On Continuous Eigenvalues in Neutron Thermalization

Abstract

The continuous eigenvalue spectra of the linearized Boltzmann operator describing the energy distribution of neutrons in an infinite Einstein crystal are studied. This operator consists of two terms: a multiplication operator and an integral operator with δ-function-type singular kernel. The eigenvalue problem is transformed into the solution of an inhomogeneous integral equation by applying Case's method on the one-velocity transport equation. The existence of the solution of the integral equation is examined by the Neumann-series expansion. It is found that for sufficiently low temperature the range of numerical values of the multiplication operator forms the continuous eigenvalue spectra of the Boltzmann operator and the corresponding eigenfunctions are of δ-function type.