Group-Theoretical Treatment of Time- and Energy-Dependent Multiple Scattering with Application to the Slowing-Down of Neutrons

Abstract

Wigner developed a group‐theoretical method for those problems of multiple scattering in which the elementary scattering law is invariant under a group of transformations. The integral transforms, used in more standard treatments for the reduction of convolutions, come in here more naturally through the representations of the groups of transformations. Wigner's work is extended here to include the time‐dependent slowing‐down of neutrons. In this case a group of linear transformations comes in, which does not yield orthogonality. Nevertheless, it is possible to determine all positive time moments of the distribution function and from them the distribution function itself. The conditions for the existence of the group are that the total scattering cross section is proportional to v^γ (γ: any real number) and that either the ratio of absorption to scattering cross section is constant (including zero) or that the absorption cross section varies as 1/v. Moreover, it is assumed that the angular dependence can be anisotropic, but does not depend on energy in the center of mass system. For the special case of no absorption and spherically symmetric elastic scattering in center of mass system, our solution reduces to Waller's recent exact expression. As a further generalization, we discuss the group which, with the same assumptions about the cross sections, exists for the case of time‐energy‐space‐direction dependence. Here also, the group‐theoretical method yields naturally all positive moments of the distribution functions.