Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

371in the unperturbed system, the weights are changed so that they account for a migration inthe perturbed system. [In brief, the perturbed fission neutrons are the progenies of theperturbed particles in the correlated game that estimate the first term in Equation (6.139), jThe second term is then estimated in the next generation, where the particles are thoseproduced by the previous generation. Their initial weights are equal to the difference of theweights of the perturbed and unperturbed particles born together, and the game is played inthe perturbed system, i.e., according to the kernel Z(r,r')-Note that, making use of the approximation in Equation (6.140), the perturbation inEquation (6.139) can be rewritten asSk = (f,ZS) - (f.ZS) + (f,Z 2 S)/(f,ZS) - (f,ZS)= (f,Z 2 S)/(f,ZS) - (f,ZS) (6,141)One might object that it is unnecessary to bother with correlated games to estimate the twoterms in Equation (6.139) when the two terms in Equation (6.141) can be directly estimatedin two successive generations. However, the objection is groundless since Equation (6.141)is only an approximate reformulation of the expression 8k = k elT- k efr, where k ctTis replacedby (f,Z 2 S)/(f,ZS) and, thus, even if the approximation in Equation (6.140) is very good,8k in Equation (6.141) is estimated as the expectedly small difference of two essentiallyindependent estimates, both of the order of magnitude.1. In contrast, the two terms in Equation (6.139) are both of the same order of magnitudeas the final estimate 8k.Note that the assumption of normalized densities, as formulated in Equation (6.138), isnot essential. The third term in the expression (6.137) of the reactivity perturbation (whichvanishes in the case of normalized densities) can also be estimated in the second step of thesimulation procedure above. 612. An important feature of the above method is that It may only work for small perturbationssince assumption (6.140) certainly fails to work if the fission-neutron densities inthe two systems are very different. Polevoi 63proposes a scheme which is valid for arbitraryperturbations. The price one pays for a more exact treatment is that more than one generationmust be simulated to determine the perturbed fission-neutron density. This, however, isdone in such a way that every generation gives a contribution to the estimate of 8k.Let us assume again that an asymptotic, unperturbed fission-neutron density, S(r), isreached in a preliminary stage of the simulation. A specific scheme Is followed from thispoint. Let a first-generation "distribution" be determined according to the relationA,(r) = jdr'8Z(r,r')S(r') (6.142)and let the distribution, A n + 1 (r), of the successive generations be simulated according tothe equationA n + 1 (r) = jdr'Z(r,r')A„(r') (6.143)The last equation defines a procedure analogous to the iterative simulation of the perturbedfission-neutron density defined in method three of Section A. The main difference between,the two simulations is that in the procedure defined by Equations (6.142) and (6.143), theinitial distribution is not necessarily everywhere positive fA,(f) is the difference of the