On the Initial-Value Problem of the Transport Equation for a Moderator Slab

Abstract

A mathematical analysis of the time-dependent neutron transport equation is presented for the case of a moderator slab. The scattering of neutrons is treated on the basis of the Van Hove theory and is assumed to be isotropic. First, the spectrum of the corresponding transport operator is investigated and it is shown that the complex λ-plane is decomposed by the operator into a band spectrum Re(λ) ≤ — (v∑t)min, a discrete spectrum on the real axis λ≤— (v∑t), min a resolvent set Re(λ)> — (v∑t)min deleted by the discrete spectrum. The discrete spectrum is carefully studied and is found to consist of isolated eigenvalues or otherwise empty. For all categories of moderator, there exists an upper limit to the thickness of a slab having empty discrete spectrum. A slab exceeding this limit has a finite number of eigenvalues if the moderator material is gas or solid. This is also true for a certain class of liquid moderators. For other liquids, there is another critical thickness for the slab thickness such that if it is exceeded, the set of eigenvalues turns out to be denumerable with an accumulation point at — (v∑t)min. Each eigenvalue possesses a finite multiplicity, and the index is one. Finally the related initial value problem is considered, and it proves to have a unique solution upon application of the Hille-Yosida theorem. The contribution of the band spectrum to the solution is evaluated and is shown to decay faster than exp{ — (v∑t)mint}.