Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

497expected score is put into the formulas. Using Equation (7.148), we obtainC(P',F) = C(P',P")o-(P')/[o-(P') - Xu,] (7.162)Although this form of the biased collision kernel conforms with Theorem 7.1, it seems tobe less favorable than the simple survival-biased kernel for two reasons. First, its dependenceon the postcollision coordinates is not altered, compared to the analog kernel. Second, thenumber of secondaries per collision is not unity, while both these requirements are essentialin a zero-variance scheme. Approximate collision kernels possessing these properties willbe introduced in the next section.Recall that the approximate optimum path-stretching scheme in Equation (7.160) wasderived in Section 7.1.1.) under the heuristically founded assumption that the path-stretchingprocedure is optimum if the transformed moment M 1(P) is independent of the position ofthe starter. As was seen in the derivation of the first scheme in the previous section, theheuristic assumption is justified; the zero-variance scheme defines a transformed game withM 1(P) - 1.When we introduced the zero-variance scheme with the expectation estimator (secondscheme in the previous section) we pointed out that this scheme does not define a real pathstretchingprocedure since the biasing factor in the transition kernel, Equation (7.153), hasthe form[I 1(P) + JdP"C(P',P")M 1(P")]/M,(P)which cannot be written in the general path-stretching biasing formcr(P') exp[b(P) -b(P')]/cr(P')[cf. Equation (7.138)]. This fact might suggest that the expectation estimator, when usedin approximate optimum path-stretching games, would be less efficient for leakage estimationsthan the last-event estimator. Although in certain cases it is indeed so, it has beenshown that for medium and strong absorption, the expectation estimator is definitely competitive.17Approximate optimum realizations of path stretching with expectation estimators arebased on the observation that in the case of deep penetration, the first-flight score is considerablysmaller than the total expected score, and thereforeI 1(P) «JdP"C(P',P")M,(P")since the RFIS is the expected total score due to a particle emerging from a collision at P'after a flight from P to P'. With this approximation, the biasing factor of the transitionkernel will beJdP"C(P',P')M 1(F')/M,(P)for both schemes (with the last-event and expectation estimators), i.e., either estimatorhe used in the same approximately optimum path-stretched game.canC PRACTICAL APPLICATIONS IN DEEP-PENETRATION CALCULATIONSWhen optimizing the path-stretching procedure, one tries to realize the zero-variancescheme derived in Section A. This is done by using appropriate approximate expressions