3. Spatial Effects in Modeling Neutron Diffusion: One-Group Models

The fundamental aspect of keeping a reactor critical was discussed in Chap. 2, and we found out that the most principle evaluation quantity of the nuclear design calculation is the effective multiplication factor (\( \overline{\sigma}\left(v,T\right)=\frac{1}{v}{\displaystyle \int {d}^3V\left|v-V\right|\sigma \left(\left|v-V\right|\right)}M\left(V,T\right) \)) and neutron flux distribution. We also so far have noticed that the excess reactivity, control rod worth, reactivity coefficient, power distribution, etc., are undoubtedly inseparable from the nuclear design calculation. Some quantities among them can be derived by secondary calculations from the effective multiplication factor or neutron flux distribution that was also discussed in Sect. 2.15 of Chap. 2. In this chapter we treat the theory and mechanism to be able to analysis and calculate the effective multiplication factor and neutron flux distribution and possibly show numerical analysis and computer codes involved with solving the diffusion equation in a one-dimensional and one-group models. The goal of this chapter is also for readers to understand simple reactor systems, the notion of criticality, what it means both physically and mathematically, how to analytically solve steady-state flux for simple geometries, and finally how to numerically solve the steady state for more arbitrarily complex geometries.