
\section{Energy loss fluctuations}  \label{gen_fluctuations}


\subsection{Fluctuations in thick absorbers}

The total continuous energy loss of charged particles is a stochastic
quantity with a distribution described in terms of a straggling function.
The straggling is partially taken into account in the simulation
of energy loss by the production of $\delta$-electrons with energy
$T > T_c$.  However, continuous energy loss also has fluctuations.
Hence in the current GEANT4 implementation two different models of
fluctuations are applied depending on the value of the parameter $\kappa$
which is the lower limit of the number of interactions of the particle in
a step.  The default value chosen is $\kappa = 10$. In the case of a high
range cut (i.e. energy loss without delta ray production)
for thick absorbers the following condition should be fulfilled:
\begin{equation}
\Delta E > \kappa \ T_{max}
\label{cond}
\end{equation}
where $\Delta E$ is the mean continuous energy loss in a track segment of
length $s$, and $T_{max}$ is the maximum kinetic energy that can be 
transferred to the atomic electron. If this condition holds the fluctuation 
of the total (unrestricted) energy loss follows a  Gaussian distribution. It 
is worth noting that this condition can be true only for heavy particles, 
because for electrons, $T_{max}=T/2$, and for positrons, $T_{max}=T$, where 
$T$ is the kinetic energy of the particle.
 In order to
simulate the fluctuation of the continuous (restricted) energy loss, the
condition should be modified. After a study, the following conditions
have been chosen:
\begin{equation}
\Delta E > \kappa \ T_c
\label{cond2}
\end{equation}

 and

\begin{equation}
T_{max} <=    2  \ T_c
\label{cond3}
\end{equation}
where  $T_c$ is the cut kinetic energy of $\delta$-electrons.  For thick
absorbers the
straggling function approaches the Gaussian distribution with Bohr's
variance \cite{eloss.ICRU49}:
\begin{equation}
\Omega^2 = 2\pi r^2_e m_e c^2 N_{el}\frac{Z_h^2}{\beta^2} T_c s
\left(1 - \frac{\beta^2}{2} \right),
\label{sig.fluc}
\end{equation}
where $r_e$ is the classical electron radius, $N_{el}$ is the electron
density of the medium, $Z_{h}$ is the charge of the incident particle in
units of positron charge, and $\beta$ is the relativistic velocity.


\subsection{Fluctuations in thin absorbers}
If the conditions \ref{cond2} and \ref{cond3} are not satisfied the model of
energy
fluctuations in thin absorbers is applied.  The formulae used to compute
the energy loss fluctuation (straggling) are based on a very simple physics
model of the atom.  It is assumed that the atoms have only two energy levels
with binding energies $E_1$ and $E_2$. The particle-atom interaction can be
an excitation with energy loss $E_1$ or $E_2$, or ionization with energy
loss distributed according to a function $g(E) \sim 1/E^2$ :
\begin{equation} \label{fluct.eqn0}
\int_{E_0}^{T_{up}} g(E)\ dE = 1 \Longrightarrow
     g(E) = \frac{E_0 T_{up}}{T_{up}-E_0} \frac{1}{E^2} .
\end{equation}

\noindent
The macroscopic cross section for excitation $(i=1,2)$ is
\begin{equation} \label{fluct.eqn1}
  \Sigma_i = C \frac{f_i}{E_i}
               \frac{\ln[2mc^2 \ (\beta\gamma)^2/E_i]-\beta^2}
                    {\ln[2mc^2 \ (\beta\gamma)^2/I]-\beta^2}\ (1-r)
\end{equation}
and the ionization cross section is
\begin{equation}  \label{fluct.eqn2}
     \Sigma_3 = C \frac{T_{up}-E_0}
         {E_0 T_{up}\ln(\frac{T_{up}}{E_0})}\ r
\end{equation}
where $E_0$ denotes the ionization energy of the atom, $I$ is the mean
ionization energy, $T_{up}$ is the
production threshold for delta ray production (or the maximum energy
transfer if this value smaller than the
production threshold), $E_i$ and $f_i$ are the energy levels and
corresponding oscillator strengths of the atom, and $C$ and $r$ are model
parameters.
\noindent
The oscillator strengths $f_i$ and energy levels $E_i$ should satisfy the
constraints
   \begin{equation}  \label{fluct.eqn3}
        f_1 + f_2 = 1
   \end{equation}
   \begin{equation}   \label{fluct.eqn4}
        f_1 \cdotp lnE_1 + f_2 \cdotp lnE_2 = lnI .
   \end{equation}
The cross section formulae \ref{fluct.eqn1},\ref{fluct.eqn2} and the sum
rule equations \ref{fluct.eqn3},\ref{fluct.eqn4} can be found e.g. in
Ref. \cite{straggling.bichsel}.
\noindent
The model parameter $C$ can be defined in the following way. The numbers of
collisions ($n_i$, $i=1,2$ for excitation and $3$ for ionization)
follow the Poisson distribution with a mean value $\langle n_i \rangle$.
In a step of length $\Delta x$ the mean number of collisions is given by
   \begin{equation}
        \langle n_i \rangle = \Delta x \ \Sigma_i
   \end{equation}
The mean energy loss in a step is the sum of the excitation and ionization
contributions and can be written as
   \begin{equation}
     \frac{dE}{dx} \cdotp \Delta x =
     \left \{ \Sigma_1 E_1 + \Sigma_2 E_2 +
        \int_{E_0}^{T_{up}} E g(E) dE \right \} \Delta x .
   \end{equation}
From this, using eq. \ref{fluct.eqn1} - \ref{fluct.eqn4}, one can see that
   \begin{equation}
         C = dE/dx .
   \end{equation}

\noindent
The other parameters in the fluctuation model have been chosen
in the following way. $Z \cdotp f_1$ and $Z \cdotp f_2$ represent in
the model the number of loosely/tightly bound electrons
   \begin{equation}
         f_2 = 0 \hspace {15 pt} for \hspace {15 pt} Z \leq 2.8
   \end{equation}
   \begin{equation}
         f_2 = 2.8/Z \hspace {15 pt} for \hspace {15 pt} Z > 2.8
   \end{equation}
   \begin{equation}
         E_2 = 10 \mbox{ eV } Z^2
   \end{equation}
   \begin{equation}
         E_0 = 10 \mbox{ eV } .
   \end{equation}
Using these parameter values, $E_2$ corresponds approximately to the
K-shell energy of the atoms.
The parameters $f_1$ and $E_1$ can be obtained from Eqs.~ \ref{fluct.eqn3}
and \ref{fluct.eqn4}.
\noindent
The parameter $r$ is the only variable in the model which can be tuned.
This parameter determines the relative contribution of ionization and
excitation to the energy loss. Based on comparisons of simulated energy
loss distributions to experimental data, its value has been parametrized
 as
   \begin{equation}
      r = 0.03 + 0.23 \cdotp ln(ln(\frac{T_{up}}{I}))
   \end{equation}

 This simple parametrization in the model gives good 
 values for the most probable energy loss. The width (FWHM) of the energy loss
 distribution is good too, if the absorber layer is not too thin.
 In order to get good FWHM in thin absorbers a width correction has been
 applied. The form of this correction is

  \begin{equation}
    w = \sqrt{\frac{10 E_1}{\Delta E}} \hspace{25 mm} for \Delta E > 10 E_1  
   \end{equation}
  \begin{equation}
    w = 1      \hspace{35 mm}           for \Delta E \leq 10 E_1  
   \end{equation}

 and the transformation
   $\langle n_1 \rangle \longrightarrow \langle n_1 \rangle \cdot w$ , $E_1 \longrightarrow \frac{E_1}{w}$ should be applied.
 This correction is a kind of scaling for the energy of the lower level and for the mean
 number of excitations with lower energy.

\paragraph{Sampling the energy loss.}
The energy loss is computed in the model under the assumption that
the step length (or relative energy loss) is small and, in consequence, the
cross section can be  considered constant along the step.  The loss due to
the excitation is
   \begin{equation}
         \Delta E_{exc} = n_1 E_1 + n_2 E_2
   \end{equation}
where $n_1$ and $n_2$ are sampled from a Poisson distribution. The energy
loss due to ionization can be generated from the distribution $g(E)$
by the inverse transformation method :
   \begin{equation}
      u = F(E) = \int_{E_0}^E g(x) dx
   \end{equation}
   \begin{equation}
      E = F^{-1}(u) = \frac {E_0}{1-u \frac{T_{up}-E_0}{T_{up}}}
   \end{equation}
where $u$ is a uniformly distributed random number $\in [0,\ 1]$.
The contribution coming from the ionization will then be
   \begin{equation}
     \Delta E_{ion} = \sum_{j = 1}^{n3} \frac {E_0}
                      {1-u_j \frac{T_{up}-E_0}{T_{up}}}
   \end{equation}
where $n_3$ is the number of ionizations sampled from the Poisson
distribution. The total energy loss in a step will be
$ \Delta E = \Delta E_{exc} + \Delta E_{ion}$ and the energy loss
fluctuation comes from fluctuations in the number of collisions $n_i$
 and from the sampling of the ionization loss.

\paragraph{Thick layers}
If the mean energy loss and step are in the range of validity of the
Gaussian approximation of the fluctuation, the much faster Gaussian
sampling is used to compute the actual energy loss.

\paragraph{Conclusions}
This simple model of energy loss fluctuations is rather fast and can
be used for any thickness of material.  This has been verified by performing
many simulations and comparing the results with experimental data, such
as those in Ref.\cite{straggling.lassila}. \\
As the limit of validity of Landau's theory is approached, the loss
distribution approaches the Landau form smoothly.

\subsection{Status of this document}
 30.01.02  created by L. Urb\'an.\\
 28.08.02  updated by V.Ivanchenko.\\
 17.08.04  moved to common to all charged particles (mma) \\
 04.12.04  spelling and grammar check by D.H. Wright \\
 04.05.05  updated by L. Urb\'an.\\ 
 09.12.05  updated by V.Ivanchenko.\\
 29.03.07  updated by L. Urb\'an.\\
 
\begin{latexonly}

\begin{thebibliography}{99}
\bibitem{straggling.bichsel} H.Bichsel
   {\em Rev.Mod.Phys. 60 (1988) 663}
\bibitem{straggling.lassila} K.Lassila-Perini, L.Urb\'an
   {\em Nucl.Inst.Meth. A362(1995) 416}
\bibitem{eloss.geant3} {\sc geant3} manual
  {\em Cern Program Library Long Writeup W5013 (1994)}
\bibitem{eloss.ICRU49}ICRU (A.~Allisy et al),
Stopping Powers and Ranges for Protons and Alpha
Particles,
ICRU Report 49, 1993.
\end{thebibliography}

\end{latexonly}

\begin{htmlonly}
\subsection{Bibliography}

\begin{enumerate}
\item H.Bichsel {\em Rev.Mod.Phys. 60 (1988) 663}
\item K.Lassila-Perini, L.Urb\'an
   {\em Nucl.Inst.Meth. A362(1995) 416}
\item {\sc geant3} manual
  {\em Cern Program Library Long Writeup W5013} (1994).
\item ICRU (A.~Allisy et al),
Stopping Powers and Ranges for Protons and Alpha
Particles, ICRU Report 49, 1993.
\end{enumerate}

\end{htmlonly}

