A model of neutron multiplication for aggregates of randomly placed fissile spheres with random material properties in a background medium is presented in terms of the transport equation. Two distinct problems are examined: (1) small spheres in an infinite bulk medium in which the total cross section in the spheres and bulk medium are the same and (2) small spheres in a void or purely absorbing medium but with different total cross sections in sphere and medium. In both cases we consider criticality in which there are random material properties of the spheres and random positions in the container. The random sphere problem is studied statistically by calculating the multiplication factor for many thousands of cases with different positions and material properties and, from the results, constructing a probability distribution function for the multiplication factor. Some of the results are also calculated using diffusion theory and therefore we are able to give guidance on the likely errors caused by diffusion theory in this type of problem.

Although the problems are restricted to the one speed approximation, they may be applicable to fast neutron problems and we apply the work to spheres composed of random proportions of 235U and 238U. The work also has some bearing on the physical behaviour of pebble bed reactors which are of current interest, and in the storage of fissile waste. We have also discussed some of the underlying statistical problems associated with random arrays of spheres in a uniform lattice. In formulating our problem, we use the collision probability technique and as a by-product derive some new inter-lump collision probabilities for two spheres.