In the process of photons scattering in air (e.g. in lightning discharges), the most important interaction is the scattering of photons at the nuclei of atoms or molecules.
The full quantum mechanical process of pair production can be described by the quadruply differential cross section given here:[1] with This expression can be derived by using a quantum mechanical symmetry between pair production and Bremsstrahlung.
the reduced Planck constant and
of the positron and electron relate to their total energies
via Conservation of energy yields The momentum
is the momentum of the incident photon.
In order to analyse the relation between the photon energy
between photon and positron, Köhn and Ebert integrated [2] the quadruply differential cross section over
The double differential cross section is: with and This cross section can be applied in Monte Carlo simulations.
An analysis of this expression shows that positrons are mainly emitted in the direction of the incident photon.