Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

468 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsthe collision density Q is connected to the transmission probabilities asC 1= C(I - t,) = C(T 1., ~ T 1)ZT 1This relation states that the number of collisions in a region is approximately proportionalto the nontransmission probability through the region. Inserting this approximation intoEquation (7.73), the quasi-optimum importances becomeI 1= a(T 1.. 1T 1)- 1 ' 2 (7.75)Here we have left m bulk parameters to be fitted. If we further assume that from preliminaryruns or by other means we know that the (physical) particle flux in the system drops like4> ~ exp( —Xx)with some positive X, then the transmission probability can be approximated asT 1== e- X x iand the importance of the region becomesI 1= aexp[X(x, + x, „0/2] (7.76)The number of split fragments due to a particle entering region i follows from Equation(7.59) asa, = 1/V 1= CXp[X(X 1- x,_V2] (7-77)Thus, for an approximate optimization of the splitting procedure, the only quantity to bedetermined in test runs is the exponent in the spatial flux-drop. Note that Equation (7.77)is a direct generalization of the approximate single-surface result in Section 7.1.D. Numericalexperiments show (cf. Section F) that the optimization method proposed in this section givesreliable results, and considerable gain in efficiency is obtained by its use in realistic problems.37Equations (7.73) and (7.76) define an easy-to-use splitting procedure applicable in mostof the deep-penetration problems. The strategy based on Equation (7.76) takes into accountthe overall characteristics of the system (through the average exponential drop of the flux).In the method defined by Equation (7.73), the number of split fragments is also influencedby the specific properties of the region entered.C PROPERTIES AND REFINEMENTS OF THE METHODIn the derivations above, we have assumed that no biasing technique other than splittingand Russian roulette is applied in the game. Survival biasing, however, is almost alwaysapplied in practical calculations, and therefore it may seem doubtful that the splitting schemeabove is also optimal in practical situations. This doubt, however, is groundless as it canbe shown 37that for splitting surfaces situated equidistantly in a homogeneous slab, the quasioptimumimportances are exactly the same as the ones given in Equation (7.73), irrespectiveof whether survival biasing is used or not. It can also be proven that even if the splittingsurfaces are located at arbitrary distances, introduction of survival biasing hardly influencesthe quasi-optimum importances. Note that equality of regionwise importances does not meanthat the number of split fragments in games with and without survival biasing are equal. In