Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

305Chapter 6SPECIAL GAMESIn the majority of Monte Carlo applications, the final goal of the procedure is theestimation of a reaction rate of the type in Equation (5.2). The weighting function in theexpression of the reaction rate is usually some cross section or a smooth functional of across section. In such cases, the moment equations investigated in Chapter 5 describe themain characteristics of the Monte Carlo schemes applicable to the estimation. There are,however, certain fields of Monte Carlo applications where either the weighting function inthe reaction rate is singular (e.g., estimation of the particles' flux at a point) or the resultof the simulation is not a reaction rate of the conventional form (e.g., eigenvalue estimationor sensitivity analysis). On the other hand, in certain cases, although reaction rates are tobe determined, it is more efficient to use special procedures rather than the conventionalschemes. This is the case when the effect of small changes in the material properties onsome reaction rates is to be estimated. For this purpose, the use of the so-called correlatedMonte Carlo, discussed in Chapter 6.1, is an efficient tool. Procedures for the estimation ofderivatives and sensitivity of a reaction rate with respect to some system parameters followfrom the correlated Monte Carlo procedures. Unbiased estimation schemes of such derivativesare discussed in Chapter 6.II. Special problems associated with the estimation of eigenvaluesand especially of the effective multiplication factor of a fissile system are considered inChapter 6.III. Techniques for the estimation of flux (or collision density) at a given spatia!point are reviewed in Chapter 6.IV.A number of Monte Carlo techniques developed for special purposes, such as conditionalMonte Carlo, 10 contribution theory, 11 ' 13,26 ' 71 or recursive Monte Carlo, 30 will not be detailedhere, partly because of their limited use and because their theory is not yet fully elaborated.Introduction of such new techniques make the transport Monte Carlo methods still moreversatile and bring up fresh ideas in the theory and practice of particle transport simulation.On the other hand, the present trend in the development of Monte Carlo methods, characterized by the introduction of the more and more sophisticated techniques mentioned above,will very likely find a powerful rival due to the appearance and spread of parallel computers.The extreme speed of such computers will motivate the users to prepare simple, easilyprogramable codes that take full advantage of the potentialities offered by the parallelprocessors. Economization of the programing effort will gain importance at the expense ofcomputing time. 6I. CORRELATED MONTE CARLO: PKRIT RBAlIONCALCULATIONSMonte Carlo methods are usually applied when fast deterministic methods fail to workfor some reason. Thus, the use of Monte Carlo simulations for the estimation of reactionrates in subregions of a complex system is indicated. Assume, that in such a system not onlyreaction rates, but also their changes due to some small alteration of the system are to bedetermined. It would seem logical to perform the Monte Carlo calculation twice, first in theoriginal (unperturbed) system and then in the altered (perturbed) system, and to calculatethe difference of the two estimates. In the case of small perturbations, however, the separateestimates must be very accurate if we wish to obtain a reliable estimate of the reaction rateperturbation; otherwise, the uncertainties in the independent estimates override the effect ofthe perturbation. On the other hand, by using the same trajectories for the estimation ofboth the unperturbed and perturbed reaction rates, the respective estimates will be stronglycorrelated and are expected to deviate from their expectations in the same direction.