Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

322 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsunless other tricks are introduced. An alternative solution proposed by Rief and Fioretti" 7involves the introduction of delta scattering into an unperturbed analog game. Yet the problemis not completely solved by either method. Let us realize that the estimated effect of theperturbation will he biased to some extent, since the scattering (and possibly multiplicative)properties of the material, removed from the unperturbed system, will have no effect on thefinal result (no scattering is played there).An exact treatment of such problems cannot be performed by completely correlatedhistories. A possible solution is that the particles' histories are separated if a collision occursin the perturbed region, the perturbed particle suffers a delta scattering, or does not suffera collision at all, whereas the unperturbed particle enters the collision and then follows anindependent path. This makes the simulation unbiased, but weakens the correlation betweenthe two histories (which, in turn, tends to increase the variance of the estimated perturbation).A procedure of this type was proposed by Takahashi 81for the calculation of reactivityperturbations in pulsed fast reactors. A satisfactory solution of the problem is expected fromfurther investigations. 28The danger of a biased correlated estimate also is real in cases where the perturbedregion is not completely voided, but its optical dimensions are very small, and thus theprobability of a collision inside the region is low. In this case, only a minute fraction of thehistories will carry information on the difference of the scattering properties of the perturbedand unperturbed regions and therefore a bias, universally associated with the estimation ofrare but important events, is expected. This difficulty may be remedied by the method offorced collisions. 19 In this method, the history of the correlated particles is split into twobranches when it enters the perturbed region for the first time. The probabilities of acollisionless free flight through the region are calculated for both the perturbed and unperturbedparticles, and the first branch of the common history is restarted from behind theperturbed region; the weights of the restarted particles are multiplied by the respective freeflightprobabilities. In other words, the fractions of the particles that are expected to crossthe perturbed region without collision are deterministically transmitted through the region.The particles in the second branch will carry weights equal to the difference of the originalweights and those of the transmitted particles, and they are forced to enter a collision insidethe perturbed region. This is performed by selecting a free flight from a truncated transitionkernel concentrated on the perturbed region, as is done in the ELP method (cf. Section5.VIII.D). Let us note in passing that care must be taken in properly altering the weight ofthe perturbed particles when they are restarted from the boundary of the perturbed regionsince their weights were altered after the previous collision according to a free flight interruptedat the boundary, (cf. Sections 5.IV.A and B).In this method, particles migrating in the second branch of the history will carry informationon the scattering properties of the perturbed region. Therefore, these particles are tobe kept in existence until they contribute to the final score. They should not, for example,be split or Russian rouletted any more. 19It may be questionable whether scattering shouldbe forced at the first crossing only or perhaps also at repeated entries to the perturbed region.Nevertheless, it seems logical to limit the forced collision to the first entry as repeated forcedcollisions are likely to deteriorate the efficiency of the method, for two reasons. First, everyrepetition increases the number of histories and thus the number of collisions to be playedper starter. Second, the weight of a particle that has been forced to collide several timestends to be very low, for the method is used for optically transparent regions when theprobability of a free flight through the region is close to one.As mentioned in the previous section, in certain applications, a reference system is usedwhere the game is played analog in order to make the estimation feasible in both the perturbedand unperturbed systems. A special technique used in conjunction with correlated games inreference systems is discussed briefly below.