  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.
