Convergence and Error Estimate of Perturbation Method in Reactor Calculations

Abstract

A general theory of obtaining the higher order perturbation terms up to any desired order was given previously by the present author. However, the perturbation series is not always convergent in all problems of practical interest, and it is generally not so easy to obtain the convergence criteria for the perturbation method in a strict mathe-matical sense. When Kato's theorem developed for linear operators in Hilbert space is applied to the problem of convergence, it can then be treated rigorously in one-group diffusion approximation. The resulting convergence criterion takes a simple form con-taining only the basic parameters of the reactor system. The perturbation series is convergent if the conditions 1>{2|ρ(1)f|/d+3|ρ(1)fs|} and 1>2|ρ(1)c/d are satisfied for fission-able and absorbing materials respectively. When only the capture cross section is changed, the higher order perturbation series with the first n terms has the error en ?? (2/d)n|ρ(1)c|n+1/(1-2 |ρ(1)c |/d). In the formulae, d is the level distance of an eigenvalue in the unper-turbed system, and ρ(1)f and ρ(1)c are the first order reactivity worths when fissionable and absorbing materials are inserted into the system, respectively. And, ρ(1)fS is the first order reactivity worth when the value of the fission source term is changed. The numerical calculations for prototype and 1, 000 MWe fast reactors are presented to show practical applicability of the higher order perturbation method.