Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

432 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationswhereA = S y.y,and VJ-N(O 1S) independently of y m(m ^ j). AccordinglyD 3 = (AQ)V(n - I) 3 - 3(AQ)7(n - I) 2 + 3AQ/(n - 1 ) - 1 (B.3)Let us denoteB 1= VjvTQ(B.4)then!1 - 1AQ = 2 B 1(B.5)and the following relations hold(AQ) 2 = V B 2 + VVB j B kJJk(AQ) 3 = S Bf + 2 2 (BjBk + BfBJ + 2 2 2 B 1BkB 1111 j k j k mjV-kj*k.m.n>*kSince the Bs are identically distributed and independent of each other, we have{AQ) = (n - I)(B) = (n - I)I((AQ) 2 ) = (n - I)(B 2 ) + (n - l)(n - 2)1and((AQ) 3 ) = (n - I)(B 2 ) + 3(n - l)(n - 2)(B 2 ) + (n - l)(n - 2)(n - 3)1whereB = yy r Q(B.6)and y:N(0,S). Inserting the above expectations into the expectation of Equation (B.3), wehave(D 3 ) = — — . T ; ((B 3 ) - 3