Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

104 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationswhere q x< 1, q l()< 1. (The possiblity of the use of the same q values for all 1-s is aconsequence of the physical fact that the process is Markovian, i.e., the laws of transportare independent of the number of previous collisions.) Subcriticality means also that theNeumann series' for the collision densities are convergent.The Monte Carlo realization of the recurrence in Equations (4.39) through (4.42) isgiven in the following steps:1. Set i = 0. Select initial coordinates (r 0,E„) from Q(r,E). This gives a sample fromX 0(r 0,E l() in Equation (4.40).2. Select the next collision site r i +, fromT(SY-He 1) drThis will define a sample from (Jj-(IV, ,,E 1) in Equations (4.40) and (4.42).3. Select the post-collision coordinates in the (i + I)-St collision fromC(E,^E|r,+OdEwhich gives a sample from X 11i( r i +1,E 1 + 1) in Equation (4.41).4. Set 1 = i + 1 and return to Step 2.There are, however, two problems with the procedure given above:• Selection of new coordinates can be made only from probability density functions,i.e. from functions whose integrals over their whole range is unity, however thiscondition is not a priori fullfilled.• The cycle of Steps 2 through 4 above is infinite i.e., for practical applications aterminating criterion must be found.As it has already been seen at the heuristical level and will be demonstrated in SectionF. the two problems are not independent of each other.The next Sections are devoted to a discussion of these questions. But before turning tothis a last — and very important for the future (Chapter 5) — remark is to be made here.If one compares Equations (4.39), (4.40), and (4.38) then the equalityOJ 1•-E) = Wr,E) (4.43)is obtained, i.e., the first-flight collision source is just the zeroth term in the Neumann seriesof the ingoing density function. Hence, inserting Equation (4.43) into Equation (4.37) theresulting equation reads(Jj(P) = i(P) + JdP'Vj)(P') K(P',P) (4.44)and this form will be the starting equation in Chapter 5.1.E. NORMALIZATIONS OF THE TRANSITION AND COLLISION KERNELSIf one assumes that both the collision and the transition kernels are normalized to unitythen the selection procedure mentioned in the previous Section needs no more modification.However, this is seldom the case. Let us now analyze the two kernels separately, since thephysical phenomena described by them are substantially different — though the mathematicalproblem is the same.