[1208] | 1 | % GEANT4 Physics Reference Manual - Cerenkov Process |
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| 2 | % in LaTex 2e - adopted from GEANT3 manual by P. Gumplinger |
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| 3 | |
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| 4 | %\documentclass[11pt,twoside,a4page]{article} |
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| 5 | %\usepackage{epsfig} |
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| 7 | %\setlength{\parindent}{0pt} |
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| 8 | |
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| 9 | %\begin{document} |
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| 10 | |
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| 11 | %\title{\v{C}erenkov Process} |
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| 12 | %\author{P.~Gumplinger} |
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| 13 | %\date{December 7, 1998} |
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| 14 | %\maketitle |
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| 15 | |
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| 16 | |
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| 17 | \section{\v{C}erenkov Effect} |
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| 18 | |
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| 19 | The radiation of \v{C}erenkov light occurs when a charged particle moves |
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| 20 | through a dispersive medium faster than the speed of light in that medium. |
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| 21 | A dispersive medium is one whose index of refraction is an increasing function |
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| 22 | of photon energy. Two things happen when such a particle slows down: |
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| 23 | \begin{enumerate} |
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| 24 | \item a cone of \v{C}erenkov photons is emitted, with the cone angle (measured |
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| 25 | with respect to the particle momentum) decreasing as the particle loses |
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| 26 | energy; |
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| 27 | |
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| 28 | \item the momentum of the photons produced increases, while the number of |
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| 29 | photons produced decreases. |
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| 30 | \end{enumerate} |
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| 31 | When the particle velocity drops below the local speed of light, photons are |
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| 32 | no longer emitted. At that point, the \v{C}erenkov cone collapses to zero. |
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| 33 | \\ |
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| 34 | |
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| 35 | \noindent |
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| 36 | In order to simulate \v{C}erenkov radiation the number of photons per track |
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| 37 | length must be calculated. The formulae used for this calculation can be |
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| 38 | found below and in \cite{Jackson98, pdg}. Let $n$ be the refractive index of |
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| 39 | the dielectric material acting as a radiator. Here $n=c/c'$ where $c'$ is the |
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| 40 | group velocity of light in the material, hence $1 \leq n$. In a dispersive |
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| 41 | material $n$ is an increasing function of the photon energy $\epsilon$ |
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| 42 | ($dn/d\epsilon \geq 0$). A particle traveling with speed $\beta = v/c$ will |
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| 43 | emit photons at an angle $\theta$ with respect to its direction, where |
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| 44 | $\theta$ is given by |
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| 45 | \begin{displaymath} |
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| 46 | \cos \theta = \frac{1}{\beta n} . |
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| 47 | \end{displaymath} |
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| 48 | From this follows the limitation for the momentum of the emitted |
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| 49 | photons: |
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| 50 | \begin{displaymath} |
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| 51 | n(\epsilon_{min}) = \frac{1}{\beta} . |
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| 52 | \end{displaymath} |
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| 53 | Photons emitted with an energy beyond a certain value are immediately |
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| 54 | re-absorbed by the material; this is the window of transparency of the |
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| 55 | radiator. As a consequence, all photons are contained in a cone of |
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| 56 | opening angle $\cos \theta_{max} = 1/(\beta n(\epsilon_{max}))$. \\ |
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| 57 | |
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| 58 | \noindent |
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| 59 | The average number of photons produced is given by the relations : |
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| 60 | \begin{eqnarray*} |
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| 61 | dN &=& \frac{\alpha z^{2}}{\hbar c}\sin^{2}\theta d\epsilon dx = |
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| 62 | \frac{\alpha z^{2}}{\hbar c}(1 - \frac{1}{n^{2}\beta^2}) d\epsilon dx |
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| 63 | \\ |
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| 64 | & \approx & 370z^{2} |
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| 65 | \frac{photons}{eV\,cm}(1 - \frac{1}{n^{2}\beta^{2}})d\epsilon dx |
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| 66 | \end{eqnarray*} |
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| 67 | and the number of photons generated per track length is |
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| 68 | \begin{displaymath} |
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| 69 | \frac{dN}{dx} \approx 370z^{2} \int_{\epsilon_{min}}^{\epsilon_{max}} |
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| 70 | d\epsilon \left(1 - \frac{1}{n^{2}\beta^2} \right) |
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| 71 | = 370z^{2} \left \lbrack \epsilon_{max} |
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| 72 | - \epsilon_{min} - \frac{1}{\beta^{2}} |
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| 73 | \int_{\epsilon_{min}}^{\epsilon_{max}} \frac{d\epsilon} |
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| 74 | {n^2 (\epsilon)}\right \rbrack |
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| 75 | \end{displaymath} . \\ |
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| 76 | |
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| 77 | \noindent |
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| 78 | The number of photons produced is calculated from a Poisson distribution with |
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| 79 | a mean of $\langle n \rangle = \mbox{StepLength}\ dN/dx$. The energy |
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| 80 | distribution of the photon is then sampled from the density function |
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| 81 | \begin{displaymath} |
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| 82 | f(\epsilon)=\left \lbrack 1 - \frac{1}{n^{2}(\epsilon)\beta^{2}} \right \rbrack |
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| 83 | \end{displaymath} . |
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| 84 | |
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| 85 | \subsection{Status of this document} |
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| 86 | 07.12.98 created by P.Gumplinger \\ |
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| 87 | 11.12.01 SI units (mma) \\ |
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| 88 | 08.05.02 re-written by D.H. Wright \\ |
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| 89 | |
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| 90 | \begin{latexonly} |
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| 91 | |
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| 92 | \begin{thebibliography}{99} |
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| 93 | \bibitem{Jackson98} |
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| 94 | J.D.Jackson, Classical Electrodynamics, John Wiley and Sons (1998) |
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| 95 | |
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| 96 | \bibitem{pdg} |
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| 97 | D.E. Groom et al. |
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| 98 | Particle Data Group . Rev. of Particle Properties. |
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| 99 | Eur. Phys. J. C15,1 (2000) http://pdg.lbl.gov/ |
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| 100 | \end{thebibliography} |
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| 101 | |
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| 102 | \end{latexonly} |
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| 103 | |
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| 104 | \begin{htmlonly} |
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| 105 | |
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| 106 | \subsection{Bibliography} |
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| 107 | |
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| 108 | \begin{enumerate} |
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| 109 | \item J.D.Jackson, Classical Electrodynamics, John Wiley and Sons (1998) |
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| 110 | |
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| 111 | \item D.E. Groom et al. |
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| 112 | Particle Data Group . Rev. of Particle Properties. |
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| 113 | Eur. Phys. J. C15,1 (2000) http://pdg.lbl.gov/ |
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| 114 | \end{enumerate} |
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| 115 | |
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| 116 | \end{htmlonly} |
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