Multiple Scattering in the Diffusion Approximation

Abstract

The passage of classical particles through a grainy scattering medium can be described by a line‐arized Boltzmann equation. A discussion is given of the physical conditions which justify the use of the Fokker‐Planck diffusion approximation to this equation. Some limiting properties of the solutions of the diffusion equation are first discussed for the initial‐value problem in an infinite medium characterized by a diffusion length D. For a total path length$\ell$ ≪ D convenient formulas are given for the distribution of scattering angles θ and, for given θ, the first few moments of the final position vector are computed. These results are taken as a basis for approximate treatment of steady‐state boundary‐value problems. The case of a particle beam incident on a thin plane parallel slab of thickness d ≪ D is considered. Approximate formulas are given for the angular distribution of the transmitted beam and for the (very small) fraction of the beam which emerges from the entrance face. Errors are assessed, and the behavior for grazing angles of incidence or exit is discussed in a conjectural way.