Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

366 Monte Carlo Particle Transport Methods: Neutron and Photon Calculations[i.e., V(r) is adjoint to S(r)j, then it follows from Equations (6.130) and (6.131) that thefollowing integral relation holdsdr,T"'(r) = dr dr'V(r)Z(r,r')^'"- n (r')/V(r')dr'|drV(r)Z(r,r')CfXn -l>( r')/V(r')= k effJdr'a !| "-'>(r')Accordingly, the n-th iterate of the multiplication factor in Equation (6.132) isK = Ktt (6-134)i.e., the exact value of the effective multiplication factor is reproduced in every iteration.This scheme would have zero variance if the reaction ratesdr',? (k) (r'), (k = n - 1, n)were estimated without statistical fluctuations. Although this can also be attained (at leastin principle), the main consequence of the scheme is that knowledge of the adjoint functionV(f) makes a one-step (iteration-free) estimation possible. Therefore, approximate valuesof V(r) may accelerate the source iteration. Before turning to a discussion of practicalapplications, let us realize that it is not necessary to perform the importance-sampled gameabove in order to obtain a one-step estimation procedure. Instead, appropriate weightingfunctions of the reaction rates defining k nmay be used in an analog game, as is stated inthe following theorem.Theorem 6.4 — Let an analog game be played according to Equation (6.127), i.e., letS