Stochastic Invariant Imbedding Theory for a Distributed Internal Source

Invariant imbedding theory is an alternative formulation of particle transport theory. Until very recently, this theory was used only for deterministic calculations, i.e., for calculations of the first moment of the particle distribution. In a previous paper we set up a probability balance equation in the invariant imbedding approach. An equation was also obtained for the probability generating functional (pgfl) of reflected particles from which equations for the first- and second-order densities were derived. The approach was illustrated by a simple forward-backward scattering model with and without incorporating energy dependence to describe sputtering due to an external source of energetic particles on a medium. In this paper we extend these results to the case of a distributed internal source of particles. Among the possible applications, we discuss the problem of internal sputtering. We derive equations for the pgfl and the first- and second-order densities and show their connection with the external source problem. We treat the finite slab problem in addition to the semi-infinite slab geometry considered in our previous paper.