\section[Ionization]{Ionization}

\subsection{Method}
 
 The class $G4hIonisation$ provides the continuous energy loss due to 
ionization and simulates the 'discrete' part of the ionization, that is, 
delta rays produced by charged hadrons.  The class $G4ionIonisation$ is
intended for the simulation of energy loss by ions and 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{hion.c}
T_{max} =\frac{2mc^2(\gamma^2 -1)}{1+2\gamma (m/M)+(m/M)^2 },
\end{equation}
where $m$ is the electron mass and $M$ is the mass of the incident particle.
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{hion.pdg} :
\begin{equation}
\label{hion.d}
\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{hion.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 energies because media have a tendency to become polarized 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{hion.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{hion.barkas}:
\begin{equation}
\label{hion.dd}
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) and 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{hion.ddd}
\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{hion.ICRU49} is used in the 
$G4BraggModel$ class. The Bethe-Bloch model is applied for higher kinetic 
energies of incident particles
\begin{equation}
\label{muion.lowen1}
T > 2 * M/M_{proton} MeV,
\end{equation}
where $M$ is the particle mass.  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
\begin{equation}
\label{hion.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}
\cite{hion.pdg}.  In {\sc Geant4} $T_{cut} \geq 1$ keV.  Integrating from
$T_{cut}$ to $T_{max}$ gives the total cross section per atom :
\begin{eqnarray}
\label{hion.j}
\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}
The last term is for spin $1/2$ only.  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 for all kinds of charged particles.

\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{hion.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 last term in $g(T)$ is for spin $1/2$ only.  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 the successful sampling of the energy, the direction
of the scattered electron is generated with respect to the direction of the
incident particle. 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.

\subsubsection{Ion Effective Charge}

As ions penetrate matter they exchange electrons with the medium. In the 
implementation of $G4ionIonisation$ the effective charge approach is 
used \cite{hion.Ziegler85}. 
A state of equilibrium between the ion and the medium is assumed, so that 
the ion's effective charge can be calculated as a function of its kinetic 
energy in a given material.  This is done according to the approximation 
described in Section \ref{le_had_ion}. Before and after each step the dynamic 
charge of the ion is recalculated and saved in $G4DynamicParticle$, where 
it can be used not only for energy loss calculations but also for the
sampling of transportation in an electromagnetic field.


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

\begin{latexonly}

\begin{thebibliography}{99}

\bibitem{hion.pdg}
  Particle Data Group. Rev. of Particle Properties.
   Eur. Phys. J. C15. (2000) 1. http://pdg.lbl.gov  
\bibitem{hion.icru1} 
  ICRU Report No. 37 (1984)
\bibitem{hion.sternheimer}
  R.M.Sternheimer. Phys.Rev. B3 (1971) 3681.
\bibitem{hion.barkas}
  W. H. Barkas. Technical Report 10292,UCRL, August 1962.
\bibitem{hion.ICRU49}ICRU (A.~Allisy et al),
Stopping Powers and Ranges for Protons and Alpha
Particles,
ICRU Report 49, 1993.
\bibitem{hion.Ziegler85}J.F.~Ziegler, J.P.~Biersack, U
.~Littmark, The Stopping
and Ranges of Ions in Solids. Vol.1, Pergamon Press, 1985.
\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.
\item J.F.~Ziegler, J.P.~Biersack, U.~Littmark, The Stopping
and Ranges of Ions in Solids. Vol.1, Pergamon Press, 1985.
\end{enumerate}

\end{htmlonly}


