Scattering by Many-Body Systems in Terms of Two-Body Collision Parameters

Abstract

The scattering of a particle by a system of $N$ particles is described without approximation by a set of equations not containing the interaction Hamiltonian, but an infinite set of two-body collision parameters which can be obtained from solving two-body problems. These equations, obtained with the assumption of two-body potential interaction between particles, can be applied to the case where the scattering particles are themselves complex systems, provided that the relevant two-body collision parameters are known or inferred from experiment. In the simplest case where the only relevant parameter is the scattering length, the first approximation is Fermi's result for the scattering length, the first approximation is Fermi's result for the scattering of slow neutrons by protons. The "impulse approximation" is shown to be an approximate form of the present first approximation. It is shown that previous approximation schemes, in general, fail to give convergent higher approximations for the limiting case of point scatterers, whereas the present equations give convergent higher terms. It can be concluded that the latter are particularly appropriate for short-range interaction. The matrix $\alpha$ which contains all necessary two-body collision parameters is expressed in terms of solutions of the ordinary free two-body scattering problem. The result is applied to derive a revised theory of refraction and diffraction of slow neutrons in crystals by including the effect of zero-point motion of the nuclei and the electrostatic interaction between neutrons and electrons.