Classical Small-Angle Scattering from Anisotropic Potentials

Abstract

A simple, direct method of calculating the small-angle scattering from an arbitrary anisotropic potential is derived. The approximations are valid for large energies or large impact parameters. Two methods are used, both yielding identical results. The first, an impulse approximation, involves integrating the force over the trajectory which would be followed if there were no interaction. The second method involves a perturbation solution of Hamilton's equations of motion. The first method is conceptually simpler, but the second yields a solution of the trajectory and is more easily generalized to include second-order effects. For scattering of ions and fast neutrals, molecular rotation can be neglected, and a simple expression is obtained for the differential cross section at small angles. This is applied to ion—dipole scattering where the potential is completely anisotropic, being repulsive for half the orientations and attractive for the other half. It is found that the differential cross section behaves exactly like that for a spherically symmetric potential given by μe/2r2. However, when the ion—dipole potential is combined with a repulsive term in the potential, deviations from the behavior of spherically symmetric potentials are found. For scattering at thermal energies, the molecular rotation averages out the anisotropic terms in the potential, introducing further deviations from the behavior of spherically symmetric potentials.