Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

46 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsIn other words there is no need at all to select the type of element hit, thus method 1. ofSection 3.1.C is preferred.The PDF to be sampled isp(a u,x)f(a u,x)0"f(a o)where f and a* are defined by Equations (3.9) and (3.10), respectively, and values of theenergy change x are to be picked up with given a ()incident energies.Concrete recipes are given in Appendix 3A.In reality, the electrons of an element or compound are neither free nor at rest whenthe photons hit them. The binding corrections are usually taken into account by applying amultiplicative correction, the so-called incoherent scattering function S(q,z):or lm(a 0,d,z) = (T 1(OL 1,,®) • S(q,z)The first argument of S is the momentum transfer:q = (a 2 + a 2 - 2aa 0cosft)" 2 (3.12)the second is the atomic number.Tables of incoherent scattering factors are given e.g., by Hubbell et al. 24while Biggset al. 3 developed analytical approximations for them. Recently, Persliden 41 developed asampling method, where first the Klein-Nishina density function is sampled and an additionalaccept or reject game is applied for the correction for electron binding.Data for the incoherent scattering factors and total cross-sections derived with them areincluded in most large cross-section libraries mentioned earlier.Trivially, if the electron binding effect is not neglected, the scattering angle (or energychange) distributions are not element independent anymore.The formula for incoherent scattering can be further refined:• one can take into account that the electrons hit are not at rest but in motion• the radiative correction reflects the emission and reabsorption of virtual photons• there is a minor probability that double Compton effect occurs, in which an additionalphoton is emittedAll these corrections are described in details by Hubbell et al. 24In practical transportcalculations, however, most frequently the use of the Klein-Nishina approximation is satisfactory.3. Pair-ProductionIn this interaction the incident photon disappears in the field of either the nucleus or anelectron, and an electron-photon pair is created. The threshold of the effect is 2m l!c 2(thecombined rest energy of the created particles). The cross-section rises monotonically fromzero at the threshold. With the exception of the lowest atomic number elements, pairproduction in the field of electrons is much less probable than that in the field of the nucleus.The cross-section for pair-production in the field of the nucleus is proportional, roughly, tothe square of the atomic number.The full phenomenon of pair-production is quite complicated, readers interested in detailsare advised to read the overview of Hubbell et al. 24Cross-sections for particular elements can be found in the libraries already mentioned.