Normal Mode Expansion in Neutron Thermalization Theory with a Simple Scattering Kernel

Abstract

Elementary solutions to the energy-dependent Boltzmann equation with a one-term degenerate scattering kernel are derived in plane geometry, and the weight function W(z) is obtained which makes these solutions mutually orthogonal over the half-range interval of the continuum. The weight function greatly facilitates determination of the expansion coefficients in general solutions and is applied to the problems infinite half space.
The diffusion length (discrete space eigenvalue) υ₀ is exactly expressed by using the halfrange characteristic function X(z). In a 1/ν-absorbing medium, as the absorption concentration q increases from zero to a critical value q^*, the diffusion length decreases from infinity to the end of the continuum. 1/Σmin. For q≥q^*, υ₀ vanishes and the neutron density can be represented by the transient term alone, whose exact expression is obtained.