Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

496 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsscheme, the path-stretching function and the expected score are related asd\x — 1Og 6M 1(X 5Pi 5E) - cr(x,p,)b(u,,E)dxi.e., the expected score (leakage rate) due to a particle started from P = (x,u,,E) has theformM,(x,p,,E) = expj- T(X,X) j (7.158)In practice, Equation (7.158) never holds exactly, but sometimes the expected score can beapproximated fairly well by a function of this form.As an example, we recall that in the straight-ahead scattering model, Equation (7.158)holds and therefore the specific form of path stretching in Equation (7.157) together withthe survival biasing were seen to yield a zero-variance scheme (cf. Section 7.I.B). Morerealistic approximate schemes are introduced in the next section.The results above also justify the aproximate optimization discussed in Section 7.1. D.There we assumed that with some constant value X, the expected score could be written asM,(x,p,,E) = Ae- M X " x ) (7.159)and we choserj(x,u,,E) — CT(X,E) - XfX (7.160)Obviously, this choice corresponds to the approximationb(p,,E) T(x,X) =X(X-x)or, equivalentlyor(x,E)b(p.,E) ~ X(XNow, accepting the approximate form of the expected score in Equation (7.159), the optimumcollision kernel follows from Equation (7.151) asC(P',F') - C(P',F') / dP"C(P',P") (7.161)i.e., approximation (7.159) calls forth an optimized game where simple survival biasingrepresents the alteration of the collision kernel.The arguments above explain the success of schemes that apply pure path stretchingwith survival biasing, but no angular biasing, of the collision kernel. 22 ' 2930 - 43 ' 47Note, however,the little trick involved when introducing Equation (7.161) on the basis of Equations(7.159) and (7.151). Indeed, if the approximation in Equation (7.159) to the expected scoreis accepted, then the stretched cross section in Equation (7.160) does follow from Theorem7.1. However, the corresponding collision kernel should be derived from Equation (7.148),not Equation (7.151), since Equations (7.148) and (7.151) are equivalent only if the exact