Compton scattering is a type of scattering that X-rays and gamma rays undergo in matter. The elastic scattering – implying conservation of energy - of photons in matter results in a decrease in energy (increase in wavelength)of an X-ray or gamma ray photon, called the Compton effect. Part of the energy of the X – Ray or gamma ray, which in the diagram below enters from the left with wavelength
is transferred to a scattering electron, indicated in the diagram below by the black arrow directed to the bottom right, which recoils and is ejected from its atom, and the rest of the energy is taken by the scattered, degraded photon which travels towards the top right.
The Compton effect is governed by the formula
Proof: If the electron is initially at rest then it's energy is it's rest mass energy, and afterwards it's energy is determined by the relativistic equation
Conservation of energy implies
rearrangement of which gives
(1)
Conservation of momentum implies
or
The dot product property gives
for a photon hence
(2)
Put
After some simplification we obtain
Divide both sides by
to obtain
and because
we have