Subcriticality for transport of multiplying particles in a slab

Abstract

A commonly used method for theoretically establishing subcriticality of a given physical system is to show that the corresponding time-independent linear integral transport equation has a nonnegative solution for any nonnegative inhomogeneous term. Olhoeft [l] used this approach to establish subcriticality for the physically obvious case of nonmultiplying transport (see Section 2 for precise definition of terms). Case and Zweifel [2] extended this result to permit the possibility of multiplying particles, provided the underlying region has a finite optical diameter. The latter restriction excludes the computationally important case of transport in a slab. Keller [3] used this method to establish a subcriticality theorem which permits the possibility of multiplying transport in a slab of finite thickness, but is restricted to monoenergetic transport with isotropic distribution of secondary particles. Our aim here is to apply this method to establish subcriticality theorems for multiplying transport of variable ennergy particles in a slab of finite thickness, with a general law of distribution of secondary particles. The basic definitions and notation are given in Section 2. Some preliminary propositions are collected in Section 3. An immediate corollary of these propositions is the above-mentioned theorem of Olhoeft in its restriction to slab geometry. The objective in Section 4 is to show that existence of a nonnegative solution of the integral transport equation for every nonnegative inhomogeneous term is equivalent to uniform convergence of the Neumann series of the integral transport operator. In Section 5 we use this equivalence