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-\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}