Spectral Analysis of the Anisotropic Neutron Transport Kernel in Slab Geometry with Applications

Abstract

A spectral analysis of the transport kernel for anisotropic scattering in finite slabs is achieved by first solving a type of generalized scattering problem for a subcritical slab. Initially, the scattering problem is stated as an inhomogeneous integral transport equation with a complex-valued source function. This is readily transformed to singular integral equations and linear constraints in which the space and angle variables enter as parameters. Dual singular equations appear in applications of Case's method to transport problems, but we cannot yet completely explain this duality. The singular equations are transformed to Fredholm equations by an extension of Muskhelishvili's standard method and by analytic continuation. It is shown that, for a wide class of scattering functions, this particular Fredholm reduction yields equations which converge rapidly under iteration for all neutron productions and slab thicknesses. The ultimate solution of the singular equations contains arbitrary constants which, when evaluated by the aforementioned linear constraints, display explicitly the Fredholm determinant and the eigenfunctions of the transport kernel. An immediate consequence of this result is the criticality condition and the associated neutron distribution. Specific applications to linear anisotropic and isotropic scattering in slab geometry are discussed. In addition, it is seen that the case of isotropic scattering in spheres can be treated with this method, and, in fact, the spectral analysis of the kernel for the slab problem immediately applies to the sphere kernel.