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.
See Cherenkov radiation and nonradiation condition for an explanation of this effect.
are the frequency-dependent permeability and index of refraction of the medium respectively,
Cherenkov radiation does not have characteristic spectral peaks, as typical for fluorescence or emission spectra.
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:
is greater than speed of light of the media
The integral is convergent (finite) because at high frequencies the refractive index becomes less than unity and for extremely high frequencies it becomes unity.
Start with Maxwell's equations (in Gaussian units) in the wave forms (also known as the Lorenz gauge condition) and take the Fourier transform:
is the elementary charge) moving with velocity
, taking the Fourier transform [note 4] gives:
Substituting this density and charge current into the wave equation, we can solve for the Fourier-form potentials:
To find the radiated energy, we consider electric field as a function of frequency at some perpendicular distance from the particle trajectory, say, at
is in the form of a modified (Macdonald) Bessel function, giving the evaluated parallel component in the form:
One can follow a similar pattern of calculation for the other fields components arriving at: We can now consider the radiated energy
It can be expressed through the electromagnetic energy flow
through the surface of an infinite cylinder of radius
around the path of the moving particle, which is given by the integral of the Poynting vector
{\displaystyle \left({\frac {dE}{dx_{\text{particle}}}}\right)_{\text{rad}}={\frac {1}{v}}P_{a}=-{\frac {c}{4\pi v}}\int _{-\infty }^{\infty }2\pi aB_{3}E_{1}\,dx}
{\displaystyle \left({\frac {dE}{dx_{\text{particle}}}}\right)_{\text{rad}}=-{\frac {ca}{2}}\int _{-\infty }^{\infty }B_{3}(t)E_{1}(t)\,dt}
To go into the domain of Cherenkov radiation, we now consider perpendicular distance
much greater than atomic distances in a medium, that is,
With this assumption we can expand the Bessel functions into their asymptotic form:
has a positive real part (usually true), the exponential will cause the expression to vanish rapidly at large distances, meaning all the energy is deposited near the path.
is real, Cherenkov radiation has the condition that
This is the statement that the speed of the particle must be larger than the phase velocity of electromagnetic fields in the medium at frequency