Particle flux in an annular gap about a sphere

We develop a method for joining diffusion theory to a void region in a consistent manner. In particular, we apply the theory to a three region problem with a central solid core, surrounded by a concentric void shell and this in turn is surrounded by a concentric shell of another solid material. The sources can be either incident on the outer surface or uniformly distributed in the inner and outer regions. The void flux is calculated from the free streaming Boltzmann equation and is linked to the diffusion equations by assuming that the angular distribution on the surfaces follows the form A + B cosϑ, where A and B are related to the diffusion theory fluxes in the inner and outer regions. This procedure allows us to calculate the flux throughout the solid and void regions without making any diffusion theory assumptions in the void. Numerical results are given to illustrate the method and comparisons with exact transport calculations are given to establish the accuracy. An extension of the method to deal with axially symmetric systems is also given in Appendix B. The theory will apply to neutrons and photons.