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