\section[Ionization]{Muon Ionization}
 

The class {\it G4MuIonisation} provides the continuous energy loss due to 
ionization and simulates the 'discrete' part of the ionization, that is, 
delta rays produced by muons. 
Inside this class the following models are used:
\begin{itemize}
\item 
{\it G4BraggModel} (valid for protons with $T < 0.2 \;MeV$)
\item
{\it G4BetherBlochModel} (valid for protons with $0.2 \;MeV < T < 1\; GeV$)
\item
{\it G4MuBetherBlochModel} (valid for protons with $T > 1\; GeV$)
\end{itemize}
The limit energy $0.2\; MeV$ is equivalent to the proton limit energy $2 MeV$ because of
scaling relation  (\ref{enloss.sc}), which allows
simulation for muons with energy below $1\; GeV$
in the same way as for point-like hadrons with spin 1/2 described in the section \ref{en_loss}.

For higher energies the {\it G4MuBetherBlochModel} is applied, in which 
leading radiative corrections are taken into account \cite{muion.br}.
Simple analytical formula for the cross section, derived with the logarithmic
are used. Calculation results appreciably differ from usual elastic $\mu -e$
scattering in the region of high energy transfers $m_e << T < T_{max}$
and give non-negligible correction to the total average energy loss of 
high-energy muons. The total cross section is written as following:
\begin{equation}
\label{muion.c}
\sigma (E,\epsilon ) = \sigma_{BB}(E, \epsilon ) \left[ 1 + \frac{\alpha}{2\pi}
ln\left( 1 + \frac{2\epsilon}{m_e} \right) 
ln\left(\frac{4m_e E(E - \epsilon )}{m_{\mu}^2( 2\epsilon + m_e) }\right) \right] ,
\end{equation}
here $\sigma(E,\epsilon)$ is the differential cross sections,
$\sigma(E,\epsilon)_{BB}$ is the Bethe-Bloch cross section (\ref{hion.i}), 
$m_e$ is the electron mass, $m_{\mu}$ is the muon mass,
$E$ is the muon energy, $\epsilon$
is the energy transfer, $\epsilon = \omega + T$, where
T is the electron kinetic energy and $\omega$ is the energy of radiative gamma.  

For computation of the truncated mean energy loss (\ref{comion.a}) the partial integration 
of the expression (\ref{muion.c}) is performed
\begin{equation}
\label{muion.d}
S(E,\epsilon_{up}) = S_{BB}(E,\epsilon_{up}) + S_{RC}(E,\epsilon_{up}), \;\; \epsilon_{up} = min(\epsilon_{max},\epsilon_{cut}),
\end{equation}
where term $S_{BB}$ is the Bethe-Bloch truncated energy loss (\ref{hion.d}) for the interval
of energy transfer $(0 - \epsilon_{up})$
and term $S_{RC}$ is a correction due to radiative effects.
The function become smooth after  log-substitution and 
is computed by numerical integration 
\begin{equation}
\label{muion.e}
S_{RC}(E,\epsilon_{up}) = 
\int^{\ln\epsilon_{up}}_{\ln\epsilon_1}\epsilon^2(\sigma(E,\epsilon) - \sigma_{BB}(E, \epsilon))d(\ln\epsilon),
\end{equation}
where lower limit $\epsilon_1$ does not
effect result of integration in first order and
in the class {\it G4MuBetheBlochModel} the default value $\epsilon_1 = 100 keV$ is used.

For computation of the discrete cross section (\ref{comion.b}) another substitution is used
in order to perform numerical integration of a smooth function 
\begin{equation}
\label{muion.f}
\sigma(E) = \int^{1/\epsilon_{up}}_{1/\epsilon_{max}}\epsilon^2\sigma (E,\epsilon )d(1/\epsilon ).
\end{equation}
The sampling of energy transfer is performed between $1/\epsilon_{up}$ and $1/\epsilon_{max}$ using
rejection constant for the function $\epsilon^2\sigma(E,\epsilon)$.
After the successful sampling of the energy transfer, the direction
of the scattered electron is generated with respect to the direction of the
incident particle. The energy of radiative gamma is neglected. 
The azimuthal electron 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     \\
  22.06.07 rewritten by V. Ivanchenko     \\

\begin{latexonly}

\begin{thebibliography}{99}

\bibitem{muion.br}
S.R.~Kelner, R.P.~Kokoulin, A.A.~Petrukhin, 
Phys. Atomic Nuclei 60 (1997) 576.
\end{thebibliography}

\end{latexonly}

\begin{htmlonly}

\subsection{Bibliography}

\begin{enumerate}
\item 
S.R.~Kelner, R.P.~Kokoulin, A.A.~Petrukhin, 
Phys. Atomic Nuclei 60 (1997) 576.
\end{enumerate}

\end{htmlonly}