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author | Karl Berry <karl@freefriends.org> | 2009-05-23 00:23:51 +0000 |
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committer | Karl Berry <karl@freefriends.org> | 2009-05-23 00:23:51 +0000 |
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tree | 424ab223921f85fd3f167a4ccd0e2d37d05c2927 /Master/texmf-dist/doc/latex/latex-web-companion/apa/latexexa.ltx | |
parent | 5beb5368a684995153c8566797ba054f21c666af (diff) |
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diff --git a/Master/texmf-dist/doc/latex/latex-web-companion/apa/latexexa.ltx b/Master/texmf-dist/doc/latex/latex-web-companion/apa/latexexa.ltx new file mode 100644 index 00000000000..2e334aa4cc5 --- /dev/null +++ b/Master/texmf-dist/doc/latex/latex-web-companion/apa/latexexa.ltx @@ -0,0 +1,109 @@ +\documentclass{article} +\usepackage{graphicx} +\usepackage{url} +\title{Simulation of Energy Loss Straggling} +\author{Maria Physicist} +\newcommand{\Emax}{\ensuremath{E_{\mathrm{max}}}} +\newcommand{\GEANT}{\texttt{GEANT}} +\begin{document} +\maketitle + +\section{Introduction} + +Due to the statistical nature of ionisation energy loss, large +fluctuations can occur in the amount of energy deposited by a particle +traversing an absorber element. Continuous processes such as multiple +scattering and energy loss play a relevant role in the longitudinal +and lateral development of electromagnetic and hadronic +showers, and in the case of sampling calorimeters the +measured resolution can be significantly affected by such fluctuations +in their active layers. The description of ionisation fluctuations is +characterised by the significance parameter $\kappa$, which is +proportional to the ratio of mean energy loss to the maximum allowed +energy transfer in a single collision with an atomic electron +\[ +\kappa =\frac{\xi}{\Emax} +\] +\Emax{} +is the maximum transferable energy in a single collision with +an atomic electron. + +....... + +\section{Vavilov theory} +\label{vavref} + +Vavilov\cite{bib-VAVI} derived a more accurate straggling distribution +by introducing the kinematic limit on the maximum transferable energy +in a single collision, rather than using $ \Emax = \infty $. +Now we can write\cite{bib-SCH1}: +\begin{eqnarray*} +f \left ( \epsilon, \delta s \right ) & = & \frac{1}{\xi} \phi_{v} +\left ( \lambda_{v}, \kappa, \beta^{2} \right ) +\end{eqnarray*} +where +\begin{eqnarray*} +\phi_{v} \left ( \lambda_{v}, \kappa, \beta^{2} \right ) & = & +\frac{1}{2 \pi i} \int^{c+i\infty}_{c-i\infty}\phi \left( s \right ) +e^{\lambda s} ds \hspace{2cm} c \geq 0 \\ +\phi \left ( s \right ) & = & +\exp \left [ \kappa ( 1 + \beta^{2}\gamma ) \right ] +~ \exp \left [ \psi \left ( s \right ) \right ], \\ +\psi \left ( s \right ) & = & s \ln \kappa + ( s + \beta^{2} \kappa ) +\left [ \ln (s/\kappa) + E_{1} (s/\kappa) \right ] - \kappa e^{-s/\kappa}, +\end{eqnarray*} +and +\begin{eqnarray*} +E_{1}(z) & = & \int^{\infty}_{z} t^{-1} e^{-t} dt +\mbox{\hspace{1cm} (the exponential integral)} \\ +\lambda_v & = & \kappa \left [ \frac{\epsilon - \bar{\epsilon}}{\xi} +- \gamma' - \beta^2 \right] +\end{eqnarray*} + +The Vavilov parameters are simply related to the Landau parameter by +$\lambda_L = \lambda_v/\kappa - \ln\kappa $. It can be shown that as +$\kappa \rightarrow 0$, the distribution of the variable $\lambda_L$ +approaches that of Landau. For $\kappa \leq 0.01$ the two +distributions are already practically identical. Contrary to what many +textbooks report, the Vavilov distribution \emph{does not} approximate +the Landau distribution for small $\kappa$, but rather the +distribution of $\lambda_L$ defined above tends to the distribution of +the true $\lambda$ from the Landau density function. Thus the routine +\texttt{GVAVIV} samples the variable $\lambda_L$ rather than +$\lambda_v$. For $\kappa \geq 10$ the Vavilov distribution tends to a +Gaussian distribution (see next section). +.... + +\begin{thebibliography}{10} +\bibitem{bib-LAND} +L.Landau. +\newblock On the Energy Loss of Fast Particles by Ionisation. +\newblock Originally published in \emph{J. Phys.}, 8:201, 1944. +\newblock Reprinted in D.ter Haar, Editor, \emph{L.D.Landau, Collected + papers}, page 417. Pergamon Press, Oxford, 1965. + +\bibitem{bib-SCH1} +B.Schorr. +\newblock Programs for the Landau and the Vavilov distributions and the + corresponding random numbers. +\newblock \emph{Comp. Phys. Comm.}, 7:216, 1974. + +\bibitem{bib-SELT} +S.M.Seltzer and M.J.Berger. +\newblock Energy loss straggling of protons and mesons. +\newblock In \emph{Studies in Penetration of Charged Particles in + Matter}, Nuclear Science Series~39, Nat. Academy of Sciences, + Washington DC, 1964. + +\bibitem{bib-TALM} +R.Talman. +\newblock On the statistics of particle identification using ionization. +\newblock \emph{Nucl. Inst. Meth.}, 159:189, 1979. + +\bibitem{bib-VAVI} +P.V.Vavilov. +\newblock Ionisation losses of high energy heavy particles. +\newblock \emph{Soviet Physics JETP}, 5:749, 1957. +\end{thebibliography} + +\end{document} |