Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

457andH(x,y) = y/(l + xy)The quality factor in the analog game isQ = (M 2- M 2 )N = M 2M 1[I - H(c,P7)[Obviously, the Russian roulette procedure increases the efficiency ifS(n) = Q/Q < 1i.e., ifS(n) = [G„(c,cP c) - H(c,P c)]/{G n(c,P c)[l - H(c,P c)]} < 1 (7.38)The ratio S(n) was evaluated 34 - 35for various values of the first-flight collision probabilityP and survival probability c, and it was found that for not-too-large regions (P c< 0.6), theincrease of efficiency offered by Russian roulette does not exceed 5%. This means that forregions with characteristic dimensions less that about 1 to 2 mean free paths, Russianroulette does not pay off. It has also been seen that for such regions, the efficiency is verysensitive to the value of n (i.e., to the number of collisions before Russian roulette), whichis connected to the Russian roulette parameter according to Equation (7.28). Therefore,there is a risk that with an improper choice of the parameter, the efficiency is decreasedeven in cases where an optimum parameter would guarantee a moderate increase. Furthermore,in medium-sized bodies, Russian roulette is not efficient in heavy absorbers and,again, the efficiency varies quite drastically with the variation of the survival probabilityThe calculations show that survival biasing with Russian roulette may result in a considerableefficiency increase in large bodies (Pc> 0.8) and especially for not-too-strong absorbers(c > 0.4). In these cases, the efficiency is a slowly varying function of the Russian rouletteparameter. 34In Table 7.2, the calculated efficiency ratio S(n) in Equation (7.38) is comparedto numerical experimental values. The latter were obtained from a Monte Carlo simulationof the collision rate in a sphere of optical radius 3.61 (P c= 0.8). The calculations andexperiments were carried out for selected values of the survival probability c and numberof collisions n before Russian roulette. The Russian roulette parameter is deduced from thesequantities according to the relationw th=c nTABLE 7.2C0.3 0.5 0.7 9.911 S(n) Exp li S(n) Exp n S(n) Exp n S(n) Exp2 0.90 0.90 2 0.86 0.83 3 0.83 0,85 6 0.88 0.853 0.98 0.99 3 0.82 0.80 5 0.79 0.78 Ii 0.85 0.824 1.12 1.15 4 0.84 0.81 7 0.8! 0.80 17 0.85 0.82There is a rule of thumb well known by practitioners for choosing the Russian rouletteparameter. According to this popular wisdom, w th= 0.1 is a reasonable choice. Theapproximate results above seem to corroborate this guess in the sense that the optimum