Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

482 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsrate, theno- a(i) « A 1ZF 1On the other hand, the absorption rate in region i is the number of particles entering, butnot leaving, the region, i.e., it can be calculated from the partial currents asA 1= Mc (i - 1) - iVP(i) + M (i) - M"(i - 1)Hence,t, + r, = -2(J 11(I) = -2LM + (i - 1) - M + O) + M-(D + M"(i - I)]ZF 1(7.122)where, as before, F) is the flux integral in region i. Two remarks are in order here. First,for the sake of brevity, we shall call M- (i) the partial currents through the surface i, althoughin a strict sense they are not currents, but numbers of crossing particles. Nevertheless, inmost practical cases they are, to a fairly good approximation, proportional to the currents,and since all the formulas to be derived contain only ratios of M +, estimated currents canalso be used in place of the estimated number of crossing particles. Second, the absorptionrate introduced above is comprised not only of those particles that are effectively capturedin a collision, but also of those which leave the domain of simulation through other processes,e.g., by leaving the energy domain considered or by escaping from the system in a directionperpendicular to the x axis (if the system is not infinite in this direction).Equation (7.122) provides us with a relation between f, r i;and the estimated quantities.Another such relation is deduced from Equation (7.110). Making use of the approximateform of the particle numbers M 1+(x) in Equation (7.118), the first-moment equation reducesto-X + M + O - 1) = t,M + (i - 1) + F 1M-(I - 1) (7.123)The solution of Equations (7.122) and (7.123) for f and r ;is^ = -X + - M 0 - I)[X + - 2(T 11 O)]Z[M + O - 1) - M'-(i - 1)] (7.124)andT 1= M + (I - I)[X + - 2(T a(i)]Z[M + (i - 1) - M-(i - 1)] (7-125)where a a(i) is given in Equation (7.122).In conclusion, the regionwise optimized importances are given by Equations (7.119),(7.120), (7.122), (7.124), and (7.125) in terms of the estimated partial currents M + (i),collision rates N 1, and flux integrals F 1. The method is easily implemented in any productioncode prepared for regionwise splitting, and it defines an approximately optimum splittingscheme for deep-penetration calculations In nonmultiplying media composed of homogeneousslabs or other geometrical forms that can be approximated by slabs (e.g., concentric cylindersof large radii).Note that Equation (7.123), together with its counterpart,X 1M-(I - 1) = I 1M-(I ~ 1) + T 1M + (I ~ 1) (7.126)