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The '''Frank–Tamm formula''' yields the amount of [[Cherenkov radiation]] emitted on a given frequency as a charged particle moves through a medium faster than the [[phase speed]] of light in that medium. It is named for Russian physicists [[Ilya Frank]] and [[Igor Tamm]] who developed the theory of the Cherenkov effect in 1937, for which they were awarded a [[Nobel Prize in Physics]] in 1958.

==Equation==
The [[energy]] <math>dE</math> emitted per unit length travelled by the particle <math>dx</math> per unit of [[angular frequency]] <math>d\omega</math> is:

:<math>dE = \frac{q^2}{4 \pi} \mu(\omega) \omega {\left(1 - \frac{c^2} {v^2 n^2(\omega)}\right)} dx d\omega</math>

provided that <math>\beta = \frac{v}{c} > \frac{1}{n(\omega)}</math>. Here <math>\mu(\omega)</math> and <math>n(\omega)</math> are the frequency-dependent [[Permeability (electromagnetism)|permeability]] and [[index of refraction]] of the medium, <math>q</math> is the [[electric charge]] of the particle, <math>v</math> is the speed of the particle, and <math>c</math> is the [[speed of light]] in vacuum.

Cherenkov radiation does not have characteristic spectral peaks, as typical for [[fluorescence]] or emission spectra. The relative intensity of one frequency is approximately proportional to the frequency. That is, higher frequencies (shorter wavelengths) are more intense in Cherenkov radiation. This is why visible Cherenkov radiation is observed to be brilliant blue. In fact, most Cherenkov radiation is in the ultraviolet spectrum; the sensitivity of the human eye peaks at green, and is very low in the violet portion of the spectrum.

The total amount of energy radiated per unit length is:

:<math>\frac{dE}{dx} = \frac{q^2}{4 \pi} \int_{v > \frac{c}{n(\omega)}} \mu(\omega) \omega {\left(1 - \frac{c^2} {v^2 n^2(\omega)}\right)} d\omega</math>

This integral is done over the frequencies <math>\omega</math> for which the particle's speed <math>v</math> is greater than speed of light of the media <math>\frac{c}{n(\omega)}</math>. The integral is non-divergent because at high frequencies the refractive index becomes less than unity.<ref>The refractive index n is defined as the ratio of the speed of electromagnetic radiation in vacuum and the ''phase speed'' of electromagnetic waves in a medium and can, under specific circumstances, become less than one. See [[Refractive index#Refractive index below 1|refractive index]] for further information.</ref>

==Notes==
{{reflist}}

==References==
{{Refbegin}}
* {{cite journal | first1 = C. A. | last1 = Mead | journal=[[Physical Review]]| volume = 110 | page = 359 | year = 1958 | title = Quantum Theory of the Refractive Index | issue = 2 | doi = 10.1103/PhysRev.110.359 |bibcode = 1958PhRv..110..359M }}
* {{cite journal | first1 = P.A. | last1 = Cerenkov | journal=[[Physical Review]]| volume = 52 | page = 378 | year = 1937 | title = Visible Radiation Produced by Electrons Moving in a Medium with Velocities Exceeding that of Light |bibcode = 1937PhRv...52..378C |doi = 10.1103/PhysRev.52.378 }}
{{Refend}}

{{DEFAULTSORT:Frank-Tamm Formula}}
[[Category:Particle physics]]
[[Category:Experimental particle physics]]