Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

377iterate of the derivative of the effective multiplication factor follows from Equation (6.155)asda(f,d n+s)/K tt(f,S) (6,164)since lim S n= 0. In practical cases, it is reasonable to assume thatA^Olim ^ = ^ (6.)65)da dawhich means that Equation (6.164) can be considered as an estimate of dk/d« in the n-thgeneration. Equation (6.164) and expressions of the higher-order derivatives of k effwerederived by Mikhailov 57 as early as 1967.An alternative method proposed by Takahashi 81is based on a direct estimation of thereaction rate-~ fdP W ) c f(P) v(P)da da Jwhich follows from Equation (6.93) by differentiation. The simulation is equivalent to a.fixed-source differential game.IV. ESTIMATION OF FLUX AT A CERTAIN POINTIn certain (mainly radiation shielding) problems, It is necessary to estimate the neutronflux at the sites of a number of detectors. If the finite dimensions of the detectors areaccounted for, the Monte Carlo procedure can be (in principle) based on the reaction-rateestimation methods reviewed previously. In the majority of such problems, however, it isdesirable to neglect the presence of the detectors, and the particle's flux at given spatialpoints is to be determined.At first glance, estimation of pointwise quantities would appear to be inconsistent withthe capabilities of transport Monte Carlo methods. Nevertheless, these methods can be madesuitable to flux-at-a-point estimations. A possible method of pointwise calculations, theadjoint Monte Carlo simulation, was introduced in Chapter 4. VI!. If the particles originatefrom an extended source and the flux (or some related quantity) is to be estimated at a singlepoint, then the adjoint game fits the problem. However, if the flux, has to be determined atseveral points, then an adjoint game is to be played as many times as the number of detectorpoints. Moreover, if the flux at a point due to a. point source is the quantity of interest, thenthe adjoint simulation faces the same difficulty as the direct game (the detector in the adjointgame is equivalent to the source in the direct game).A special estimator of flux at a point was first proposed by Kaios. 37This estimator canbe easily incorporated into any direct game without considerable alterations of the simulation.The estimator is derived in Section A, and it is shown to have the disadvantage of resultingin an unbounded variance. Divergence of the variance, in turn, yields a slower convergenceof the average score to its expectation than in estimates with finite variances (Section B).In order to avoid or relax the difficulties related to the singular variance of the simplestestimator, several alternative estimators were proposed, some of them using special nonanalogestimation procedures. 22 ' 23 ' S8 - 69 ' 76 " 8 " Details of newer developments along with anextensive list of earlier results are given in the report by Kalli and Cashwell. 39