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The Frank–Tamm formula yields the amount of Cherenkov radiation emitted as a charged particle moves through a medium at superluminal velocity. 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.

When a charged particle moves faster than the phase speed of light in a medium, electrons interacting with the particle can emit coherent photons while conserving energy and momentum, this process can be viewed as a decay. See Cherenkov radiation and nonradiation condition for an explanation of this effect.

[edit] Equation

The energy emitted per unit length (travelled by the particle) per unit frequency is:

$dE = \frac{\mu(\omega) q^2} {4 \pi} \omega {\left(1 - \frac{c^2} {v^2 n^2(\omega)}\right)} dx d\omega$ ,

where $μ$ and $n (ω)$ are the permeability and index of refraction of the medium, $q$ is the electric charge of the particle, $v$ is the speed of the particle, $c$ is the speed of light in vacuum, and $ω$ is the angular frequency of radiation.

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:

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

This integral is done over the frequencies $ω$ for which the particle's speed $v$ is greater than speed of light of the media $\frac{c}{n(\omega)}$ . The integral is non-divergent because at high frequencies the refractive index becomes less than unity^[1]

[edit] References

^ The refractive index n is defined as the ratio of the the speed of electromagnetic radiation in vacuum and the phase speed of electromagnetic waves in a medium and can under specific circumstances become less then one. Referring to article on refractive index

[0] The refractive index n is defined as the ratio of the the speed of electromagnetic radiation in vacuum and the phase speed of electromagnetic waves in a medium and can under specific circumstances become less then one. Referring to article on refractive index

[1]