Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

116 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsLet us now substitute Equation (4.34) into Equation (4.72), the reaction rate then reads:drdEa- '(r (1,E)T(r-*rjE) Q(r,E) + dE'C(E'->E|r)iKr.E') (4,75)The total flux in Equation (4.75) is a sum of two terms (R = R„ 4- R'). The first oneis the direct contribution from the uncollided source particles:R 0= j J drdEa" '(r„,E)T(r->rjE)Q(r,E) (4.76)The second term describes the contribution of the collided particles:R' = JJdrdEa-\r„3)TU-*rjE) |dE'C(E'-^Ejr)i|>(r,E') (4.77)By interchanging the order of integrations over E' and E, and then the two symbols themselves,Equation (4.77) becomes:R' = JJdrdEi]i(r,E)JdE'«r '(r 0,E')T(r-^r 0SE')C(E-^E'jr) (4. 78)In the majority of practical cases (in every case in photon transport) the post-scatteringenergy and the scattering angle are uniquely related. If so, the collision kernel can befactorized as followsC(E->E'jr) = C(Oi-W|E,r)S[E' - G(WW 1E)] (4.79)where the first factor describes the change in the direction of flight, whereas the relationE' = G(o>w',E)gives the new energy of a particle entering a collision with energy E and being scattered byan angle -9, cos-9 = too).Now by substituting the actual forms of the transition and collision kernels as given byEquations (4.31) and (4.79), respectively into (4.78), the contribution of the collided particlesto the flux isR' = drdEi|j(r,E) dw'dE'cxpcr(r',E)ds, r„ — r8 io — TL R