Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

182 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsand we have made use of Equation (5.97). A little algebra yields an alternative form of theequation:W 2 M 2(P) = dP'T(P,P'H c a(P')[W'f(P,P') + Wf 1(P')] 22 |dP"C(P',P")tW'f(P,P') + W"f 5(P',P")] 2 r (W") r M r(P")ldP'T(P,P')JdP"C(P',P"){[W 2-(W") 2 ]M 2(P")+ (W 1' - W")M 2 (P")} (5.99)Comparing this equation to Equation (5.58) that describes the second moment of the scorein the original game with no splitting, it is apparent that the effect of the splitting on thevariance basically is determined by the last term in Equation (5.99).E. ALTERNATIVE FORMS OF THE COLLISION KERNELThe collision kernel in Equation (5.74) accounts for the possible physical processes:absorption, scattering, or multiplication.* Notice that multiplication may also yield a singleprogeny (with a probability q,), and in this case multiplication and scattering are not differentfrom a simulational point of view. The distinction between the two processes is justifiedwhen their contributions to the score, f sand fj, respectively, are different. On the otherhand, if f s= f,, the collision kernel can be used in a simpler form asC(P',F') = c a(P')8(P" - P) + [1 - c a(P')] y Hq n(P') C n(P',F) (5.100)where, by puttingC 1(P',?") = [£ S(P')C S(P',P") + 6XPOq 1(POC 1(P',F)/[£ S(P') +CXPOq 1(P')]andC] 1(P') = RBiP') + £ f(P') q,(P')Mc,(P') + c t