Frank–Tamm formula explained

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

dE emitted per unit length travelled by the particle per unit of frequency d\omega is:$$\backslash frac\; =\; \backslash frac\; \backslash mu(\backslash omega)\; \backslash omega\; \backslash left(1\; -\; \backslash frac\; \backslash right)$$provided that \beta = (v)/(c) > (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:$$\backslash frac\; =\; \backslash frac\; \backslash int\_\; \backslash mu(\backslash omega)\; \backslash omega\; \backslash left(1\; -\; \backslash frac\; \backslash right)\; \backslash ,\; d\backslash omega$$

This integral is done over the frequencies

\omega for which the particle's speed v is greater than speed of light of the media $\backslash frac$. 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]}

Derivation of Frank–Tamm formula

Consider a charged particle moving relativistically along

x -axis in a medium with refraction index $n(\backslash omega)\; =\; \backslash sqrt$^[3] with a constant velocity \vec v = (v, 0 , 0 ) . Start with Maxwell's equations (in Gaussian units) in the wave forms (also known as the Lorenz gauge condition) and take the Fourier transform:$$\backslash left\; (k^2\; -\; \backslash frac\; \backslash varepsilon(\backslash omega)\; \backslash right)\; \backslash Phi(\backslash vec\; k,\backslash omega)\; =\; \backslash frac\; \backslash rho(\backslash vec\; k,\; \backslash omega)$$$$\backslash left\; (k^2\; -\; \backslash frac\; \backslash varepsilon(\backslash omega)\; \backslash right)\; \backslash vec\; A(\backslash vec\; k,\backslash omega)\; =\; \backslash frac\; \backslash vec\; J(\backslash vec\; k,\; \backslash omega)$$

For a charge of magnitude

ze (where e is the elementary charge) moving with velocity v , the density and charge density can be expressed as \rho(\vec x, t) = q \delta(\vec x - \vec v t) and \vec J(\vec x,t) = \vec v \rho(\vec x,t) , taking the Fourier transform ^[4] gives:$$\backslash rho(\backslash vec\; k,\; \backslash omega)\; =\; \backslash frac\; \backslash delta(\backslash omega\; -\; \backslash vec\; k\; \backslash cdot\; \backslash vec\; v)$$$$\backslash vec\; J(\backslash vec\; k,\; \backslash omega)\; =\; \backslash vec\; v\; \backslash rho\; (\backslash vec\; k,\backslash omega)$$

Substituting this density and charge current into the wave equation, we can solve for the Fourier-form potentials:$$\backslash Phi(\backslash vec\; k,\; \backslash omega)\; =\; \backslash frac\; \backslash frac$$ and $$\backslash vec\; A(\backslash vec\; k,\backslash omega)\; =\; \backslash varepsilon(\backslash omega)\; \backslash frac\; \backslash Phi(\backslash vec\; k,\backslash omega)$$

Using the definition of the electromagnetic fields in terms of potentials, we then have the Fourier-form of the electric and magnetic field:$$\backslash vec\; E(\backslash vec\; k,\backslash omega)\; =\; i\; \backslash left(\backslash frac\; \backslash frac\; -\; \backslash vec\; k\; \backslash right)\; \backslash Phi(\backslash vec\; k,\backslash omega)$$ and $$\backslash vec\; B(\backslash vec\; k,\backslash omega)\; =\; i\; \backslash varepsilon(\backslash omega)\; \backslash vec\; k\; \backslash times\; \backslash frac\; \backslash Phi(\backslash vec\; k,\backslash omega)$$

To find the radiated energy, we consider electric field as a function of frequency at some perpendicular distance from the particle trajectory, say, at

( 0 ,b, 0 ) , where b is the impact parameter. It is given by the inverse Fourier transform:$$\backslash vec\; E(\backslash omega)\; =\; \backslash frac\; \backslash int\; d^3k\; \backslash ,\; \backslash vec\; E(\backslash vec\; k,\backslash omega)\; e^$$

First we compute

x -component E _{1} of the electric field (parallel to \vec v ):$$E\_1(\backslash omega)\; =\; \backslash frac\; \backslash int\; d^3k\; \backslash ,\; e^\; \backslash left(\backslash frac\; -\; k\_1\; \backslash right)\; \backslash frac$$

For brevity we define $\backslash lambda^2\; =\; \backslash frac\; -\; \backslash frac\; \backslash varepsilon(\backslash omega)\; =\; \backslash frac\; \backslash left\; (1\; -\; \backslash beta^2\; \backslash varepsilon(\backslash omega)\; \backslash right)$. Breaking the integral apart into

k _{1,} k _{2,} k _{3} , the k _{1} integral can immediately be integrated by the definition of the Dirac Delta:$$E\_1(\backslash omega)\; =\; -\; \backslash frac\; \backslash left(\backslash frac\; -\; \backslash beta^2\; \backslash right)\; \backslash int\_^\backslash infty\; dk\_2\; \backslash ,\; e^\; \backslash int\_^\backslash infty\; \backslash frac$$

The integral over

k _{3} has the value $\backslash frac$, giving:$$E\_1(\backslash omega)\; =\; -\; \backslash frac\; \backslash left(\backslash frac\; -\; \backslash beta^2\; \backslash right)\; \backslash int\_^\backslash infty\; dk\_2\; \backslash frac$$

The last integral over

k _{2} is in the form of a modified (Macdonald) Bessel function, giving the evaluated parallel component in the form:$$E\_1(\backslash omega)\; =\; -\; \backslash frac\; \backslash left(\backslash frac\; \backslash right)^\; \backslash left(\backslash frac\; -\; \backslash beta^2\; \backslash right)\; K\_0(\backslash lambda\; b)$$

One can follow a similar pattern of calculation for the other fields components arriving at:

E _{2(\omega)} = (q)/(v) \left( (2)/(\pi) \right) ^{1/2} (λ)/(\varepsilon(\omega)) K _{1(λ} b), E _{3} = 0 and B _{1} = B _{2} = 0 , B _{3(\omega)} = \varepsilon(\omega) \beta E _{2(\omega)}

We can now consider the radiated energy

dE per particle traversed distance dx _{particle} . It can be expressed through the electromagnetic energy flow P _{a} through the surface of an infinite cylinder of radius a around the path of the moving particle, which is given by the integral of the Poynting vector S = c / ( 4 \pi) [ E x H] over the cylinder surface:$$\backslash left(\backslash frac\; \backslash right)\_\; =\; \backslash frac\; P\_a\; =\; -\; \backslash frac\; \backslash int\_^\; 2\; \backslash pi\; a\; B\_3\; E\_1\; \backslash ,\; dx$$

The integral over

dx at one instant of time is equal to the integral at one point over all time. Using dx = v dt :$$\backslash left(\backslash frac\; \backslash right)\_\; =\; -\; \backslash frac\; \backslash int\_^\backslash infty\; B\_3(t)\; E\_1(t)\; \backslash ,\; dt$$

Converting this to the frequency domain:$$\backslash left(\backslash frac\; \backslash right)\_\; =\; -c\; a\; \backslash operatorname\; \backslash left(\backslash int\_0^\backslash infty\; B\_3^*(\backslash omega)\; E\_1(\backslash omega)\; \backslash ,\; d\backslash omega\; \backslash right)$$

To go into the domain of Cherenkov radiation, we now consider perpendicular distance

b much greater than atomic distances in a medium, that is, | λ b | \gg 1 . With this assumption we can expand the Bessel functions into their asymptotic form:$$E\_1(\backslash omega)\; \backslash rightarrow\; \backslash frac\; \backslash left(1\; -\; \backslash frac\; \backslash right)\; \backslash frac$$E _{2(\omega)} → (q)/(\sqrt{ v \varepsilon(\omega)) (λ)/(b) } e^ and B _{3(\omega)} = \varepsilon(\omega) \beta E _{2(\omega)}

Thus:

$$\backslash left(\backslash frac\; \backslash right)\_\; =\; \backslash operatorname\; \backslash left(\backslash int\_0^\backslash infty\; \backslash frac\; \backslash left(-i\; \backslash sqrt\backslash right)\; \backslash omega\; \backslash left(1\; -\; \backslash frac\; \backslash right)\; e^\; \backslash ,\; d\backslash omega\; \backslash right)$$

If

λ 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. However, this isn't true when λ is purely imaginary – this instead causes the exponential to become 1 and then is independent of a , meaning some of the energy escapes to infinity as radiation – this is Cherenkov radiation.λ is purely imaginary if \varepsilon(\omega) is real and \beta ^{2} \varepsilon(\omega) > 1 . That is, when \varepsilon(\omega) is real, Cherenkov radiation has the condition that $v\; >\; \backslash frac\; =\; \backslash frac$. 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 \omega in order to have Cherenkov radiation. With this purely imaginary λ condition, $\backslash sqrt\; =\; i$ and the integral can be simplified to:$$\backslash left(\backslash frac\; \backslash right)\_\; =\; \backslash frac\; \backslash int\_\; \backslash omega\; \backslash left(1\; -\; \backslash frac\; \backslash right)\; \backslash ,\; d\backslash omega\; =\; \backslash frac\; \backslash int\_\; \backslash omega\; \backslash left(1\; -\; \backslash frac\; \backslash right)\; \backslash ,\; d\backslash omega$$

This is the Frank–Tamm equation in Gaussian units.^[5]

References

• C. A. . Mead . Physical Review. 110 . 359 . 1958 . Quantum Theory of the Refractive Index . 2 . 10.1103/PhysRev.110.359 . 1958PhRv..110..359M .
• P.A. . Cerenkov . Physical Review. 52 . 4 . 378 . 1937 . Visible Radiation Produced by Electrons Moving in a Medium with Velocities Exceeding that of Light . 1937PhRv...52..378C . 10.1103/PhysRev.52.378 .

External links

• Cherenkov radiation (Tagged ‘Frank-Tamm formula’)

Notes and References

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.
3. For simplicity we consider magnetic permeability \mu(\omega) = 1 .
4. We use engineering notation for the Fourier transform, where 1 /\sqrt{ 2 \pi} factors appear both in direct and inverse transforms.
5. Book: Jackson, John. Classical Electrodynamics. limited. John Wiley & Sons, Inc. 1999. 978-0-471-30932-1. 646–654.