Importance Function Approach

This method is utilized to calculate perturbations in the reactivity of a multiplying system due to changes in cross sections. This formulation makes full use of the properties of the importance (adjoint) function [Mat72, Hof72a, Hof78]. Development of this perturbation theory is started from two equations, proposed by Ussachoff [Uss55], representing the homogeneous transport equation for the unperturbed system and the corresponding adjoint equation for the perturbed system. From these two equations an expression for $\Delta$K is derived in terms of the importance function of a fission neutron, the fundamental mode fission neutron production function, and the fission kernel. These three kernels are then expressed in terms of the Green's functions for the nonmultiplying systems. Application of the difference flux concept [Bra70, Hof72b], to the Boltzmann transport equations satisfied by these Green's function, allows the expression of $\Delta$K in terms of the unperturbed and perturbed fission cross sections, the angular flux generated in the unperturbed system by the fission source, and the angular flux generated in the perturbed system by the perturbation source [Mat72].

A Monte Carlo simulation starts by guessing the initial fission source distribution in the unperturbed system and the importance function in the perturbed system. Simulations of the fission neutrons in the unperturbed system gives the angular flux in the unperturbed system, the perturbation source, an uncorrected importance function, and the fission source distribution for the next generation. Simulation of the perturbed source particles gives the angular flux in the perturbed system. After several of these iterations, the fundamental mode eigenfunction is achieved, and estimations for $\Delta$K can be accumulated.