In pair production, a photon creates an electron positron pair. 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.
is the atomic number,
the fine structure constant,
the reduced Planck's constant and
the speed of light. The kinetic energies
of the positron and electron relate to their total energies
and momenta
via
Conservation of energy yields
The momentum
of the virtual photon between incident photon and nucleus is:
where the directions are given via:
where
is the momentum of the incident photon.
In order to analyse the relation between the photon energy
and the emission angle
between photon and positron, Köhn and Ebert integrated [2] the quadruply differential cross section over
and
. 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.
References[]
- ^ Bethe, H.A., Heitler, W., 1934. On the stopping of fast particles and on the creation of positive electrons. Proc. Phys. Soc. Lond. 146, 83–112
- ^ Koehn, C., Ebert, U., Angular distribution of Bremsstrahlung photons and of positrons for calculations of terrestrial gamma-ray flashes and positron beams, Atmos. Res. (2014), vol. 135-136, pp. 432-465