\section[Ionization]{Ionization}
 
\subsection{Method}

 The class $G4MuIonisation$ provides the continuous energy loss due to 
ionization and simulates the 'discrete' part of the ionization, that is delta
rays produced by muons. The approach described in Section \ref{en_loss} is used.
The value of the maximum energy transferable to a free electron $T_{max}$ 
is given by the following relation:
\begin{equation}
\label{muion.c}
T_{max} =\frac{2mc^2(\gamma^2 -1)}{1+2\gamma (m/M)+(m/M)^2 } .
\end{equation}
Here $m$ is the electron mass and $M$ the muon mass.  The method of 
calculation of the continuous energy loss and the total cross section are 
explained below. 

\subsection{Continuous Energy Loss}

The integration of \ref{comion.a} leads to the Bethe-Bloch restricted energy 
loss formula \cite{muion.pdg} :
\begin{equation}
\label{muion1}
\left. \frac{dE}{dx} \right]_{T < T_{cut}} =
       2 \pi r_e^2 mc^2 n_{el} \frac{(z_p)^2}{\beta^2}
       \left [\ln \left(\frac{2mc^2 \beta^2 \gamma^2 T_{up}} {I^2} \right)
       - \beta^2 \left( 1 + \frac{T_{up}}{T_{max}} \right)
       - \delta - \frac{2C_e}{Z} \right ]
\end{equation}
 where
\[
\begin{array}{ll}
r_e          & \mbox{classical electron radius:} 
                  \quad e^2/(4 \pi \epsilon_0 mc^2 )        \\
mc^2         & \mbox{mass-energy of the electron}           \\
n_{el}       & \mbox{electrons density in the material}     \\
I            & \mbox{mean excitation energy in the material}\\
\gamma       & \mbox{$E/mc^2$}                              \\
\beta^2      & 1-(1/\gamma^2)                               \\
T_{up}       & \min(T_{cut},T_{max})                        \\
\delta       & \mbox{density effect function}               \\
C_e          & \mbox{shell correction function}
\end{array}
\]
In a single element the electron density is
$$ n_{el} = Z \: n_{at} = Z \: \frac{\mathcal{N}_{av} \rho}{A} $$
($\mathcal{N}_{av}$: Avogadro number, $\rho$: density of the material, 
 $A$: mass of a mole).  In a compound material
$$ 
n_{el} = \sum_i Z_i \: n_{ati} 
       = \sum_i Z_i \: \frac{\mathcal{N}_{av} w_i \rho}{A_i} . 
$$
$w_i$ is the proportion by mass of the $i^{th}$ element, with molar mass 
$A_i$.


The mean excitation energy, $I$, for all elements is tabulated according to the
ICRU recommended values \cite{muion.icru1}.

\subsubsection{Density Correction}

$\delta$ is a correction term which takes into account the reduction
in energy loss due to the so-called {\it density effect}. This becomes
important at high energy because media have a tendency to become
polarised as the incident particle velocity increases. As a consequence,
the atoms in a medium can no longer be considered as isolated. To correct
for this effect the formulation of Sternheimer~\cite{muion.sternheimer}
is used:
\input{electromagnetic/utils/densityeffect}

\subsubsection{Shell Correction}

$2C_e/Z$ is the so-called {\it shell correction term} which accounts for the 
fact that, at low energies for light elements and at all energies for heavy 
ones, the probability of collision with the electrons of the inner atomic 
shells (K, L, etc.) is negligible.  The semi-empirical formula used in
{\sc Geant4}, applicable to all materials, is due to 
Barkas \cite{muion.barkas}:
\begin{equation}
\label{muion2}
C_e(I, \beta\gamma) = \frac{a(I)}{(\beta\gamma)^2}
                     +\frac{b(I)}{(\beta\gamma)^4}
                     +\frac{c(I)}{(\beta\gamma)^6} .
\end{equation}
The functions a(I), b(I), c(I) can be found in the source code.  This formula
breaks down at low energies, and is valid only when 
$\beta\gamma > 0.13$ ($T > 7.9$ MeV for a proton).  For $\beta\gamma \leq
0.13$ the shell correction term is calculated as:
\begin{equation}
\label{muion3}
\left . C_{e}(I,\beta\gamma) \rule{0mm}{5mm} \right |_{\beta\gamma \leq 0.13}
 = C_{e}(I,\beta\gamma=0.13)\frac{\ln(T/T_{2l})}{\ln(7.9 \: \rm MeV/T_{2l})}
\end{equation}
i.e. the correction is switched off logarithmically from $T=7.9$ MeV
to $T=T_{2l}=2$ MeV.


\subsubsection{Parameterization}

The mean energy loss can be described by the Bethe-Bloch formula
(\ref{muion1}) only if the projectile velocity is larger than that of the
orbital electrons.  In the low energy region this is not the case, and the 
parameterization from the ICRU'49 report \cite{muion.ICRU49}
is used in the $G4BraggModel$ class. The Bethe-Bloch model is applied to 
muons of higher kinetic energies
\begin{equation}
\label{muion.1}
T > 2 * M_{\mu}/M_{proton} MeV.
\end{equation}
The details of the low energy parameterization are described in 
Section \ref{le_had_ion}.

\subsection{Total Cross Section per Atom and Mean Free Path}

For $T \gg I $ the differential cross section can be written as 
\cite{muion.pdg} 
\begin{equation}
\label{muion.i}
\frac{d\sigma }{dT} = 2\pi r_e^2 mc^2 Z \frac{z_p^2}{\beta^2} \frac{1}{T^2}
     \left[ 1 - \beta^2 \frac{T}{T_{max}} + \frac{T^2}{2E^2} \right] . 
\end{equation}
In {\sc Geant4} $T_{cut} \geq 1$ keV.  Integrating from $T_{cut}$ to 
$T_{max}$ gives the total cross-section per atom :
\begin{eqnarray}
\sigma (Z,E,T_{cut}) & = & \frac {2\pi r_e^2 Z z_p^2}{\beta^2}mc^2 \times 
  \\ & &     \left[ \left( \frac{1}{T_{cut}} - \frac{1}{T_{max}} \right)
                   - \frac{\beta^2}{T_{max}} \ln \frac{T_{max}}{T_{cut}}
                   + \frac{T_{max} - T_{cut}}{2E^2}
             \right] . \nonumber    
\end{eqnarray}
In a given material the mean free path is
\begin{equation}
\begin{array}{lll} 
\lambda = (n_{at} \cdot \sigma)^{-1} & or &
\lambda = \left( \sum_i n_{ati} \cdot \sigma_i \right)^{-1} . 
\end{array}
\end{equation}
The mean free path is tabulated during initialization as a function of the
material and of the energy of the incident muon.

\subsection{Simulating Delta-ray Production}

A short overview of the sampling method is given in Chapter \ref{secmessel}. 
Apart from the normalization, the cross section \ref{muion.i} can be 
factorized :
\begin{eqnarray}
\frac{d\sigma}{dT}=f(T) g(T) &with& T \in [T_{cut}, \ T_{max}]
\end{eqnarray}
where
\begin{eqnarray}
f(T) &=& \left(\frac{1}{T_{cut}} - \frac{1}{T_{max}} \right) \frac{1}{T^2} \\
g(T) &=& 1 - \beta^2 \frac{T}{T_{max}} + \frac{T^2}{2E^2} .
\end{eqnarray}
The energy $T$ is chosen by
\begin{enumerate}
\item sampling $T$ from $f(T)$
\item calculating the rejection function $g(T)$ and accepting the
      sampled $T$ with a probability of $g(T)$.
\end{enumerate}
After successful sampling of the energy, the direction of the scattered 
electron is generated with respect to the direction of the incident muon. 
The azimuthal angle $\phi$ is generated isotropically.  The polar angle
$\theta$ is calculated from energy-momentum conservation.  This information
is used to calculate the energy and momentum of both scattered
particles and to transform them into the {\em global} coordinate system.

\subsection{Status of this document}
  09.10.98  created by L. Urb\'an. \\
  14.12.01  revised by M.Maire \\
  30.11.02  re-worded by D.H. Wright \\
  01.12.03 revised by V. Ivanchenko     \\

\begin{latexonly}

\begin{thebibliography}{99}

\bibitem{muion.pdg}
  Particle Data Group. Rev. of Particle Properties.
   Eur. Phys. J. C15. (2000) 1. http://pdg.lbl.gov  
\bibitem{muion.icru1} 
  ICRU Report No. 37 (1984)
\bibitem{muion.sternheimer}
  R.M.Sternheimer. Phys.Rev. B3 (1971) 3681.
\bibitem{muion.barkas}
  W. H. Barkas. Technical Report 10292,UCRL, August 1962.
\bibitem{muion.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 Particle Data Group. Rev. of Particle Properties.
   Eur. Phys. J. C15. (2000) 1. http://pdg.lbl.gov  
\item ICRU Report No. 37 (1984)
\item R.M.Sternheimer. Phys.Rev. B3 (1971) 3681.
\item W.H. Barkas. Technical Report 10292,UCRL, August 1962.
\item ICRU (A.~Allisy et al),
Stopping Powers and Ranges for Protons and Alpha
Particles, ICRU Report 49, 1993.
\end{enumerate}

\end{htmlonly}
