A relativistic quantum-mechanical treatment is given to explain the observations of Smith and Purcell on the emission of light by an electron beam in the presence of a diffraction grating. The dependence of the fundamental wavelength and its harmonics upon the grating spacing, the velocity of the electron, and the direction of the emitted photon are obtained. Intensity and polarization of the emitted light are discussed.

• Received 7 December 1967

DOI:https://doi.org/10.1103/PhysRev.174.57