Small correlation length solutions for planar symmetry beam transport in a stochastic medium

This paper considers planar symmetry, steady-state, monoenergetic linear transport of a normally incident particle beam through a purely scattering thin slab, composed of a stochastic mixture of two immiscible materials. The scattering process is assumed sufficiently forward peaked so that the Fokker-Planck description is valid. If the material mixing obeys Markovian statistics, and ignoring particle backscattering, this situation is a joint Markov process. As such, the stochastic Liouville master equation is valid, and we obtain a set of two coupled linear transport equations describing exactly the ensemble-averaged solution for the particle intensity. An asymptotic expansion is used to reduce these two equations, in the small correlation length limit, to a single renormalized equation. This equation is treated by an angular moments method, exploiting the assumed peakedness in angle of the solution. An analytic solution of these moment equations provides a simple and explicit solution for the spatial distribution of the particle density. The effect of stochasticity is to increase the solution over the corresponding nonstochastic result recently reported in the literature.