Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

124 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsIt has been shown 8 - 23that, estimation can be carried out by correlated tracking. Thismeans that the differential game can be played identical with the ordinary game, that is thegame with the nominal value of a is considered as the unperturbed game. For the calculationof the differential score (the perturbed game) weights are applied.Without proof (which is given in Section 6.II.B.) we cite here the result that the statisticalweight is additive in the differential game, that is after n-steps:W„Vwhere the i-th weight term is expressed as:1^K(TV ,,P,)W: = • (4.92)K(P 1. ,,P 1) da . v /Since K = CT the weight term is a sum of two terms:^ = - ^ + --1 (4.93)Kda Cdct TdaLet us illustrate the derivation of the weight terms again with the example where thesensitivity to the density is investigated, i.e., we replace the parameter a in Equations (4.92)and (4.93) by p.As we already discussed, the collision kernel is density independent:^dp= Oand thus- i , K - = *L (4.94)Kdp TapFor calculating dT/dp let us consider that the macroscopic cross section is a product ofthe density and a density independent factor (cr):(T = perand thus the transport kernel can be written asT(Iv 1 1IV 1) = pd-exp(-pd-R) (4.95)From Equations (4.92), (4.94), and (4.95) the weight term is expressed as:w, = -(1 - crR.)Pif R: = r, - r,Up to now we have met only multiplicative weights. If the unperturbed game is nonanalogthen its multiplicative weight is to be multiplied by the additive weight in the estimation of