Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

190 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsflight from P to P'. [In fact, it was mentioned in Chapter 3 and also will be seen in Section5. V.B. that the simplest choice of w(P,P') is the ratio of the nonanalog and analog transitionkernels at the argument (P,P').J Making use of this rule, the weight in the game withoutsplitting is changed asW= W- w(P,P')Similarly, in the game with splittingW' = W • w(P,,P')Thus, in view of Equation (5.108), condition 1 reads1. - ~kW(P,P')W = W(P 11P') 2 &(PnW) 2 W ( o kk= 1 i= 1Condition 2 amounts to saying that the expected contribution due to an intercollisionflight must not be influenced by a possible splitting.Note that the theorem has an important practical consequence. Recall that the statisticalweight of a particle is changed in a free flight because the transition kernel T. which is usedto select the length of the free flight, is different from the analog kernel. Let us realize thatcondition 1 leaves some freedom in choosing the postsplitting weights since the conditionconcerns the total weight of the split fragments at the next collision, while this weight isdetermined by the weights of the fragments after the splitting and by their change due tothe reselection of the free flight at the site of splitting. In practical realizations of theprocedure, this uncertainty must be excluded. Now it is physically reasonable to fix therules according to the following scheme.1. The starter is at P and its weight is W. LetP 0= P, W 0= W2. Select a point Q from f(P 0,Q): Q = (r c+ Dto.E).3. Select a point P 1from t(P 0,P,): P 1= (r c+ D,co,E).4. If D 15- D, then let the next collision point of the particle be P' - Q and let its weightand contribution be determined as in a game without splitting for a free flight from Pto P', i.e., they will be W 0and W' 0f(P,P'), respectively.5. If D 1< D, then split the particle into k fragments with a probability g k(Pi,W 0).6- If k = 0, let the contribution be W 0f g(P,P,) as defined by Equation (5.114) and startto process a new particle.7. If k > 0, let the weight of the i-th fragment be W (1)ksuch thatk2 Bk(P 1-W 1,) I W (]>k= W 0k= 1 i= 1Take the fragments one after another. For the i-th fragment, set P H= P 1and W n=and repeat steps 2 through 6 until every fragment reaches a collision point or is eliminatedfrom the system.W (1)k