Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

244 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsIn the case of a general monoenergetic moment equation, the separation assumption willbe used in the following formdPCA(P) JdP'T(P,P')A(P,P') JdP"C(P',P")M(P")dr dw dTe ~ T A(r,w;r + Dw,w)c(r + Dw) M (5.257)Where we put X = V, a x= 1/4-rrV for a volume source and X = S, a x= 1/2-rrS for asurface source. A(P,P') is some given function, c(P) is the mean number of secondaries percollision at P, and M stands for the total score moment in questionM =4TTV JdPM(P)C. ON THE QUALITY OF THE APPROXIMATIONThe form of the separation assumption as introduced above suggests that it may onlybe used in monoenergetic, isotropic transport in bodies of more or less regular shape.Although the assumption might be generalized to more realistic cases, such a generalizationdoes not seem to offer any advantage, for two reasons. First, in complex problems, theevaluation of the integrals on the RHS of Equation (5.257) would be almost as laborious asthe application of more precise deterministic solutional schemes. Second, the approximationsproposed here are primarily intended for semiquantitative (and usually comparative) analysisof various Monte Carlo methods and, except for very special cases, such an analysis canbe performed on elementary transport problems with satisfactory results.Therefore, when judging the quality of the approximation, the most meaningful questionis, how well does Equation (5.253) approximate the exact collision rate of monoenergeticparticles in a homogeneous medium with isotropic scattering? In other words, what can besaid about the difference of the exact and approximate solutions'? Extensive numerical testsshow 31that the approximate and Monte Carlo values of collision rates and related quantitiesare in very good agreement for a not-too-high number of secondaries per collision (c < 1),and the approximation consistently underestimates the exact collision rates.Although no general proof has so far been found, the following arguments make it morethan probable that underestimation is an inherent feature of the approximation. Let usintroduce the following notationsM 1(r) =-- ~4irJdCoM 1(P)andI ,Or) = •— Idw IdP'T(P.P') = —- ldc*[l e" < r D < H "4ir J •'• 4TT JThus, from Equation (5.251), the first-flight collision probability readsPc == ~ jdrl,(r) (5.258)