\section{Gamma Conversion}

\subsection{Total cross-section}

The total cross-section of the Gamma Conversion process 
is determined from the data as described in section \ref{subsubsigmatot}. 

\subsection{Sampling of the final state}

For low energy incident photons, the simulation of the Gamma Conversion
final state is performed according to ~\cite{gc-geant3}.

The secondary $e^\pm$ energies are sampled using the Bethe-Heitler cross-sections with Coulomb correction.

The Bethe-Heitler differential cross-section with the Coulomb correction for a photon of energy $E$ to produce a pair with one of the particles having energy $\epsilon E$ ($\epsilon$ is the fraction of the photon energy carried by one particle of the pair) is given by~\cite{gc-slac}:

\begin{eqnarray*}
  \frac{d \sigma(Z,E,\epsilon)}{d \epsilon} & = & \frac{r_0^2\alpha Z (Z + \xi(Z))}{E^2}\left[ (\epsilon^2 + ( 1 -\epsilon)^2) \left( \Phi_1(\delta) - \frac{F(Z)}{2}\right) \right. +\\ 
& & + \left. \frac{2}{3}\epsilon (1-\epsilon) \left( \Phi_2(\delta) - \frac{F(Z)}{2} \right) \right] 
\end{eqnarray*}

where $\Phi_i(\delta)$ are the screening functions depending on the screening variable $\delta$~\cite{gc-geant3}. 

The value of $\epsilon$ is sampled using composition and rejection Monte Carlo methods \cite{gc-geant3,gc-butch,gc-messel}. 

After the successful sampling of $\epsilon$, the process generates the polar angles of the electron with respect to an axis defined along the
direction of the parent photon. The electron and the positron are assumed to
have a symmetric angular distribution. The energy-angle distribution is given 
by\cite{gc-tsai}:

\begin{eqnarray*}
\frac{d \sigma}{dp d\Omega} & = & \frac{2 \alpha^2 e^2}{\pi k m^4}\left[ \left(\frac{2x(1-x)}{(1+l)}^2 - \frac{12 lx(1-x)}{(1+l)^4} \right)(Z^2+Z) + \right. \\ 
& & +  \left. \left( \frac{ 2x^2 - 2x+1}{(1+l)^2} + \frac{4lx(1-x)}{(1+l)^4} \right) (X-2Z^2 f((\alpha Z)^2)) \right] 
\end{eqnarray*}

where $k$ is the photon energy, $p$ the momentum and $E$ the energy of the electron of the $e^\pm$ pair $x=E/k$ and $l = E^2\theta^2/m^2$. The sampling of this cross-section is obtained according to \cite{gc-geant3}.

The azimuthal angle $\phi$  is generated isotropically. 

This information together with the momentum conservation is used to calculate the momentum vectors of both decay products and to transform them to the GEANT coordinate system. The choice of which particle in the pair is the electron/positron is made randomly. 

\subsection{Status of the document}

\noindent
18.06.2001 created by Francesco Longo\\

\begin{latexonly}

\begin{thebibliography}{99}
\bibitem{gc-geant3}Urban L., in Brun R. et al. (1993), {\it Geant. Detector Description and Simulation Tool}, CERN Program Library, section {\tt Phys/211} 
\bibitem{gc-slac} R. Ford and W. Nelson., {\em SLAC-210, UC-32} (1978)    
\bibitem{gc-butch} J.C. Butcher and H. Messel.
   {\em Nucl. Phys. 20} 15 (1960)   
\bibitem{gc-messel} H. Messel and D. Crawford.
   {\em Electron-Photon shower distribution, Pergamon Press} (1970)
\bibitem{gc-tsai}Y. S. Tsai, {\em Rev. Mod. Phys. 46} 815 (1974), Y. S. Tsai, {\em Rev. Mod. Phys. 49} 421 (1977)

\end{thebibliography}

\end{latexonly}

\begin{htmlonly}

\subsection{Bibliography}

\begin{enumerate}
\item Urban L., in Brun R. et al. (1993), {\it Geant. Detector Description and Simulation Tool}, CERN Program Library, section {\tt Phys/211} 
\item R. Ford and W. Nelson., {\em SLAC-210, UC-32} (1978)    
\item J.C. Butcher and H. Messel.
   {\em Nucl. Phys. 20} 15 (1960)   
\item H. Messel and D. Crawford.
   {\em Electron-Photon shower distribution, Pergamon Press} (1970)
\item Y. S. Tsai, {\em Rev. Mod. Phys. 46} 815 (1974), Y. S. Tsai, {\em Rev. Mod. Phys. 49} 421 (1977)
\end{enumerate}

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