Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

236 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsNevertheless, from a practical Monte Carlo point of view, their usefulness lies in the factthat in different problems their relative efficiency is different. Investigation of the relativevariances of the estimators is postponed to Chapter 5.VIII. Nevertheless, in order to obtainsome insight into the main characteristics of the various estimators, the following Sectionis devoted to investigation of the variances of various estimators in a highly idealized transportmodel.C ANALYSIS OF VARIANCES IN THE STRAIGHT-AHEAD SCATTERINGMODELThe straight-ahead scattering model is a favorite tool of approximate analytical MonteCarlo and transport theoretical calculations. In this model, the particles are assumed topropagate along a straight line and a collision may result in either an absorption or anemission of one or more particles with a direction identical to the direction of the incidentparticle. In nonmultiplying cases, it is also called the delta-scattering model. 42It is a modelone step simpler than the Fermi scattering model introduced in Section 5.11.1). Its mainadvantage is that most of the equations appearing in our treatment can be solved analyticallyin the straight-ahead approach; at the same time, the solutions reflect the basic characteristicsof the exact ones. We shall here consider the first and second moments of the score providedby the different estimators when the absorption rate is estimated in a finite homogenousslab.Let the particles start at x = 0 in a positive direction along the x axis and let the slabbe situated between x = 0 and x = X. Assume that the total cross section of the materialin the slab is unity, the probability of absorption is c a, and the mean number of secondariesper collision is c. Let the slab be surrounded by a purely absorbing medium of total crosssection1. We do not require that the medium be nonmultiplying; however, for the sake ofsimplicity, we assume that the nonmultiplying game equivalent to the multiplying processis applied (cf. Section 5.III.C). (Remember that for c < 1, it is equivalent to survivalbiasing.) Otherwise, the game is assumed to be analog. Then the expected score, accordingto Equation (5.86), satisfies the equationM,(x) = I dxV x xl [f(x,x') + cM,(x')]= I,(x) + cj dx'e- , x '-^M 1(X') (5.233)where f(x,x') is any of the partially unbiased estimators. As the absorption rate is to beestimated,R •--fxdxijj(x)c athe simplest estimator is the weighting functionf(x) - c aif O S x ^ Xand zero otherwise. With this estimator, the expected partial score (the expected score in afree flight and collision) isI,(x) = c ae- (x '-*>dx' = c a[l - e- (X -*>] (5.234)