Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

471As for the first question, let us realize that the integrity of a real particle manifests itselfonly when the particle crosses geometrical boundaries or when it enters a collision. Therefore,the simulation particles of the continuous splitting game can be converted to "real" particlesat every important instance by forming "entire" particles out of them at real collision pointsand at geometrical surfaces of interest. Specifically, if x is the site of a real collision or aboundary crossing and k(x) denotes the number of hypothetical particles at x, then one canrelate the hypothetical continuous splitting game to a real game by setting the number ofreal particles at x equal tok -ent[k(x)jwith a probability k + 1 — k(x) and to k + 1 with the complementary probability. In otherwords, the equivalent real particles materialize after a Russian roulette played with thefractional part of the hypothetical simulation particles.The mathematical description of the continuous splitting game would be troublesomein terms of the usual transition and collision kernels since the latter define discrete jumpsin the phase space and may not account for a continuous variation of the number of simulationparticles. An additional difficulty associated with the idea of the continuous game is thatsplitting is always followed by the reselection of the free flights of the fragments, which isclearly not feasible in the case of continuous splitting. The problems above can be bypassedby using a transport model which does not account for the elementary processes of collisionand transition explicitly, but is built up in terms of transmission and reflection probabilitiesin infinitesimally thin slabs. The idea is analogous to the invariant embedding approach tothe transport equation. In this concept, (a possibly noninteger number of) simulation particlesappear independently at the faces of the infinitesimal slab and the averaged result of thephysical processes inside the slab is described by the transmission and reflection probabilities.The treatment will necessitate the introduction of certain approximations, but will excludethe necessity of a detailed collision-by-collision description of the migration. Scoring in thecontinuous splitting game is analogous to that in a normal game, i.e., the score due to astarter is the product of the number and the weight of the particles that reach the surface atx = X. The weight of a particle at X is always equal to 1/I(X); the number of particlesreaching X is a random variable. Various moments of the score are obtained by taking the,expectation of powers of the score with respect to the probability density of reaching thesurface at X. The probability density function of the number of transmitted particles, inturn, follows from a master equation built up in terms of the transmission and reflectionprobabilities.Regarding the question of practical realization, we mention in advance that the optimumimportance function I(x) will be expressed in terms of functions of x which, in turn, aredetermined by the material composition of the system. If these functions can be determinedin preliminary runs or by approximate calculations, then one may define a splitting surfaceanywhere along the x axis or, alternatively, one can use collisionwise splitting, and a particleat x is split into a number of fragments in such a way that the weights of the fragments willbe equal to 1/I(x). The more surfaces that are introduced, the better the approximation ofthe optimum continuous scheme.If the x-dependent functions composing I(x) are not available explicitly, one can proceedin the usual way by defining fixed splitting surfaces and regions between them. The continuousimportance function I(x) is then replaced by region importances after averaging I(x)over the regions. The bulk parameters appearing in the averaged importances are thendetermined in a preliminary run. Practical realization of the optimized scheme will bediscussed in more detail in Section F.Let us note two peculiarities of the continuous splitting game. First, since the weights