On the exact resolution of the transport equation for an anisotropic scattering medium into a system of diffusive equations

The classical expansion for the angular flux in the Boltzmann transport equ ation for an anisotropic scattering medium in terms of surface spherical ha rmonics leads to a complicated set of equations for the moments of the expa nsion as functions of position. Here, the Boltzmann equation is generalized so that it has its solution a generalized angular flux with a solid harmon ic expansion in the components of an arbitrary vector lambda replacing the unit vector Omega. When \lambda\ = 1 this generalized specification of neut ron transport reduces to the conventional treatment. The generalized, or relaxed transport equation can be resolved readily for arbitrary lambda as a system of simple coupled equations in the solid harmo nics of the expansion for the generalized angular flux. The dependence of t he harmonics on the components of lambda is removed to give a set of couple d diffusion equations with coefficients and impressed sources which depend solely on the nuclear data. The solution of the Boltzmann transport equatio n therefore satisfies the system of coupled diffusion equations. The SHPN diffusion equations of the method for a P-N solid harmonic expansi on have (N+1)/2 moments to be determined at each node of a three-dimensiona l spatial mesh, in contrast to the N(N+1)/2 coefficients per node for trans port solutions required with a surface spherical harmonic expansion. (C) 19 99 Published by Elsevier Science Ltd. All rights reserved.