% GEANT4 Physics Reference Manual - Cerenkov Process
% in LaTex 2e - adopted from GEANT3 manual by P. Gumplinger

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%\title{\v{C}erenkov Process}
%\author{P.~Gumplinger}
%\date{December 7, 1998}
%\maketitle


\section{\v{C}erenkov Effect}

The radiation of \v{C}erenkov 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:
\begin{enumerate} 
\item a cone of \v{C}erenkov photons is emitted, with the cone angle (measured
with respect to the particle momentum) decreasing as the particle loses
energy;

\item the momentum of the photons produced increases, while the number of 
photons produced decreases. 
\end{enumerate} 
When the particle velocity drops below the local speed of light, photons are 
no longer emitted.  At that point, the \v{C}erenkov cone collapses to zero.
\\

\noindent
In order to simulate \v{C}erenkov radiation the number of photons per track
length must be calculated.  The formulae used for this calculation can be
found below and in \cite{Jackson98, pdg}.  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 
\begin{displaymath}
\cos \theta = \frac{1}{\beta n} .
\end{displaymath}
From this follows the limitation for the momentum of the emitted
photons: 
\begin{displaymath}
n(\epsilon_{min}) = \frac{1}{\beta} .
\end{displaymath}
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 $\cos \theta_{max} = 1/(\beta n(\epsilon_{max}))$. \\ 

\noindent
The average number of photons produced is given by the relations :
\begin{eqnarray*}
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{photons}{eV\,cm}(1 - \frac{1}{n^{2}\beta^{2}})d\epsilon dx
\end{eqnarray*}
and the number of photons generated per track length is 
\begin{displaymath}
\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 
\end{displaymath} . \\

\noindent
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 
\begin{displaymath}
f(\epsilon)=\left \lbrack 1 - \frac{1}{n^{2}(\epsilon)\beta^{2}} \right \rbrack
\end{displaymath} .

\subsection{Status of this document}
07.12.98 created by P.Gumplinger \\
11.12.01 SI units (mma) \\
08.05.02 re-written by D.H. Wright \\

\begin{latexonly}

\begin{thebibliography}{99}
\bibitem{Jackson98}
J.D.Jackson, Classical Electrodynamics, John Wiley and Sons (1998)

\bibitem{pdg}
  D.E. Groom et al.
  Particle Data Group . Rev. of Particle Properties.
  Eur. Phys. J. C15,1 (2000) http://pdg.lbl.gov/        
\end{thebibliography}

\end{latexonly}

\begin{htmlonly}

\subsection{Bibliography}

\begin{enumerate}
\item J.D.Jackson, Classical Electrodynamics, John Wiley and Sons (1998)

\item D.E. Groom et al.
  Particle Data Group . Rev. of Particle Properties.
  Eur. Phys. J. C15,1 (2000) http://pdg.lbl.gov/        
\end{enumerate}

\end{htmlonly}