Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

176 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsComparison of Equations (5.79) and (5.86) shows that the two equations are identical if thefollowing two conditions hold.1. The statistical weight of a particle emerging from a collision is independent of thenumber of secondaries in the collision and is the same in both the physical andhypothetical games. Denoting this weight by W", the condition is formulated asw: = w; = w (5.87)2. The contribution functions in the two games are related asJdP"C(P',P")W"f,(P',P") = c a(P')W a f a(P') + c s(P')|dP"C s(P',P'')W''f s(P',P'')+ 6 f(P') 2 nq„(P') fdP"C„(P',P")WT n(P',P") (5.88)H= I •>Condition 2 requires that the expected score contributed by a particle entering a collision atP be equal in the two games. We have thus proven the following.Theorem 5.3 — Under conditions 1 and 2 the hypothetical nonmultiplying game resultsin the same expected score as the (possibly multiplying) physical game.•For the sake of illustration let us consider a couple of cases when conditions 1 and 2are satisfied. Obviously, if the games are analog (with respect to the physical and hypotheticaltransport processes, respectively), then all the statistical weights are unity and Equation(5.87) is automatically met. The rules of generating the statistical weights in an unbiasednonanalog game will be derived in Section 5.V.B; nevertheless, it is heuristically obviouswithout any derivation that if in the nonanalog game only the transition kernel t is differentfrom the analog kernel, but the nonanalog and analog collision kernels are identical, thenthe statistical weights of the particles are not changed in a collision [except for the multiplicationby c(P') in the hypothetical game] and Equation (5.87) is again satisfied. Finally,if the nonanalog and analog kernels differ only in the mean number of secondaries percollision, i.e., if Equation (5.83) can be rewritten asC(P',F') = c(P')C,(P',P") =c(P')C(P',P")/c(P)then the postcollision weight is not affected by the type of the collision and condition 1holds. Note that in the three examples above, Equation (5.87) is satisfied in such a way thatthe pre- and postcollision weights are related asW" = W" = a(P')W' (5,89)with some function a(P') independent of the postcollision coordinates P". This relation holdsin most practical cases and if so, the postcollision coordinates P" are selected from the analogdensity, which also means that there is no need to change the statistical weight of the particleswhen they emerge from a collision. The rigorous proof of the reasoning above follows fromTheorem 5.8 in Section 5.V.B.The following examples demonstrate that condition 2 is also met in most practical cases.Let us first consider the case when the contribution functions f rdo not depend on the