CERENKOV EFFECT

The Cerenkov electromagnetic radiation, usually bluish light, is emitted by a beam of high-energy charged particles passing through a transparent medium at a speed greater than the speed of light in that medium.

The effect is similar to that of a sonic boom when an object moves faster than the speed of sound; in this case the radiation is a shock wave set up in the electromagnetic field.

In fig.1, a charged particle moves with speed , where  through a depressive medium with the refraction index,  is the group velocity of light in the material and c is the velocity of light in vacuum.

Fig.1 Cerenkov effect

The particle traveling with speed , will emit photons contained in a cone, at an angle q with respect to its direction, where qcan be computed with relation:

              (2.1)

or          , where                   (2.2)

  When the charged particle reaches a speed slower than the local speed of light, photons are no longer emitted.

If we compute the speed of light in a depressive medium, such as water for which  n=1,3, we get:

      (2.3)

so the particles must have a velocity higher than this value.

For beta particles with the rest mass , one can calculate the minimum value of the total energy at which Cerenkov radiation is emitted, using the following formula:

             (2.4)

Computing (4) for  gives

The Cerenkov photoemission may be studied as a relativistic effect in which the conservation of energy and linear momentum must both be satisfied.

In Fig.2, there is represented a charged particle with initial momentum , rest mass m₀ and velocity , moving through a medium with refraction index n, faster than light in the same medium. A photon with frequency n or wavelength l and momentum occurs. The charged particle travels with the final momentum  , after the photon emission.

One can write:

Fig.2

Conservation of linear momentum gives:

           (2.5)

and                     (2.6)

Conservation of total energy gives:

             (2.7)

where  is the photon energy.

The fundamental relation between the total energy and momentum gives:

   and        (2.8)

Plugging (2.8) into (2.7) gives:

or                (2.9)

and              (2.10)

where the final momentum p  is unknown.

This one can be eliminated by using (2.6) and (2.10)

       (2.11)

The photon momentum  can be written:

        (2.12)

where the wavelength is expressed as:

       (2.13)

Replacing (2.13) into (2.11) gives:

        (2.14)

Eq. (2.14) becomes:

        (2.15)

where the particle energy E_i is:

            (2.16)

Therefore:

         (2.17)

so             (2.18)

Calculating  from (2.4) and (2.8) and knowing n from the blue light, one can estimate the last term in (2.18) which is very small and can be neglected.

Therefore:

  (2.19)

This effect was used for building up the Cerenkov counters for detecting and counting high-energy charged particles. A photo multiplier tube registers the resulted radiation.

According to the CPT theorem, one can exchange the places of particles in the Cerenkov process ( ). Through the charge conjugation, we get:

where  and   are a pair of particles both an electron and positron obtained from a single gamma ray:

Thus, the antiparticle generation is equivalent with the particle annihilation.

The conservation laws (energy and linear momentum) are satisfied in the pair production, and allow us to determine the momentum, direction and magnitude of the gamma ray.

In fig.3  are momenta of gamma-ray photon, electron and positron.

Having the same rest masses, one can assume that resulted particles will have   and , so:

   (2.20)

where E₁ and E₂ are total energies of the resulted particles.

     (2.21)

where 

Using the Cerenkov radiation to measure the trajectory of the pairs, we can determine the momentum of the gamma-ray.

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