Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

445and on the optical thickness X. Analytical minimization of Equation (7.7) is cumbersomebut it can be easily minimized by simply calculating Q(n) at successive integer values of itIn Table 7.1, optimum splitting ratios are given for selected values of c and X.TABLE 7.1CX 0.1 0.2 0.3 0.4 0.5 0.62.5 1 1 1 1 1 [3.0 2 1 1 1 1 14.0 2 2 2 1 1 15.0 2 2 2 1 1 110 3 ? 2 2 2 115 3 3 2 2 2 120 4 3 2 ? 230 4 3 3 2 2 240 5 3 3 2 2 250 5 4 3 3 2 2The numerical results obtained from the mode! are certainly not realistic; however, theyreflect the main tendencies expected from more rigorous models. Thus, it is seen that theoptimum number of split fragments per collision increases with increasing absorption (decreasingc) and also with increasing thickness of the slab. It is also observed that the optimumsplitting ratio varies very slowly as the thickness of the slab is increased and therefore thereal optimum is very likely a noninteger value of n. In other words, collisionwise splittingcan only be changed in rough steps and does not seem to be a sufficiently fine tool ofefficiency maximization. (One could object that a noninteger splitting ratio can also berealized by allowing for two possible outcomes of every splitting with appropriate probabilites. It can, however, be seen that even in this case, collisionwise splitting is less flexibleand efficient than geometrical splitting.)Geometrical (or surface) splitting can also be optimized analytically in the straightaheadmodel. Let us consider again the transmission of particles through a slab of thicknessX in which the mean number of secondaries per collision is equal to c. Analytical treatment,of the problem is especially simple in purely absorbing media. Efficiency of surface splittingin this case was investigated in Reference 15. However, in most practical cases, optimizationof splitting is related to deep-penetration calculations with mild absorption and thereforeresults from a purely absorbing model are not very enlightening. The straight-ahead scatteringmodel offers a simple tool for investigating the geometrical splitting procedure.Let a splitting surface be situated at x = x», i.e., assume that a particle is split into nfragments whenever it crosses this surface. With the notations of Section 5.IV.B, this meansthat the only nonvanishing splitting probability is g n(x) and it is nonzero at x = x, only:§k.n if x = x.0 otherwiseFurthermore, the weights of the split fragments areW (l)„ = 1/n, (i = 1,2, n)