Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

325It is seen from Equation (6.37) that 8i|/(P) satisfies a transport equation analogous to Equation(6.36) with a first collision density 8«|/„(P) defined in Equation (6.39). Rewriting Equation(5.10) in terms of the collision density perturbation, we have a relation between the directand adjoint collision densities as(6,40)This relation suggests the following procedure. In the first step, let us simulate the unperturbedcollision density in a direct game with the unperturbed kernels and determine thequantity 8\|* A(P). In practice, integrals of8«|»„(P) over small phase-space regions are estimated,which implies the assumption that 8i)/ (,(P) is approximately constant over the separate regions.If the perturbations are small, this approximation causes a negligible error. The first twoterms in Equation (6.39) can often be determined analytically. If analytical integration isnot possible, they can be estimated as first-flight collision densities. To see this, let us writethe first two terms in the formObviously, this expression defines the first-flight collision density at P due to a starter atP' selected from the source density Q(P') when the starter has an initial weightW : - 8Q(P')QiP')+8T(P',P)T(P',P)This contribution to Sifi,, (or better, to its integral over the small phase-space regions) canbe estimated parallel to the simulation of the first flights in the unperturbed direct game.The integral of the third term in Equation (6.39) over a small region is an ordinary reactionrate in the unperturbed game with a weighting function equal to the integral of the kernel8K(P",P) over the region. Such integrals can be estimated in the course of the direct simulation.In the second step, an adjoint game is played again in the unperturbed system whichsimulates I|J*(P) (cf. Section 4.VII) and the adjoint reaction rate of the RHS of Equation(6.40) is estimated. The weighting function in the adjoint reaction rate is 8i|s 0(P), calculated,in the first step.Although the considerations above are valid only in first-order perturbation approximation,this limitation can be easily removed. Indeed, if the direct game is played in theperturbed system, then the exact value of the perturbation in the collision density, 8ifi t, inEquation (6.38), can be estimated and used as the weighting function of the reaction ratein the adjoint game.The same result is obtained in the alternative method, where the direct game is playedin the unperturbed system and the adjoint game in the perturbed. 34 - 35These methods havethe common drawback that if the perturbed system is considerably more complicated thanthe unperturbed (i.e., if small but complex structures are inserted into a simple geometry),the simulation of the perturbed game may require considerably more effort than that of theunperturbed game.The techniques treated in this Section were introduced mainly for estimation of reactivityperturbations. This concern will be revisited in Section 6.111.An alternative method that does not require the introduction of an adjoint game is based