\section[Ionization]{Ionization} \label{sec:em.eion}

\subsection{Method}

The $G4eIonisation$ class provides the continuous and discrete
energy losses of electrons and positrons due to ionization in a material
according to the approach described in Section \ref{en_loss}. 
The value of the maximum energy transferable to a free electron $T_{max}$ 
is given by the following relation:
\begin{equation}
\label{eion.c}
T_{max} = \left\{ \begin{array}{ll}
             E-mc^2 & {for \hspace{.2cm} e^+}  \\
             (E-mc^2)/2 & {for \hspace{.2cm} e^- } \\
              \end{array} \right .
\end{equation}
where $mc^2$ is the electron mass.  
Above a given threshold energy the energy loss is simulated by the 
explicit production of delta rays by M\"{o}ller scattering ($e^- e^-$), or 
Bhabha scattering ($e^+ e^-$).  Below the threshold the soft electrons 
ejected are simulated as continuous energy loss by the incident
${e^{\pm}}$.

\subsection{Continuous Energy Loss} \label{seceloss}

The integration of \ref{comion.a} leads to the Berger-Seltzer 
formula \cite{eion.messel}:
\begin{equation}
\label{eion.d e}
\left. \frac{dE}{dx} \right]_{T < T_{cut}} =
       2 \pi r_e^2 mc^2 n_{el} \frac{1}{\beta^2}
       \left [\ln \frac{2(\gamma + 1)} {(I/mc^2)^2}+ F^{\pm} (\tau , \tau_{up})
       - \delta \right ]
\end{equation}
with
\[
\begin{array}{ll}
r_e          & \mbox{classical electron radius:} 
                  \quad e^2/(4 \pi \epsilon_0 mc^2 )        \\
mc^2         & \mbox{mass energy of the electron}           \\
n_{el}       & \mbox{electron density in the material}      \\
I            & \mbox{mean excitation energy in the material}\\
\gamma       & \mbox{$E/mc^2$}                              \\
\beta^2      & 1-(1/\gamma^2)                               \\
\tau         & \gamma-1                                     \\
T_{cut}      & \mbox{minimum energy cut for $\delta$ -ray production} \\
\tau_c       & \mbox{$T_{cut}/mc^2$}                        \\
\tau_{max}   & \mbox{maximum energy transfer:
                     $\tau$ for $e^+$, $\tau/2$ for $e^-$}  \\
\tau_{up}    & \min(\tau_c,\tau_{max})                      \\
\delta       & \mbox{density effect function} .
\end{array}
\]
In an elemental material the electron density is
$$ n_{el} = Z \: n_{at} = Z \: \frac{\mathcal{N}_{av} \rho}{A} . $$
$\mathcal{N}_{av}$ is Avogadro's number, $\rho$ is the material density, 
and $A$ is the mass of a mole.  In a compound material
$$ 
n_{el} = \sum_i Z_i \: n_{ati} 
       = \sum_i Z_i \: \frac{\mathcal{N}_{av} w_i \rho}{A_i} ,
$$
where $w_i$ is the proportion by mass of the $i^{th}$ element, with molar 
mass $A_i$ .
\par
\noindent 
The mean excitation energies $I$ for all elements are taken from 
\cite{ioni.icru1}.
\par
\noindent 
The functions $ F^{\pm}$  are given by :
\begin{eqnarray}
F^+ (\tau,\tau_{up}) & = &\ln(\tau\tau_{up} ) \\ 
 & & -\frac{\tau_{up}^2}{\tau}\left[\tau + 2 \tau_{up} -
\frac{3\tau_{up}^2 y } {2} -\left(\tau_{up} - \frac{\tau_{up}^3 }{3} \right) y^2
- \left (\frac{\tau_{up}^2}{2} - \tau
       \frac{\tau_{up}^3}{3} + \frac{\tau_{up}^4 } {4} \right)
          y^3  \right]  \nonumber
\end{eqnarray}

\begin{eqnarray}
F^- (\tau,\tau_{up} ) & = & -1 -\beta^2 \\
 & & +\ln \left [(\tau - \tau_{up})
\tau_{up} \right ] + \frac{\tau}{\tau -\tau_{up}}
+ \frac{1}{\gamma^2} \left [
\frac{\tau_{up}^2}{2} + ( 2\tau +1) \ln
\left (1- \frac{\tau_{up}}{\tau} \right ) \right ]   \nonumber
\end{eqnarray}
where $y = 1/(\gamma+1)$.

 
The density effect correction is calculated according to the formalism of 
Sternheimer \cite{eion.sternheimer}: 
\input{electromagnetic/utils/densityeffect}

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

The total cross section per atom for M\"{o}ller scattering ($e^- e^-$) and
Bhabha scattering ($e^+ e^-$) is obtained by integrating Eq.~\ref{comion.b}.
In {\sc Geant4} $T_{cut}$ is always 1 keV or larger.  For delta ray energies
much larger than the excitation energy of the material ($T \gg I$), the 
total cross section becomes \cite{eion.messel} for M\"{o}ller scattering, 
\begin{eqnarray}
\sigma ( Z,E,T_{cut} ) & = & \frac {2 \pi r_e^2 Z}{\beta^2(\gamma -1)} \times \\
 & &     \left[\frac{(\gamma-1)^2} {\gamma^2}\left(\frac{1}{2}-x\right)
         +\frac{1}{x}-\frac{1}{1-x}-\frac{2\gamma-1}{\gamma^2}\ln
         \frac{1-x}{x}\right] , \nonumber
\end{eqnarray}
and for Bhabha scattering ($e^+ e^-$),
\begin{eqnarray}
\sigma (Z,E,T_{cut}) & = & \frac{ 2 \pi r_e^2 Z }{(\gamma -1)} \times \\
  & &  \left [\frac {1 }{\beta^2}  \left(\frac{1}{x}-1\right)
       + B_1 \ln x + B_2 (1-x) -
       \frac {B_3 } {2} ( 1-x^2 ) +\frac{B_4}{3}(1-x^3)\right] .  \nonumber
\end{eqnarray}
Here
\[
\begin{array}{lcllcl}
 \gamma  & = & E/mc^2                &
 B_1     & = & 2-y^2                \\
 \beta^2 & = & 1-(1/\gamma^2)        &
 B_2     & = & (1-2y)(3+y^2 )       \\
 x       & = & T_{cut}/(E-mc^2)      &
 B_3     & = & (1-2y)^2+(1-2y)^3    \\
 y       & = & 1/(\gamma + 1)        &
 B_4     & = & (1-2y)^3 .  
\end{array}
\]
The above formulas give the total cross section for scattering above the 
threshold energies

\begin{equation}
T_{\rm Moller}^{\rm thr} =2T_{cut}  \mbox{\hspace{2cm}and\hspace{2cm}}
T_{\rm Bhabha}^{\rm thr} = T_{cut} .
\end{equation}

\noindent
In a given material the mean free path is then
\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}

\subsection{Simulation of Delta-ray Production}
\subsubsection{Differential Cross Section}

For $T \gg I$ the differential cross section per atom becomes 
\cite{eion.messel} for M\"{o}ller scattering,
\begin{eqnarray}
\label{eion.i}
\frac{d\sigma }{d \epsilon } &=& \frac{2 \pi r_e^2 Z}{\beta^2 (\gamma -1)}
\times \\
 & & \left[ \frac{(\gamma -1 )^2}  {\gamma^2 }+\frac{1}{\epsilon}
     \left(\frac{1}{\epsilon}-\frac{2 \gamma -1 } {\gamma^2 } \right) +
     \frac{1}{1- \epsilon}\left(\frac{1} {1- \epsilon} - \frac{2 \gamma - 1}
     {\gamma^2 }\right)  \right] \nonumber
\end{eqnarray}
and for Bhabha scattering,
\begin{equation}
\label{eion.j}
\frac{d \sigma}{d \epsilon}=\frac{2 \pi r_e^2 Z}{(\gamma -1)}\left[
\frac{1} {\beta^2 \epsilon^2}-\frac{B_1}{\epsilon}+B_2 - B_3 \epsilon
+ B_4 \epsilon^2\right] .
 \end{equation}
Here $\epsilon = T/(E-mc^2)$.  The kinematical limits of $\epsilon$ are
\[
\epsilon_0 = \frac{T_{cut}}{E-mc^2} \leq \epsilon \leq \frac{1}{2}
\mbox{\hspace{.2cm} for $e^- e^-$} \hspace{2cm}
\epsilon_0 = \frac{T_{cut}}{E-mc^2} \leq \epsilon \leq 1
\mbox{\hspace{.2cm} for $e^+ e^-$} .
\]

\subsubsection{Sampling}
The delta ray energy is sampled according to methods discussed in 
Chapter \ref{secmessel}.  Apart from normalization, the cross section can 
be factorized as
\begin{equation}
\frac{d\sigma}{d\epsilon}=f(\epsilon) g(\epsilon) .
\end{equation}
For $e^- e^-$ scattering
\begin{eqnarray}
f(\epsilon)&=&\frac{1}{\epsilon^2} \frac{\epsilon_0 }{1- 2\epsilon_0} \\
g(\epsilon)&=&\frac{4}{9\gamma^2 - 10 \gamma + 5}\left[(\gamma -1)^2
\epsilon^2 - (2 \gamma^2 +2\gamma -1) \frac{\epsilon} {1- \epsilon }+
\frac{\gamma^2}{(1- \epsilon )^2 }\right]
\end{eqnarray}
and for $e^+ e^-$ scattering
\begin{eqnarray}
  f(\epsilon)&=&\frac{1}{\epsilon^2} \frac{\epsilon_0}{1- \epsilon_0 } \\
  g(\epsilon)&=&\frac{B_0 -B_1 \epsilon +B_2 \epsilon^2
       -B_3 \epsilon^3 +B_4 \epsilon ^4}{B_ 0-B_1\epsilon_0
       +B_2\epsilon_0^2
       -B_3 \epsilon_0^3 +B_4 \epsilon_0^4} .
\end{eqnarray}
Here $ B_0=\gamma^2/(\gamma^2-1)$ and all other quantities have been defined
above.


To choose $\epsilon$, and hence the delta ray energy,
\begin{enumerate}
\item $\epsilon$ is sampled from $f(\epsilon)$
\item the rejection function $g(\epsilon)$ is calculated using the sampled
      value of $\epsilon$
\item $\epsilon$ is accepted with probability $g(\epsilon)$.
\end{enumerate}
After the successful sampling of $\epsilon$, the direction of the ejected 
electron is generated with respect to the direction of the incident 
particle.  The azimuthal angle $\phi$ is generated isotropically and the 
polar angle $\theta$ is calculated from energy-momentum conservation.
This information is used to calculate the energy and momentum of both the
scattered incident particle and the ejected electron, and to transform them 
to the global coordinate system.

\subsection{Status of this document}
 \ 9.10.98 created by L. Urb\'an. \\
  29.07.01 revised by M.Maire.     \\
  13.12.01 minor cosmetic by M.Maire.  \\
  24.05.02 re-written by D.H. Wright. \\
  01.12.03 revised by V. Ivanchenko.    \\

\begin{latexonly}

\begin{thebibliography}{99}

\bibitem{eion.messel}
  H.~Messel and D.F.~Crawford. {\em Pergamon Press, Oxford (1970).}
\bibitem{ioni.icru1} 
  ICRU (A.~Allisy et al), Stopping Powers for Electrons and Positrons,
  {\em ICRU Report No.37 (1984).}
\bibitem{eion.sternheimer}
  R.M.~Sternheimer. {\em Phys.Rev. B3 (1971) 3681.}
\end{thebibliography}

\end{latexonly}

\begin{htmlonly}

\subsection{Bibliography}

\begin{enumerate}
\item H.~Messel and D.F.~Crawford. {\em Pergamon Press, Oxford (1970).}
\item ICRU (A.~Allisy et al), Stopping Powers for Electrons and Positrons,
  {\em ICRU Report No.37 (1984).}
\item R.M.~Sternheimer. {\em Phys.Rev. B3 (1971) 3681.}
\end{enumerate}

\end{htmlonly}