Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

158 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsvations is performed in the greatest possible generality in order to enable the reader to usethe results for newly invented schemes of his own. The very nature of the derivations isquite simple; nevertheless, the great number of functions and quantities appearing due tothe general treatment makes the resulting formulas sometimes complicated. In case one isinterested in well-defined specific cases, it is recommended that the formulas be rephrasedafter having specified all the quantities (contribution functions, weights, and probabilities)characteristic of the special cases. By doing so, for simple specific cases, the momentequations become equally simple.II. MOMENT EQUATIONS IN NONMULTIPLYIN(i GAMESEquations describing various moments of the score in a nonmultiplying game were firstderived by Amster and Djomehri 1for analog simulations where the contributions to the finalscore may originate from scattering and absorption events. The theory has been generalizedto track-length estimators by Booth and Amster 3 and to nonanalog games by Lux. 28 In thefollowing sections, a general derivation is presented which results in equations that accountfor the expectation of an almost arbitrary function of the score in a nonanalog nonmultiplyingsimulation.* Here, and in the following Sections, it will be assumed that the nonanaloggame at hand is feasible, i.e., that the total weight of the particle in the system tends tozero if the number of collisions increases. In Chapter 5.V, conditions are derived underwhich a nonanalog game which is unbiased with respect to a feasible analog game (i.e.,which results in the same expected score as the analog game) is also feasible.A. SCORE PROBABILITY EQUATIONSFirst, we derive the equation that describes the score probability IT(P,W,S) defined inEquation (5.29). Let us introduce truncated score probabilities by the following definition:let ir K(P,W,s)ds be the probability that a history that starts at P with a weight W and consistsof exactly k collisions will yield a total score in ds about s. With this definition, the scoreprobability in Equation (5.29) readsTT(P,W,S) = V Tr k(P,W,s) (5.42)k~ IThe probability density of a score s from a history of exactly one collision satisfies therelationTT,(P,W,S) = JdP'f(P,P')p(P,P',W',s) * p A(P',W a ,s)c A(P') (5.43)where the asterisk denotes convolution with respect to s, i.e.,a(s) * b(s) = J ds'a(s - s')b(s')Note that in nonmultiplying gamesc a(P') - 1 -- JdP''C(P',P'') - 1 - c s(P') = I - c(P')The meaning of Equation (5.43) is obvious: it is the probability that the shortest possible* Further generalizations of the moment equations by Booth, 4 Booth and Cashwell, 5 Sarkar and Prashad, 41 andothers, will be discussed in subsequent chapters.