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Angular distribution of Bremsstrahlung photons and of positrons for calculations of terrestrial gamma-ray flashes and positron beams

Within thunderstorms electrons can gain energies of up to hundred(s) of MeV. These electrons can create X-rays and gamma-rays as Bremsstrahlung when they collide with air molecules. Here we calculate the distribution of angles between incident electrons and emitted photons as a function of electron and photon energy. We derive these doubly differential cross-sections by integrating analytically over the triply differential cross-sections derived by Bethe and Heitler; this is appropriate for light atoms like nitrogen and oxygen (Z=7,8) if the energy of incident and emitted electron is larger than 1 keV. We compare our results with the approximations and cross section used by other authors. We also discuss some simplifying limit cases, and we derive some simple approximation for the most probable scattering angle. We also provide cross sections for the production of electron positron pairs from energetic photons when they interact with air molecules. This process is related to the Bremsstrahlung process by some physical symmetry. Therefore the results above can be transferred to predictions on the angles between incident photon and emitted positron, again as a function of photon and positron energy. We present the distribution of angles and again a simple approximation for the most probable scattering angle. Our results are given as analytical expressions as well as in the form of a C++ code that can be directly be implemented into Monte Carlo codes.