Editing Smith–Purcell effect (section)

=== Dispersion relation ===
When a charged particle travels above a periodic grating (or periodic media inhomogeneity), a current is induced on the surface of the grating. This induced current then emits radiation at the discontinuities of the grating due to the scattering of the Coulomb field of the induced charges at the grating boundaries. The [[dispersion relation]] for the Smith–Purcell effect (SPE) is given as follows:<ref>{{Cite journal| vauthors = Andrews HL, Boulware CH, Brau CA, Jarvis JD |date=2005-05-20|title=Dispersion and attenuation in a Smith–Purcell free electron laser |journal=Physical Review Special Topics - Accelerators and Beams|language=en|volume=8|issue=5|pages=050703|doi=10.1103/PhysRevSTAB.8.050703|issn=1098-4402|doi-access=free|bibcode=2005PhRvS...8e0703A }}</ref>

:<math>\lambda = \frac{L}{n}\left(\frac{1}{\beta}-\cos{\theta}\right)</math>,

where the wavelength <math>\lambda</math> is observed at an angle <math>\theta</math> to the direction of the electron beam for the <math>n^{th}</math> order reflection mode, and <math>L</math> is the grating period and <math>\beta </math> is the relative electron velocity (<math>v/c</math>). This relation can be derived through considering energy and momentum conservation laws.