Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

252 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationswhereFh n(P') = dP"C n(P',P")M,(P") (5.275)This form of the moment equation will be used for the definition of the nonanalog zerovariancekernels below. Let us consider a nonanalog game played by the transition kernelT(P 1P') and by the collision kernel C(P',P") above. Assume that the analog contributionfunction f(P,P') and the weight generation rules in Equations (5.142) through (5.149) areapplied; then this game is partially unbiased. The second moment of the score due to astarter with unit weight at P reads, according to Equation (5,81), asM 2(P) =JdP'T(P,P'){(W') 2 f 2 (P,P')+ 2Wf(P 1P') 2 nc n(P') I C n(IVr)W nM 1(P")+ 2 1 H" - Dc n(P')dP" C n(P', F)W nM 1(P")dP'T(P 1P') 2 nc n(P') |dP"C n(P',P")(W;;) 2 M 2(P")n " 1J(5.276)where, in view of the weight generation rulesW = T(P,P')/f(P,P')(5.277)andW n- -W'c„(P') C n(P',P 1 Vc n(P') C n(P',P")(5.278)Equation (5.276) follows from Equation (5.81) by putting W == 1 andf. = f = f = 0Then, reordering the terms in Equation (5.81) according to the powers of f(P,P') and makinguse of the relation above of the quantities c tand C 1(i = 1,2,...) to the reaction probabilitiesc rand kernels C 1., Equation (5.276) is obtained. In the derivation, the identitiesCa(P) + E 4(P) = 1anddP"C,(P',P") = 1have also been exploited.The game will result in zero variance ifM 2(P) - [M 1(P)] 2