Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

201in an n-fold multiplication from P' to P" where the weight factor in Equation (5.149) wassplit up asw,„(P') =wJP'.P") =c,(P')q n(P')/£ f(P')q n(P')C n(P',FyC n(PVP")and2. if the nonanalog and analog contribution functions (estimators) are related as|dP'T(P,P')|f(P,P') - f(P,P') + c a(P')[f,(P') - f a(P')j+ c s(P')jdP"C 5(PVF)K(PVF) - f s(P',F)]+ c f(P') E nq„(P')JdFC n(PVF)[f n(P',P") - f„(P',P")]} - 0Proof. Simple substitution of Equations (5.142) through (5.149) into Equations (5.12?)through (5.131) shows that the equality (5.140) is satisfied, while condition 2 is just thedetailed form of Equation (5.141) after substitution of the weights. Thus, the conditionsabove imply Equations (5.140) and (5.141) as was to be shown.LlEquations (5.142) through (5.149) will be called the weight generation rules of a nonanaloggame. The rules have an obvious interpretation. In a nonanalog game, the coordinatesof an event are selected from a probability density different from the analog (physical) one.The weight of the particle participating in the event, however, is multiplied by the ratio ofthe analog and nonanalog probability densities, i.e., if the probability of an event in theanalog game is lower than in the nonanalog, only a "fraction" of an analog particle takespart in the event of increased probability and vice versa. Equation (5.140) expresses the factthat the effective number of particles undergoing various events in a collision that followsa free flight is the same in both games.It is to be emphasized that the weight generation rules of the theorem represent sufficientbut not necessary conditions of the fulfilment of Equation (5.140), In some applications,e.g., the multiplying part of the collision kernels in the nonanalog and analog games cannotbe related as in Equations (5.148) and (5.149) because the possible number of secondariesare different in the two games. In such cases, the weight generation rules in Equations(5.148) and (5.149) can be replaced by the implicit relationW" °° , w"C 1(PVCfPVF)^ 7 7+ C 1(P') 2 nq„(P')C n(PVP")W n- 1 VV= c,(P')C 8(PVP") + C 1(P') 2 Hq n(PVC n(PVF) (5,150)n= iEquation (5.150) follows from Equation (5.140) by making use of conditions (5.142) ana(5.143) to obtain— C(PVP") -= C(PVF)