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The Smith–Purcell effect describes the interaction of a free electron and radiation in the presence of a periodic medium (such as a diffraction grating). It was first predicted by Ilya Frank in 1941 and experimentally demonstrated in 1953 by Steve Smith and the Nobel laureate, Edward Purcell. Dircetly related to Cherenkov radiation, the Smith-Purcell effect was the precursor of a few important inventions, such as Free-Electron Lasing and Dielectric Laser Acceleration.

History
In his 1941 paper*, entitled "Doppler effect in a refractive medium", Frank suggested that an electric dipole moving at a constant speed in a periodic medium will experience a periodic contraction, which will cause it to radiate.

Mathematical Description

 * $$\lambda= \frac{a}{m}\left(\frac{1}{\beta}-\cos\theta \right) $$