Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

193where we use the notational conventionTr(P,W (0)k)s) = 5(s) (5 J 20)The connection between •d and -n is due to the collision process and it follows from Equation(5.76) in Section 5.III.A ast1(P',W',s) = c a(P')p a(P',W a ,s) +c s(P')|dP"C s(P',P")p s(P',P",W",s)* T](P'\N",S) + c f(P') E 4,(P') H * JdP",,C n(P',P",))p n(P',P^15 W(J) 1S)* T 1(P 1^1W(V 1S) (5.1.21)Finally, TJ and -ft are interdependent due to collisionwise splitting according to Equation•(5.95) in Section 5.III.D asTS(P",W",S) = X z m(P",W") N*ir(P",W' 0). m)s) (5.122)m = 0J = Owhere again the convention of Equation (5.120) is valid.Equations (5.119) through (5.122) describe the score probability in a general MonteCarlo game. Their moments can be obtained according to the procedures followed in theprevious chapters. Here we leave the description of the general game since all its relevantinstances have already been discussed. On the other hand, the system [(5.119) through(5.122)] as a whole is mainly of theoretical interest since, in practical cases, one or anotherspecific trick is to be investigated at a time and very seldom a general procedure.I). INCLUSION OF TIME DEPENDENCEWe have so far limited our discussion to problems where time dependence plays no roleeither because the quantity of interest is the cumulative effect of a certain amount of particlesor because the distribution of particles is assumed to be constant in time and the estimatedquantity is some reaction rate per unit time. In the sequel, we refer to such problems asstationary.On the other hand, simulation of particle transport inherently involves modeling ofevents at successive times and therefore estimation of time-dependent quantities is not incontrast to the very nature of Monte Carlo methods.Time-dependent problems may be classified into two main types. In the first type ofcalculation, evolution of a certain quantity, e.g., reaction rate in or escape rate from a givenregion, is investigated as a function of the time elapsed since the start of the particle's batch.The second type of time-dependent estimation concerns the investigation of the variation ofsome quantity due to the change in time of some characteristic function [e.g., the weightingfunction in the RHS of Equation (5.2) or the kernels that govern the transport]. Monte Carlosimulation of the second type of time dependence is only occasional (e.g., for investigationof reactivity change due to the moving components of a nuclear reactor) and will not bediscussed here.Moment equation accounting for evolutional-type time dependence were first derivedby Booth and Cashwell 4 and by Booth 5 for multiplying nonanalog games with geometricalsplitting. To illustrate the method to be used when deriving time-dependent moment equations,we consider a nonanalog nonmultiplying game. Extension to more complicated cases(with multiplication or splitting) goes along the lines of the previous Chapters.