Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

13!The flux-at-a point pay-off given in Equation (4.80) was obtained via a relatively longderivation and could be used in the direct simulation — at least if the point of interest wasnot imbedded in media having non-zero cross sections.At the same time this pay-off formula is useless for the adjoint simulation, since itssampling is complicated.There may be problems where it is easier to determine the pay-off ofparticles. If the form of fx (r,E) is more convenient for adjoint source samj). •>start the simulation cycle from it. Taking into account Equation (4.53) andgiven in Section B. such a simulation leads to the omission of the very first stethat there will be no points representing the adjoint source particles, but theof all further movements of the pseudo-particles is numerically simulated. Astitute for the missing adjoint source term is given in Section F.I). THE COLLISION KERNEL OF THE VALUE EQUATIONIn the definition of the collision kernel given in Equation (4.33) both the ithe denominator have their physical meaning. If we integrate C(E'-*E|r) ovenergies after the collision:C*the result ("* is the expected number of particles leaving a collision at the site r,If there is no multiplicative event, C* is the non-absorption probability, i.e., less thanunity. For non-multiplicative, nonabsorbing materials C* = 1. If multiplicative interactionsand events leading to absorption of the incident particle without emission of secondaries ofthe same type can take place in the same medium, C* may be either smaller or greater thanunity. However, it follows from the physics of the interactions that C* cannot be substantiallygreater than two. Thus, the statistical weight corrections applied in parallel with the normalizationof the collision kernel (see Section 4.IV.E.2.) do not cause extremely large weigh-:fluctuations.This is not the case with the adjoint collision kernel. First of all, it should be emphasizedthat the integral(4, HO)that should be used for the normalization has no physical meaning. It follows from Equation(4.108) that the evaluation of the integral given in Equation (4.110) means the calculationof terms of the form(4..1 II)and such integrals may have extremely high values. Consequently, very large fluctuationsmay occur in the statistical weight and thus in the score contributions.There are, in fact, cases when the integrals given by Equation (4.111) are even divergentLet us show this in two examples. First, consider the Klein-Nishina formula describingthe Compton-scattering of photons.