An integral equation arising in neutron transport theory

A classic problem in nuclear reactor physics is the calculation of the spatial distribution of fissile material to make the associated neutron flux distribution spatially constant. We examine a special case of that problem for an infinite slab of fissile material which is infinitely reflected on both sides by a non-multiplying material. The conditions for a constant flux are derived and lead to a singular integral equation. This equation is reduced analytically to a non-singular integral equation and the solution thereby obtained is compared with that from a direct numerical method. Some of the physical implications are examined. We also note that, contrary to a theorem for multi-group diffusion theory, the resulting total fissile loading of the system is not a minimum but rather a maximum. An important aspect of the present work is that transport theory is used and not diffusion theory. Indeed, we note that no solution exists for the corresponding diffusion theory model unless it is specially modified by the addition of generalised functions, and hence we note that the problem is intrinsically governed by transport effects.