# Diffusion Theory

In previous section we dealt with the multiplication system and we defined the infinite and finite multiplication factor. This section was about conditions for a stable, self-sustained fission chain reaction and how to maintain such conditions. This problem contains no information about the spatial distribution of neutrons, because it is a point geometry problem. We have characterized the effects of the global distribution of neutrons simply by a nonleakage probability (thermal or fast), which as stated earlier increases toward a value of one as the reactor core becomes larger. In order to design a nuclear reactor properly, the prediction how the neutrons will be distributedthroughout the system is of the highest importance. This is a very difficult problem, because the neutrons interacts with differently with different environments (moderator, fuel, etc.) that are in a reactor core. Neutrons undergo various interactions, when they migrate through the multiplying system. To a first approximation the overall effect of these interactions is that the neutrons undergo a kind of diffusion in the reactor core, much like the diffusion of one gas in another. This approximation is usually known as the diffusion approximation and it is based on the neutron diffusion theory. This approximation allows solving such problem using the diffusion equation.

In this chapter we will introduce the neutron diffusion theory and we will examine the spatial migration of neutrons to understand the relationships between reactor size, shape, and criticality, and to determine the spatial flux distributions within power reactors. The diffusion theory provides theoretical basis for a neutron-physical computing of nuclear cores. It must be added there are many neutron-physical codes that are based on this theory.

First, we will analyze the spatial distributions of neutrons and we will consider a one group diffusion theory (monoenergetic neutrons) for a uniform non-multiplying medium. That means that the neutron flux and cross sections have already been averaged over energy. Such a relatively simple model has the great advantage of illustrating many important features of spatial distribution of neutrons without the complexity introduced by the treatment of effects associated with the neutron energy spectrum.