Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

The contribution functions f(P,P') are usually called track length-ty < ., .most commonly used such estimator is proportional to the optical track 1« • «(h .. i , .between two successive collision points (cf. Chapter 4. V). f s(P',P"), anecollision-type estimators, for obvious reasons (cf. Chapter 4.Y). In the >applications, they do not depend on the postcollision coordinates P". f.,ithe absorption or last-event estimator which scores at the collisions thatIf the contribution function assigned to a free flight depends only ( i*P, i.e.,f(P,P') ^f(P)then it is called an expectation-type or next-event estimator because the cothe flight does not depend on the actual length of the free flight and henceexpectation over the next collision points (cf. Chapter 3.II), Obviously, tpectation-type partially unbiased estimator is the source term I(P) of EquatioEquation (5.134) becomes especially simple in special cases. If, for ex;nonvanishing estimator is the track length-type estimator, then it. is partial!;JdP'T(P,P')f(P,P') = JdP'T(P,P')f(P')On the other hand, if collisions also contribute to the score but their contributions areindependent of the postcollision coordinates and of the number of secondaries emergingfrom the collision, i.e., iff s(P',P") = f„(P',P") = f c(P')then Equation (5.134) reduces to|dP'T(P,P')[f(P,P') + c a(P')f a(P') + c(P')f c(P')] = [dP'T(P,P')f(P')where c(P') is the mean number of secondaries per collision at P' as defined in Equation(5.19).In the derivations of this chapter, we have assumed that the contributions to tbfrom various events in a history depend only on the coordinates characteristic to the e , i «and are independent of the sequence number of the collision point in the history tothey are related. In other words, we assume that identical events give identical contrilirrespective of the stage of the simulation at which they occur. It is this assumptimakes it possible to construct integral equations for the score moments, and on!estimators belong to the class of the partially unbiased estimators. Although in the ITof the practical cases partially unbiased estimators are applied, it is obvious that theyexhaust all the possible unbiased estimators. In fact, it is possible to define a wide cunbiased estimators, the forms of which explicitly depend on the sequence numberactual collision point in the history. Such estimators are treated in full generaKhisamutdinov 19and also by Mikhailov." Certain special forms of such estimators ztreated in Reference 40. As these special estimators have limited application in usualCarlo problems, they will not be detailed here. A special application of such estim;demonstrated in Section 5.IX.A.B. WEIGHT GENERATION RULESThe second important open question of a general Monte Carlo simulation is how to