
\section{Compton Scattering by Linearly Polarized Gamma Rays}


\subsection{The Cross Section}

The quantum mechanical Klein - Nishina differential cross section for 
polarized photons is [Heitler 1954]:

\[\frac{d\sigma}{d\Omega} = \frac{1}{4}r_0^2 \frac{h\nu^2}{h\nu_o^2} \frac{h\nu_o^2}{h\nu^2} \left[\frac{h\nu_o}{h\nu}+\frac{h\nu}{h\nu_o}-2+4 cos^2\Theta \right] \]

\noindent
where $\Theta$ is the angle between the two polarization vectors.  In terms 
of the polar and azimuthal angles $ (\theta, \phi) $ this cross section can 
be written as

\[\frac{d\sigma}{d\Omega} = \frac{1}{2}r_0^2 \frac{h\nu^2}{h\nu_o^2} \frac{h\nu_o^2}{h\nu^2} \left[\frac{h\nu_o}{h\nu}+\frac{h\nu}{h\nu_o}-2 cos^2\phi sin^2\theta \right] \] .


\subsection{Angular Distribution}


The integration of this cross section over the azimuthal angle produces the 
standard cross section.  The angular and energy distribution are then 
obtained in the same way as for the standard process.  Using these values 
for the polar angle and the energy, the azimuthal angle is sampled from the 
following distribution:

\[ P(\phi)= 1 - \frac{a}{b} cos^2\phi \]

\noindent
where $a = sin^2\theta $ and $b = \epsilon + 1/\epsilon$.  $\epsilon$ is
the ratio between the scattered photon energy and the incident photon 
energy.


\subsection{Polarization Vector}

The components of the vector polarization of the scattered photon are
calculated from

\[ \vec{\epsilon'_\bot} = \frac{1}{N} \left( \hat{j} cos\theta - \hat{k} sin\theta sin\phi \right) sin\beta \]

 
\[ \vec{\epsilon'_\|} = \left[ N \hat{i}- \frac{1}{N} \hat{j} sin^2\theta sin\phi cos\phi - \frac{1}{N} \hat{k} sin\theta cos\theta cos\phi \right] cos\beta \]

\noindent
where \[ N = \sqrt{1-sin^2\theta cos^2\phi} . \]

\noindent
$cos\beta$ is calculated from $cos\Theta = N cos\beta $, while $cos\Theta$ 
is sampled from the Klein - Nishina distribution.

The binding effects and the Compton profile are neglected.
The kinetic energy and momentum of the recoil electron are then

\[ T_{el} = E - E' \]
\[ \vec{P_{el}} = \vec{P_\gamma} - \vec{P_\gamma '} . \]

The momentum vector of the scattered photon $\vec{P_\gamma}$ and its 
polarization vector are transformed into the {\tt World} coordinate system. 
The polarization and the direction of the scattered gamma in the final 
state are calculated in the reference frame in which the incoming photon is 
along the $z$-axis and has its polarization vector along the $x$-axis.  The 
transformation to the {\tt World} coordinate system performs a linear 
combination of the initial direction, the initial poalrization and the cross
product between them, using the projections of the calculated quantities 
along these axes. 

\subsection{Unpolarized Photons}

A special treatment is devoted to unpolarized photons.  In this case a 
random polarization in the plane perpendicular to the incident photon is 
selected.

\subsection{Status of this document}

18.06.2001 created by Gerardo Depaola and Francesco Longo \\
10.06.2002 revision by Francesco Longo \\
26.01.2003 minor re-wording and correction of equations by D.H. Wright

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\begin{thebibliography}{99}

\bibitem{Heitler} W. Heitler {\em The Quantum Theory of Radiation, Oxford Clarendom Press } (1954)

\end{thebibliography}

\end{latexonly}

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\subsection{Bibliography}

\begin{enumerate}
\item W. Heitler {\em The Quantum Theory of Radiation, Oxford Clarendom Press } (1954)
\end{enumerate}

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