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\author{Antykwa Torunska}

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\title{Simulation of  Energy Loss  Straggling}

\newcommand{\Emax}{\ensuremath{E\sb{\mathrm{max}}}}

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\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. \ldots

\section{Vavilov theory}

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-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-VAVI}
P.V.Vavilov.
\newblock Ionisation losses of high energy heavy particles.
\newblock \emph{Soviet Physics JETP}, 5:749, 1957.
\end{thebibliography}

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