source: trunk/documents/UserDoc/UsersGuides/PhysicsReferenceManual/latex/electromagnetic/lowenergy/rayleigh.tex@ 1222

\section{Rayleigh Scattering}

\subsection{Total Cross Section}
The total cross section for the Rayleigh scattering process 
%(also called coherent scattering~\footnote{Coherent scattering 
%is usually described as an interaction
%between a photon and the inner most, most tightly bound electrons of an atom.}) 
is determined from the data as described in section \ref{subsubsigmatot}. 

\subsection{Sampling of the Final State}

The coherent scattered photon angle $\theta$ is sampled according to the 
distribution obtained from the product of the Rayleigh formula $(1+\cos^2\theta)\sin\theta $ and the square of Hubbel's form factor
$FF^2(q)$~\cite{re-hubbel2}~\cite{re-reda}
\begin{equation}
\Phi(E, \theta) = [ 1+\cos^2 \theta] \sin \theta  \times FF^2(q) ,
\end{equation}  
where $q = 2 E \sin(\theta/2)$ is the momentum transfer.

Form factors introduce a dependency on the initial energy $E$ of the photon 
that is not taken into account in the Rayleigh formula.  At low energies, 
form factors are isotropic and do not affect angular distribution, while at 
high energies they are forward peaked.

The sampling procedure is as follows~\cite{re-stepanek}:
\begin{enumerate}
\item $cos\theta$ is chosen from a uniform distribution between -1 and 1
\item the form factor $FF$ is extracted from the data table for the 
considered element, using logarithmic data interpolation, 
for $q = 2 E \cdot\sin(\theta/2)$
\item if the value obtained for $\Phi(E, \theta)$ is larger than a random
number uniformly distributed between 0 and $Z^2$, the procedure is repeated
from step 1, otherwise $\theta$ is taken as the photon scattering angle 
with respect to its incident direction.
\item the azimuthal direction of the scattered photon is chosen at random. 
\end{enumerate}


\subsection{Status of this 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\\
10.06.2002 modified by Francesco Longo and Gerardo Depaola\\
26.01.2003 minor re-write and correction of equations by D.H. Wright

\begin{latexonly}

\begin{thebibliography}{99}
\bibitem{re-hubbel2}
  "Relativistic Atom Form Factors and Photon Coherent Scattering Cross Sections",
  J.H. Hubbell et al., J.Phys.Chem.Ref.Data, 8,69(1979)
\bibitem{re-reda}
  "A simple model of photon transport", 
  D.E. Cullen, Nucl. Instr. Meth. in Phys. Res. B 101(1995)499-510 
\bibitem{re-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
  "Relativistic Atom Form Factors and Photon Coherent Scattering Cross 
   Sections", J.H. Hubbell et al., J.Phys.Chem.Ref.Data, 8,69(1979)
\item
  "A simple model of photon transport", 
  D.E. Cullen, Nucl. Instr. Meth. in Phys. Res. B 101(1995)499-510 
\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}