Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

475In the equation above we have introduced the notationst(k,k'|x) = — TXk.k'lx.y)!,.,ay(7.92)andr(k,k'jx) = — R(k,k'|x,y)U, (7.95Inserting the equations above into Equation (7.90) the probability density equation becomesf) dP + (M) + — P + (k,x)dx = P + (M) - A.(x) — [kP^(k,x)|dxdxelkdk,P + (k 1,x)t(k,,k|x)dx (7.94)dk, dk 2P + (k,,x)r(k 2, k - k,|x)P-(k 2,x)dx 4- 0(dx 2)Note that in the algebraic manipulations that lead to Equation (7,94) we have putdk 2P (k 2,x) =that follows from Equation (7.81). Now, dividing Equation (7.94) by dx and letting dx tendto zero, we haved a "A(x) - k + -ak ax P'(M) = Jd^P+ (Mx)I(MkIx)|dk,jdk 2P + (k.,x)r(k 2 ,k - k,|x)P~(k 2,x) (7.95)By similar arguments, the counterpart of Equation (7.95) follows from Equation (7.86) asa a i fX(x)~rk + — P-(M) = dk.P-(Mx)t(Mk|x)ak ax J+ Jdk,Jdk 2P (k,,x)r(k 2,k - k,|x)P + (k 2,x)Assume that exactly one particle enters the slab at x = 0 and none at xboundary conditions for Equations (7.95) and (7.96) areX.P + (k,0) = 8(k - 1), P (k,X) == 8(k)(7 97)Solution of the equation system in Equations (7.95) through (7.97) mayfortunately, we are only interested in the first and second moments of the p;in Equation (7.88). Taking the v~th moment of Equations (7.95) and (7.