1 | |
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2 | \section{Compton Scattering by Linearly Polarized Gamma Rays} |
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3 | |
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4 | |
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5 | \subsection{The Cross Section} |
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6 | |
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7 | The quantum mechanical Klein - Nishina differential cross section for |
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8 | polarized photons is [Heitler 1954]: |
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9 | |
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10 | \[\frac{d\sigma}{d\Omega} = \frac{1}{4}r_0^2 \frac{h\nu^2}{h\nu_o^2} \frac{h\nu_o^2}{h\nu^2} \left[\frac{h\nu_o}{h\nu}+\frac{h\nu}{h\nu_o}-2+4 cos^2\Theta \right] \] |
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11 | |
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12 | \noindent |
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13 | where $\Theta$ is the angle between the two polarization vectors. In terms |
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14 | of the polar and azimuthal angles $ (\theta, \phi) $ this cross section can |
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15 | be written as |
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16 | |
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17 | \[\frac{d\sigma}{d\Omega} = \frac{1}{2}r_0^2 \frac{h\nu^2}{h\nu_o^2} \frac{h\nu_o^2}{h\nu^2} \left[\frac{h\nu_o}{h\nu}+\frac{h\nu}{h\nu_o}-2 cos^2\phi sin^2\theta \right] \] . |
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18 | |
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19 | |
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20 | \subsection{Angular Distribution} |
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21 | |
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22 | |
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23 | The integration of this cross section over the azimuthal angle produces the |
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24 | standard cross section. The angular and energy distribution are then |
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25 | obtained in the same way as for the standard process. Using these values |
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26 | for the polar angle and the energy, the azimuthal angle is sampled from the |
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27 | following distribution: |
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28 | |
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29 | \[ P(\phi)= 1 - \frac{a}{b} cos^2\phi \] |
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30 | |
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31 | \noindent |
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32 | where $a = sin^2\theta $ and $b = \epsilon + 1/\epsilon$. $\epsilon$ is |
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33 | the ratio between the scattered photon energy and the incident photon |
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34 | energy. |
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35 | |
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36 | |
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37 | \subsection{Polarization Vector} |
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38 | |
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39 | The components of the vector polarization of the scattered photon are |
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40 | calculated from |
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41 | |
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42 | \[ \vec{\epsilon'_\bot} = \frac{1}{N} \left( \hat{j} cos\theta - \hat{k} sin\theta sin\phi \right) sin\beta \] |
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43 | |
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44 | |
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45 | \[ \vec{\epsilon'_\|} = \left[ N \hat{i}- \frac{1}{N} \hat{j} sin^2\theta sin\phi cos\phi - \frac{1}{N} \hat{k} sin\theta cos\theta cos\phi \right] cos\beta \] |
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46 | |
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47 | \noindent |
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48 | where \[ N = \sqrt{1-sin^2\theta cos^2\phi} . \] |
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49 | |
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50 | \noindent |
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51 | $cos\beta$ is calculated from $cos\Theta = N cos\beta $, while $cos\Theta$ |
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52 | is sampled from the Klein - Nishina distribution. |
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53 | |
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54 | The binding effects and the Compton profile are neglected. |
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55 | The kinetic energy and momentum of the recoil electron are then |
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56 | |
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57 | \[ T_{el} = E - E' \] |
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58 | \[ \vec{P_{el}} = \vec{P_\gamma} - \vec{P_\gamma '} . \] |
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59 | |
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60 | The momentum vector of the scattered photon $\vec{P_\gamma}$ and its |
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61 | polarization vector are transformed into the {\tt World} coordinate system. |
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62 | The polarization and the direction of the scattered gamma in the final |
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63 | state are calculated in the reference frame in which the incoming photon is |
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64 | along the $z$-axis and has its polarization vector along the $x$-axis. The |
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65 | transformation to the {\tt World} coordinate system performs a linear |
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66 | combination of the initial direction, the initial poalrization and the cross |
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67 | product between them, using the projections of the calculated quantities |
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68 | along these axes. |
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69 | |
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70 | \subsection{Unpolarized Photons} |
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71 | |
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72 | A special treatment is devoted to unpolarized photons. In this case a |
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73 | random polarization in the plane perpendicular to the incident photon is |
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74 | selected. |
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75 | |
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76 | \subsection{Status of this document} |
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77 | |
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78 | 18.06.2001 created by Gerardo Depaola and Francesco Longo \\ |
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79 | 10.06.2002 revision by Francesco Longo \\ |
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80 | 26.01.2003 minor re-wording and correction of equations by D.H. Wright |
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81 | |
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82 | \begin{latexonly} |
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83 | |
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84 | \begin{thebibliography}{99} |
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85 | |
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86 | \bibitem{Heitler} W. Heitler {\em The Quantum Theory of Radiation, Oxford Clarendom Press } (1954) |
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87 | |
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88 | \end{thebibliography} |
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89 | |
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90 | \end{latexonly} |
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91 | |
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92 | \begin{htmlonly} |
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93 | |
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94 | \subsection{Bibliography} |
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95 | |
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96 | \begin{enumerate} |
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97 | \item W. Heitler {\em The Quantum Theory of Radiation, Oxford Clarendom Press } (1954) |
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98 | \end{enumerate} |
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99 | |
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100 | \end{htmlonly} |
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101 | |
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102 | |
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103 | |
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104 | |
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105 | |
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