Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

505let us determine explicitly the weight of a particle that leaves the domain of simulation. Letthe particle be started at P 0= (x 0,|x 0,E 0) and let the pre- and postcoliision coordinates ofthe i-th collision in the history be J = (x,, (X 1_ ,,E 1-i) and P 1= (X^1Ix 11E 1), respectively 0= 1, 2,..., n 4- 1), where x„ < X and x„t, ^ X. Then the weight of the escaping particleisW = W 1(P n)Hw(P 1. ,,PI)W 1XPJ 1P 1)W(P n^+1)Q 1X(XJX 01EJq(X 01E 1)[I -RX 0B(X o,E„)] Tlc(x„E,_,)C(X 11E 1)2C 1XfX 1..., —> IxJx 11E 1.,,E 1)* 1 - (XiIKx 1,E 1)11 —,—~—: exp,-1 1 (X 1^B(X 1,£,_,)Xdlb(t,E,_ JoXt 1E 1,) [1 — (X nS(X 11, j,E„)pNow, in a monoenergetic, homogeneous isotropic transport c = c, andW = q • e _1 " r]~1Og 0(I - b)/bThe above result has two interesting consequences. First, in this simple case, one maycompletely forget about statistical weights during the simulation, and the weight is to beintroduced in the last flight only. Thus, (he homogeneous, monoenergetic, isotropic versionof our scheme is analogous to an exponential transformed game. Second, since the mediumat x > X is irrelevant from the point of view of the simulation, one may assume that it isfilled with a black absorber of a very large cross section. Then X N + 1= X, andW = qe - bu(X - x„) (7.186)i.e., every particle that is transmitted from x Dto X reaches X with the same weight W. Itis tempting to attribute the low variance capability of the game to the fact that the contributionsof all the leaking particles are nearly equal (sometimes it is so stated in the literature);however, recall that the analog game has the same property (as every transmitted particlescores unity) and the latter seldom results in zero variance. In fact, uniform scores yieldlow variance only if the number of transmitted particles due to a starter does not fluctuateheavily, i.e., if an approximately constant number of particles reach X in every history.Now, if b is chosen such that the probability M 1of the transmission of an analog particlestarted from x Din a positive direction isM 1then the weight in Equation (7.186) is related to this probability asW = qM,On the other hand, if k denotes the expected number of particles transmitted in the biased