1 | % GEANT4 Physics Reference Manual - Cerenkov Process |
---|
2 | % in LaTex 2e - adopted from GEANT3 manual by P. Gumplinger |
---|
3 | |
---|
4 | %\documentclass[11pt,twoside,a4page]{article} |
---|
5 | %\usepackage{epsfig} |
---|
6 | |
---|
7 | %\setlength{\parindent}{0pt} |
---|
8 | |
---|
9 | %\begin{document} |
---|
10 | |
---|
11 | %\title{\v{C}erenkov Process} |
---|
12 | %\author{P.~Gumplinger} |
---|
13 | %\date{December 7, 1998} |
---|
14 | %\maketitle |
---|
15 | |
---|
16 | |
---|
17 | \section{\v{C}erenkov Effect} |
---|
18 | |
---|
19 | The radiation of \v{C}erenkov light occurs when a charged particle moves |
---|
20 | through a dispersive medium faster than the speed of light in that medium. |
---|
21 | A dispersive medium is one whose index of refraction is an increasing function |
---|
22 | of photon energy. Two things happen when such a particle slows down: |
---|
23 | \begin{enumerate} |
---|
24 | \item a cone of \v{C}erenkov photons is emitted, with the cone angle (measured |
---|
25 | with respect to the particle momentum) decreasing as the particle loses |
---|
26 | energy; |
---|
27 | |
---|
28 | \item the momentum of the photons produced increases, while the number of |
---|
29 | photons produced decreases. |
---|
30 | \end{enumerate} |
---|
31 | When the particle velocity drops below the local speed of light, photons are |
---|
32 | no longer emitted. At that point, the \v{C}erenkov cone collapses to zero. |
---|
33 | \\ |
---|
34 | |
---|
35 | \noindent |
---|
36 | In order to simulate \v{C}erenkov radiation the number of photons per track |
---|
37 | length must be calculated. The formulae used for this calculation can be |
---|
38 | found below and in \cite{Jackson98, pdg}. Let $n$ be the refractive index of |
---|
39 | the dielectric material acting as a radiator. Here $n=c/c'$ where $c'$ is the |
---|
40 | group velocity of light in the material, hence $1 \leq n$. In a dispersive |
---|
41 | material $n$ is an increasing function of the photon energy $\epsilon$ |
---|
42 | ($dn/d\epsilon \geq 0$). A particle traveling with speed $\beta = v/c$ will |
---|
43 | emit photons at an angle $\theta$ with respect to its direction, where |
---|
44 | $\theta$ is given by |
---|
45 | \begin{displaymath} |
---|
46 | \cos \theta = \frac{1}{\beta n} . |
---|
47 | \end{displaymath} |
---|
48 | From this follows the limitation for the momentum of the emitted |
---|
49 | photons: |
---|
50 | \begin{displaymath} |
---|
51 | n(\epsilon_{min}) = \frac{1}{\beta} . |
---|
52 | \end{displaymath} |
---|
53 | Photons emitted with an energy beyond a certain value are immediately |
---|
54 | re-absorbed by the material; this is the window of transparency of the |
---|
55 | radiator. As a consequence, all photons are contained in a cone of |
---|
56 | opening angle $\cos \theta_{max} = 1/(\beta n(\epsilon_{max}))$. \\ |
---|
57 | |
---|
58 | \noindent |
---|
59 | The average number of photons produced is given by the relations : |
---|
60 | \begin{eqnarray*} |
---|
61 | dN &=& \frac{\alpha z^{2}}{\hbar c}\sin^{2}\theta d\epsilon dx = |
---|
62 | \frac{\alpha z^{2}}{\hbar c}(1 - \frac{1}{n^{2}\beta^2}) d\epsilon dx |
---|
63 | \\ |
---|
64 | & \approx & 370z^{2} |
---|
65 | \frac{photons}{eV\,cm}(1 - \frac{1}{n^{2}\beta^{2}})d\epsilon dx |
---|
66 | \end{eqnarray*} |
---|
67 | and the number of photons generated per track length is |
---|
68 | \begin{displaymath} |
---|
69 | \frac{dN}{dx} \approx 370z^{2} \int_{\epsilon_{min}}^{\epsilon_{max}} |
---|
70 | d\epsilon \left(1 - \frac{1}{n^{2}\beta^2} \right) |
---|
71 | = 370z^{2} \left \lbrack \epsilon_{max} |
---|
72 | - \epsilon_{min} - \frac{1}{\beta^{2}} |
---|
73 | \int_{\epsilon_{min}}^{\epsilon_{max}} \frac{d\epsilon} |
---|
74 | {n^2 (\epsilon)}\right \rbrack |
---|
75 | \end{displaymath} . \\ |
---|
76 | |
---|
77 | \noindent |
---|
78 | The number of photons produced is calculated from a Poisson distribution with |
---|
79 | a mean of $\langle n \rangle = \mbox{StepLength}\ dN/dx$. The energy |
---|
80 | distribution of the photon is then sampled from the density function |
---|
81 | \begin{displaymath} |
---|
82 | f(\epsilon)=\left \lbrack 1 - \frac{1}{n^{2}(\epsilon)\beta^{2}} \right \rbrack |
---|
83 | \end{displaymath} . |
---|
84 | |
---|
85 | \subsection{Status of this document} |
---|
86 | 07.12.98 created by P.Gumplinger \\ |
---|
87 | 11.12.01 SI units (mma) \\ |
---|
88 | 08.05.02 re-written by D.H. Wright \\ |
---|
89 | |
---|
90 | \begin{latexonly} |
---|
91 | |
---|
92 | \begin{thebibliography}{99} |
---|
93 | \bibitem{Jackson98} |
---|
94 | J.D.Jackson, Classical Electrodynamics, John Wiley and Sons (1998) |
---|
95 | |
---|
96 | \bibitem{pdg} |
---|
97 | D.E. Groom et al. |
---|
98 | Particle Data Group . Rev. of Particle Properties. |
---|
99 | Eur. Phys. J. C15,1 (2000) http://pdg.lbl.gov/ |
---|
100 | \end{thebibliography} |
---|
101 | |
---|
102 | \end{latexonly} |
---|
103 | |
---|
104 | \begin{htmlonly} |
---|
105 | |
---|
106 | \subsection{Bibliography} |
---|
107 | |
---|
108 | \begin{enumerate} |
---|
109 | \item J.D.Jackson, Classical Electrodynamics, John Wiley and Sons (1998) |
---|
110 | |
---|
111 | \item D.E. Groom et al. |
---|
112 | Particle Data Group . Rev. of Particle Properties. |
---|
113 | Eur. Phys. J. C15,1 (2000) http://pdg.lbl.gov/ |
---|
114 | \end{enumerate} |
---|
115 | |
---|
116 | \end{htmlonly} |
---|
117 | |
---|
118 | |
---|
119 | |
---|
120 | |
---|
121 | |
---|
122 | |
---|
123 | |
---|
124 | |
---|
125 | |
---|
126 | |
---|