\section{Photoelectric effect}\label{secphoto}

\subsection{Total cross-section}

The total photoelectric cross-section at a given energy, E, is calculated as 
described in section~\ref{subsubsigmatot}. Note that for this process the 
{\it MeanFreePathTable} is not built, since the cross-section is not a  
smooth function of the energy, therefore in all calculations the 
cross-section is used directly.

\subsection{Sampling of the final state}

The incident photon is absorbed and an electron is emitted.

The electron kinetic energy is the difference between the incident photon 
energy and the binding energy of the electron before the interaction. 
The sub-shell, from which the electron is emitted, is randomly selected 
according to the relative cross-sections of all subshells, 
determined at the given energy, $T$, by interpolating the evaluated 
cross-section data from the EPDL97 data bank~\cite{pe-EPDL97}.

The interaction leaves the atom in an excited state. 
The deexcitation of the atom is simulated as described in section~\ref{relax}. 

\subsection{Angular distribution of the emmited photoelectron}

Three models are available to describe the direction of the emmited photoelectron:
G4\-Photo\-Electric\-Angular\-Generator\-Simple, G4\-Photo\-Electric\-Angular\-Generator\-Sauter\-Gavrila and
G4PhotoElectricAngularGeneratorPolarized.

\subsubsection{G4PhotoElectricAngularGeneratorSimple}

The default model assumes that the photoelectron direction is emmited in the same 
direction as the incident photon. 

\subsubsection{G4PhotoElectricAngularGeneratorSauterGavrila}

This model implements the Sauter--Gavrilla distribution has 
presented in the Standard Photoelectric effect.

\subsubsection{G4PhotoElectricAngularGeneratorPolarized}

This model models the double differential cross section (for angles $\theta$ and $\phi$) and
thus it is capable of account for polarization of the incident photon. The developed
generator was based in the research of Sauter in 1931\cite{Sauter:1931}. The Sauter's formula was recalculated by Gavrila in 
1959 for the K-shell~\cite{Gavrila:1959} and in 1961 for the L-shells~\cite{Gavrila:1961}. These new double differential formulas have some limitations, $\alpha$Z$<<$1 and have a range between 0.1$<\beta<$0.99 c.  

\subsubsection*{K--shell}

The double differential photoeffect for K--shell can be written as~\cite{Gavrila:1959}:
\begin{equation}
\frac{d\sigma}{d \omega}(\theta,\phi) = \frac{4}{m^2}{\alpha^6}{Z^5}\frac{\beta^3(1-\beta^2)^3}{\left[1-(1-\beta^2)^{1/2}\right]}
\left(F\left(1-\frac{\pi\alpha Z}{\beta}\right)+ \pi\alpha Z G\right)
\end{equation}
where
\begin{eqnarray*}
F &=& \frac{\sin^2 \theta \cos^2 \phi}{(1-\beta \cos \theta)^4} - \frac{1-(1-\beta^2)^{1/2}}{2(1-\beta^2)}\frac{\sin^2\theta\cos^2\phi}{(1-\beta\cos\theta)^3} \nonumber \\ &+&\frac{\left[1-(1-\beta^2)^{1/2}\right]^2}{4(1-\beta^2)^{3/2}}\frac{\sin^2\theta}{(1-\beta\cos\theta)^3}
\end{eqnarray*}
\begin{eqnarray*}
G &=& \frac{[1-(1-\beta^2)^{1/2}]^{1/2}}{2^{7/2} \beta^2 (1-\beta \cos \theta)^{5/2}}\left[\frac{4\beta^2}{(1-\beta^2)^{1/2}} \frac{\sin^2 \theta \cos^2 \phi}{1-\beta\cos\theta} + \frac{4\beta}{1-\beta^2}\cos \theta \cos^2 \phi - \right.{} \nonumber \\
&-&4  \left.\frac{1-(1-\beta^2)^{1/2}}{1-\beta^2}(1-\cos^2\phi)-\beta^2\ \frac{1-(1-\beta^2)^{1/2}}{1-\beta^2} \frac{\sin^2 \theta}{1-\beta \cos \theta} - \right.{} \nonumber \\
&+& \left.4\beta^2\frac{1-(1-\beta^2)^{1/2}}{(1-\beta^2)^{3/2}} - 4\beta \frac{\left[ 1-(1-\beta^2)^{1/2}\right]^2}{(1-\beta^2)^{3/2}}\right] \nonumber \\
&+&\frac{1-(1-\beta^2)^{1/2}}{4\beta^2(1-\beta\cos\theta)^2}\left[\frac{\beta}{1-\beta^2}-\frac{2}{1-\beta^2}\cos\theta\cos^2\phi + \frac{1-(1-\beta^2)^{1/2}}{(1-\beta^2)^{3/2}}\cos\theta \right.{} \nonumber \\
&-& \left.\beta \frac{1-(1-\beta^2)^{1/2}}{(1-\beta^2)^{3/2}}\right]
\end{eqnarray*}
where $\beta$ is the electron velocity, $\alpha$ is the fine--structure constant, $Z$ is the atomic number of the material and $\theta$, $\phi$ are the emission angles with respect to the electron initial direction.

\subsubsection*{L1--shell}

The double differential photoeffect distribution for L1--shell is the same as 
for K--shell despising a constant~\cite{Gavrila:1961}:
\begin{equation}
B = \xi \frac{1}{8}
\end{equation}
where $\xi$ is equal to 1 when working with unscreened Coulomb wave functions as it is done in this development. 

\subsubsection*{The generation of the photoelectron distribution}

Since the polarized Gavrila cross--section is a 2--dimensional non--factorized distribution an acceptance--rejection technique was the adopted~\cite{Peralta:2003}. For the Gravrila distribution, two functions 
were defined $g_1(\phi)$ and $g_2(\theta)$:
\begin{eqnarray}
g_1(\phi) &=& a \\
g_2(\theta) &=& \frac{\theta}{1+c\theta^2}
\end{eqnarray}
such that:
\begin{equation}
A g_1(\phi)g_2(\theta) \ge \frac{d^2 \sigma}{d\phi d\theta}
\end{equation}
where A is a global constant. The method used to calculate the distribution is
the same as the one used in Low Energy 2BN Bremsstrahlung Generator, being the
difference $g_1(\phi) = a$.

\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\\
13.05.2002 modified by Vladimir Ivanchenko
01.05.2006 modified by Ana Farinha, Andreia Trindade, Lu\'{\i}s Peralta and Pedro Rodrigues\\

\begin{latexonly}

\begin{thebibliography}{99}
\bibitem{pe-EPDL97} 
  %http://reddog1.llnl.gov/homepage.red/photon.htm
  ``EPDL97: the Evaluated Photon Data Library, '97 version",
  D.Cullen, J.H.Hubbell, L.Kissel,
  UCRL--50400, Vol.6, Rev.5
\bibitem{Sauter:1931}
``K--Shell Photoelectric Cross Sections from 200 keV to 2 MeV", R H Pratt, R D Levee, 
 R L Pexton and W Aron, Phys. Rev. 134 (1964) 4A 
\bibitem{Gavrila:1959}
``Relativistic K--Shell Photoeffect", M. Gavrila, Phys. Rev. 113 (1959) 2
\bibitem{Gavrila:1961}
``Relativistic L--Shell Photoeffect", M. Gavrila, Phys. Rev. 124 (1961) 4
\bibitem{Peralta:2003}
``Monte Carlo Generation of 2BNBremsstrahlung Distribution",  
  L. Peralta, P. Rodrigues, A. Trindade
  CERN EXT--2004--039 (July, 2003)
\end{thebibliography}

\end{latexonly}

\begin{htmlonly}

\subsection{Bibliography}

\begin{enumerate}
\item 
  %http://reddog1.llnl.gov/homepage.red/photon.htm
  ``EPDL97: the Evaluated Photon Data Library, '97 version",
  D.Cullen, J.H.Hubbell, L.Kissel,
  UCRL--50400, Vol.6, Rev.5
\item
``K--Shell Photoelectric Cross Sections from 200 keV to 2 MeV", R H Pratt, R D Levee, 
 R L Pexton and W Aron, Phys. Rev. 134 (1964) 4A 
\item
``Relativistic K--Shell Photoeffect", M. Gavrila, Phys. Rev. 113 (1959) 2
\item
``Relativistic L--Shell Photoeffect", M. Gavrila, Phys. Rev. 124 (1961) 4
\item
``Monte Carlo Generation of 2BNBremsstrahlung Distribution",  
  L. Peralta, P. Rodrigues, A. Trindade
  CERN EXT--2004--039 (July, 2003)
\end{enumerate}

\end{htmlonly}