Klein-Nishina Formula for Photon-Electron Scattering

Low-energy (Thomson) scattering of a photon by an electron is approximated by the differential scattering cross section , where cm, the classical electron radius. The corresponding total scattering cross section is given by . The Thomson formula is, however, inadequate to treat the higher-energy photoelectric and Compton effects. Klein and Nishina (1929) derived the scattering cross-section according to Dirac's relativistic theory of the electron: , where and , the incident photon energy in units of the electron rest energy (0.511 MeV/). The formulas pertain to the average of the two photon polarizations. A polar plot of the differential scattering cross section is shown in the graphic, with photon energy selectable in the range 0-50 MeV. The directions of the incident and scattered photons are shown by red arrows, while the scattered electron (initially at rest) follows the blue arrow. Numerical values are given for the photon differential and total cross sections, the electron scattering angle and the electron kinetic energy . (Subtract angle labels greater than 180° from 360°).

At higher energies, radiative corrections to the Klein–Nishina formula become significant. With energies in excess of 100 MeV, pair production becomes dominant.