Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

460 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationswhereas the number of collisions is expressed asN 1= N 0(I - e- Xx -), N 2= N 0[I - e- X(X -"-»] (7.44)The optimum splitting ratio then follows from Equations (7.42) through (7.44) asn opl= (P 1P 2T" 2= e^ x' 2 (7.45)In this case, we have a single parameter, X, to be estimated in preliminary calculations.Notice that the optimum n value in Equation (7.45) is analogous to that in Equation(7.17), obtained from the straight-ahead model, since X = 1 — c in this model. The similarityis due to the approximation in Equation (7.43), which implicitly neglects the effect of particlesleaving and reentering the respective half-slabs, as in the straight-ahead model.A slightly more general presentation of the method outlined here can be found inReference 14; an elaborated theory based on the same principles was worked out by Dubiet al. 9 " The considerations will be extended to several splitting surfaces in Chapter 7.11Let us also note that this simplified model reflects the danger of over splitting. Thequality factor in Equation (7.41) decreases from n = 1 to n = n opIand then it increasesessentially linearly with increasing n. Therefore, when choosing a splitting ratio n less thann upI, the worst that can happen is that the efficency of the game will not be considerablyhigher than that of the analog game. On the other hand, if the splitting ratio is much largerthan its optimum value, the efficiency of the game may be even lower than it would bewithout splitting. This is due to the fact that for low values of n the variance of the scoreremains finite, whereas the number of collisions tends to infinity with increasing n.A direct statistical approach to optimization of the path-stretching parameter can bededuced from the original idea of the exponential transformation. Assume that we wish toemphasize the free flights of the particles along a given direction and also that the positionof a particle can be characterized by a unique distance value measured along this direction.For example, in the case of penetration through a slab, the direction is the one perpendicularto the surfaces of the slab and the distance is the depth of the point in the slab. In calculatingthe escape from cylindrical or spherical bodies, the favored direction may, for example,point outward along the radius, and the position is characterized by the radial coordinate.Let (a, denote the favored direction for a particle at P = (r, o>, E) = (r, E) and let x bethe distance value corresponding to r. Specifically let x be the projection of r on the direction(a,. The expectation of the score in the analog game satisfies the now very familiar Equation(5.80). With the notation of Equations (5.7) and (5.8), this equation readsM,(r,E) =[dr'Tfr + r'|E)f(r,r'|E)+ Jdr'T(r-+ r'|E) JdE 1 C(E -» E'|r')M,(r',E') (7.46)where, according to Equation (5.32)T(r-> r'|E)dr' = cr(r',E)exp| - J' ' dtcr(r + tto.E) js^,~_ T^ - l^d|r' - r| (7.47)Let us now apply the exponential transformation in the formM 1(P) = e" " 1 M 1(P) (7.48)