Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

58 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationson the balance between the time gain achieved by the introduction of the simpler densityfunction and the increase of the statistical uncertainty caused by the fluctuation of the initial(and inherited to all successive) weights.B. REPLACEMENT OF MULTIPLICATION BY INCREASE OF THE WEIGHTIn neutron transport in presence of fissioning materials v progeny neutrons with independentlydistributed energies and isotropic direction distributions can be released by fission.In the analog game first an actual integer value of n (see Section 3.I.E. point 5.) then nenergies and directions are selected and all the n neutrons are followed individually. Instead,one can choose a single energy and direction and follow a single progeny with a weight ofv — the expected value of n — times the weight the incident neutron that caused the fission(times the non-absorption probability, if absorption is replaced by weight reduction).Similarly, at pair-production interactions of photons, from the annihilation of the producedpositron two photons with the same energy are emitted. Here, again it is moreconvenient to follow a single photon with a starting weight of twice the weight of the incidentphoton (times the nonabsorption factor — if it is applied).C. RUSSIAN ROULETTE AND SPLITTINGThe purpose in introducing the weight was twofold. In Section A loss of paths byabsorption or leakage was prevented, in Section B unnecessary multiplications of simulationswere avoided. There may, however, be cases when just the opposite is profitable for theuser: unimportant paths should be stopped, or — in important regions — more independenthistories are needed than are at disposal.By the absorption and/or leakage replacement method the weight of the particle monotonicallydecreases and may reach very small values: following it any further is more orless a waste of time, since it will surely have only small contributions to the quantity to bedetermined (see the scoring formula [3.29]). In such cases the Russian roulette method canhelp. Let us decide that if a particle's weight (W) falls below a preset minimum W tn, weeither restore its starting (or any other preset) value, W 0, and continue its path, or we killit. The process is trivially correct if the survival probability (p) isp =Ww:and the history is terminated with a probability 1-p. In an alternative way, one can fix thesurviving probability p (), and increase the weight of the survivors toWW„ ew= -PoSince the importance of a particle depends not only on its weight but also, for example,on the region where (e.g., how near to the detector) it is, different minimum weights, orsurviving probabilities can be set to different regions.On the other hand, in very important regions we can increase the number of historiesby splitting the entering particle into several "fragments" which are afterwards followedindependently. If the particle is split into n new ones, "n for one splitting", the new weightis triviallyfor each progeny.Ww new= -n