Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

499The constant A follows from the condition that the total expected score due to a particlestarted from x = X in a positive direction is unity, i.e., (for isotropic scattering)j o dp,M,(X,jx) = 1which yields A =1.By inserting Equation (7.167) into the optimum biased functions inEquations (7.146) and (7.151), we obtain the stretched cross section aso- = CT(I - bp,) (7.168)and the biased collision kernel asC(M,- u/|x) = UW) = ^ T - = V T ^ 169)2 1 — bp, 2cr(fx )Note that although C is proportional to c, the survival probability, it is normalized to unitysincec 1 + bJ ^ix'Qjx -- |x'|x) = ^ iog e1 - bfor b satisfies the Placzek Equation (7.165). If Q(x,p.) denotes the analog source density,then the transformed source density follows from Equation (7.143) asQ(x,p7) = e tofr - x >Q(x,p,)cb b(p.) (7.170)The scheme defined by Equations (7.168) through (7.170) can only be applied forhomogeneous, monoenergetic isotropic transport. With heterogeneous and/or energy-dependentproblems, the cross section cr and the survival probability c depend on the position andenergy of the particle. Therefore, the asymptotic solution in Equation (7.167) of the momentequation does not apply directly to such cases. Approximate optimization of the pathstretchingparameter can be obtained by this method e.g., by requiring that the leakagedetermined by Equation (7.167) with some homogenized material constants be equal to theestimated (i.e., real) leakage rate. Thus, in the case of the isotropic unit source at x = 0,i.e., ifQ(x,|x,E) = 8(x)Q E(E);jxeJO,]]a value (bcr) is to be chosen such that the homogeneous expectation in Equation (7,167)satisfies the equalityd UJVi 1(O,^) = M 1where M 1is a preliminary estimate of the leakage rate in the real system. Accordingly, werequire that