\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}
