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
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^ 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