In the classical Compton scattering, the probability of a photon to scatter in a certain direction depends only on the scattered photon and electron. However, since photons are bosons, the probability of scattering in a certain direction increases with the number of photons already moving in that direction. Let be the distribution function of photons, such that the number density of photons (photons per unit volume) is given by , and is the wave - number (and the momentum of the photons). Let the distribution functions of the electrons be , where is the momentum of the electron. Let be the probability of scattering an electron with momentum and a photon of momentum to a state where the momentum of the photon is (and the momentum of the electron is ). The rate equation for the photons is given by
If the electron does not recoil, then the incoming and outgoing photons have the momenta, so detailed balance reads
Substituting into the rate equation eliminates the extra terms (due to symmetry) and only the regular scattering is left.
However, if recoil is taken into account, then detailed balance reads
where . Substituting this into the equation, assuming and applying a Taylor expansion yields
The first integral is just the classical Compton scattering. The second term is a correction due to recoil and induced emission.