In particle physics, the Klein–Nishina formula gives the differential cross section (i.e. the "likelihood" and angular distribution) of photons scattered from a single free electron, calculated in the lowest order of quantum electrodynamics. It was first derived in 1928 by Oskar Klein and Yoshio Nishina, constituting one of the first successful applications of the Dirac equation. The formula describes both the Thomson scattering of low energy photons (e.g. visible light) and the Compton scattering of high energy photons (e.g. x-rays and gamma-rays), showing that the total cross section and expected deflection angle decrease with increasing photon energy. Formula For an incident unpolarized photon of energy , the differential cross section is: where is the classical electron radius (~2.82 fm, is about 7.94 × 10−30 m2 or 79.4 mb) is the ratio of the wavelengths of the incident and scattered photons is the scattering angle (0 for an undeflected photon). The angular dependent photon wavelength (or energy, or frequency) ratio is as required by the conservation of relativistic energy and momentum (see Compton scattering). The dimensionless quantity expresses the energy of the incident photon in terms of the electron rest energy (~511 keV), and may also be expressed as , where is the Compton wavelength of the electron (~2.42 pm). Notice that the scatter ratio increases monotonically with the deflection angle, from (forward scattering, no energy transfer) to (180 degree backscatter, maximum energy transfer). In some cases it is convenient to express the classical electron radius in terms of the Compton wavelength: , where is the fine structure constant (~1/137) and is the reduced Compton wavelength of the electron (~0.386 pm), so that the constant in the cross section may be given as: Polarized photons If the incoming photon is polarized, the scattered photon is no longer isotropic with respect to the azimuthal angle. For a linearly polarized photon scattered with a free electron at rest, the differential cross section is instead given by: where is the azimuthal scattering angle. Note that the unpolarized differential cross section can be obtained by averaging over . Limits Low energy For low energy photons the wavelength shift becomes negligible () and the Klein–Nishina formula reduces to the classical Thomson expression: which is symmetrical in the scattering angle, i.e. the photon is just as likely to scatter backwards as forwards. With increasing energy this symmetry is broken and the photon becomes more likely to scatter