Editing Smith–Purcell effect (section)

== Background ==
Charged particles usually radiate/generate radiation via two different mechanisms:

# Acceleration or change of direction of motion: e.g. [[Bremsstrahlung]] radiation (e.g. in [[X-ray tube]]s), [[synchrotron radiation]] (as in [[Free-electron laser|FEL]] due to electron beams going through [[Wiggler (synchrotron)|wiggler]]/ [[undulator]] set-ups, or a beam energy-loss mechanism in [[Particle accelerator|circular colliders]]).
# Polarisation: A moving charge has a dynamic [[Coulomb's law|Coulomb field]]. For a conducting/[[Polarization density|polarisable]] material, the interaction between this field and the charges in the material/ medium could generate radiation. This includes [[Cherenkov radiation|Cherenkov]] and [[transition radiation]], where the particle moves within the medium which generates the radiation, but also diffraction radiation,<ref>{{Cite journal|title=On the theory of diffraction radiation|journal= Journal of Experimental and Theoretical Physics|first1=D. V.|last1=Karlovets|first2=A. P.|last2=Potylitsyn|year=2008 |volume=107 |issue=5 |pages=755–768 |language=en|doi=10.1134/s1063776108110058|bibcode= 2008JETP..107..755K|s2cid=121821580 }}</ref> where (usually relativistic) particles move in the vicinity of the target material, generating for example, optical diffraction radiation (ODR)<ref>{{Cite journal|last1=Fiorito|first1=R. B.|last2=Shkvarunets|first2=A. G.|last3=Watanabe|first3=T.|last4=Yakimenko|first4=V.|last5=Snyder|first5=D.|date=2006-05-24|title=Interference of diffraction and transition radiation and its application as a beam divergence diagnostic|journal=Physical Review Special Topics - Accelerators and Beams|volume=9|issue=5|pages=052802|doi=10.1103/PhysRevSTAB.9.052802|arxiv=physics/0605110|issn=1098-4402|doi-access=free|bibcode=2006PhRvS...9e2802F }}</ref> and Smith–Purcell radiation (SPR).

The benefit of using polarisation radiation in particular is the lack of direct effect on the original beam; the beam inducing the radiative emission can continue its original path unaltered and having induced [[Electromagnetic radiation|EM]] radiation. This is unlike the bremsstrahlung or synchrotron effects which actually alter or bend the incoming beam. Due to this non-destructive feature, SPR has become an interesting prospect for beam diagnostics, also offering the possibility of reliable technologies due to theoretically no contact or scattering interactions between the beam and the target.
[[File:The Smith-Purcell Effect .png|thumb|300x300px|The Smith–Purcell effect]]

=== 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.