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+\documentclass{article}
+\usepackage{graphicx}
+\usepackage{url}
+\title{Simulation of Energy Loss Straggling}
+\author{Maria Physicist}
+\usepackage{ae}
+\usepackage[T1]{fontenc}
+\usepackage[latin1]{inputenc}
+\usepackage{fancyhdr,pifont}
+\pagestyle{fancy}
+\cfoot{\NavigationBar}
+\def\NavigationBar{{\Large
+ \Acrobatmenu{FontsInfo}{\reflectbox{\ding{227}}}
+ \Acrobatmenu{NextPage}{\ding{227}}
+ \Acrobatmenu{FirstPage}{\reflectbox{\ding{224}}}
+ \Acrobatmenu{LastPage}{\ding{224}}
+ \Acrobatmenu{GoBack}{\reflectbox{\ding{249}}}
+ \Acrobatmenu{Quit}{\ding{54}}%
+}}
+\usepackage[colorlinks, backref,
+ pdfpagemode=FullScreen,
+ pdfauthor={Maria Physicist},
+ pdftitle={Simulation of Energy Loss Straggling},
+ pdfcreator={pdfTeX},
+ pdfsubject={Energy Loss},
+ pdfkeywords={physics,energy},
+ pdfpagetransition={Blinds /Dm /V}
+]{hyperref}
+\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.
+\[
+\Emax =\frac{2 m_e \beta^2\gamma^2 }
+{1 + 2\gamma m_e/m_x + \left ( m_e/m_x\right)^2},
+\]
+where $\gamma = E/m_x$, $E$ is energy and
+$m_x$ the mass of the incident particle,
+$\beta^2 = 1 - 1/\gamma^2$ and $m_e$ is the electron mass.
+$\xi$ comes from the Rutherford scattering cross section
+and is defined as:
+\begin{eqnarray*} \xi = \frac{2\pi z^2 e^4 N_{Av} Z \rho \delta x}
+ {m_e \beta^2 c^2 A} = 153.4 \frac{z^2} {\beta^2} \frac{Z}{A}
+ \rho \delta x \quad\mathrm{keV},
+\end{eqnarray*}
+where
+
+\begin{tabular}{ll}
+$z$ & charge of the incident particle \\
+$N_{Av}$ & Avogadro's number \\
+$Z$ & atomic number of the material \\
+$A$ & atomic weight of the material \\
+$\rho$ & density \\
+$ \delta x$ & thickness of the material \\
+\end{tabular}
+
+$\kappa$ measures the contribution of the collisions with energy
+transfer close to \Emax. For a given absorber, $\kappa$ tends
+towards large values if $\delta x$ is large and/or if $\beta$ is
+small. Likewise, $\kappa$ tends towards zero if $\delta x $ is small
+and/or if $\beta$ approaches 1.
+
+The value of $\kappa$ distinguishes two regimes which occur in the
+description of ionisation fluctuations :
+
+\begin{enumerate}
+\item A large number of collisions involving the loss of all or most
+ of the incident particle energy during the traversal of an absorber.
+
+ As the total energy transfer is composed of a multitude of small
+ energy losses, we can apply the central limit theorem and describe
+ the fluctuations by a Gaussian distribution. This case is
+ applicable to non-relativistic particles and is described by the
+ inequality $\kappa > 10 $ (i.e. when the mean energy loss in the
+ absorber is greater than the maximum energy transfer in a single
+ collision).
+
+\item Particles traversing thin counters and incident electrons under
+ any conditions.
+
+ The relevant inequalities and distributions are $ 0.01 < \kappa < 10
+ $, Vavilov distribution, and $\kappa < 0.01 $, Landau distribution.
+\end{enumerate}
+
+An additional regime is defined by the contribution of the collisions
+with low energy transfer which can be estimated with the relation
+$\xi/I_0$, where $I_0$ is the mean ionisation potential of the atom.
+Landau theory assumes that the number of these collisions is high, and
+consequently, it has a restriction $\xi/I_0 \gg 1$. In \GEANT{}
+(see URL \url{http://wwwinfo.cern.ch/asdoc/geant/geantall.html}), the
+limit of Landau theory has been set at $\xi/I_0 = 50$. Below this
+limit special models taking into account the atomic structure of the
+material are used. This is important in thin layers and gaseous
+materials. Figure \ref{fg:phys332-1} shows the behaviour of $\xi/I_0$
+as a function of the layer thickness for an electron of 100 keV and 1
+GeV of kinetic energy in Argon, Silicon and Uranium.
+
+\begin{figure}
+ \centering
+ \includegraphics[width=.6\linewidth]{phys332-1}
+ \caption{The variable $\xi/I_0$ can be used to measure the
+ validity range of the Landau theory. It depends
+ on the type and energy of the particle, $Z$, $A$
+ and the ionisation potential of the material and
+ the layer thickness.
+ }
+ \label{fg:phys332-1}
+\end{figure}
+
+In the following sections, the different theories and models for the
+energy loss fluctuation are described. First, the Landau theory and
+its limitations are discussed, and then, the Vavilov and Gaussian
+straggling functions and the methods in the thin layers and gaseous
+materials are presented.
+
+\section{Landau theory}
+\label{sec:phys332-1}
+
+For a particle of mass $m_x$ traversing a thickness of material
+$\delta x $, the Landau probability distribution may be written in
+terms of the universal Landau function $\phi(\lambda)$
+as\cite{bib-LAND}:
+\begin{eqnarray*}
+f( \epsilon , \delta x ) & = &\frac{1}{\xi} \phi ( \lambda )
+\end{eqnarray*}
+where
+\begin{eqnarray*}
+\phi(\lambda )& = & \frac{1} {2 \pi i}\int^{c+i\infty}_{c-i\infty}
+\exp \left ( u \ln u + \lambda u \right ) du \hspace{2cm} c \geq 0 \\
+\lambda & = & \frac{\epsilon -\bar{\epsilon} }{\xi}
+ - \gamma' - \beta^2 - \ln \frac{\xi}{\Emax} \\
+\gamma' & = & 0.422784\dots = 1 - \gamma \\
+\gamma & = & 0.577215\dots \mbox{(Euler's constant)} \\
+\bar{\epsilon} & = & \mbox{average energy loss} \\
+\epsilon & = & \mbox{actual energy loss}
+\end{eqnarray*}
+
+\subsection{Restrictions}
+
+The Landau formalism makes two restrictive assumptions :
+\begin{enumerate}
+\item The typical energy loss is small compared to the maximum energy
+ loss in a single collision. This restriction is removed in the
+ Vavilov theory (see section \ref{vavref}).
+\item The typical energy loss in the absorber should be large compared
+ to the binding energy of the most tightly bound electron. For
+ gaseous detectors, typical energy losses are a few keV which is
+ comparable to the binding energies of the inner electrons. In such
+ cases a more sophisticated approach which accounts for atomic energy
+ levels\cite{bib-TALM} is necessary to accurately simulate data
+ distributions. In \GEANT, a parameterised model by L. Urb\'{a}n is
+ used (see section \ref{urban}).
+\end{enumerate}
+
+In addition, the average value of the Landau distribution is infinite.
+Summing the Landau fluctuation obtained to the average energy from the
+$dE/dx$ tables, we obtain a value which is larger than the one coming
+from the table. The probability to sample a large value is small, so
+it takes a large number of steps (extractions) for the average
+fluctuation to be significantly larger than zero. This introduces a
+dependence of the energy loss on the step size which can affect
+calculations.
+
+A solution to this has been to introduce a limit on the value of the
+variable sampled by the Landau distribution in order to keep the
+average fluctuation to 0. The value obtained from the \texttt{GLANDO}
+routine is:
+\[
+\delta dE/dx = \epsilon - \bar{\epsilon} = \xi ( \lambda - \gamma'
++\beta^2 +\ln \frac{\xi}{\Emax})
+\]
+In order for this to have average 0, we must impose that:
+\[
+\bar{\lambda} = -\gamma' - \beta^2 -\ln \frac{\xi}{\Emax}
+\]
+
+This is realised introducing a $\lambda_{\mathrm{max}}(\bar{\lambda})$
+such that if only values of $\lambda \leq \lambda_{\mathrm{max}}$ are
+accepted, the average value of the distribution is $\bar{\lambda}$.
+
+A parametric fit to the universal Landau distribution has been
+performed, with following result:
+\[
+\lambda_{\mathrm{max}} = 0.60715 +
+ 1.1934\bar{\lambda}+(0.67794+0.052382\bar{\lambda})
+ \exp(0.94753+0.74442\bar{\lambda})
+\]
+only values smaller than $\lambda_{\mathrm{max}}$ are accepted, otherwise the
+distribution is resampled.
+
+
+
+\newpage
+\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).
+
+\section{Gaussian Theory}
+
+Various conflicting forms have been proposed for Gaussian straggling
+functions, but most of these appear to have little theoretical or
+experimental basis. However, it has been shown\cite{bib-SELT} that
+for $\kappa \geq 10 $ the Vavilov distribution can be replaced by a
+Gaussian of the form :
+\begin{eqnarray*}
+f( \epsilon , \delta s) \approx \frac{1}
+{\xi \sqrt{\frac{2 \pi}{\kappa} \left( 1 - \beta^2/2 \right)}}
+ \exp \left [ \frac{( \epsilon - \bar{\epsilon} )^2}{2} \frac{\kappa}
+ {\xi^2 (1- \beta^2/2)}\right ]
+\end{eqnarray*}
+thus implying
+\begin{eqnarray*}
+\mathrm{mean} & = & \bar{\epsilon} \\
+\sigma^2 & = & \frac{\xi^2}{\kappa} (1-\beta^2/2) = \xi
+ \Emax (1-\beta^2/2)
+\end{eqnarray*}
+
+\section{Urb\'an model}
+\label{urban}
+
+The method for computing restricted energy losses with $\delta$-ray
+production above given threshold energy in \GEANT{} is a Monte
+Carlo method that can be used for thin layers. It is fast and it can
+be used for any thickness of a medium. Approaching the limit of the
+validity of Landau's theory, the loss distribution approaches smoothly
+the Landau form as shown in Figure \ref{fg:phys332-2}.
+\begin{figure}
+ \centering
+ \includegraphics[width=.6\linewidth]{phys332-2}
+ \caption{Energy loss distribution for a 3 GeV electron in
+ Argon as given by standard \GEANT. The width of the layers is
+ given in centimeters.}
+ \label{fg:phys332-2}
+\end{figure}
+
+It is assumed that the atoms have only two energy levels with binding
+energy $E_1$ and $E_2$. The particle--atom interaction will then be
+an excitation with energy loss $E_1$ or $E_2$, or an ionisation with
+an energy loss distributed according to a function $g(E) \sim 1/E^2$:
+\begin{equation}
+g(E) = \frac{(\Emax + I)I}{\Emax} \frac{1}{E^2}
+\end{equation}
+
+The macroscopic cross-section for excitations ($i=1,2$) is
+\begin{equation}
+\label{eq:sigex}
+\Sigma_i = C \frac{f_i}{E_i} \frac{\ln (2 m \beta^2 \gamma^2/E_i) - \beta^2}
+ {\ln (2 m \beta^2 \gamma^2/ I) - \beta^2}(1-r)
+\end{equation}
+and the macroscopic cross-section for ionisation is
+\begin{equation}
+\label{eq:sigion}
+\Sigma_3 = C \frac{\Emax}{I(\Emax+I) \ln(\frac{\Emax+I}{I})}
+ ~ r
+\end{equation}
+\Emax{} is the \GEANT{} cut for $\delta$-production, or the maximum
+energy transfer minus mean ionisation energy, if it is smaller than
+this cut-off value. The following notation is used:
+
+\begin{tabular}{ll}
+$r, C$ & parameters of the model \\
+$E_i$ & atomic energy levels \\
+$I$ & mean ionisation energy \\
+${f_i}$ & oscillator strengths
+\end{tabular}
+
+The model has the parameters $f_i$, $E_i$, $C$ and $r ~(0\leq r\leq
+1)$. The oscillator strengths $f_i$ and the atomic level energies
+$E_i$ should satisfy the constraints
+\begin{eqnarray}
+f_1 + f_2 & = & 1 \label{eq:fisum}\\
+f_1 \ln E_1 + f_2 \ln E_2 & = & \ln I \label{eq:flnsum}
+\end{eqnarray}
+The parameter $C$ can be defined with the help of the mean energy loss
+$dE/dx$ in the following way: The numbers of collisions ($n_i$, i =
+1,2 for the excitation and 3 for the ionisation) follow the Poisson
+distribution with a mean number $ \langle n_i \rangle $. In a step
+$\Delta x$ the mean number of collisions is
+\begin{equation}
+\langle n_i \rangle = \Sigma_i \Delta x
+\end{equation}
+The mean energy loss $dE/dx$ in a step is the sum of the excitation
+and ionisation contributions
+\begin{equation}
+\frac{dE}{dx} \Delta x = \left[ \Sigma_1 E_1 + \Sigma_2 E_2 +
+ \Sigma_3 \int_{I}^{\Emax+I} E~g(E)~dE \right]
+ \Delta x
+\end{equation}
+From this, using the equations (\ref{eq:sigex}), (\ref{eq:sigion}),
+(\ref{eq:fisum}) and (\ref{eq:flnsum}), one can define the parameter
+$C$
+\begin{equation}
+C = \frac{dE}{dx}
+\end{equation}
+
+The following values have been chosen in \GEANT{} for the other
+parameters:
+\[
+\begin{array}{lcl}
+f_2 = \left\{ \begin{array}{ll}
+ 0 & \mathrm{if}\, Z \leq 2 \\
+ 2/Z & \mathrm{if}\, Z > 2 \\
+ \end{array} \right. & \Rightarrow & f_1 = 1 - f_2 \\
+E_2 = 10 Z^2 \mathrm{eV} & \Rightarrow & E_1 = \left(\frac{I}{E_{2}^{f_2}}
+ \right)^{\frac{1}{f_1}} \\
+r = 0.4 & & \\
+\end{array}
+\]
+With these values the atomic level $E_2$ corresponds approximately
+the K-shell energy of the atoms and $Zf_2$ the number of K-shell
+electrons. $r$ is the only variable which can be tuned freely. It
+determines the relative contribution of ionisation and
+excitation to the energy loss.
+
+The energy loss is computed with the assumption that the step length
+(or the relative energy loss) is small, and---in consequence---the
+cross-section can be considered constant along the path length. The
+energy loss due to the excitation is
+\begin{equation}
+\Delta E_e = n_1 E_1 + n_2 E_2
+\end{equation}
+where $n_1$ and $n_2$ are sampled from Poisson distribution as
+discussed above. The loss due to the ionisation can be generated from
+the distribution $g(E)$ by the inverse transformation method:
+\begin{eqnarray}
+u = F(E) & = & \int_{I}^E g(x) dx \nonumber \\
+E = F^{-1}(u) & = & \frac{I}{1 - u \frac {\Emax}{\Emax+I}} \\
+\end{eqnarray}
+where $u$ is a uniform random number between $F(I)=0$ and
+$F(\Emax+I)=1$. The contribution from the ionisations will be
+\begin{equation}
+\Delta E_i = \sum_{j=1}^{n_3} \frac{I}
+ {1 - u_j \frac {\Emax}{\Emax + I}}
+\end{equation}
+where $n_3$ is the number of ionisation (sampled from Poisson
+distribution). The energy loss in a step will then be $\Delta E =
+\Delta E_e + \Delta E_i$.
+
+\subsection{Fast simulation for $n_3 \geq 16$}
+
+If the number of ionisation $n_3$ is bigger than 16, a faster sampling
+method can be used. The possible energy loss interval is divided in
+two parts: one in which the number of collisions is large and the
+sampling can be done from a Gaussian distribution and the other in
+which the energy loss is sampled for each collision. Let us call the
+former interval $[I, \alpha I]$ the interval A, and the latter
+$[\alpha I,\Emax]$ the interval B. $\alpha$ lies between 1 and
+$\Emax/I$. A collision with a loss in the interval A happens with
+the probability
+\begin{equation}
+\label{eq:phys332-5}
+P(\alpha) = \int_I^{\alpha I} g(\!E\!) \, dE =
+ \frac {( \Emax + I) (\alpha - 1)}{\Emax \alpha}
+\end{equation}
+The mean energy loss and the standard deviation for this type
+of collision are
+\begin{equation}
+\langle \Delta E(\alpha) \rangle = \frac{1}{P(\alpha)}
+ \int_I^{\alpha I} E \, g(\!E\!) \, dE =
+ \frac{I \alpha \ln \alpha}{\alpha - 1}
+\end{equation}
+and
+\begin{equation}
+\sigma^2(\alpha) = \frac{1}{P(\alpha)}
+ \int_I^{\alpha I} E^2 \, g(\!E\!) \, dE =
+ I^2 \alpha \left( 1 - \frac{\alpha \ln \! ^2 \alpha}{(\alpha - 1)^2} \right)
+\end{equation}
+If the collision number is high , we assume that the number of the
+type A collisions can be calculated from a Gaussian distribution
+with the following mean value and standard deviation:
+\begin{eqnarray}
+\label{eq:phys332-1}
+\langle n_A \rangle & = & n_3 P(\alpha) \\
+\label{eq:phys332-2}
+\sigma_A^2 & = & n_3 P(\alpha) ( 1 - P(\alpha))
+\end{eqnarray}
+It is further assumed that the energy loss in these collisions
+has a Gaussian distribution with
+\begin{eqnarray}
+\label{eq:phys332-3}
+\langle \Delta E_A \rangle & = & n_A \langle \Delta E(\alpha) \rangle \\
+\label{eq:phys332-4}
+\sigma_{E,A}^2 & = & n_A \sigma^2(\alpha)
+\end{eqnarray}
+The energy loss of these collision can then be sampled from the
+Gaussian distribution.
+
+The collisions where the energy loss is in the interval B are sampled
+directly from
+\begin{equation}
+\Delta E_B = \sum_{i=1}^{n_3 - n_A} \frac{\alpha I}
+ {1 - u_i \frac{\Emax + I - \alpha I}{\Emax + I}}
+\end{equation}
+The total energy loss is the sum of these two types of collisions:
+\begin{equation}
+\Delta E = \Delta E_A + \Delta E_B
+\end{equation}
+
+The approximation of equations ((\ref{eq:phys332-1}),
+(\ref{eq:phys332-2}), (\ref{eq:phys332-3}) and (\ref{eq:phys332-4})
+can be used under the following conditions:
+\begin{eqnarray}
+\label{eq:phys332-6}
+\langle n_A \rangle - c \, \sigma_A & \geq & 0 \\
+\label{eq:phys332-7}
+\langle n_A \rangle + c \, \sigma_A & \leq & n_3 \\
+\label{eq:phys332-8}
+\langle \Delta E_A \rangle - c \, \sigma_{E,A} & \geq & 0
+\end{eqnarray}
+where $c \geq 4$. From the equations (\ref{eq:phys332-5}),
+(\ref{eq:phys332-1}) and (\ref{eq:phys332-3}) and from the conditions
+(\ref{eq:phys332-6}) and (\ref{eq:phys332-7}) the following limits can
+be derived:
+\begin{equation}
+\alpha_{\mathrm{min}} = \frac{(n_3 + c^2)(\Emax+I)}
+ {n_3 (\Emax + I) + c^2 I} \; \leq \alpha \; \leq
+\alpha_{\mathrm{max}} = \frac{(n_3 + c^2)(\Emax+I)}
+ {c^2 (\Emax + I) + n_3 I}
+\end{equation}
+This conditions gives a lower limit to number of the ionisations $n_3$
+for which the fast sampling can be done:
+\begin{equation}
+n_3 \; \geq \; c^2
+\end{equation}
+As in the conditions (\ref{eq:phys332-6}), (\ref{eq:phys332-7}) and
+(\ref{eq:phys332-8}) the value of $c$ is as minimum 4, one gets $n_3
+\; \geq 16$. In order to speed the simulation, the maximum value is
+used for $\alpha$.
+
+The number of collisions with energy loss in the interval B (the
+number of interactions which has to be simulated directly) increases
+slowly with the total number of collisions $n_3$. The maximum number
+of these collisions can be estimated as
+\begin{equation}
+n_{B,max} = n_3 - n_{A,min} \approx n_3 (\langle n_A \rangle
+ - \sigma_A)
+\end{equation}
+From the previous expressions for $\langle n_A \rangle$ and $\sigma_A$
+one can derive the condition
+\begin{equation}
+n_B \; \leq \; n_{B,max} = \frac{2 n_3 c^2}{n_3+c^2}
+\end{equation}
+The following values are obtained with $c=4$:
+
+\begin{tabular}{llcrr}
+$n_3$ & $n_{B,max}$ & & $n_3$ & $n_{B,max}$\\ \hline
+16 & 16 & & 200 & 29.63\\
+20 & 17.78 & & 500 & 31.01 \\
+50 & 24.24 & & 1000 & 31.50 \\
+100 & 27.59 & & $\infty$ & 32.00
+\end{tabular}
+
+\subsection{Special sampling for lower part of the spectrum}
+
+If the step length is very small ($\leq 5$ mm in gases, $\leq$ 2-3
+$\mu$m in solids) the model gives 0 energy loss for some events. To
+avoid this, the probability of 0 energy loss is computed
+\begin{equation}
+P( \Delta E=0) = e^{-( \langle n_1 \rangle + \langle n_2 \rangle
+ + \langle n_3 \rangle )}
+\end{equation}
+If the probability is bigger than 0.01 a special sampling is done,
+taking into account the fact that in these cases the projectile
+interacts only with the outer electrons of the atom. An energy level
+$E_0 = 10$ eV is chosen to correspond to the outer electrons. The mean
+number of collisions can be calculated from
+\begin{equation}
+\langle n \rangle = \frac{1}{E_0} \frac{dE}{dx} \Delta x
+\end{equation}
+The number of collisions $n$ is sampled from Poisson distribution.
+In the case of the thin layers, all the collisions are considered as
+ionisations and the energy loss is computed as
+\begin{equation}
+\Delta E = \sum_{i=1}^n \frac{E_0}
+ {1 - \frac {\Emax}{\Emax + E_0} u_i}
+\end{equation}
+
+\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}