Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

370 Monie Carlo Particle Transport Methods: Neutron and Photon Calculations3. In the first two forms of termination, the neutron yields no contribution to the reactionrate. If a fission occurs at the phase-space point P" = (r'.E') then the contributionof the history isW(r)v(Pjf(r')where y(P") is the mean number of fission neutrons leaving a fission at P".A rigorous proof of the unbiasedness of the estimation procedure in steps 1 through 3 forestimating the reaction rate (f,ZS) is given in Appendix 6A.A number of the simulation and estimation techniques introduced in Chapter 5 can alsobe applied here to modify the above procedure. The most obvious possible alteration of step3 follows from the theory of partially unbiased estimators. Accordingly, the scoreW(r)v(P")f(r') per fissions can be replaced by the sum of the scores W(F)C 1(P 1MP 1)Rr 1) overall the collision points in the history, where c,(P () is the probability of a fission in a collisionat P 1. Let us now consider some of the possible ways of estimating the perturbation in theeffective multiplication factor.1. We shall first assume that the converged, unperturbed fission-neutron density andthe corresponding (but unknown) perturbed density are normalized as(f,S) = (f,S) - 1 (6.138)Then the perturbation m k t:11in Equation (6.137) reduces took = (LSZS) + (f,ZSS) (6.139)The first term on the RHS is easily estimated in a correlated game since it is of the formof a reaction-rate perturbation8R = (L(Z-Z)S) = J drf(r)