\section{Compton Scattering}

\subsection{Total Cross Section}

The total cross section for the Compton scattering process 
%(also called incoherent
%scattering~\footnote{Incoherent scattering is usually described as an interaction
%between a photon and the outer most, most loosely bound electrons.}) 
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 Compton scattering
process is performed according to the same procedure used for the
``standard" Compton scattering simulation, with the addition that
Hubbel's atomic form factor~\cite{ce-hubbel} or scattering function, $SF$,
is taken into account.  The angular and energy distribution of the 
incoherently scattered photon is then given by the product of the 
Klein-Nishina formula $\Phi(\epsilon)$ and the scattering function, 
$SF(q)$~\cite{ce-reda}
\begin{equation}
P(\epsilon, q ) = \Phi( \epsilon ) \times SF(q) .
\end{equation}
$\epsilon$ is the ratio of the scattered photon energy $E'$, and the 
incident photon energy $E$.  The momentum transfer is given by 
$q = E \times \sin^2(\theta/2)$, where $\theta$ is the polar angle of the 
scattered photon with respect to the direction of the parent photon.
$\Phi(\epsilon)$ is given by
\begin{equation}
  \Phi(\epsilon) \cong {[{1\over\epsilon} + \epsilon] [1-{\epsilon \over{1+\epsilon^2}} sin^2\theta]} .
\end{equation}

The effect of the scattering function becomes significant at low energies,
especially in suppressing forward scattering~\cite{ce-reda}.

The sampling method of the final state is  
based on composition and rejection Monte Carlo methods \cite{ce-butch,ce-messel,ce-egs4}, 
with the $SF$ function included in the rejection function
\begin{equation}\label{en-samp-comp}
g(\epsilon) = \left[1-\frac{\epsilon}{1+\epsilon^2} \sin^2\theta \right] \times SF(q) ,
\end{equation}
with $0<g(\epsilon)<Z$.
Values of the scattering functions at each momentum transfer, $q$, are 
obtained by interpolating the evaluated data for the corresponding atomic 
number, $Z$. 

The polar angle $\theta$ is deduced from the sampled $\epsilon$ value.
In the azimuthal direction, the angular distributions of both the scattered 
photon and the recoil electron are considered to be 
isotropic~\cite{ce-stepanek}.

Since the incoherent scattering occurs mainly on the outermost electronic 
subshells, the binding energies can be neglected, as stated in
reference~\cite{ce-stepanek}.  The momentum vector of the scattered photon, 
$\overrightarrow{P'_{\gamma}}$,
is transformed into the {\tt World} coordinate system.
The kinetic energy and momentum of the recoil electron are then
\begin{eqnarray*}
T_{el} & = & E - E' \\
\overrightarrow{P_{el}} & = & 
              \overrightarrow{P_{\gamma}} - \overrightarrow{P'_{\gamma}} .
\end{eqnarray*}


\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\\
26.01.2003 minor re-write by D.H. Wright

\begin{latexonly}

\begin{thebibliography}{99}
\bibitem{ce-hubbel}
  ``Summary of Existing Information on the Incoherent Scattering of Photons
  particularly on the Validity of the Use of the Incoherent Scattering
  Function",
  Radiat. Phys. Chem. Vol. 50, No 1, pp 113-124 (1997)
\bibitem{ce-reda}
  ``A simple model of photon transport", 
  D.E. Cullen, Nucl. Instr. Meth. in Phys. Res. B 101(1995)499-510 
\bibitem{ce-butch} J.C. Butcher and H. Messel.
   {\em Nucl. Phys. 20} 15 (1960)   
\bibitem{ce-messel} H. Messel and D. Crawford.
   {\em Electron-Photon shower distribution, Pergamon Press} (1970)
\bibitem{ce-egs4} R. Ford and W. Nelson.
   {\em SLAC-265, UC-32} (1985)    
\bibitem{ce-stepanek}
   ``New Photon, Positron and Electron Interaction Data for Geant in Energy 
Range from 1 eV to 10 TeV",
   J. Stepanek, Draft to be submitted for publication

\end{thebibliography}

\end{latexonly}

\begin{htmlonly}

\subsection{Bibliography}

\begin{enumerate}
\item
  ``Summary of Existing Information on the Incoherent Scattering of Photons
  particularly on the Validity of the Use of the Incoherent Scattering
  Function",
  Radiat. Phys. Chem. Vol. 50, No 1, pp 113-124 (1997)
\item
  ``A simple model of photon transport", 
  D.E. Cullen, Nucl. Instr. Meth. in Phys. Res. B 101(1995)499-510 
\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 R. Ford and W. Nelson.
   {\em SLAC-265, UC-32} (1985)    
\item
   ``New Photon, Positron and Electron Interaction Data for Geant in Energy 
Range from 1 eV to 10 TeV",
   J. Stepanek, Draft to be submitted for publication

\end{enumerate}

\end{htmlonly}
