Frank–Tamm formula

The Frank–Tamm formula yields the amount of Cherenkov radiation emitted on a given frequency 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.

Equation

The energy $dE$ emitted per unit length travelled by the particle per unit of frequency $d\omega$ is:

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

provided that $\beta = \frac{v}{c} > \frac{1}{n(\omega)}$. Here $\mu(\omega)$ and $n(\omega)$ are the frequency-dependent permeability and index of refraction of the medium respectively, $q$ is the electric charge of the particle, $v$ is the speed of the particle, and $c$ 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:

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

This integral is done over the frequencies $\omega$ for which the particle's speed $v$ is greater than speed of light of the media $\frac{c}{n(\omega)}$. The integral is convergent (finite) because at high frequencies the refractive index becomes less than unity and for extremely high frequencies it becomes unity.^[1][2]

Notes

1. 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 for further information.
2. The refractive index can become less than unity near the resonance frequency but at extremely high frequencies the refractive index becomes unity.

References

• Mead, C. A. (1958). "Quantum Theory of the Refractive Index". Physical Review 110 (2): 359. Bibcode:1958PhRv..110..359M. doi:10.1103/PhysRev.110.359.
• Cerenkov, P.A. (1937). "Visible Radiation Produced by Electrons Moving in a Medium with Velocities Exceeding that of Light". Physical Review 52: 378. Bibcode:1937PhRv...52..378C. doi:10.1103/PhysRev.52.378.