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 $dE$ emitted per unit length travelled by the particle per unit of frequency $d\omega$ is:

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

provided that $\beta ={\frac {v}{c}}>{\frac {1}{n(\omega )}}$ . 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:

{\frac {dE}{dx}}={\frac {q^{2}}{4\pi }}\int _{{v>{\frac {c}{n(\omega )}}}}\mu (\omega )\omega {\left(1-{\frac {c^{2}}{v^{2}n^{2}(\omega )}}\right)}d\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 ${\frac {c}{n(\omega )}}$ . 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 in a medium with a constant velocity. Start with Maxwell's equations (in Gaussian units) in the wave form and take the Fourier transformation:

${\bigg (}k^{2}-{\frac {\omega ^{2}}{c^{2}}}\epsilon (\omega ){\bigg )}\Phi ({\vec {k}},\omega )={\frac {4\pi }{\epsilon (\omega )}}\rho ({\vec {k}},\omega )$

${\bigg (}k^{2}-{\frac {\omega ^{2}}{c^{2}}}\epsilon (\omega ){\bigg )}{\vec {A}}({\vec {k}},\omega )={\frac {4\pi }{c}}{\vec {J}}({\vec {k}},\omega )$

For a charge moving with velocity $v$ , the density and charge density can be expressed as $\rho ({\vec {x}},t)=ze\delta ({\vec {x}}-{\vec {v}}t)$ and ${\vec {J}}({\vec {x}},t)={\vec {v}}\rho ({\vec {x}},t)$ , taking the Fourier transformation gives:

$\rho ({\vec {k}},\omega )={\frac {ze}{2\pi }}\delta (\omega -{\vec {k}}\cdot {\vec {v}})$

${\vec {J}}({\vec {k}},\omega )={\vec {v}}\rho ({\vec {k}},\omega )$

Substituting this density and charge current into the wave equation, we can solve for the Fourier-form potentials:

$\Phi ({\vec {k}},\omega )={\frac {2ze}{\epsilon (\omega )}}{\frac {\delta (\omega -{\vec {k}}\cdot {\vec {v}})}{k^{2}-{\frac {\omega ^{2}}{c^{2}}}\epsilon (\omega )}}$ and ${\vec {A}}({\vec {k}},\omega )=\epsilon (\omega ){\frac {\vec {v}}{c}}\Phi ({\vec {k}},\omega )$

Using the definition of the electromagnetic fields in terms of potentials, we then have the Fourer-form of the electric and magnetic field:

${\vec {E}}({\vec {k}},\omega )=i{\bigg (}{\frac {\omega \epsilon (\omega )}{c}}{\frac {\vec {v}}{c}}-{\vec {k}}{\bigg )}\Phi ({\vec {k}},\omega )$ and ${\vec {B}}({\vec {k}},\omega )=i\epsilon (\omega ){\vec {k}}\times {\frac {\vec {v}}{c}}\Phi ({\vec {k}},\omega )$

To find the energy loss, we're interested in the electric field as a function of wavelength at a observation at some perpendicular distance at, say, $(0,b,0)$ , where $b$ is the impact parameter. To remove the wave number dependence, we integrate our electric field expression as:

${\vec {E}}(\omega )={\frac {1}{(2\pi )^{3/2}}}\int d^{3}k{\vec {E}}({\vec {k}},\omega )e^{ibk_{2}}$

To have a general form, we need to complete this integral. Let's first find the electric field at a distance parallel to $v$ , so we can take just one component of electric field expression, and integrate over all $k$ . Substituting this in:

$E_{1}(\omega )={\frac {2ize}{\epsilon (\omega )(2\pi )^{3/2}}}\int d^{3}ke^{ibk_{2}}{\bigg (}{\frac {\omega \epsilon (\omega )v}{c^{2}}}-k_{1}{\bigg )}{\frac {\delta (\omega -vk_{1})}{k^{2}-{\frac {\omega ^{2}}{c^{2}}}\epsilon (\omega )}}$

To simply, we can define $\lambda ^{2}={\frac {\omega ^{2}}{v^{2}}}-{\frac {\omega ^{2}}{c^{2}}}\epsilon (\omega )={\frac {\omega ^{2}}{v^{2}}}{\big (}1-\beta ^{2}\epsilon (\omega ){\big )}$ . 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:

$E_{1}(\omega )={\frac {2ize\omega }{v^{2}(2\pi )^{3/2}}}{\bigg (}{\frac {1}{\epsilon (\omega )-\beta ^{2}}}{\bigg )}\int _{-\infty }^{\infty }dk_{2}e^{ibk_{2}}\int _{-\infty }^{\infty }{\frac {dk_{3}}{k_{2}^{2}+k_{3}^{2}+\lambda ^{2}}}$

The integral over $k_{3}$ has the value ${\frac {\pi }{(\lambda ^{2}+k_{2}^{2})^{1/2}}}$ , giving:

$E_{1}(\omega )=-{\frac {ize\omega }{v^{2}{\sqrt {2\pi }}}}{\bigg (}{\frac {1}{\epsilon (\omega )-\beta ^{2}}}{\bigg )}\int _{-\infty }^{\infty }dk_{2}{\frac {e^{ibk_{2}}}{(\lambda ^{2}+k_{2}^{2})^{1/2}}}$

The last integral over $k_{2}$ is in the form of a modified Bessel function, giving the evaluated parallel component in the form:

$E_{1}(\omega )=-{\frac {ize\omega }{v^{2}}}{\big (}{\frac {2}{\pi }}{\big )}^{1/2}{\bigg (}{\frac {1}{\epsilon (\omega )}}-\beta ^{2}{\bigg )}K_{0}(\lambda b)$

One can follow a similar pattern of calculation for the other fields, which will be quoted here:

$E_{2}(\omega )={\frac {ze}{v}}{\big (}{\frac {2}{\pi }}{\big )}^{1/2}{\frac {\lambda }{\epsilon (\omega )}}K_{1}(\lambda b)$ and $B_{3}(\omega )=\epsilon (\omega )\beta E_{2}(\omega )$

We can now consider the energy loss. Consider the electromagnetic energy flow through a cylinder of radius $a$ around the path of the incident particle. By conservation of energy, this can be expressed as:

${\bigg (}{\frac {dE}{dx}}{\bigg )}_{b>a}={\frac {1}{v{\frac {dE}{dt}}}}=-{\frac {c}{4\pi v}}\int _{-\infty }^{\infty }2\pi aB_{3}E_{1}dx$

The integral over $dx$ at one instant of time is equal to the integral at one point over all time. Using $dx=vdt$ :

${\bigg (}{\frac {dE}{dx}}{\bigg )}_{b>a}=-{\frac {ca}{2}}\int _{-\infty }^{\infty }B_{3}(t)E_{1}(t)dt$

Converting this to the frequency domain:

${\bigg (}{\frac {dE}{dx}}{\bigg )}_{b>a}=-ca*{\text{Re}}{\bigg (}\int _{0}^{\infty }B_{3}^{*}(\omega )E_{1}(\omega )d\omega {\bigg )}$

To go into the domain of Cherenkov radiation, we now consider the radiation being emitted much further than atomic distances in a medium, that is, $|\lambda a|\gg 1$ . With this assumption we can expand the Bessel functions into their asymptotic form:

$E_{1}(\omega )\rightarrow {\frac {ize\omega }{c^{2}}}{\big (}{\frac {2}{\pi }}{\big )}^{1/2}{\bigg (}1-{\frac {1}{\beta ^{2}\epsilon (\omega )}}{\bigg )}{\frac {e^{-\lambda b}}{\sqrt {\lambda b}}}$

$E_{2}(\omega )\rightarrow {\frac {ze}{v}}{\sqrt {\frac {\lambda }{b}}}e^{-\lambda b}$ and $B_{3}(\omega )=\epsilon (\omega )\beta E_{2}(\omega )$

Substituting these in:

${\bigg (}{\frac {dE}{dx}}{\bigg )}_{rad}={\text{Re}}{\bigg (}\int _{0}^{\infty }{\frac {z^{2}e^{2}}{c^{2}}}{\bigg (}-i{\sqrt {\frac {\lambda ^{*}}{\lambda }}}{\bigg )}\omega {\bigg (}1-{\frac {1}{\beta ^{2}\epsilon (\omega )}}{\bigg )}e^{-(\lambda +\lambda ^{*})a}d\omega {\bigg )}$

Consider the real portion of the integral over all frequencies. If $\lambda$ has a positive, real portion (generally 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 $\lambda$ 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.

$\lambda$ is purely imaginary if $\epsilon (\omega )$ is real and $\beta ^{2}\epsilon (\omega )>1$ . That is, when $\epsilon (\omega )$ is real, Cherenkov radiation has the condition that $v>{\frac {c}{{\sqrt {\epsilon (\omega }})}}$ . This is the statement that the speed of the particle much be larger than the phase velocity of electromagnetic fields at frequency $\omega$ in order to have Cherenkov radiation. With this purely imaginary $\lambda$ condition, ${\sqrt {\frac {\lambda ^{*}}{\lambda }}}=i$ and the integral can be simplified to:

${\bigg (}{\frac {dE}{dx}}{\bigg )}_{rad}={\frac {z^{2}e^{2}}{c^{2}}}\int _{\epsilon (\omega )>{\frac {1}{\beta ^{2}}}}\omega {\bigg (}1-{\frac {1}{\beta ^{2}\epsilon (\omega )}}{\bigg )}d\omega$

This is the Frank Tamm equation in Gaussian units. This derivation follows Jackson 3rd Edition^[3]

Notes

↑ 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.
↑ The refractive index can become less than unity near the resonance frequency but at extremely high frequencies the refractive index becomes unity.
↑ Jackson, John (1999). Classical Electrodynamics. John Wiley & Sons, Inc. pp. 646–654. ISBN 0-471-30932-X.

References

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

External links

Cherenkov radiation (Tagged ‘Frank-Tamm formula’)

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[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] Jackson, John (1999). Classical Electrodynamics. John Wiley & Sons, Inc. pp. 646–654. ISBN 0-471-30932-X.