# A Derivation of the Compton Scatter Equations

### Purpose

To better understand the Compton scattering equations, and to get a little math practice.

### Question

Compton scattering consists of a photon with initial energy E colliding with an atomic electron, which is ejected from the atom. During the collision the photon is scattered at an angle \phi relative to its initial direction with a new energy E' which is less than or equal to its initial energy. Any energy lost by the scattered photon is transferred to the electron as kinetic energy T; the electron travels at an angle \theta relative to the initial photon direction. A kinematic diagram of Compton scattering is shown in the figure below. Derive the equation to calculate the scattered photon energy in terms of the initial photon energy and the scattering angle:

E' = \frac{E}{1 + \frac{E}{m_e c^2}(1 - cos\phi)}