[1208] | 1 | \section{Rayleigh Scattering} |
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| 2 | |
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| 3 | \subsection{Total Cross Section} |
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| 4 | The total cross section for the Rayleigh scattering process |
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| 5 | %(also called coherent scattering~\footnote{Coherent scattering |
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| 6 | %is usually described as an interaction |
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| 7 | %between a photon and the inner most, most tightly bound electrons of an atom.}) |
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| 8 | is determined from the data as described in section \ref{subsubsigmatot}. |
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| 9 | |
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| 10 | \subsection{Sampling of the Final State} |
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| 11 | |
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| 12 | The coherent scattered photon angle $\theta$ is sampled according to the |
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| 13 | distribution obtained from the product of the Rayleigh formula $(1+\cos^2\theta)\sin\theta $ and the square of Hubbel's form factor |
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| 14 | $FF^2(q)$~\cite{re-hubbel2}~\cite{re-reda} |
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| 15 | \begin{equation} |
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| 16 | \Phi(E, \theta) = [ 1+\cos^2 \theta] \sin \theta \times FF^2(q) , |
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| 17 | \end{equation} |
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| 18 | where $q = 2 E \sin(\theta/2)$ is the momentum transfer. |
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| 19 | |
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| 20 | Form factors introduce a dependency on the initial energy $E$ of the photon |
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| 21 | that is not taken into account in the Rayleigh formula. At low energies, |
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| 22 | form factors are isotropic and do not affect angular distribution, while at |
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| 23 | high energies they are forward peaked. |
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| 24 | |
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| 25 | The sampling procedure is as follows~\cite{re-stepanek}: |
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| 26 | \begin{enumerate} |
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| 27 | \item $cos\theta$ is chosen from a uniform distribution between -1 and 1 |
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| 28 | \item the form factor $FF$ is extracted from the data table for the |
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| 29 | considered element, using logarithmic data interpolation, |
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| 30 | for $q = 2 E \cdot\sin(\theta/2)$ |
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| 31 | \item if the value obtained for $\Phi(E, \theta)$ is larger than a random |
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| 32 | number uniformly distributed between 0 and $Z^2$, the procedure is repeated |
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| 33 | from step 1, otherwise $\theta$ is taken as the photon scattering angle |
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| 34 | with respect to its incident direction. |
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| 35 | \item the azimuthal direction of the scattered photon is chosen at random. |
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| 36 | \end{enumerate} |
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| 37 | |
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| 38 | |
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| 39 | \subsection{Status of this document} |
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| 40 | |
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| 41 | \noindent |
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| 42 | 30.09.1999 created by Alessandra Forti\\ |
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| 43 | 07.02.2000 modified by V\'eronique Lef\'ebure\\ |
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| 44 | 08.03.2000 reviewed by Petteri Nieminen and Maria Grazia Pia\\ |
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| 45 | 10.06.2002 modified by Francesco Longo and Gerardo Depaola\\ |
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| 46 | 26.01.2003 minor re-write and correction of equations by D.H. Wright |
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| 47 | |
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| 48 | \begin{latexonly} |
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| 49 | |
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| 50 | \begin{thebibliography}{99} |
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| 51 | \bibitem{re-hubbel2} |
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| 52 | "Relativistic Atom Form Factors and Photon Coherent Scattering Cross Sections", |
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| 53 | J.H. Hubbell et al., J.Phys.Chem.Ref.Data, 8,69(1979) |
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| 54 | \bibitem{re-reda} |
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| 55 | "A simple model of photon transport", |
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| 56 | D.E. Cullen, Nucl. Instr. Meth. in Phys. Res. B 101(1995)499-510 |
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| 57 | \bibitem{re-stepanek} |
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| 58 | "New Photon, Positron and Electron Interaction Data for Geant in Energy Range |
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| 59 | from 1 eV to 10 TeV", |
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| 60 | J. Stepanek, Draft to be submitted for publication |
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| 61 | \end{thebibliography} |
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| 62 | |
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| 63 | \end{latexonly} |
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| 64 | |
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| 65 | \begin{htmlonly} |
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| 66 | |
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| 67 | \subsection{Bibliography} |
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| 68 | |
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| 69 | \begin{enumerate} |
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| 70 | \item |
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| 71 | "Relativistic Atom Form Factors and Photon Coherent Scattering Cross |
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| 72 | Sections", J.H. Hubbell et al., J.Phys.Chem.Ref.Data, 8,69(1979) |
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| 73 | \item |
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| 74 | "A simple model of photon transport", |
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| 75 | D.E. Cullen, Nucl. Instr. Meth. in Phys. Res. B 101(1995)499-510 |
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| 76 | \item |
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| 77 | "New Photon, Positron and Electron Interaction Data for Geant in Energy |
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| 78 | Range from 1 eV to 10 TeV", |
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| 79 | J. Stepanek, Draft to be submitted for publication |
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| 80 | \end{enumerate} |
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| 81 | |
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| 82 | \end{htmlonly} |
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