On a probability distribution function arising in stochastic neutron transport theory

In a recently developed method for understanding the transport of neutrons in spatially stochastic media, it is necessary to prescribe a random distribution for the fissile lumps. In most practical cases the lumps are assumed to be uniformly distributed within the container. However, the associated centre-to-centre separation distances are not uniformly distributed and this paper describes a method for calculating that distribution function. Two methods are used: an analytical approach which is exact for lumps of zero size and a numerical simulation which accounts for finite size and excludes the possibility of more than one particle occupying the same space. The numerical simulations are compared with the analytical expression and we find excellent agreement for small spheres, which in the limit will be exact. Deviations from the theoretical curve become larger as the sphere becomes larger and size effects become significant. Results are presented for a rectangular box, spherical container, square and one-dimensional rod.