The Klein–Nishina formula gives the differential cross section of photons scattered from a single free electron in lowest order of quantum electrodynamics. At low frequencies (e.g., visible light) this yields Thomson scattering; at higher frequencies (e.g., x-rays and gamma-rays) this yields Compton scattering.
For an incident photon of energy
where
Note that this result may also be expressed in terms of the classical electron radius
The Klein–Nishina formula was derived in 1928 by Oskar Klein and Yoshio Nishina, and was one of the first results obtained from the study of quantum electrodynamics. Consideration of relativistic and quantum mechanical effects allowed development of an accurate equation for the scattering of radiation from a target electron. Before this derivation, the electron cross section had been classically derived by the British physicist and discoverer of the electron, J.J. Thomson. However, scattering experiments showed significant deviations from the results predicted by the Thomson cross section. Further scattering experiments agreed perfectly with the predictions of the Klein–Nishina formula.
Note that if
The final energy of the scattered photon,