Convergence and Error Estimate of Perturbation Method in Reactor Calculations

Abstract

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 mathematical 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 containing 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 fissionable 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 th...