Solving the Transport Equation in Heterogeneous Media Using First-Flight Green's Functions for Homogeneous Media

The integral neutron-transport equation is solved for the space-dependent mono-energetic neutron density in a unit cell. By using step functions to represent the spatial dependence of the collision probabilities, one may rearrange the integro-differential transport equation in a special way such that the left-hand side contains only the leakage term and the term describing the total collision probability for the homogeneous medium of one region, k′, of the original problem. The Green's-function technique is then used to convert the integro-differential equation to an integral equation. Thus, although the resulting equation may be applied to a heterogeneous cell, the kernel of the equation depends only on the total collision probability in the particular region k′. Numerical results are presented for a two-region unit cell in slab geometry and compared with published results of DSN, P_N double-P_N and variational calculations. For unit cells that are of the order of two mean free paths or less in thickness, the zeroth-order spherical harmonic approximation for this method yields results comparable to very high order DSN, P_N and double-P_N calculations. Further, once the Green's function has been computed, additional cell calculations can be performed with relatively little additional computational effort.