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

d

E

{\displaystyle dE}

emitted per unit length travelled by the particle per unit of frequency

d

ω

{\displaystyle d\omega }

is:

∂

2

E

∂

x

∂

ω

=

q

2

4

π

μ

(

ω

)

ω

(

1

−

c

2

v

2

n

2

(

ω

)

)

{\displaystyle {\frac {\partial ^{2}E}{\partial x\,\partial \omega }}={\frac {q^{2}}{4\pi }}\mu (\omega )\omega \left(1-{\frac {c^{2}}{v^{2}n^{2}(\omega )}}\right)}

provided that

β

=

v

c

>

1

n

(

ω

)

{\displaystyle \beta ={\frac {v}{c}}>{\frac {1}{n(\omega )}}}

. Here

μ

(

ω

)

{\displaystyle \mu (\omega )}

and

n

(

ω

)

{\displaystyle n(\omega )}

are the frequency-dependent permeability and index of refraction of the medium respectively,

q

{\displaystyle q}

is the electric charge of the particle,

v

{\displaystyle v}

is the speed of the particle, and

c

{\displaystyle 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.