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\section[Photoabsorption ionization model]{Photoabsorption Ionization Model}
  \label{secpai}
  
\subsection{Cross Section for Ionizing Collisions}

The Photoabsorption Ionization (PAI) model describes the ionization energy 
loss of a relativistic charged particle in matter.  For such a particle, the
differential cross section $d\sigma_i/d\omega$ for ionizing collisions with 
energy transfer $\omega$ can be expressed most generally by the following
equations \cite{pai.asosk}:

\begin{eqnarray}
\label{PAI1}
\frac{d\sigma_i}{d\omega} 
& = & \frac{2\pi Ze^4}{mv^2}    
\left\{
\frac{f(\omega)}{\omega\left|\varepsilon(\omega)\right|^2}
\left[
\ln\frac{2mv^2}{\omega\left|1-\beta^2\varepsilon\right|} -  
\right. \right. \nonumber \\
&& \left. \left.
- \frac{\varepsilon_1 - \beta^2\left|\varepsilon\right|^2}{\varepsilon_2}
\arg(1-\beta^2\varepsilon^*)
\right] + 
 \frac{\tilde{F}(\omega)}{\omega^2}
\right\} ,
\end{eqnarray}

\[
\tilde{F}(\omega) = \int_{0}^{\omega}\frac{f(\omega')}
{\left|\varepsilon(\omega')\right|^2}d\omega' ,
\]

\[
f(\omega) = \frac{m\omega\varepsilon_2(\omega)}{2\pi^2ZN\hbar^2} .
\]
Here $m$ and $e$ are the electron mass and charge, $\hbar$ is Planck's 
constant, $\beta = v/c$ is the ratio of the particle's velocity $v$ to
the speed of light $c$, $Z$ is the effective atomic number, $N$ is the 
number of atoms (or molecules) per unit volume, and 
$\varepsilon = \varepsilon_1 + i\varepsilon_2$ is the complex dielectric 
constant of the medium.  In an isotropic non-magnetic medium the dielectric 
constant can be expressed in terms of a complex index of refraction, 
$n(\omega) = n_1 + in_2$, $\varepsilon(\omega) = n^2(\omega)$.  In the 
energy range above the first ionization potential $I_1$ for all cases of 
practical interest, and in particular for all gases, $n_1 \sim 1$. 
Therefore the imaginary part of the dielectric constant can be expressed in 
terms of the photoabsorption cross section $\sigma_{\gamma}(\omega)$:

\[
\varepsilon_2(\omega) = 2n_1n_2 \sim 2n_2 = \frac{N\hbar c}{\omega}
\sigma_{\gamma}(\omega) .
\]
The real part of the dielectric constant is calculated in turn from the 
dispersion relation

\[
\varepsilon_1(\omega) - 1 = \frac{2N\hbar c}{\pi}V.p.\int_{0}^{\infty}
\frac{\sigma_{\gamma}(\omega')}{\omega'^2 - \omega^2}d\omega'  ,
\]
where the integral of the pole expression is considered in terms of the
principal value.  In practice it is convenient to calculate the contribution
from the continuous part of the spectrum only.  In this case the normalized
photoabsorption cross section

\[
\tilde{\sigma}_{\gamma}(\omega) = \frac{2\pi^2\hbar e^2Z}{mc}
\sigma_{\gamma}(\omega)
\left[
\int_{I_1}^{\omega_{max}}\sigma_{\gamma}(\omega')d\omega'
\right]^{-1}, \  \omega_{max} \sim 100 \ keV 
\]
is used, which satisfies the quantum mechanical sum rule \cite{pai.fano}:

\[
\int_{I_1}^{\omega_{max}}\tilde{\sigma}_{\gamma}(\omega')d\omega' = 
\frac{2\pi^2\hbar e^2Z}{mc} .
\]

\noindent
The differential cross section for ionizing collisions is expressed by the 
photoabsorption cross section in the continuous spectrum region:

\begin{eqnarray}
\frac{d\sigma_i}{d\omega}
& = & \frac{\alpha}{\pi\beta^2}
\left\{
\frac{\tilde{\sigma}_{\gamma}(\omega)}
{\omega\left|\varepsilon(\omega)\right|^2}
\left[
\ln\frac{2mv^2}{\omega\left|1-\beta^2\varepsilon\right|} - 
\right. \right.   \nonumber \\
&   & \left. \left.
- \frac{\varepsilon_1-\beta^2\left|\varepsilon\right|^2}{\varepsilon_2}
\arg(1-\beta^2\varepsilon^*)
\right] 
 + \frac{1}{\omega^2}\int_{I_1}^{\omega}\frac{\tilde{\sigma}_{\gamma}(\omega')}
{\left|\varepsilon(\omega')\right|^2}d\omega'
\right\} ,
\end{eqnarray}

\[
\varepsilon_2(\omega) = \frac{N\hbar c}{\omega}
\tilde{\sigma}_{\gamma}(\omega) ,
\]

\[
\varepsilon_1(\omega) - 1 = \frac{2N\hbar c}{\pi}V.p.\int_{I_1}^{\omega_{max}}
\frac{\tilde{\sigma}_{\gamma}(\omega')}{\omega'^2 - \omega^2}d\omega'  .
\] 
\\

\noindent
For practical calculations using Eq.~\ref{PAI1} it is convenient to
represent the photoabsorption cross section as a polynomial in $\omega^{-1}$ 
as was proposed in \cite{sandia}:

\[
\sigma_{\gamma}(\omega) = \sum_{k=1}^{4}a_{k}^{(i)}\omega^{-k} ,
\]
where the coefficients, $a_{k}^{(i)}$ result from a separate least-squares 
fit to experimental data in each energy interval $i$.  As a rule the 
interval borders are equal to the corresponding photoabsorption edges.  The 
dielectric constant can now be calculated analytically with elementary 
functions for all $\omega$, except near the photoabsorption edges where 
there are breaks in the photoabsorption cross section and the integral for 
the real part is not defined in the sense of the principal value. \\

\noindent 
The third term in Eq. (\ref{PAI1}), which can only be integrated
numerically, results in a complex calculation of $d\sigma_i/d\omega$. 
However, this term is dominant for energy transfers $\omega > 10\ keV$, 
where the function $\left|\varepsilon(\omega)\right|^2 \sim 1$.  This is 
clear from physical reasons, because the third term represents the 
Rutherford cross section on atomic electrons which can be considered as 
quasifree for a given energy transfer \cite{allis}.  In addition, for high 
energy transfers, 
$\varepsilon(\omega) = 1 - \omega_{p}^{2}/\omega^2 \sim 1$, 
where $\omega_{p}$ is the plasma energy of the material.  Therefore the 
factor $\left|\varepsilon(\omega)\right|^{-2}$ can be removed from under the 
integral and the differential cross section of ionizing collisions can be 
expressed as:

\begin{eqnarray}
\frac{d\sigma_i}{d\omega}
& = &\frac{\alpha}
{\pi\beta^2\left|\varepsilon(\omega)\right|^2}
\left\{
\frac{\tilde{\sigma}_{\gamma}(\omega)}{\omega}
\left[
\ln\frac{2mv^2}{\omega\left|1-\beta^2\varepsilon\right|} - 
\right. \right. \nonumber \\ 
&   & \left. \left.
- \frac{\varepsilon_1-\beta^2\left|\varepsilon\right|^2}{\varepsilon_2}
\arg(1-\beta^2\varepsilon^*)
\right]
 + \frac{1}{\omega^2}\int_{I_1}^{\omega}\tilde{\sigma}_{\gamma}(\omega')d\omega'
\right\} .
\end{eqnarray}
This is especially simple in gases when 
$\left|\varepsilon(\omega)\right|^{-2} \sim 1$  for all $\omega > I_1$
\cite{allis}.

\subsection{Energy Loss Simulation}

For a given track length the number of ionizing collisions is simulated by
a Poisson distribution whose mean is proportional to the total cross
section of ionizing collisions:

\[
\sigma_i = \int_{I_1}^{\omega_{max}}\frac{d\sigma(\omega')}{d\omega'}d\omega' .
\]
The energy transfer in each collision is simulated according to a
distribution proportional to

\[
\sigma_i(>\omega) = \int_{\omega}^{\omega_{max}}
\frac{d\sigma(\omega')}{d\omega'}d\omega' .
\]
The sum of the energy transfers is equal to the energy loss. PAI ionisation is implemented 
according to the model approach (class G4PAIModel) allowing a user to select specific 
models in different regions. Here is an example physics list:
\begin{verbatim}
  const G4RegionStore* theRegionStore = G4RegionStore::GetInstance();
  G4Region* gas = theRegionStore->GetRegion("VertexDetector"); 
  ...
  if (particleName == "e-") 
  { 
    G4eIonisation* eion = new G4eIonisation();
    G4PAIModel*     pai = new G4PAIModel(particle,
                                                 "PAIModel");
    // set energy limits where 'pai' is active
    pai->SetLowEnergyLimit(0.1*keV);
    pai->SetHighEnergyLimit(100.0*TeV);

    // here 0 is the highest priority in region 'gas'
    eion->AddEmModel(0,pai,pai,gas);

    pmanager->AddProcess(eion,-1, 2, 2);
    pmanager->AddProcess(new G4MultipleScattering, -1, 1,1);
    pmanager->AddProcess(new G4eBremsstrahlung,-1,1,3);           
  } 
\end{verbatim}
It shows how to select the G4PAIModel to be the preferred ionisation model for electrons 
in a G4Region named VertexDetector.  The first argument in AddEmModel is 0 which means 
highest priority. 
 
The class G4PAIPhotonModel generates both $\delta$-electrons and photons as secondaries 
and can be used for more detailed descriptions of ionisation space distribution around 
the particle trajectory.

\subsection{Status of  this document}

01.12.05 expanded discussion by V. Grichine \\
08.05.02 re-written by D.H. Wright \\
16.11.98 created by V. Grichine \\

\begin{latexonly}

\begin{thebibliography}{99}
\bibitem{pai.asosk} Asoskov V.S., Chechin V.A., Grichine V.M. at el,
{Lebedev Institute annual report, v. 140, p. 3} (1982)
\bibitem{pai.fano} Fano U., and Cooper J.W.
{Rev.Mod.Phys., v. 40, p. 441} (1968)
\bibitem{sandia} Biggs F., and Lighthill R.,
{Preprint Sandia Laboratory, SAND 87-0070} (1990)
\bibitem{allis} Allison W.W.M., and Cobb J.
{Ann.Rev.Nucl.Part.Sci., v.30,p.253} (1980)
\end{thebibliography}

\end{latexonly}

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\subsection{Bibliography}

\begin{enumerate}
\item Asoskov V.S., Chechin V.A., Grichine V.M. at el,
{Lebedev Institute annual report, v. 140, p. 3} (1982)
\item Fano U., and Cooper J.W.
{Rev.Mod.Phys., v. 40, p. 441} (1968)
\item Biggs F., and Lighthill R.,
{Preprint Sandia Laboratory, SAND 87-0070} (1990)
\item Allison W.W.M., and Cobb J.
{Ann.Rev.Nucl.Part.Sci., v.30,p.253} (1980)
\end{enumerate}

\end{htmlonly}