Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

490 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsEquation (5.80):M 1(P) = I 1(P) + J rPn dP'T(P,P')jdP"C(P',P")M,(P") (7.134)where, according to Equations (7.130) through (7.132)I 1(P) = exp[-T(P 1P 1,)] (7.135)Let us now recall the steps in Capter 5.V.D that lead to an unbiased exponential transformedgame. The expected score, M 1(P), in the transformed game is obtained from the analogmoment by the transformationTl 1(P) = expj WP)]M ,(P) (7.136)where b(P) is the path-stretching function to be chosen such that the second moment of thescore is minimum. As is shown in Chapter 5.V.D, the transformed transition and collisionkernels in an unbiased transformed game take on the formst(P,P') = rx(P')exp[b(P) -b(P')]T(P,P')/cr(P')= o-(P')expi>T(P,P')J (7.137)andC(P',P") - o-(P')explb(P') - b(P")]C(P',P")/o-(P') (7.138)wherecr(P) = o-(P) + w Vb(P) (7.139)is the stretched cross section. Now, according to Equation (5.165), the transformed momentsatisfies the equationM 1(P) = SC 1(P) + |dP't(P,P')[dP"C(P',P")i1,(P") (7.140)whereSy(P) == exp[b(P)]I,(P) (7,141)and I 1(P) is the analog first-flight score in Equation (7.135). Let us now assume that thepath-stretching function b(P) vanishes at the boundary of V for directions pointing outward,i.e., letb(P b) = 0 for ton, s 0where n bis the outer normal of V at r„. Then Equation (7.141) with Equation (7.135) reads9F 1(P) = expLb(P)]exp["T(P,P')] = exp[ - T(P,P')]