On the evaluation of kinetic-theory collision integrals

Abstract

Expressions are obtained using the Distorted-Wave Born Approximation (DWBA) for a number of kinetic-theory collision cross sections. In particular, DWBA results are given for the (production) collision integrals that couple angular-momentum directional polarizations with the hydrodynamic fluxes. These integrals as well as the full relaxation rate (kinetic-theory collision) matrix of angular-momentum polarizations involve energetically inelastic collisions. Except for this energy inelasticity, the DWBA results can be written as a product of an internal-state dependent factor and a translational factor. The simplest approximation is to ignore the dependence of the translational factor on energy inelasticity, or if the integral vanishes if that is done, to include the energy inelasticity to first order. Such a procedure yields an explicit estimate of the complete dependence of the collision integrals on internal-state quantum numbers. A crude approximation for the translational factors is also investigated. This amounts to assuming that the transition operator depends (translationally) only on the momentum transfer. Some relations between relaxation and production collision integrals then follow. Throughout this work, reduced matrix elements (in the sense of the 3-dimensional rotation group) are stressed, so that the results are applicable to any molecular species and most results are valid for any form of intermolecular potential.