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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 Ilya Mikhailovich Frank (russian: Илья́ Миха́йлович Франк; 23 October 1908 – 22 June 1990) was a Soviet winner of the Nobel Prize for Physics in 1958 jointly with Pavel Alekseyevich Cherenkov and Igor Y. Tamm, also of t ...
and
Igor Tamm Igor Yevgenyevich Tamm ( rus, И́горь Евге́ньевич Тамм , p=ˈiɡərʲ jɪvˈɡʲenʲjɪvitɕ ˈtam , a=Ru-Igor Yevgenyevich Tamm.ogg; 8 July 1895 – 12 April 1971) was a Soviet physicist who received the 1958 Nobel Prize i ...
who developed the theory of the Cherenkov effect in 1937, for which they were awarded a
Nobel Prize in Physics ) , image = Nobel Prize.png , alt = A golden medallion with an embossed image of a bearded man facing left in profile. To the left of the man is the text "ALFR•" then "NOBEL", and on the right, the text (smaller) "NAT•" then " ...
in 1958. When a charged particle moves faster than the
phase speed The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, ...
of light in a medium, electrons interacting with the particle can emit coherent
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are Massless particle, massless ...
s while conserving
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
and
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
. This process can be viewed as a decay. See Cherenkov radiation and
nonradiation condition Classical nonradiation conditions define the conditions according to classical electromagnetism under which a distribution of accelerating charges will not emit electromagnetic radiation. According to the Larmor formula in classical electromagnetis ...
for an explanation of this effect.


Equation

The
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
dE emitted per unit length travelled by the particle per unit of
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from '' angular frequency''. Frequency is measured in hertz (Hz) which is ...
d\omega is: \frac = \frac \mu(\omega) \omega \left(1 - \frac \right) provided that \beta = \frac > \frac. Here \mu(\omega) and n(\omega) are the frequency-dependent permeability and
index of refraction In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
of the medium respectively, q is the
electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respecti ...
of the particle, v is the speed of the particle, and c is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
in vacuum. Cherenkov radiation does not have characteristic spectral peaks, as typical for
fluorescence Fluorescence is the emission of light by a substance that has absorbed light or other electromagnetic radiation. It is a form of luminescence. In most cases, the emitted light has a longer wavelength, and therefore a lower photon energy, ...
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 = \frac \int_ \mu(\omega) \omega \left(1 - \frac \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. The integral is convergent (finite) because at high frequencies the refractive index becomes less than unity and for extremely high frequencies it becomes unity.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 In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, o ...
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.


Derivation of Frank–Tamm formula

Consider a charged particle moving relativistically along x-axis in a medium with refraction index n(\omega) = \sqrtFor simplicity we consider magnetic permeability \mu(\omega) = 1. with a constant velocity \vec v = (v,0,0) . Start with
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. Th ...
(in
Gaussian units Gaussian units constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the Gaussian unit system, Gaussian-cgs uni ...
) in the wave forms (also known as the
Lorenz gauge condition In electromagnetism, the Lorenz gauge condition or Lorenz gauge, for Ludvig Lorenz, is a partial gauge fixing of the electromagnetic vector potential by requiring \partial_\mu A^\mu = 0. The name is frequently confused with Hendrik Lorentz, who h ...
) and take the Fourier transform: \left ( k^2 - \frac \varepsilon(\omega) \right) \Phi(\vec k,\omega) = \frac \rho(\vec k, \omega) \left ( k^2 - \frac \varepsilon(\omega) \right) \vec A(\vec k,\omega) = \frac \vec J(\vec k, \omega) For a charge of magnitude ze (where e is the
elementary charge The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a funda ...
) 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 We use engineering notation for the Fourier transform, where 1/\sqrt factors appear both in direct and inverse transforms. gives: \rho(\vec k, \omega) = \frac \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 \frac and \vec A(\vec k,\omega) = \varepsilon(\omega) \frac \Phi(\vec k,\omega) Using the definition of the electromagnetic fields in terms of potentials, we then have the Fourier-form of the electric and magnetic field: \vec E(\vec k,\omega) = i \left( \frac \frac - \vec k \right) \Phi(\vec k,\omega) and \vec B(\vec k,\omega) = i \varepsilon(\omega) \vec k \times \frac \Phi(\vec k,\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: \vec E(\omega) = \frac \int d^3k \, \vec E(\vec k,\omega) e^ First we compute x-component E_1 of the electric field (parallel to \vec v): E_1(\omega) = \frac \int d^3k \, e^ \left( \frac - k_1 \right ) \frac For brevity we define \lambda^2 = \frac - \frac \varepsilon(\omega) = \frac \left ( 1 - \beta^2 \varepsilon(\omega) \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(\omega) = - \frac \left( \frac - \beta^2 \right) \int_^\infty = dk_2 \, e^ \int_^\infty \frac The integral over k_3 has the value \frac, giving: E_1(\omega) = - \frac \left( \frac - \beta^2 \right) \int_^\infty dk_2 \frac The last integral over k_2 is in the form of a modified (Macdonald)
Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...
, giving the evaluated parallel component in the form: E_1(\omega) = - \frac \left( \frac \right)^ \left( \frac - \beta^2 \right) K_0(\lambda b) One can follow a similar pattern of calculation for the other fields components arriving at: : E_2(\omega) = \frac \left( \frac \right)^ \frac K_1(\lambda b), \quad E_3 = 0 \quad and \quad B_1 = B_2 = 0, \quad B_3(\omega) = \varepsilon(\omega) \beta E_2(\omega) We can now consider the radiated energy dE per particle traversed distance dx_ . 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 In physics, the Poynting vector (or Umov–Poynting vector) represents the directional energy flux (the energy transfer per unit area per unit time) or ''power flow'' of an electromagnetic field. The SI unit of the Poynting vector is the watt ...
\mathbf S = c / (4 \pi) \mathbf E \times \mathbf H over the cylinder surface: \left( \frac \right)_ = \frac P_a = - \frac \int_^ 2 \pi a B_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 = v \, dt: \left( \frac \right)_ = - \frac \int_^\infty B_3(t) E_1(t) \, dt Converting this to the frequency domain: \left( \frac \right)_ = -c a \operatorname \left( \int_0^\infty B_3^*(\omega) E_1(\omega) \, d\omega \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, , \lambda b , \gg 1. With this assumption we can expand the Bessel functions into their asymptotic form: E_1(\omega) \rightarrow \frac \left( 1 - \frac \right) \frac : E_2(\omega) \rightarrow \frac \sqrt e^ and B_3(\omega) = \varepsilon(\omega) \beta E_2(\omega) Thus: : \left( \frac \right)_ = \operatorname \left( \int_0^\infty \frac \left(-i \sqrt\right) \omega \left( 1 - \frac \right) e^ \, d\omega \right) If \lambda 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 \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 \varepsilon(\omega) is real and \beta^2 \varepsilon(\omega) > 1. That is, when \varepsilon(\omega) is real, Cherenkov radiation has the condition that v > \frac = \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 \lambda condition, \sqrt = i and the integral can be simplified to: \left( \frac \right)_ = \frac \int_ \omega \left( 1 - \frac \right) \, d\omega = \frac \int_ \omega \left( 1 - \frac \right) \, d\omega This is the Frank–Tamm equation in Gaussian units.


Notes


References

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External links


Cherenkov radiation (Tagged ‘Frank-Tamm formula’)
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