Frank-Tamm formula

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

When a charged particle moves faster than the speed of light in a medium, it can emit a photon while conserving energy and momentum, this process can be viewed as a decay.

Equation

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

:$dE\; =\; frac\{mu(omega)\; q^2\}\; \{4\; pi\}\; omega\; (1\; -\; frac\{c^2\}\; \{v^2\; n(omega)^2\})\; dx\; domega$,

where $mu$ and $n(omega)$ 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 $omega$ is the angular frequency of radiation.

The total amounted of energy radiated per unit length is:

:$frac\{dE\}\{dx\}\; =\; frac\{q^2\}\{4\; pi\}\; int\_\{v>c/n(omega)\}\; mu(omega)\; omega\; (1\; -\; frac\{c^2\}\; \{v^2\; n(omega)^2\})\; domega$

This integral is done over the frequencies which the particle's speed $v$ is greater than speed of light of the media $frac\{c\}\{n(omega)\}$.