1 | \section{Compton Scattering} |
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2 | |
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3 | \subsection{Total Cross Section} |
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4 | |
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5 | The total cross section for the Compton scattering process |
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6 | %(also called incoherent |
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7 | %scattering~\footnote{Incoherent scattering is usually described as an interaction |
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8 | %between a photon and the outer most, most loosely bound electrons.}) |
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9 | is determined from the data as described in section \ref{subsubsigmatot}. |
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10 | |
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11 | \subsection{Sampling of the Final State} |
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12 | |
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13 | For low energy incident photons, the simulation of the Compton scattering |
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14 | process is performed according to the same procedure used for the |
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15 | ``standard" Compton scattering simulation, with the addition that |
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16 | Hubbel's atomic form factor~\cite{ce-hubbel} or scattering function, $SF$, |
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17 | is taken into account. The angular and energy distribution of the |
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18 | incoherently scattered photon is then given by the product of the |
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19 | Klein-Nishina formula $\Phi(\epsilon)$ and the scattering function, |
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20 | $SF(q)$~\cite{ce-reda} |
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21 | \begin{equation} |
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22 | P(\epsilon, q ) = \Phi( \epsilon ) \times SF(q) . |
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23 | \end{equation} |
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24 | $\epsilon$ is the ratio of the scattered photon energy $E'$, and the |
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25 | incident photon energy $E$. The momentum transfer is given by |
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26 | $q = E \times \sin^2(\theta/2)$, where $\theta$ is the polar angle of the |
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27 | scattered photon with respect to the direction of the parent photon. |
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28 | $\Phi(\epsilon)$ is given by |
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29 | \begin{equation} |
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30 | \Phi(\epsilon) \cong {[{1\over\epsilon} + \epsilon] [1-{\epsilon \over{1+\epsilon^2}} sin^2\theta]} . |
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31 | \end{equation} |
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32 | |
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33 | The effect of the scattering function becomes significant at low energies, |
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34 | especially in suppressing forward scattering~\cite{ce-reda}. |
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35 | |
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36 | The sampling method of the final state is |
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37 | based on composition and rejection Monte Carlo methods \cite{ce-butch,ce-messel,ce-egs4}, |
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38 | with the $SF$ function included in the rejection function |
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39 | \begin{equation}\label{en-samp-comp} |
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40 | g(\epsilon) = \left[1-\frac{\epsilon}{1+\epsilon^2} \sin^2\theta \right] \times SF(q) , |
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41 | \end{equation} |
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42 | with $0<g(\epsilon)<Z$. |
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43 | Values of the scattering functions at each momentum transfer, $q$, are |
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44 | obtained by interpolating the evaluated data for the corresponding atomic |
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45 | number, $Z$. |
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46 | |
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47 | The polar angle $\theta$ is deduced from the sampled $\epsilon$ value. |
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48 | In the azimuthal direction, the angular distributions of both the scattered |
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49 | photon and the recoil electron are considered to be |
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50 | isotropic~\cite{ce-stepanek}. |
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51 | |
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52 | Since the incoherent scattering occurs mainly on the outermost electronic |
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53 | subshells, the binding energies can be neglected, as stated in |
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54 | reference~\cite{ce-stepanek}. The momentum vector of the scattered photon, |
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55 | $\overrightarrow{P'_{\gamma}}$, |
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56 | is transformed into the {\tt World} coordinate system. |
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57 | The kinetic energy and momentum of the recoil electron are then |
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58 | \begin{eqnarray*} |
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59 | T_{el} & = & E - E' \\ |
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60 | \overrightarrow{P_{el}} & = & |
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61 | \overrightarrow{P_{\gamma}} - \overrightarrow{P'_{\gamma}} . |
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62 | \end{eqnarray*} |
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63 | |
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64 | |
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65 | \subsection{Status of the document} |
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66 | |
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67 | \noindent |
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68 | 30.09.1999 created by Alessandra Forti\\ |
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69 | 07.02.2000 modified by V\'eronique Lef\'ebure\\ |
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70 | 08.03.2000 reviewed by Petteri Nieminen and Maria Grazia Pia\\ |
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71 | 26.01.2003 minor re-write by D.H. Wright |
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72 | |
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73 | \begin{latexonly} |
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74 | |
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75 | \begin{thebibliography}{99} |
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76 | \bibitem{ce-hubbel} |
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77 | ``Summary of Existing Information on the Incoherent Scattering of Photons |
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78 | particularly on the Validity of the Use of the Incoherent Scattering |
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79 | Function", |
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80 | Radiat. Phys. Chem. Vol. 50, No 1, pp 113-124 (1997) |
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81 | \bibitem{ce-reda} |
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82 | ``A simple model of photon transport", |
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83 | D.E. Cullen, Nucl. Instr. Meth. in Phys. Res. B 101(1995)499-510 |
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84 | \bibitem{ce-butch} J.C. Butcher and H. Messel. |
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85 | {\em Nucl. Phys. 20} 15 (1960) |
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86 | \bibitem{ce-messel} H. Messel and D. Crawford. |
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87 | {\em Electron-Photon shower distribution, Pergamon Press} (1970) |
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88 | \bibitem{ce-egs4} R. Ford and W. Nelson. |
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89 | {\em SLAC-265, UC-32} (1985) |
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90 | \bibitem{ce-stepanek} |
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91 | ``New Photon, Positron and Electron Interaction Data for Geant in Energy |
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92 | Range from 1 eV to 10 TeV", |
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93 | J. Stepanek, Draft to be submitted for publication |
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94 | |
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95 | \end{thebibliography} |
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96 | |
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97 | \end{latexonly} |
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98 | |
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99 | \begin{htmlonly} |
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100 | |
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101 | \subsection{Bibliography} |
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102 | |
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103 | \begin{enumerate} |
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104 | \item |
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105 | ``Summary of Existing Information on the Incoherent Scattering of Photons |
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106 | particularly on the Validity of the Use of the Incoherent Scattering |
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107 | Function", |
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108 | Radiat. Phys. Chem. Vol. 50, No 1, pp 113-124 (1997) |
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109 | \item |
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110 | ``A simple model of photon transport", |
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111 | D.E. Cullen, Nucl. Instr. Meth. in Phys. Res. B 101(1995)499-510 |
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112 | \item J.C. Butcher and H. Messel. |
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113 | {\em Nucl. Phys. 20} 15 (1960) |
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114 | \item H. Messel and D. Crawford. |
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115 | {\em Electron-Photon shower distribution, Pergamon Press} (1970) |
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116 | \item R. Ford and W. Nelson. |
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117 | {\em SLAC-265, UC-32} (1985) |
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118 | \item |
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119 | ``New Photon, Positron and Electron Interaction Data for Geant in Energy |
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120 | Range from 1 eV to 10 TeV", |
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121 | J. Stepanek, Draft to be submitted for publication |
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122 | |
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123 | \end{enumerate} |
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124 | |
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125 | \end{htmlonly} |
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