[1208] | 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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| 105 | |
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