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.

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

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

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