\section[Photoelectric Effect]{PhotoElectric effect}

The photoelectric effect is the ejection of an electron from a material after
a photon has been absorbed by that material.  It is simulated by using a
parameterized photon absorption cross section to determine the mean free path,
atomic shell data to determine the energy of the ejected electron, and the 
K-shell angular distribution to sample the direction of the electron.
 
\subsection{Cross Section and Mean Free Path}

The parameterization of the photoabsorption cross section proposed by 
Biggs et al. \cite{ph.sandia} was used :
\begin{equation}  \label{eqsandia}
\sigma(Z,E_{\gamma}) = \frac{a(Z,E_{\gamma})}{E_{\gamma}} +
                       \frac{b(Z,E_{\gamma})}{E_{\gamma}^2} +
                       \frac{c(Z,E_{\gamma})}{E_{\gamma}^3} +
                       \frac{d(Z,E_{\gamma})}{E_{\gamma}^4}
\end{equation}

\noindent 
Using the least-squares method, a separate fit of each of the coefficients 
$a,b,c,d$ to the experimental data was performed in several energy intervals 
\cite{ph.sandia.grich}.  As a rule, the boundaries of these intervals were 
equal to the corresponding photoabsorption edges.
 
\noindent
In a given material the mean free path, $\lambda$, for a photon to interact 
via the photoelectric effect is given by :
\begin{equation}  \label{lambda}
\lambda(E_{\gamma}) =
 \left( \sum_i n_{ati} \cdot \sigma (Z_i,E_{\gamma}) \right)^{-1}
\end{equation}
where $n_{ati}$ is the number of atoms per volume of the $i^{th}$ element
of the material.  The cross section and mean free path are 
discontinuous and must be computed 'on the fly' from the formulas 
 \ref{eqsandia} and \ref{lambda}. 

\subsection{Final State}
\subsubsection{Choosing an Element}
The binding energies of the shells depend on the atomic number $Z$ of the
material.  In compound materials the $i^{th}$ element is chosen randomly 
according to the probability:
\[
  Prob(Z_i,E_{\gamma}) = 
                      \frac{n_{ati} \sigma(Z_i,E_{\gamma})}
                      {\sum_i [ n_{ati} \cdot \sigma_i (E_{\gamma})]} .
\]
\subsubsection{Shell}
A quantum can be absorbed if $E_{\gamma} > B_{shell}$ where the shell 
energies are taken from {\tt G4AtomicShells} data: the closest available 
atomic shell is chosen.  The photoelectron is emitted with kinetic energy :
\begin{equation}
T_{photoelectron} = E_{\gamma}-B_{shell}(Z_i)
\end{equation}

\subsubsection{Theta Distribution of the Photoelectron}
The polar angle of the photoelectron is sampled from the Sauter-Gavrila
distribution (for K-shell) \cite{ph.cost}, which is correct only to zero order 
in $\alpha Z$ :
\begin{equation}
\frac{d\sigma}{d(\cos\theta)} \sim \frac{\sin^2\theta}{(1-\beta\cos\theta)^4}
\left\lbrace 1 + \frac{1}{2} \gamma (\gamma-1)(\gamma-2)(1-\beta\cos\theta)
\right\rbrace
\end{equation}
where $\beta$ and $\gamma$ are the Lorentz factors of the photoelectron.

\noindent
$\cos\theta$ is sampled from the probability density function :
\begin{equation}
f(\cos\theta) = \frac{1-\beta^2}{2\beta} \frac{1}{(1-\beta\cos\theta)^2}
\hspace{5mm} \Longrightarrow \hspace{5mm}
\cos\theta = \frac{(1-2r)+\beta}{(1-2r)\beta+1}
\end{equation}
The rejection function is :
\begin{equation}
g(\cos\theta) = \frac{1-\cos^2\theta}{(1-\beta\cos\theta)^2}
\left\lbrack 1+b(1-\beta\cos\theta) \right\rbrack
\end{equation}
with $b=\gamma(\gamma-1)(\gamma-2)/2$  \\
It can be shown that $g(\cos\theta)$ is positive $\forall \cos\theta \in
[-1,\ +1]$, and can be majored by : 
\begin{eqnarray}
gsup&=&\gamma^2 \ \lbrack 1+b(1-\beta) \rbrack \mbox{ if } \gamma \in \ ]1,2] \\
    &=&\gamma^2 \ \lbrack 1+b(1+\beta) \rbrack \mbox{ if } \gamma > 2 \nonumber
\end{eqnarray}
The efficiency of this method is $\sim 50\%$ if $\gamma < 2$, $\sim 25\%$ if
$\gamma \in [2,\ 3]$. 

\subsubsection{Relaxation} 
In the current implementation the relaxation of the atom is not simulated,
but instead is counted as a local energy deposit.

\subsection{Status of this document}
09.10.98 created by M.Maire. \\
08.01.02 updated by mma \\
22.04.02 re-worded by D.H. Wright \\
02.05.02 modifs in total cross section and final state (mma) \\
15.11.02 introduction added by D.H. Wright \\

\begin{latexonly}

\begin{thebibliography}{99}
\bibitem{ph.sandia} Biggs F., and Lighthill R.,
{Preprint Sandia Laboratory, SAND 87-0070} (1990)
\bibitem{ph.sandia.grich} Grichine V.M., Kostin A.P., Kotelnikov S.K. et al.,
{Bulletin of the Lebedev Institute no. 2-3, 34} (1994).
\bibitem{ph.cost} Gavrila M.
{Phys.Rev. 113, 514} (1959).
\end{thebibliography}

\end{latexonly}

\begin{htmlonly}

\subsection{Bibliography}

\begin{enumerate}
\item Biggs F., and Lighthill R.,
{Preprint Sandia Laboratory, SAND 87-0070} (1990)
\item Grichine V.M., Kostin A.P., Kotelnikov S.K. et al.,
{Bulletin of the Lebedev Institute no. 2-3, 34} (1994).
\item Gavrila M.
{Phys.Rev. 113, 514} (1959).
\end{enumerate}

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