
\section[Compton scattering]{Compton scattering}\label{sec:em.compton}
\subsection{Cross Section and Mean Free Path}
\subsubsection{Cross Section per Atom}
When simulating the Compton scattering of a photon from an atomic electron, an
empirical cross section formula is used, which reproduces the cross section 
data down to 10 keV:
\begin{equation}
\sigma(Z,E_{\gamma}) = \left [ P_{1}(Z) \ \frac{\log(1+2X)}{X} +
\frac{P_{2}(Z)+P_{3}(Z) X + P_{4}(Z) X^{2}}{1+aX+bX^{2}+cX^{3}} \right ] . 
\end{equation}
Here,
\begin{eqnarray*}
 Z          & = & \mbox{atomic number of the medium} \, \\
 E_{\gamma} & = & \mbox{energy of the photon}  \\
 X          & = & E_{\gamma}/mc^2  \\
 m          & = & \mbox{electron mass}  \\
 P_{i}(Z)   & = & Z (d_{i} + e_{i}Z + f_{i}Z^{2}) . \,
\end{eqnarray*}
\\ 
The values of the parameters can be found within the method which computes
the cross section per atom.  A fit of the parameters was made to over 511 data
points \cite{hubbell.comp,storm.comp} chosen from the intervals
\begin{eqnarray*}
1 \leq Z \leq 100
\end{eqnarray*}
and
\begin{eqnarray*}
 E_{\gamma} \in [10 \mbox{ keV} , 100 \mbox{ GeV}] .
\end{eqnarray*}
The accuracy of the fit was estimated to be
 
\vspace{.3cm}
$$
\frac{\Delta\sigma}{\sigma} = \left\{ 
\begin{array}{lcl}
\approx 10\% & \makebox[2cm][r]{\rm for } E_{\gamma} & \simeq 10 \mbox{ keV} 
-20 \mbox{ keV} \\ 
\leq 5-6\% & \makebox[2cm][r]{\rm for } E_{\gamma} & > 20 \mbox{ keV}
\end{array} \right.
$$
\subsubsection{Mean Free Path}
In a given material the mean free path, $\lambda$, for a photon to interact 
via Compton scattering is given by
\begin{equation}
\lambda(E_{\gamma}) =
 \left( \sum_i n_{ati} \cdot \sigma_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.
 
\subsection {Sampling the Final State} 
The quantum mechanical Klein-Nishina differential cross section per atom
is \cite{klein.comp} :
\begin{equation}
\frac{d\sigma}{d\epsilon} =\pi r_e^2 \ 
\frac{m_e c^2}{E_0} \ Z
     \left[\frac{1}{\epsilon}+\epsilon\right]
     \left[1 - \frac{\epsilon \sin^2 \theta}{1+\epsilon^2}\right]
\end{equation}
where \quad
\begin{tabular}[t]{l@{\ = \ }l}
$r_e$       & classical electron radius       \\
$m_e c^2$   & electron mass                   \\
$E_0$       & energy of the incident photon   \\
$E_1$       & energy of the scattered photon  \\
$\epsilon$  & $E_1/E_0$  .                      
\end{tabular}

\noindent 
Assuming an elastic collision, the scattering angle $\theta$ is defined by the
Compton formula: 
\begin{equation}
   E_1   = E_0 \ \frac{m_{\rm e}c^2}{ m_{\rm e}c^2 + E_0(1-\cos\theta )} .
\end{equation}

\subsubsection{Sampling the Photon Energy}
The value of $\epsilon$ corresponding to the minimum photon energy (backward 
scattering) is given by
\begin{equation}
\epsilon_0 = \frac{m_{\rm e}c^2}{m_{\rm e}c^2+2E_0} ,
\end{equation}
hence $\epsilon \in [\epsilon_0, 1]$.  Using the combined composition and 
rejection Monte Carlo methods described in
\cite{butch.comp,messel.comp,ford.comp} one may set
\begin{equation}
\Phi(\epsilon)
     \simeq  \left[ \frac{1}{\epsilon}+\epsilon \right]
                \left[ 1 - \frac{\epsilon \sin^2 \theta}{1+\epsilon^2} \right]
     = f(\epsilon) \cdot g(\epsilon)
     = \left[ \alpha_1 f_1(\epsilon) + \alpha_2 f_2(\epsilon) \right] 
              \cdot g(\epsilon) ,
\end{equation}
where
$$
\begin{array}{lcl}
\alpha_1      = \ln (1/\epsilon_0) & ; & 
f_1(\epsilon) = 1/(\alpha_1\epsilon)       \\
\alpha_2      = (1-\epsilon_0^2)/2 & ; &
f_2(\epsilon) = \epsilon/\alpha_2 .          
\end{array}
$$ 
$f_1$ and $f_2$ are probability density functions defined on the interval
$\lbrack\epsilon_0, 1\rbrack$, and 
$$
g(\epsilon) = \left[ 1 - \frac{\epsilon}{1+\epsilon^2} \sin^2\theta \right]
$$  
is the rejection function
$\forall \epsilon \in [\epsilon_0, 1] \Longrightarrow 0 < g(\epsilon) \leq 1$.\\
  
\noindent 
Given a set of 3 random numbers $r, r', r''$ uniformly distributed on the 
interval [0,1], the sampling procedure for $\epsilon$ is the following:
\begin{enumerate}
\item
decide whether to sample from $f_1(\epsilon)$ or $f_2(\epsilon)$: \\
if $ r < \alpha_1/(\alpha_1+\alpha_2)$ select $f_1(\epsilon)$, otherwise
select $f_2(\epsilon)$
\item 
sample $\epsilon$ from the distributions corresponding to $f_1$ or $f_2$: \\ 
for $f_1 : \epsilon   = \epsilon_0^{r'} \qquad (\equiv \exp(-r' \alpha_1))$ \\ 
for $f_2 : \epsilon^2 = \epsilon_0^2 + (1-\epsilon_0^2)r'$
 
\item 
calculate $\sin^2\theta = t(2-t)$ where 
$t \equiv (1-\cos\theta) = m_e c^2 (1-\epsilon)/(E_0 \epsilon)$

\item 
test the rejection function: \\
if $g(\epsilon) \geq r''$ accept $\epsilon$, otherwise go to step 1.
\end{enumerate}

\subsubsection{Compute the Final State Kinematics}
After the successful sampling of $\epsilon$, the polar angles of the 
scattered photon with respect to the direction of the parent photon
are generated.  The azimuthal angle, $\phi$, is generated isotropically and
$\theta$ is as defined in the previous section.  The momentum vector of the 
scattered photon, $\overrightarrow{P_{\gamma1}}$, is then transformed into 
the {\tt World} coordinate system.  The kinetic energy and momentum of the 
recoil electron are then
\begin{eqnarray*}
T_{el} & = & E_0 - E_1 \\
\overrightarrow{P_{el}} & = & 
              \overrightarrow{P_{\gamma0}} - \overrightarrow{P_{\gamma1}} .
\end{eqnarray*}

\subsection{Validity}
 
The differential cross-section is valid only for those collisions in which the
energy of the recoil electron is large compared to its binding energy (which 
is ignored).  However, as pointed out by Rossi \cite{rossi.comp}, this has a 
negligible effect because of the small number of recoil electrons produced at 
very low energies.

\subsection{Status of this document}
09.10.98 created by M.Maire. \\
14.01.02 minor revision (mma) \\
22.04.02 reworded by D.H. Wright \\
18.03.04 include references for total cross section (mma) \\ 
 
\begin{latexonly}

\begin{thebibliography}{99}
\bibitem{hubbell.comp} Hubbell, Gimm and Overbo.
   {\em J. Phys. Chem. Ref. Data 9} 1023 (1980)
\bibitem{storm.comp} H. Storm and H.I. Israel
   {\em Nucl. Data Tables A7} 565 (1970)   
\bibitem{klein.comp} O. Klein and Y. Nishina.
   {\em Z. Physik 52} 853 (1929)
\bibitem{butch.comp} J.C. Butcher and H. Messel.
   {\em Nucl. Phys. 20} 15 (1960)   
\bibitem{messel.comp} H. Messel and D. Crawford.
   {\em Electron-Photon shower distribution, Pergamon Press} (1970)
\bibitem{ford.comp} R. Ford and W. Nelson.
   {\em SLAC-265, UC-32} (1985)    
\bibitem{rossi.comp} B. Rossi.
   {\em High energy particles, Prentice-Hall} 77-79 (1952)       
\end{thebibliography}

\end{latexonly}

\begin{htmlonly}

\subsection{Bibliography}

\begin{enumerate}
\item Hubbell, Gimm and Overbo.
   {\em J. Phys. Chem. Ref. Data 9} 1023 (1980)
\item H. Storm and H.I. Israel
   {\em Nucl. Data Tables A7} 565 (1970)   
\item O. Klein and Y. Nishina.
   {\em Z. Physik 52} 853 (1929)
\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 B. Rossi.
   {\em High energy particles, Prentice-Hall} 77-79 (1952)       
\end{enumerate}

\end{htmlonly}
