Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

170 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsFor the sake of simplicity, we shall assume that the outcome of any collision is one ofthe following three events: (1) absorption [with a probability C 11(P')], (2) scattering [with aprobability c s(P'), or (3) multiplication [with a probability c,(P') = 1 - c a(P') - c s(P')].Thus, in accordance with the notations of Section 5.I.D, the collision kernel has the formC(P',F') = c„(P')8(P" - P) T 6 S(P')C S(P',P") + c t(P') 2 nq„(P')C„(P',P") (5.74)TL= 1where we recall that the score probability and the contribution functions vanish at P. Alternativeforms of the scattering kernel will be considered in Section E. The equations willagain be derived for a general nonanalog game which is played with the kernels f and C.The score probability equation and the moment equations are derived in Sections A andB, respectively. It is shown in Section C that almost any nonanalog multiplying game canbe replaced by an equivalent nonmultiplying game that results in an identical expected score.On the other hand, the widely used variance reduction procedure, called splitting, representsa special branch of the multiplying games (even if the physical process that is simulated isnonmultiplying), as is demonstrated in Section D.A. SCORE PROBABILITY EQUATIONLet -d(P',W',s)ds denote the probability that a particle entering a collision at P' with aweight W will yield (together with the particles it creates) a total score in ds about s. SinceTr(P, W,s) was defined as the same probability density due to a particle starting a flight, thetwo densities are related asTT(P,W,S) = JdP'f(P,P')p(P,P',W',s) * -&(P',W',s) (5.75)since the score due to the starter at P is the sum of the score in the first flight and of thescore resulting from the first collision and from the rest of the history. On the other hand,the score probability represented by -& is the sum of the probabilities associated with thepossible and mutually exclusive outcomes of the collision, i.e.,-STP',W',s) = c,(P')p,(P',W,s) + c,(P')JdP"C,(P',P")p»(P',P",W",s) * TT(P",W",S)+ C 1(P') S 4.(P') n * [dP'; ;)c„(P',p'