Diagnóstico BEWE

Cherenkov radiation is the light emitted by a charged particle, such as an electron, moving through an insulating medium at a velocity greater than the phase velocity of light in the medium. As an electron moves through an insulating medium, its electric field distorts nearby atoms by displacing the electrons and polarising the medium around the moving electron (see Figure 1.6). These distorted atoms behave like elementary electric dipoles with the negative poles pointing away from the passing electron. If the electron is moving more slowly than the phase velocity of light in the medium, then the polarisation field around the electron will be symmetric and there will be no net field at large distances and, therefore, no radiation. In the case where the electron is moving at a speed comparable to that of light, however, the polarisation field is no longer completely symmetrical, and as the atoms revert to their original state a pulse of electromagnetic radiation will be emitted (Jelley, 1958). The threshold velocity, βminc, above which Cherenkov emission

will take place in a medium with refractive index n is given by βmin = 1/n; at this velocity, the

direction of the emission will correspond with that of the particle, while higher velocities will result in a cone of radiation up to the angle θc, the Cherenkov angle, defined by:

cos θc =

1

βn (1.24)

where the velocity of the particle is βc, (Jelley, 1958).

The classical theory of Cherenkov radiation was developed by Frank and Tamm in 1937 (Frank

1_{It is important to note that recently variability has been observed at GeV energies (Tavani et al., 2011) and}

Figure 1.6: The polarisation set up in a dialectric medium by the passage of a charged particle at (a) a velocity lower than that of light in the medium, (b) a velocity greater than that of light in the medium. Taken from Jelley (1958).

& Tamm, 1937); the formulae are derived in Jelley (1958). It is found that the output of radiation per unit length, dW/dl, at a specified frequency, ω, is

dW dl = e2 c2 Z βn>1  1 − 1 β2_n2  ωdω (1.25) When deriving the spectrum, Jelley (1958) notes that no frequency cut-off is imposed, which implies that the radiation output is infinite. In reality there are two factors which set an upper limit to the frequency spectrum and cause the radiation output to remain finite. The first is that a real medium is always dispersive, which restricts radiation to those frequency bands where n(ω) > 1/β, whereas, in the treatment by Frank and Tamm, dispersion is ignored to the first order. No emission can be observed for X-rays, because n(ω) > 1 in the X-ray regime, and in media which are transparent at optical wavelengths the absorption bands are found at shorter wavelengths, limiting radiation to the near ultraviolet and longer wavelengths. A further limiting factor is the classical diameter of the electron, de = e2/2π0mec2 = 5.6 × 10−15 m, because to

satisfy coherence conditions the angular wavelengths (angular wavelength = λ/2π) of the emitted photons must be greater than this. This leads to the constraint that the radiation must have a wavelength greater than λmin = 2πde = 3.5 × 10−15 m, which falls in the γ-ray region of the

spectrum. The total energy lost by a relativistic particle per unit length via Cherenkov emission is given by:

dW dl = e2ω2₀ 2c2 ( − 1) ln  _  − 1  (1.26) where ω0is the frequency of the first resonance of the spectrum and  is the dielectric constant

of the medium. Using this it can be seen that for a relativistic particle in a typical medium where ω0= 6 × 1015s−1, dW/dl is of the order of several keV per cm, which is ∼ 0.1% of the energy lost

by ionisation for the particle.

To find the duration of the light flash, Jelley considered dispersion within the medium. In a nondispersive medium, the wavefront is infinitely thin and the duration of the light pulse must therefore be infinitely short. In a dispersive medium, however, the Cherenkov angle depends on the wavelength of the emission, and the duration, ∆t, of the light flash as seen by a given detector is

∆t = ρ

βc(tan θ2− tan θ1) (1.27) where θ1 and θ2 are the Cherenkov angles for the frequency limits of the detector, and ρ is

the distance from the path of the particle. This means that for a fast electron moving through the upper atmosphere (β = 1, n = 1.000292, θc = 1.403◦), observed from 100 m away by a

detector which can detect radiation with wavelengths between 180 nm (where n = 1.000346 and θc= 1.507◦) and 750nm (where n = 1.000275 and θc= 1.345◦), the Cherenkov emission will have

a pulse length of ∼ 10−6 s.

Thus far, the formulae given above have assumed that the relativistic particle is moving at a constant speed, but as the particle traverses the medium it will lose energy via Bremsstrahlung and ionisation, which will also affect its direction of motion (as will non-radiative Coulomb scattering). To ensure that coherence is preserved despite the change in β (and hence θc), the deceleration

must not be too rapid and must satisfy

T. dv dt

  c

n (1.28)

where T is one period of the wave emitted and dv/dt is the deceleration of the electron. At visible wavelengths, this condition is easily satisfied where ionisation is the dominant form of energy loss. As the energy loss via Cherenkov emission is so small, the energy of the emitted photon must also be small when compared to the energy of the interacting particle; this means that quantum effects can generally be ignored and the classical treatment by Frank and Tamm, is generally valid.