In physics, Compton scattering or the Compton effect, is the decrease in energy (increase in wavelength) of an X-ray or gamma ray photon, when it interacts with matter. Inverse Compton scattering also exists, where the photon gains energy (decreasing in wavelength) upon interaction with matter. The amount the wavelength increases by is called the Compton shift. Although nuclear compton scattering exists, Compton scattering usually refers to the interaction involving only the electrons of an atom. The Compton effect was observed by Arthur Holly Compton in 1923 and further verified by his graduate student Y. H. Woo in the years followed. Arthur Compton earned the 1927 Nobel Prize in Physics for the discovery.
The effect is important because it demonstrates that light cannot be explained purely as a wave phenomenon. Thomson scattering, the classical theory of an electromagnetic wave scattered by charged particles, cannot explain any shift in wavelength. Light must behave as if it consists of particles in order to explain the Compton scattering. Compton's experiment convinced physicists that light can behave as a stream of particles whose energy is proportional to the frequency.
The interaction between electrons and high energy photons results in the electron being given part of the energy (making it recoil), and a photon containing the remaining energy being emitted in a different direction from the original, so that the overall momentum of the system is conserved. If the photon still has enough energy left, the process may be repeated. If the photon has sufficient energy (in general a few eV, right around the energy of visible light), it can even eject an electron from its host atom entirely (a process known as the Photoelectric effect).

The Compton shift formula
Begin with energy and momentum conservation:

where

and are the energy and momentum of the photon and
and are the energy and momentum of the electron.

Derivation
Now we fill in for the energy part:

We solve this for p_e':



Solving (2)
Then we have two equations for (eq 3 & 4), which we equate:

Now, one simplifies. First by multiplying both sides by c and then by :

Now the left-hand side can be rewritten as simply

This is equivalent to the Compton scattering equation, but it is usually written using λ's rather than f's. To make that switch use

so that finally,

Putting it together

Applications
Compton scattering is of prime importance to radiobiology, as it happens to be the most probable interaction of high energy X rays with atomic nuclei in living beings and is applied in radiation therapy.
In material physics, Compton scattering can be used to probe the wave function of the electrons in matter in the momentum representation.
Compton Scatter is an important effect in Gamma spectroscopy which gives rise to the Compton edge, as it is possible for the gamma rays to scatter out of the detectors used. Compton suppression is used to detect stray scatter gamma rays to counteract this effect.

Inverse Compton scattering

Thomson scattering
Klein-Nishina formula
Photoelectric effect
Pair production
Timeline of cosmic microwave background astronomy
Peter Debye
Walther Bothe
List of astronomical topics
List of physics topics
Washington University in St. Louis (Site of discovery)