Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

375where P" = (r',E'). Multiplying this equation by S(Jr 0), integrating with respect to :r„, andtaking into account that, in view of Equation (6.101), S(r c) can be written asS(r') =JdE'4»(P")c t(P")v(P")/k clTwe have the following equation for 1Ji 1(P)4, (P) = [dP"4i,(P")K s(P",P) + jdP"ili(P")x |sk s(P",P) + ^-" Cf(P") V(P")8tx(E!i-')T(r' »r|E)]j(6.16s)where we have made use of the relation in Equation (6.160). Equation (6.161) is a transportequation defining the collision density IJi 1(P) in a nonmultiplying system. The first-flightcollision density in the equation is represented by the second integral on the RBS, and itcontains ty(P). The simulation of this collision density consists of two steps, In the firststep, the original collision density ip(P) is determined in an ordinary nonmultiplying gamewith a source S(f), and the first-flight collision density of Ip 1(P) defined in Equation (6.161)is also established. In the course of this step, the second term of Equation (6.159) is estimatedas a reaction rate due to the collision density i|/(P). In the second step of the simulation, thefirst-flight collision density [second term in Equation (6.161), determined in the first step!is used to simulate 4•,(P) in Equation (6.161), and the first term in the reactivity perturbation.Equation (6.159), is determined through an estimate of the corresponding reaction rate with1Ii 1(P). Note that both steps involve nonmultiplying games since Ji(P) is produced by thefixed fission source S(r), according to Equation (6.160), and IJi 1(P) obeys the nonmultiplyingtransport equation (6.161). More details of the practical realization are given by Matthes-'' 0and Hoffman et al. 355. The perturbation source method presented above is exact up to the first order of theperturbation because (V,S) in Equation (6.157) was approximated by (V,S) in the expression(6.158) of the reactivity perturbation. Hoffman et al. 35 propose a modification of the procedurenot limited to first-order perturbations. The modification is based on the fact that themultiplication factor of the perturbed system is also an eigenvalue of the adjoint perturbedequationk cffV(r) =Jdr'V(r')Z(r',r)and therefore, for arbitrary function f(r), it is expressed ask eff=(V,SZf)/(V,f)Now, putting f(f) = V(f) (the adjoint density in the perturbed system) andEquations (6.135) and (6.162), respectively, the perturbation of the effectivefactor readsSk = k eff- k e(r= [(V 7ZS) - (V,ZS)j7(V,S) =- (V,8ZS)/(V,S) (6.163)