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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}