Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

474 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsstep, let us introduceX(x) - ~ logj(x) (7.89)dxWith this notation, Equation (7.85) becomesk" - k'[l + X(x)dx] + 0(dx 2 )Next, let us expand the reflection probability R in Equation (7.84) into a Taylor series withrespect to its k" argument to yieldR(k 2,k - k"x,x 4- dx) = [1 - X(x) — k'dx]R(k„k - k'jx.x 4- dx)3kThen Equation (7.84) can be rewritten asP + (k,x + dx) = J dk,J dk 2J dk'H(k,,k 2,k',k|x,x 4- dx)X(x) JdkJdk 2jdk'H(k,,k 2,k',k|x,x 4- dx) 4 0(dx 2 ) (7.90)where we have used the shortened notationH(k,,k 2,k',k|x,y) = P +(k,,x)T(k,,k'|x,y)x R(k 2,k - k'|x,y)P~(k 2,y) (7.91)Finally let us expand Equation (7.90) into a Taylor series around x. The LHS becomesP + (k,x) + — P + (k,x)dx 4- 0(dx 2 )3xThe RHS contains the Taylor series of the function H in Equation (7.91):_3_H(k,,k 2,k',k|x,x 4- dx) = H(k,,k 2,k',k|x,x) + — H(k,,k 2,k',k|x,y)| y = xdx 4- 0(dx 2)dyIn view of (7.82), (7.83) and (7.91) we haveH(k,,k 2,k',kjx,x) = P'(k,,x)8(k, - k')8(k' - k)P-(k 2,x)whereas differentiation of Equation (7.91) yieldsdyH(k„k 2,k\k|x,y)| y = x= P^k^xMk^k'lx^k' - k)P-(k 2,x)+ P + (k,,x)8(k 1- k')r(k 2, k - k'|x)P (k 2,x)d+ P'(k,,x)8(k 1- k')o(k' - k) — Pdx(k 2,x)