conservation and the absence of boundary waves. A new formulation of the leading terms of the small-slope expansions is derived and is used to explore the conditions under which the two expansions reduce to the Kirchhoff approximation. Finally, a numerical example is presented to demonstrate that the extended approximation provides corrections that are important for near grazing scatter.     

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.     

Four-body multiple-scattering expansion of the total transition amplitude—MST

The elastic scattering differential cross section is calculated for proton scattering from ⁶He at 717 MeV, using single-scattering terms of the multiple-scattering expansion of the total transition amplitude. We analyze the effects of different scattering frameworks, specifically the factorized impulse approximation and the fixed scatterer (adiabatic) approximation. The text was submitted by the authors in English.     

Scattering on rough surfaces with alpha-stable non-Gaussian height distributions

We study the electromagnetic scattering problem on a random rough surface when the height distribution of the profile belongs to the family of alpha-stable laws. This allows us to model peaks of very large amplitude that are not accounted for by the classical Gaussian scheme. For such probability distributions with infinite variance the usual roughness parameters such as the RMS height, the correlation length or the correlation function are irrelevant. We show, however, that these notions can be extended to the alpha-stable case and introduce a set of adapted roughness parameters that coincide with the classical quantities in the Gaussian case. Then we study the scattering problem on a stationary alpha-stable surface and compute the scattering coefficient under the first-order Kirchhoff and small-slope approximations. An analytical formula is given in the high-frequency limit, which generalizes the well known geometrical optics approximation. Some numerical results are given and discussed.     

The high-energy quark-quark scattering: from Minkowskian to Euclidean theory

In this paper we consider some analytic properties of the high-energy quark-quark scattering amplitude, which, as is well known, can be described by the expectation value of two lightlike Wilson lines, running along the classical trajectories of the two colliding particles. We will show that the expectation value of two infinite Wilson lines, forming a certain hyperbolic angle in Minkowski space-time, and the expectation value of two infinite Euclidean Wilson lines, forming a certain angle in Euclidean four-space, are connected by an analytic continuation in the angular variables: the proof is given for an Abelian gauge theory (QED) in the so-called quenched approximation and for a non-Abelian gauge theory (QCD) up to the fourth order in the renormalized coupling constant in perturbation theory. This could open the possibility of evaluating the high-energy scattering amplitude directly on the lattice or using the stochastic vacuum model.     

Path integral methods for inelastic scattering

Corrections to the primitive semi-classical amplitude for multiple inelastic scattering are obtained from a path integral formulation of scattering theory. The path integrals are calculated by making an expansion about a classical orbit describing elastic scattering. Terms are collected to give a series in inverse powers of the reduced mass m of relative motion of the target and projectile. The leading term is the primitive semi-classical amplitude for multiple excitation and explicit formulae are given for the corrections of order $\text{1}\text{m}$. These are calculated in detail for a one-dimensional model. It is shown that some, but not all, of the corrections can be included by evaluating the primitive amplitude with a symmetrized orbit.     

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.     

HIGH ENERGY NUCLEUS-NUCLEUS ELASTIC SCATTERING

We calculate the differential cross section of α-α elastic scattering with the complete expansion formula of the Glauber amplitude of high energy nucleus-nucleus elastic scattering and the rigid projectile approximation. The results are satisfactory.