# Uncertain Neutron Diffusion

Generally, every system possesses uncertainties caused by measurement errors, vague data and boundary conditions. Accordingly, the involved coefficients, parameters and constants become uncertain. Neutron diffusion is the backbone of a nuclear reactor. The scattering of a neutron occurs when it undergoes diffusion, which involves uncertainty due to the reactor parameters, measurement errors and boundary conditions. These uncertainties play an important role in the problems related to reactors, which should be understood well to help in a better design. In this chapter, we discuss the factors which may be considered as uncertain in a reactor system and then incorporate the corresponding idea of modelling.

## Uncertain Factors Involved in Neutron Diffusion Theory

The neutron collision inside a reactor depends upon the geometry of the reactor, diffusion coefficient, absorption coefficient, etc. In general, these parameters are not crisp and hence we get an uncertain neutron diffusion equation. Here, these uncertain parameters are taken as fuzzy. To investigate the uncertain spectrum of neutron flux distribution, we formulate the fuzzy finite element method (FFEM) with the linear triangular fuzzy element discretizing the domain.

When neutrons undergo diffusion, they suffer scattering collisions with the nuclei, assumed to be initially stationary, and as a result, a typical neutron trajectory divides into a number of short path elements. These are scattering free paths. The average of these is the mean free path. When a large number of neutrons are considered, there is a net movement of neutrons from regions of higher to those of lower concentration. As the path of a neutron after scattering may not be known exactly, we may take the cross section and transport the mean free path as fuzzy. As a result, the diffusion coefficient lies in an uncertain region and becomes fuzzy. Similarly, the absorption coefficient may also be taken as fuzzy.