Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

374 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationswhere, again, V(r) is the function adjoint to the unperturbed fission-neutron density, i.e.it is the solution of the equationk effV(r) =Jdr'V(r')Z(r',r)Using this weighting function, the last two terms in Equation (6.137) vanish and the perturbationof the effective multiplication factor reduces to8k = (V,8ZS)/(V,S) (6.157)This expression suggests a simple estimation procedure that is exact for small perturbations.If we are given a well-converged, unperturbed fission-neutron density S(r) and a sufficientlyaccurate approximate adjoint distribution (e.g., by one of the methods presented in theprevious section), then for sufficiently small perturbations, one can neglect the differenceof (V,S) and (V,S) when estimating the denominator of Equation (6.157). An estimate ofthe reactivity perturbation exact up to the first order of the perturbation is then obtained as5k = (V,8ZS)/(V,S) (6.158)by correlated sampling of the numerator. This method has the advantage that an estimateof 8k is obtained in a single generation, but also the drawbacks that it necessitates thecalculation of the adjoint density and is limited to small perturbations.4. As an alternative to the method above, the perturbation source method 50 introducedin Section 6.1.H can also be applied here. The perturbation kernel 8Z(r,r') follows fromEquation (6.103) as8Z(r,r') = JdE8z(r,E,r')cXP)v(P) + J _dEz(r,E,r')8[ Cf(P)v(P)]where z(r.E.r') is the collision density at about P = (r,E) due to a fission neutron emergedfrom r', and 8F denotes the difference of the function F in the perturbed and unperturbedsystems. With this partition, the numerator of Eqaution (6.158) reads(V.8ZS) - JdPV(F)(CXP)V(P)^1(P) + 8[ Cf(P)V(P)]VIi(P)} (6.159)whereiji(P) = jdr'z(r,E,r')S(r') (6.160)is the collision density due to the fission source S(r') andIJi 1(P) = Jdr 08z(r,E,r 0)S(r 0)The perturbation of zir.E.r') follows from Equation (6.100) as8z(r,E,r„) =JdP"8z(r',E',r 0)K s(P",P)idP"z(r',E',r 0)8k s(P",P) +8[ x(E|r 0)T(r 0-*r|E)]