Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

196 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsthe mean number of secondaries per collision. The solution of this equation is easily obtainedascjl - e-"- c ) , ]/(l - c) if c < 1c,t if c = l(5.125)The stationary expected absorption rate follows from the solution Equation (5.125) by lettingt tend to infinity, and it is seen that the solution tends to a finite value with increasing timeif c < 1, but it is unbounded at c = 1. This is in accordance with Theorem 5.1 since theconditions of the Theorem are met only if c < 1. Finally, we note that in view of the resultsin Section 5.1I1.C, the solution for c < 1 in Equation (5.125) also defines the expectedabsorption rate for c > 1.V. ANALYSIS OF THE FIRST-MOMENT EQUATIONWe have established the moment equations that describe the expectation of variouspowers of the total score. There remains, however, a number of open questions. First, it isto be clarified what kind of contribution functions result in estimates of the required reactionrate, Equation (5.2). Second, it is not yet clear how to choose the statistical weight of aparticle in order to keep a nonanalog game unbiased, i.e., in order to ensure that a nonanaloggame does result in the same expected score as the analog game it corresponds to. Thesequestions will be answered in Sections A and B of this chapter. We consider here the firstmomentequation of such multiplying analog and nonanalog games in which the secondariesof a multiplying event are indistinguishable. The considerations below can be easily generalizedto the case of distinguishable secondaries. We remind the reader that the statisticalweights of the split fragments in a game with splitting are essentially arbitrary (except forthe conservation rules in Theorems 5.4 and 5.6) and they do not follow from the requirementof unbiasedness. Selection of the split weights will be investigated in Section 5.VIILI.For easy reference, let us recall the first-moment equation for a multiplying game. Theequation concerning an analog game is given in Equation (5.80) asM 1(P) = I 1(P) + JdP'T(P,P')JdP"C(P',P")M 1(P") (5.126)whereI 1(P) = JdP'T(P 1P') [f(P,P') +C 8(PX(P')+ C 5(P') JdP" C 8(P' ,P") f s(P' ,P")+ c f(P') 2 nq n(P')JdP"C n(P',P")f n(P',P")j (5.127)andC(P',F') - c s(P')C s(P',P") + C 1(P') i nq„(P')C„(P',P") (5.128)