The usual expressions for the Compton effect are based on the assumptions that the scattering electron is free and stationary. These assumptions are reasonable when the photon energy is large compared to the electron binding energy; however, in many situations this is not the case, and the binding energy and momentum of the electron alter Compton scattering by causing (1) line broadening, (2) a discrete spectrum, (3) a line-shift defect, (4) a decrease in incoherent scattering at low angles, (5) an increase in coherent scattering at low angles, and (6) a change in recoil electron distribution. These effects are described, and differential and average cross sections for carbon are calculated on the Thomas-Fermi model to illustrate them quantitatively.

The classic Compton equation was derived for an initially free stationary electron. Expermental results on scattering of x rays from atomic electrons are in disagreement with predictions from the Compton equation, showing a line broadening and a defect of the center of this broadened line toward a wavelength shorter than that predicted by the Compton equation. These effects have been explained separately on Doppler effect and impulse models. In this paper a relativistically covariant formalism including electron binding, described by a heavy electron, is used to derive the general Compton equation containing the line broadening and defect terms. The defect term contains explicit dependence on the scattering angle.