The Spatial Distribution of Moderated Neutrons

Abstract

In this chapter we shall calculate the space and energy distribution of neutrons during the slowing-down process. Our goal will be to obtain the flux Φ (r, E) or the slowing-down density q (r, E) arising from given sources. It will turn out that this general problem is considerably more difficult than the two special cases of it previously discussed, viz., diffusion without moderation and moderation without diffusion. We can easily see the reasons for the difficulty if we consider, for example, the neutron field due to a point source of fast neutrons in an infinite medium. At large distances from the source, the neutron flux is predominantly due to neutrons that have made no collisions or at most a few small-angle collisions that produce only a small energy loss. Their distribution of directions is strongly anisotropie, and a description of the diffusion process by Fick’s law is no longer possible. For this reason, the treatment of “deep penetration” problems is particularly difficult. They are mainly of interest in shielding calculations, and we will consider them no further here¹ (cf. Holte as well as Verde and Wick). At smaller distances — this limitation will be made more precise later — the distribution of neutron directions (with the exception of the energy range immediately below the source energy) is only weakly anisotropic, and with certain limitations Fick’s law is valid. The flux can therefore be described approximately by an energy-dependent diffusion equation. Even this equation is not soluble in general, and we are forced to make further approximations.