A closed-form solution for the two-dimensional transport equation by the LTSN nodal method in the energy range of Compton effect

In the present work we report on a closed-form solution for the two-dimensional Compton transport equation by the LTSN nodal method in the energy range of Compton effect. The solution is determined using the LTSN nodal approach for homogeneous and heterogeneous rectangular domains, assuming the Klein–Nishina scattering kernel and a multi-group model. The solution is obtained by two one-dimensional SN equation systems resulting from integrating out one of the orthogonal variables of the SN equations in the rectangular domain. The leakage angular fluxes are approximated by exponential forms, which allows to determine a closed-form solution for the photons transport equation. The angular flux and the parameters of the medium are used for the calculation of the absorbed energy in rectangular domains with different dimensions and compositions. In this study, only the absorbed energy by Compton effect is considered. We present numerical simulations and comparisons with results obtained by using the simulation platform GEANT4 (version 9.1) with its low energy libraries.