Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

242 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsLet us now consider the case of a volume source. Multiplying the equation by Q V(P)in Equation (5.247) and integrating with respect to P, the resulting equation readswhereM., = I 1+ — j dP J dTe- 1 — Idto'M.ir + Dio.co') (5.248)4"frv JV Jo 4TT JM, =T4'JTV JandlT idPM,(P) = R (5.249)= 4^V I D P I ' ( P )The last and most important approximation in our model is that M 1I -P") in Equation (5.248)is replaced by its average value M 1in Equation (5.249). Doing so, Equation (5.248) becomesM 1« I 14 CP 1M 1i.e.,M 1= I 1Z(I - cP c) (5.250)wherep=_L Jd PI ci're' T = IdP[I - e~ t r , w ]4irV Jv Jo 4TTV JV= — dr dw[l - e-" D fr^] (5.251)4LTV JV J4-HP cis the first-flight collision probability 8in V. More precisely, P, is the probability that aparticle started in V from an isotropic, uniform distribution suffers its next collision in V.The approximation above is called the separation assumption, referring to the fact that theaverage of a product is approximated by the product of the separate averages. It can be seenthat in the simple form above, the separation assumption Is equivalent to the assumptionthat the collision points are uniformly distributed over V. An analogous approximation wasintroduced through an invariance principle by Stuart 44for escape probability calculations.Note that the first-flight collision probability P cfor a body of a given shape is a uniquefunction of the optical mean chord lengthT = a? (5.252)in the body, 8whereI = 4v/Sis the geometrical mean chord length.