Čerenkov Effect

The radiation of Čerenkov light occurs when a charged particle moves through a dispersive medium faster than the speed of light in that medium. A dispersive medium is one whose index of refraction is an increasing function of photon energy. Two things happen when such a particle slows down:

  1. a cone of Čerenkov photons is emitted, with the cone angle (measured with respect to the particle momentum) decreasing as the particle loses energy;

  2. the momentum of the photons produced increases, while the number of photons produced decreases.

When the particle velocity drops below the local speed of light, photons are no longer emitted. At that point, the Čerenkov cone collapses to zero. In order to simulate Čerenkov radiation the number of photons per track length must be calculated. The formulae used for this calculation can be found below and in [JDJackson98, eal00]. Let \(n\) be the refractive index of the dielectric material acting as a radiator. Here \(n=c/c'\) where \(c'\) is the group velocity of light in the material, hence \(1 \leq n\). In a dispersive material \(n\) is an increasing function of the photon energy \(\epsilon\) (\(dn/d\epsilon \geq 0\)). A particle traveling with speed \(\beta = v/c\) will emit photons at an angle \(\theta\) with respect to its direction, where \(\theta\) is given by

\[\cos \theta = \frac{1}{\beta n} .\]

From this follows the limitation for the momentum of the emitted photons:

\[n(\epsilon_{min}) = \frac{1}{\beta} .\]

Photons emitted with an energy beyond a certain value are immediately re-absorbed by the material; this is the window of transparency of the radiator. As a consequence, all photons are contained in a cone of opening angle \(\cos \theta_{max} = 1/(\beta n(\epsilon_{max}))\). The average number of photons produced is given by the relations:

\[\begin{split}dN &= \frac{\alpha z^{2}}{\hbar c}\sin^{2}\theta d\epsilon dx = \frac{\alpha z^{2}}{\hbar c}(1 - \frac{1}{n^{2}\beta^2}) d\epsilon dx \\ & \approx 370z^{2} \frac{\rm photons}{\rm eV\,cm}(1 - \frac{1}{n^{2}\beta^{2}})d\epsilon dx\end{split}\]

and the number of photons generated per track length is

\[\frac{dN}{dx} \approx 370z^{2} \int_{\epsilon_{min}}^{\epsilon_{max}} d\epsilon \left(1 - \frac{1}{n^{2}\beta^2} \right) = 370z^{2} \left \lbrack \epsilon_{max} - \epsilon_{min} - \frac{1}{\beta^{2}} \int_{\epsilon_{min}}^{\epsilon_{max}} \frac{d\epsilon} {n^2 (\epsilon)}\right \rbrack\ .\]

The number of photons produced is calculated from a Poisson distribution with a mean of \(\langle n \rangle = \mbox{StepLength}\ dN/dx\). The energy distribution of the photon is then sampled from the density function

\[f(\epsilon)=\left \lbrack 1 - \frac{1}{n^{2}(\epsilon)\beta^{2}} \right \rbrack .\]

Bibliography

eal00

D.E. Groom et al. Particle data group, rev. of particle properties. Eur. Phys. J. C15, 1, 2000. http://pdg.lbl.gov/.

JDJackson98

J.D.Jackson. Classical Electrodynamics. John Wiley and Sons, third edition, 1998.