Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

204 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationswhich may be greater than unity, thus resulting in a multiplying game. Similarly, in caseof a general analog kernel, the transformation may result in a collision kerne! in which purescattering also leads to multiplication. This leaves some freedom to the user in defining theabsorption, scattering, and multiplication probabilities. This question will not be investigatedhere, but should be considered in practical applications. In what follows, we use the generalform of the collision kernel as given in Equation (5.74) (omitting the irrelevant absorptionterm). Accordingly, the transformed kernel from Equation (5.152) readsC(P',P") - c\(P')C s(P',P") + C 1-(P') X nq n(P')C n(P',P")V(P") c s(P')C s(P',P") + c f(P') V nq n(P')C n(P',P"),/U(P') (5.155)for every P' and P" inside the domain of simulation.We shall now derive an unbiased nonanalog game which is played by the kernels inEquations (5.151) and (5.152) and in which the statistical weight of a particle is not changed.In a sense, this game is very unusual: it is nonanalog, since the kernels differ from theanalog ones; nevertheless, it resembles an analog game because of the constant statisticalweights. Such games will be referred to as transformed games. They also differ from thenonanalog games treated in the previous section in that they are not expected to give unbiased"pointwise" estimates, i.e., the expected score due to a single starter will not necessarilybe the same as in the analog game. Instead, we require that it produce an unbiased estimateof the final score, i.e., of the integral quantity of interest. Let M 1(P) denote the expectedscore due to a unit weight starter from P and let Q(P) be the (yet undefined) nonanalogsource density. The transformed game is unbiased ifdPQ(P)IOi,(P) -JdPQ(P)M 1(P) = Rwhere W is the statistical weight of the starter from P in the transformed game. This weightis defined by the equality itself in the sense that proper choice of it will eventually ensurethe unbiased final estimate, as will be seen below.Again let f(P,P'), f a(P'), f s(P',P"), and f n(P',P") be the contribution functions assignedto a transition, absorption, scattering, and n-tuple multiplication event, respectively, in thetransformed game. Obviously, the transformed contribution functions are not expected tobe partially unbiased and, in general, will be different from the analog estimators.The equation that governs the expected score due to a unit weight starter follows fromEquations (5.129), (5.130), (5.151), and (5.152) asA(P) = i,(P)/V(P) + JdP' T(P 5P') J dP" C(P', F)Jt 1(P")= .¢, (P)/V(P) + J dP' T(P 5P') J dP"C(P' ,F') V(P")it,(P")/V(P) (5.156)where^1(P) =dP'T(P 5P') U(P')f(P,P') +c a(P')f„(P')+ E,(P')JdP"C s(P',p")f s(P',p")+ c f(P') I nq„(P') fdP"C n(P',P")f„(P',P")(5.157)