\section{Electron ionisation} \label{secioni2}

    
 The class G4LivermoreIonisationModel calculates the continuous energy loss
  due to electron ionisation and
 simulates  
 $\delta$-ray production by electrons. 
 The $delta$-electron 
production
threshold for a given material, $T_c$, is used to separate
the continuous and the discrete parts of the process. The energy loss
of an electron with the incident energy, $T$, is expressed
via the sum over all atomic shells, $s$, and the integral over the energy, $t$,
of $delta$-electrons:
\begin{equation}
{dE\over dx}=\sum_{s}\left(\sigma_s(T){{\int^{T_c}_{0.1eV}t{d\sigma\over dt}dt} \over{\int^{T_{max}}_{0.1eV}{d\sigma\over dt}dt}}\right),
\end{equation}
where $T_{max} = 0.5T$ is the maximum energy transfered to a $\delta$-electron,
$\sigma_s(T)$ is
the total cross-section for the shell, $s$, at a given 
incident kinetic energy, $T$, and $0.1eV$ is the low energy limit
of the EEDL data.
The $\delta$-electron production cross-section is a complimentary
function:
\begin{equation}
\sigma(T)=\sum_{s}\left(\sigma_s(T){{\int^{T_{max}}_{T_c}{d\sigma\over dt}dt}\over {\int^{T_{max}}_{0.1eV}{d\sigma\over dt}dt}}\right).
\end{equation}
 The partial sub-shell cross-sections, $\sigma_s$, are obtained
from an interpolation of the evaluated cross-section data in the EEDL
library~\cite{io-EEDL},
according to the formula (\ref{eqloglog}) in Section~\ref{subsubsigmatot}.

The probability of emission of a $\delta$-electron with kinetic energy,
$t$, from a sub-shell, $s$, of binding energy, $B_s$, as the 
result of the interaction of
an incoming electron with kinetic energy, $T$,
is described by: 
\begin{equation}
{d\sigma \over dt} = {P(x) \over x^2}, \;\; \mbox{with} x={t + B_s \over T + B_s}, 
\end{equation}
where the parameter $x$ is varied from $x_{min} = (0.1eV + B_s)/(T + B_s)$ to
0.5.  The function, $P(x)$,  is parametrised differently in 3 regions of $x$: from $x_{min}$
to $x_1$ the linear interpolation with linear scale of 4 points is used; from $x_1$ to $x_2$
the linear interpolation with logarithmic scale of 16 points is used; from $x_2$ to $0.5$
the following interpolation is applied:
\begin{equation}
\label{io-ff}
P(x) = 1 - gx +(1 - g)x^2 + {x^2 \over 1-x}({1 \over 1-x} - g) + A*(0.5 - x)/x,
\end{equation}
where $A$ is a fit coefficient, $g$ is expressed via the gamma factor
of the incoming electron:
\begin{equation}
\label{ff}
g = (2\gamma - 1) / \gamma^2.
\end{equation}
For the high energy case ($x >> 1$) the formula (\ref{io-ff}) is transformed
to the M\"{o}ller electron-electron scattering formula \cite{io-g3,io-messel}.

The value of the coefficient, $A$,  for each element is obtained as a result
of the fit on the spectrum from the EEDL data for those
energies which are available in the database.
The values of $x_1$ and $x_2$ are chosen for each atomic shell 
according to the spectrum of $\delta$-electrons in this shell. Note that $x_1$
corresponds to the maximum of the spectrum, if the maximum does not coincide 
with $x_{min}$.
The dependence of all 24 parameters  on the incident energy, $T$, 
is evaluated from a logarithmic interpolation (\ref{eqloglog}).

The sampling of the final state proceeds in three steps. 
First a shell is randomly selected, then the energy of the
$delta$-electron is sampled, finally 
the angle of emission of the scattered electron and of the $\delta$-ray 
is determined by energy-momentum conservation taken into account
electron motion on the atomic orbit.

The interaction leaves the atom in an excited state. 
The deexcitation of the atom is simulated as 
described in section~\ref{relax}. Sampling of the excitations is carried out 
for both the continuous and the discrete parts 
of the process.

\subsection{Status of the document}

\noindent
30.09.1999 created by Alessandra Forti\\
07.02.2000 modified by V\'eronique Lef\'ebure\\
08.03.2000 reviewed by Petteri Nieminen and Maria Grazia Pia\\
05.12.2001 modified by Vladimir Ivanchenko \\
13.05.2002 modified by Vladimir Ivanchenko

\begin{latexonly}

\begin{thebibliography}{99}
\bibitem{io-EEDL} 
  %http://reddog1.llnl.gov/homepage.red/Electron.htm
  ``Tables and Graphs of Electron-Interaction Cross-Sections from 10~eV to 100~GeV Derived from
  the LLNL Evaluated Electron Data Library (EEDL), Z=1-100"
  S.T.Perkins, D.E.Cullen, S.M.Seltzer,
  UCRL-50400 Vol.31
\bibitem{io-g3}
  G\textsc{eant3} manual ,CERN Program Library Long Writeup W5013 (October 1994).
\bibitem{io-messel}
  H.Messel and D.F.Crawford. Pergamon Press,Oxford,1970.
\end{thebibliography}

\end{latexonly}

\begin{htmlonly}

\subsection{Bibliography}

\begin{enumerate}
\item 
  %http://reddog1.llnl.gov/homepage.red/Electron.htm
  ``Tables and Graphs of Electron-Interaction Cross-Sections from 10~eV to 100~GeV Derived from
  the LLNL Evaluated Electron Data Library (EEDL), Z=1-100"
  S.T.Perkins, D.E.Cullen, S.M.Seltzer,
  UCRL-50400 Vol.31
\item
  G\textsc{eant3} manual ,CERN Program Library Long Writeup W5013 (October 1994).
\item
  H.Messel and D.F.Crawford. Pergamon Press,Oxford,1970.
\end{enumerate}

\end{htmlonly}

