Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

393Now, since bl(P) = g(P) is bounded, the expected score 1(P 0) has a 1/r singularity and thusa finite variance. Therefore, 1(P 0) itself may serve as an estimator of the flux at r*. In mostpractical cases, the integrals in Equation (6.191) cannot be evaluated analytically, andtherefore 1(P 0) is estimated by the one-sample method and with the aid of an estimatorf(P,P') with 1/b singularity in the following manner:1. The number of secondaries and the postcollision energy E of every secondary in acollision at P 0is selected from the marginal scattering kernelC,(E (,~->E|r,w 0)Then for each secondary, the next steps are to be repeated.2. The angles x and 0 are selected from a uniform distribution over [0,2Tr] and [0,ir],respectively, and the direction io(8,x) is determined. Let P = (r,«,E).3. A possible next collision point P' = (r + D w,E) is selected from the transitbrtkernel T(P,P').4. The contribution s, from an estimator f(P,P') of singularity l/b is determined accordingto steps I through 4 in the previous section.5. The quantitys„ = 2Tr 2 C 2(co 0-^co(e, X)!r,E,E 0)bs 1/r= 2TT 2 0 2- ^^^(^^(e^jIr^.EjgTD.D.lr.Eyr (6.192)is scored.Certain possible practical modifications of the procedure above are obvious. 'Thus,, ifthe angles x and 0 are sampled from some density function h(6,x) instead of the uniformdensity in step 2, then the score becomess c= [C 2(w 0-^(o(e,x)!r,E,E 0)/h(e,x)]b Sl/r (6.193)Furthermore, if one prefers scoring before the number of secondaries in the collision isdetermined, then step 1 is modified thusly:1' a possible postcollision energy E due to a collision at P 1, = (r,co 0)E 0) is selected fromthe marginal densityC,(E 0^E[r,w 0)/c(P)Accordingly, the score in step 5 becomess„ = 2Tr 2 C 2(w 0-^w(e,x)!r,E,E 0)c(P 0)bs,/r (6,194)The estimator introduced by the above procedure is called the once-more collided flux-ata-pointestimator since it scores from events which are two collisions ahead. Note that whenusing the once-more collided estimator, source particles also contribute to the collided pariof the flux. In the case of source particles, the first collision in step 1 is replaced by theselection of the initial coordinates. Thus, if Q(r,a»,E) is the source density, then the expectedscore corresponding to Equation (6.191) readsI 0= ~ JdrJdEQ,(r,E) Jf & j[ £ 2 1r 2 Q 2(w(e,x)|r,E)jM(r,«(e, x),E)]