Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

394 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationswhereQ 1Cr 1E) -jdwQlr.to.E)andQ 2(«|r,E) =Q(r,w,E)/Q,(r,E)In this case, step 1 above is replaced by the step1" The site r and energy E of a starter is selected from Q,(r,E). Furthermore, C 2in Equations(6.192) through (6.194) is to be replaced by Q 2above.All the considerations of this Chapter concern analog games and no attempt was madeto introduce nonanalog kernels appropriate for eliminating the singularities. In many of thepoint-flux estimation methods, the 1/r 2singularity of the next-event estimator is transferredto the transition kernel, thus reaching a bounded variance. 19,76 ' 78These methods have thecommon disadvantage that special measures arc to be taken to avoid unwanted fluctuationsof the statistical weights. We will not discuss such schemes; the interested reader is referredto the literature quoted.To conclude this section, we note that two problems remain unresolved. First, the direct(uncollided) distribution of the source particles to the flux at a point [cf. Equation (6.170)]still has an unbounded variance if the estimation point (detector) is embedded in the sourceregion. This problem cannot be avoided by tricks similar to those yielding bounded-varianceestimators of the collided part and seems to be persistent in any (nonadjoint) scheme. Second,although (one-sample) averaging of the score over future events eliminates the singularityof the variance, higher moments of the score will be singular, which brings up the samedifficulties in estimating the variance as arose in estimating the mean with the next-eventestimator. Higher moments can be made bounded by averaging the score over further events;however, every averaging step involves duplication of the simulation procedure in the sensethat determination of a contribution is performed through Monte Carlo procedures similarto those played for the continuation of the history. In practical realizations of the once-morecollided flux estimator, such duplication can be avoided in the majority of events, as willbe seen in the next section.E. PRACTICAL MODIFICATIONS OF THE BASIC METHODSThe motivation for all the effort invested in the derivation of new point estimatorsoriginates from the singular behavior of the next-event estimator near the detector point. Farfrom the detector, the next-event estimator behaves regularly and there is no need to applymore sophisticated estimators. For example, if the detector is situated in a vacuum surrounding,i.e., if no collision occurs in the vicinity of the detector point, the next-eventestimator can be safely applied. The same is also true for collisions far from the detector.On the other hand, for distant collision points, the contribution of the next-event estimatoris small because of the rapidly decreasing exponential and 1/r 2functions in it. Consequently,for such collisions, a considerable amount of computing time is spent with essentiallynegligible influence on the final result.Iida and Seki 36propose a very simple method for economizing computational time. Theidea is that a quantityf(P',P") =f NB(P',P")/p(r')