[1211] | 1 | |
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| 2 | % ====================================================================== |
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| 3 | |
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| 4 | %\newcommand{\bvec}[1]{{\mathbf{#1}}} |
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| 5 | \newcommand{\bvec}[1]{{\boldsymbol{#1}}} %% use boldsymbol if amsmath is available! |
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| 6 | |
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| 7 | \section{Introduction}\label{sec:pol.intro} |
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| 8 | |
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| 9 | With the EM polarization extension it is possible to track polarized |
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| 10 | particles (leptons and photons). Special emphasis will be |
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| 11 | put in the proper treatment of polarized matter and its interaction |
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| 12 | with longitudinal polarized electrons/positrons or circularly |
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| 13 | polarized photons, which is for instance essential for the simulation |
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| 14 | of positron polarimetry. The implementation is base on Stokes vectors |
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| 15 | \cite{polIntro:McMaster:1961}. Further details can be found in |
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| 16 | \cite{polIntro:Laihem:thesis}. |
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| 17 | |
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| 18 | In its current state, the following polarization |
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| 19 | dependent processes are considered |
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| 20 | \begin{itemize} |
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| 21 | \item Bhabha/M{\o}ller scattering, |
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| 22 | \item Positron Annihilation, |
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| 23 | \item Compton scattering, |
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| 24 | \item Pair creation, |
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| 25 | \item Bremsstrahlung. |
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| 26 | \end{itemize} |
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| 27 | |
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| 28 | %\subsection{Existing codes for the simulation of polarized processes} |
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| 29 | |
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| 30 | Several simulation packages for the realistic description |
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| 31 | of the development of electromagnetic showers in matter have been |
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| 32 | developed. A prominent example of such codes is EGS (Electron Gamma |
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| 33 | Shower)\cite{polIntro:Nelson:1985ec}. |
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| 34 | For this simulation framework extensions with the treatment of |
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| 35 | polarized particles exist \cite{polIntro:Floettmann:thesis,polIntro:Namito:1993sv,polIntro:Liu:2000ey}; |
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| 36 | the most complete has been developed by K.~Fl{\"o}ttmann |
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| 37 | \cite{polIntro:Floettmann:thesis}. It is based on the matrix formalism |
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| 38 | \cite{polIntro:McMaster:1961}, which enables a very general treatment of |
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| 39 | polarization. However, the Fl{\"o}ttmann extension concentrates on |
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| 40 | evaluation of polarization transfer, i.e.\ the effects of polarization |
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| 41 | induced asymmetries are neglected, and interactions with polarized |
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| 42 | media are not considered. |
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| 43 | |
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| 44 | Another important simulation tool for detector studies is \textsc{Geant3} |
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| 45 | \cite{polIntro:Brun:1985ps}. Here also some effort has been made to include |
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| 46 | polarization \cite{polIntro:Alexander:2003fh,polIntro:Hoogduin:thesis}, but these |
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| 47 | extensions are not publicly available. |
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| 48 | |
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| 49 | %\section{Definitions} |
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| 50 | |
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| 51 | In general the implementation of polarization in this EM polarization |
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| 52 | library follows very closely the approach by K.~Fl{\"o}tt\-mann |
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| 53 | \cite{polIntro:Floettmann:thesis}. The basic principle is to associate a {\em |
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| 54 | Stokes vector} to each particle and track the mean polarization from |
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| 55 | one interaction to another. The basics for this approach is the matrix |
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| 56 | formalism as introduced in \cite{polIntro:McMaster:1961}. |
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| 57 | |
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| 58 | \subsection{Stokes vector} |
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| 59 | |
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| 60 | The {\em Stokes vector} \cite{polIntro:Stokes:1852,polIntro:McMaster:1961} is a rather |
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| 61 | simple object (in comparison to e.g.\ the spin density matrix), three |
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| 62 | real numbers are sufficient for the characterization of the polarization |
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| 63 | state of any single electron, positron or photon. |
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| 64 | Using {\em Stokes vectors} {\bf all} possible polarization states can |
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| 65 | be described, i.e.\ circular and linear polarized photons can be |
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| 66 | handled with the same formalism as longitudinal |
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| 67 | and transverse polarized electron/positrons. |
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| 68 | |
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| 69 | The {\em Stokes vector} can be used also for beams, in the sense that |
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| 70 | it defines a mean polarization. |
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| 71 | |
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| 72 | In the EM polarization library the Stokes vector is defined as |
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| 73 | follows: |
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| 74 | |
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| 75 | \begin{center} |
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| 76 | %\rotatebox{90}{ Method A} |
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| 77 | \renewcommand{\arraystretch}{1.15} |
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| 78 | \begin{tabular}{|c|c|c|} |
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| 79 | \hline |
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| 80 | & Photons & Electrons \\ |
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| 81 | \hline |
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| 82 | $\xi_1$ & linear polarization & polarization in x direction \\ |
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| 83 | $\xi_2$ & linear polarization but $\pi/4$ to right |
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| 84 | & polarization in y direction \\ |
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| 85 | $\xi_3$ & circular polarization & polarization in z direction \\ |
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| 86 | \hline |
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| 87 | \end{tabular} |
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| 88 | \end{center} |
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| 89 | This definition is assumed in the {\em particle reference frame}, |
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| 90 | i.e. with the momentum of the particle pointing to the z direction, |
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| 91 | cf.\ also next section about coordinate transformations. |
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| 92 | Correspondingly a 100\% longitudinally polarized |
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| 93 | electron or positron is characterized by |
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| 94 | \begin{equation} |
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| 95 | \bvec{\xi}=\mbox{$\scriptscriptstyle\left(\begin{array}{c}0\\0\\\pm1\end{array}\right)$}, |
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| 96 | \end{equation} |
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| 97 | where $\pm1$ corresponds to spin parallel (anti parallel) to |
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| 98 | particle's momentum. |
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| 99 | % |
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| 100 | Note that this definition is similar, but not |
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| 101 | identical to the definition used in McMaster \cite{polIntro:McMaster:1961}. |
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| 102 | |
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| 103 | Many scattering cross sections of polarized processes using |
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| 104 | Stokes vectors for the characterization of initial and final states are |
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| 105 | available in \cite{polIntro:McMaster:1961}. In general a differential cross |
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| 106 | section has the form |
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| 107 | \begin{equation} |
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| 108 | \frac{d\sigma(\bvec{\zeta}^{(1)},\bvec{\zeta}^{(2)},\bvec{\xi}^{(1)},\bvec{\xi}^{(2)})}{d\Omega}\;, |
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| 109 | \end{equation} |
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| 110 | i.e.\ it is a function of the polarization states of the initial |
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| 111 | particles $\bvec{\zeta}^{(1)}$ and $\bvec{\zeta}^{(2)}$, as well as of the polarization states |
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| 112 | of the final state particles $\bvec{\xi}^{(1)}$ and $\bvec{\xi}^{(2)}$ (in addition to the |
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| 113 | kinematic variables $E$, $\theta$, and $\phi$). |
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| 114 | |
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| 115 | Consequently, in a simulation we have to account for |
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| 116 | \begin{itemize} |
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| 117 | \item Asymmetries: |
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| 118 | \item[] Polarization of beam ($\bvec{\zeta}^{(1)}$) and target ($\bvec{\zeta}^{(2)}$) can induce |
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| 119 | azimuthal and polar asymmetries, and may also influence on the total |
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| 120 | cross section ({\tt Geant4: GetMeanFreePath()}). |
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| 121 | \item Polarization transfer / depolarization effects |
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| 122 | \item[] The dependence on the final state polarizations defines a |
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| 123 | possible transfer from initial polarization to final state particles. |
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| 124 | \end{itemize} |
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| 125 | |
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| 126 | \subsection{Transfer matrix} |
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| 127 | |
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| 128 | %For asymmetries one can extent the existing standard EM physics classes, |
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| 129 | %introducing the polarization of the initial states. On the other hand |
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| 130 | %for a general simulation of polarization transfer one has to work harder. |
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| 131 | Using the formalism of McMaster, differential cross section and |
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| 132 | polarization transfer from the initial state ($\bvec{\zeta}^{(1)}$) to one final state |
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| 133 | particle ($\bvec{\xi}^{(1)}$) are combined in an interaction matrix $T$: |
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| 134 | |
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| 135 | \begin{equation} |
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| 136 | \left(\begin{array}{c} |
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| 137 | O \\ |
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| 138 | \bvec{\xi}^{(1)} |
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| 139 | \end{array}\right) |
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| 140 | = T \, |
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| 141 | \left(\begin{array}{c} |
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| 142 | I \\ |
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| 143 | \bvec{\zeta}^{(1)} |
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| 144 | \end{array}\right)\;, |
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| 145 | \end{equation} |
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| 146 | where $I$ and $O$ are the incoming and outgoing currents, respectively. |
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| 147 | % |
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| 148 | In general the $4\times4$ matrix $T$ depends on the target |
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| 149 | polarization $\bvec{\zeta}^{(2)}$ (and of course on the kinematic |
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| 150 | variables $E$, $\theta$, $\phi$). Similarly one can define |
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| 151 | a matrix defining the polarization transfer to second final state |
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| 152 | particle like |
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| 153 | \begin{equation} |
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| 154 | \left(\begin{array}{c} |
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| 155 | O \\ |
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| 156 | \bvec{\xi}^{(2)} |
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| 157 | \end{array}\right) |
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| 158 | = T' \, |
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| 159 | \left(\begin{array}{c}I\\ |
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| 160 | \bvec{\zeta}^{(1)}\end{array}\right) \;. |
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| 161 | \end{equation} |
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| 162 | % |
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| 163 | %The components $I$ and $O$ refer to the incoming and outgoing |
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| 164 | %intensities, respectively. |
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| 165 | In this framework the transfer matrix $T$ is of the form |
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| 166 | \begin{equation} |
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| 167 | T = |
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| 168 | \left( |
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| 169 | \begin{array}{llll} |
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| 170 | S & A_1 & A_2 & A_3 \\ |
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| 171 | P_1 & M_{11} & M_{21} & M_{31} \\ |
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| 172 | P_2 & M_{12} & M_{22} & M_{32} \\ |
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| 173 | P_3 & M_{13} & M_{23} & M_{33} \\ |
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| 174 | \end{array} |
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| 175 | \right) |
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| 176 | \;. |
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| 177 | \end{equation} |
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| 178 | The matrix elements $T_{ij}$ can be identified as (unpolarized) |
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| 179 | differential cross section ($S$), polarized differential cross section |
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| 180 | ($A_j$), polarization transfer ($M_{ij}$), and (de)polarization ($P_i$). |
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| 181 | In the Fl{\"o}ttmann extension the elements $A_j$ and $P_i$ have been |
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| 182 | neglected, thus concentrating on polarization transfer only. |
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| 183 | Using the full matrix takes now all polarization effects into account. |
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| 184 | |
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| 185 | |
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| 186 | The transformation matrix, i.e.\ the dependence of the mean |
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| 187 | polarization of final state particles, can be derived from the |
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| 188 | asymmetry of the differential cross section w.r.t.\ this particular |
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| 189 | polarization. |
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| 190 | Where the asymmetry is defined as usual by |
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| 191 | \begin{equation} |
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| 192 | A = \frac{\sigma(+1)-\sigma(-1)}{\sigma(+1)+\sigma(-1)} \;. |
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| 193 | \end{equation} |
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| 194 | The mean final state polarizations can be determined coefficient by |
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| 195 | coefficient. |
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| 196 | % |
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| 197 | %For instance the components of the mean Stokes vector |
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| 198 | %% following eq.\ \eqref{eq:diffxsec} |
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| 199 | %$\bvec{\hat\xi}^{(1)}$ of the first final state particle is obtained |
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| 200 | %by |
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| 201 | %\begin{equation} |
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| 202 | % \hat\xi^{(2)}_1 = \frac{\sigma(\bvec{\zeta}^{(1)},\bvec{\zeta}^{(2)}, |
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| 203 | %\bvec{\xi}^{(1)}=\mbox{\small$\left(\!\!\begin{array}{c}+1\\0\\0\end{array}\!\!\right)$}, |
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| 204 | %\bvec{\xi}^{(2)}=\mbox{\small$\left(\!\!\begin{array}{c}0\\0\\0\end{array}\!\!\right)$}) |
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| 205 | %- |
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| 206 | %\sigma(\bvec{\zeta}^{(1)},\bvec{\zeta}^{(2)}, |
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| 207 | %\bvec{\xi}^{(1)}=\mbox{\small$\left(\!\!\begin{array}{c}-1\\0\\0\end{array}\!\!\right)$}, |
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| 208 | %\bvec{\xi}^{(2)}=\mbox{\small$\left(\!\!\begin{array}{c}0\\0\\0\end{array}\!\!\right)$}) |
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| 209 | %}{\sigma(\dots)+\sigma(\dots)} |
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| 210 | %\end{equation} |
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| 211 | % |
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| 212 | %\begin{equation} |
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| 213 | % \hat\xi^{(2)}_2 = \frac{\sigma(\bvec{\zeta}^{(1)},\bvec{\zeta}^{(2)}, |
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| 214 | %\bvec{\xi}^{(1)}=\mbox{\small$\left(\!\!\begin{array}{c}0\\+1\\0\end{array}\!\!\right)$}, |
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| 215 | %\bvec{\xi}^{(2)}=\mbox{\small$\left(\!\!\begin{array}{c}0\\0\\0\end{array}\!\!\right)$}) |
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| 216 | %- |
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| 217 | %\sigma(\bvec{\zeta}^{(1)},\bvec{\zeta}^{(2)}, |
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| 218 | %\bvec{\xi}^{(1)}=\mbox{\small$\left(\!\!\begin{array}{c}0\\-1\\0\end{array}\!\!\right)$}, |
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| 219 | %\bvec{\xi}^{(2)}=\mbox{\small$\left(\!\!\begin{array}{c}0\\0\\0\end{array}\!\!\right)$}) |
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| 220 | %}{\sigma(\dots)+\sigma(\dots)} |
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| 221 | %\end{equation} |
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| 222 | % |
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| 223 | %\begin{equation} |
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| 224 | % \hat\xi^{(2)}_3 = \frac{\sigma(\bvec{\zeta}^{(1)},\bvec{\zeta}^{(2)}, |
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| 225 | %\bvec{\xi}^{(1)}=\mbox{\small$\left(\!\!\begin{array}{c}0\\0\\+1\end{array}\!\!\right)$}, |
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| 226 | %\bvec{\xi}^{(2)}=\mbox{\small$\left(\!\!\begin{array}{c}0\\0\\0\end{array}\!\!\right)$}) |
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| 227 | %- |
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| 228 | %\sigma(\bvec{\zeta}^{(1)},\bvec{\zeta}^{(2)}, |
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| 229 | %\bvec{\xi}^{(1)}=\mbox{\small$\left(\!\!\begin{array}{c}0\\0\\-1\end{array}\!\!\right)$}, |
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| 230 | %\bvec{\xi}^{(2)}=\mbox{\small$\left(\!\!\begin{array}{c}0\\0\\0\end{array}\!\!\right)$}) |
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| 231 | %}{\sigma(\dots)+\sigma(\dots)} |
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| 232 | %\end{equation} |
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| 233 | % |
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| 234 | |
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| 235 | In general, the differential cross section is a linear function |
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| 236 | of the polarizations, i.e. |
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| 237 | \begin{eqnarray} |
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| 238 | \frac{d\sigma(\bvec{\zeta}^{(1)},\bvec{\zeta}^{(2)},\bvec{\xi}^{(1)},\bvec{\xi}^{(2)})}{d\Omega} &=& |
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| 239 | \Phi_{(\zeta^{(1)},\zeta^{(2)})} |
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| 240 | + \bvec{A}_{(\zeta^{(1)},\zeta^{(2)})} \cdot\bvec{\xi}^{(1)} |
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| 241 | + \bvec{B}_{(\zeta^{(1)},\zeta^{(2)})} \cdot\bvec{\xi}^{(2)} \nonumber\\ |
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| 242 | && \quad \quad \quad |
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| 243 | +\, {\bvec{\xi}^{(1)}}^T M_{(\zeta^{(1)},\zeta^{(2)})} \,\bvec{\xi}^{(2)} |
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| 244 | \end{eqnarray} |
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| 245 | In this form, the mean polarization of the final state can be read off |
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| 246 | easily, and one obtains |
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| 247 | \begin{eqnarray} |
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| 248 | \langle\bvec{\xi}^{(1)}\rangle &=& \frac{1}{\Phi_{(\zeta^{(1)},\zeta^{(2)})}} |
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| 249 | \bvec{A}_{(\zeta^{(1)},\zeta^{(2)})} \;\; \mbox{and} \\ |
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| 250 | \langle\bvec{\xi}^{(2)}\rangle &=& \frac{1}{\Phi_{(\zeta^{(1)},\zeta^{(2)})}} |
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| 251 | \bvec{B}_{(\zeta^{(1)},\zeta^{(2)})} \;. |
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| 252 | \end{eqnarray} |
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| 253 | |
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| 254 | Note, that the {\em mean} polarization states do not depend on the |
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| 255 | correlation matrix $M_{(\zeta^{(1)},\zeta^{(2)})}$. In order to account for |
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| 256 | correlation one has to generate {\em single} particle Stokes |
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| 257 | vector explicitly, i.e.\ on an event by event basis. However, this |
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| 258 | implementation generates {\em mean} polarization states, and neglects |
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| 259 | correlation effects. |
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| 260 | |
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| 261 | %\newpage |
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| 262 | \subsection{Coordinate transformations} |
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| 263 | |
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| 264 | \begin{figure}[h!] |
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| 265 | \centerline{\includegraphics[width=8.cm]{electromagnetic/standard/plots/frames.eps}} |
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| 266 | \caption{\label{pol.interframe} |
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| 267 | The {\em interaction frame} and the {\em particle frames} for the |
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| 268 | example of Compton scattering. The momenta of all participating |
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| 269 | particle lie in the $x$-$z$-plane, the scattering plane. The |
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| 270 | incoming photon gives the $z$ direction. The outgoing photon is |
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| 271 | defined as {\em particle 1} and gives the $x$-direction, perpendicular to |
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| 272 | the $z$-axis. The $y$-axis is then perpendicular to the scattering |
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| 273 | plane and completes the definition of a right handed coordinate |
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| 274 | system called {\em interaction frame}. |
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| 275 | The {\em particle frame} is defined by the Geant4 routine |
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| 276 | {\tt G4ThreeMomemtum::rotateUz()}.} |
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| 277 | \end{figure} |
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| 278 | |
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| 279 | Three different coordinate systems are used in the evaluation of |
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| 280 | polarization states: |
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| 281 | \begin{itemize} |
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| 282 | \item {\bf World frame} |
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| 283 | %\item[] |
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| 284 | |
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| 285 | The geometry of the target, and the momenta of all particles |
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| 286 | in Geant4 are noted in the world frame $X$, $Y$, $Z$ (the {\em global |
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| 287 | reference frame}, GRF). It is the basis of the calculation of any |
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| 288 | other coordinate system. |
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| 289 | \item {\bf Particle frame} |
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| 290 | %\item[] |
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| 291 | |
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| 292 | Each particle is carrying its own coordinate system. |
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| 293 | In this system the direction of motion coincides with the |
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| 294 | $z$-direction. Geant4 provides a transformation from any particle |
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| 295 | frame to the World frame by the method |
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| 296 | {\tt G4ThreeMomemtum::rotateUz()}. Thus, the $y$-axis of the |
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| 297 | {\em particle reference frame} (PRF) lies in the $X$-$Y$-plane of |
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| 298 | the world frame. |
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| 299 | |
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| 300 | The Stokes vector of any moving particle is defined w.r.t. the |
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| 301 | corresponding particle frame. |
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| 302 | Particles at rest (e.g.\ electrons of a media) use the world frame as |
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| 303 | particle frame. |
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| 304 | \item {\bf Interaction frame} |
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| 305 | %\item[] |
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| 306 | |
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| 307 | For the evaluation of the polarization transfer another |
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| 308 | coordinate system is used, defined by the scattering plane, cf.\ |
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| 309 | fig.\ \ref{pol.interframe}. There the |
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| 310 | $z$-axis is defined by the direction of motion of the incoming |
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| 311 | particle. The scattering plane is spanned by the $z$-axis and the |
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| 312 | $x$-axis, in a way, that the direction of {\em particle~1} has a |
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| 313 | positive $x$ component. The definition of {\em particle~1} depends on |
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| 314 | the process, for instance in Compton scattering, the outgoing photon |
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| 315 | is referred as {\em particle~1}\footnote{Note, for an incoming |
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| 316 | particle travelling on the $Z$-axis (of GRF), the $y$-axis of the PRF |
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| 317 | of both outgoing particles is parallel to the $y$-axis of the |
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| 318 | {\em interaction frame}.}. |
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| 319 | \end{itemize} |
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| 320 | |
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| 321 | All frames are right handed. |
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| 322 | |
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| 323 | |
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| 324 | \subsection{Polarized beam and material} |
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| 325 | |
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| 326 | Polarization of beam particles is well established. It can be used for |
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| 327 | simulating low-energy Compton scattering of linear polarized |
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| 328 | photons. The interpretation as Stokes vector allows now the usage in a |
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| 329 | more general framework. |
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| 330 | % |
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| 331 | The polarization state of a (initial) beam particle can be fixed |
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| 332 | using standard the ParticleGunMessenger class. For example, the class {\tt |
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| 333 | G4ParticleGun} provides the method {\tt SetParticlePolarization()}, |
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| 334 | which is usually accessable via |
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| 335 | \begin{verbatim} |
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| 336 | /gun/polarization <Sx> <Sy> <Sz> |
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| 337 | \end{verbatim} |
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| 338 | in a macro file. |
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| 339 | |
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| 340 | In addition for the simulation of polarized media, a possibility |
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| 341 | to assign Stokes vectors to physical volumes is provided by a new |
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| 342 | class, the so-called {\em G4PolarizationManager}. |
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| 343 | %It also provides some helper routines for the evaluation of Stokes |
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| 344 | %vectors in different frames of reference. |
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| 345 | % |
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| 346 | The procedure to assign a polarization vector to a media, is done |
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| 347 | during the {\em detector construction}. There the {\em |
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| 348 | logical volumes} with certain polarization are made known to |
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| 349 | {\em polarization manager}. One example {\tt DetectorConstruction} |
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| 350 | might look like follows: |
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| 351 | |
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| 352 | \begin{verbatim} |
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| 353 | G4double Targetthickness = .010*mm; |
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| 354 | G4double Targetradius = 2.5*mm; |
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| 355 | |
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| 356 | G4Tubs *solidTarget = |
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| 357 | new G4Tubs("solidTarget", |
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| 358 | 0.0, |
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| 359 | Targetradius, |
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| 360 | Targetthickness/2, |
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| 361 | 0.0*deg, |
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| 362 | 360.0*deg ); |
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| 363 | |
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| 364 | G4LogicalVolume * logicalTarget = |
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| 365 | new G4LogicalVolume(solidTarget, |
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| 366 | iron, |
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| 367 | "logicalTarget", |
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| 368 | 0,0,0); |
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| 369 | |
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| 370 | G4VPhysicalVolume * physicalTarget = |
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| 371 | new G4PVPlacement(0,G4ThreeVector(0.*mm, 0.*mm, 0.*mm), |
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| 372 | logicalTarget, |
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| 373 | "physicalTarget", |
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| 374 | worldLogical, |
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| 375 | false, |
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| 376 | 0); |
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| 377 | |
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| 378 | G4PolarizationManager * polMgr = G4PolarizationManager::GetInstance(); |
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| 379 | polMgr->SetVolumePolarization(logicalTarget,G4ThreeVector(0.,0.,0.08)); |
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| 380 | \end{verbatim} |
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| 381 | Once a logical volume is known to the {\tt G4PolarizationManager}, its |
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| 382 | polarization vector can be accessed from a macro file by its name, |
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| 383 | e.g.\ the polarization of the logical volume called ``logicalTarget'' |
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| 384 | can be changed via |
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| 385 | \begin{verbatim} |
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| 386 | /polarization/volume/set logicalTarget 0. 0. -0.08 |
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| 387 | \end{verbatim} |
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| 388 | Note, the polarization of a material is stated in the world frame. |
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| 389 | |
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| 390 | \subsection{Status of this document} |
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| 391 | 20.11.06 created by A.Sch{\"a}licke\\ |
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| 392 | |
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| 393 | \begin{latexonly} |
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| 394 | |
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| 395 | \begin{thebibliography}{10} |
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| 396 | |
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| 397 | \bibitem{polIntro:McMaster:1961} |
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| 398 | W.~H.~McMaster, Rev.\ Mod.\ Phys.\ {\bf 33} (1961) 8; and references therein. |
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| 399 | |
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| 400 | \bibitem{polIntro:Laihem:thesis} |
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| 401 | K.~Laihem, PhD thesis, Measurement of the positron polarization at an |
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| 402 | helical undulator based positron source for the International Linear |
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| 403 | Collider ILC, Humboldt University Berlin, Germany, (2008). |
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| 404 | |
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| 405 | |
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| 406 | %%EGS |
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| 407 | \bibitem{polIntro:Nelson:1985ec} |
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| 408 | W.~R.~Nelson, H.~Hirayama, D.~W.~O.\ Rogers, |
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| 409 | %``The Egs4 Code System,'' |
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| 410 | SLAC-R-0265. |
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| 411 | |
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| 412 | \bibitem{polIntro:Floettmann:thesis} |
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| 413 | K.~Fl\"ottmann, PhD thesis, DESY Hamburg (1993); DESY-93-161. |
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| 414 | |
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| 415 | %kek extension |
---|
| 416 | \bibitem{polIntro:Namito:1993sv} |
---|
| 417 | Y.~Namito, S.~Ban, H.~Hirayama, |
---|
| 418 | %``Implementation of linearly polarized photon scattering into the EGS4 code,'' |
---|
| 419 | Nucl.\ Instrum.\ Meth.\ A {\bf 332} (1993) 277. |
---|
| 420 | |
---|
| 421 | \bibitem{polIntro:Liu:2000ey} |
---|
| 422 | J.~C.~Liu, T.~Kotseroglou, W.~R.~Nelson, D.~C.~Schultz, |
---|
| 423 | %``Polarization study for NLC positron source using EGS4,'' |
---|
| 424 | SLAC-PUB-8477. |
---|
| 425 | %Geant3 |
---|
| 426 | \bibitem{polIntro:Brun:1985ps} |
---|
| 427 | R.~Brun, M.~Caillat, M.~Maire, G.~N.~Patrick, L.~Urban, |
---|
| 428 | %``The Geant3 Electromagnetic Shower Program And A Comparison With The Egs3 |
---|
| 429 | %Code,'' |
---|
| 430 | CERN-DD/85/1. |
---|
| 431 | |
---|
| 432 | %% E166 |
---|
| 433 | \bibitem{polIntro:Alexander:2003fh} |
---|
| 434 | G.~Alexander {\it et al.}, |
---|
| 435 | %``Undulator-based production of polarized positrons: A proposal for |
---|
| 436 | % the 50-GeV beam in the FFTB,'' |
---|
| 437 | SLAC-TN-04-018, SLAC-PROPOSAL-E-166. |
---|
| 438 | |
---|
| 439 | \bibitem{polIntro:Hoogduin:thesis} |
---|
| 440 | J.~Hoogduin, PhD thesis, Rijksuniversiteit Groningen (1997). |
---|
| 441 | |
---|
| 442 | \bibitem{polIntro:Stokes:1852} |
---|
| 443 | G.~Stokes, |
---|
| 444 | Trans.\ Cambridge Phil.\ Soc.\ {\bf 9} (1852) 399. |
---|
| 445 | |
---|
| 446 | |
---|
| 447 | \end{thebibliography} |
---|
| 448 | |
---|
| 449 | \end{latexonly} |
---|
| 450 | |
---|
| 451 | \begin{htmlonly} |
---|
| 452 | |
---|
| 453 | \begin{enumerate}{10} |
---|
| 454 | \item |
---|
| 455 | W.~H.~McMaster, Rev.\ Mod.\ Phys.\ {\bf 33} (1961) 8; and references therein. |
---|
| 456 | |
---|
| 457 | \item |
---|
| 458 | K.~Laihem, PhD thesis, Measurement of the positron polarization at an |
---|
| 459 | helical undulator based positron source for the International Linear |
---|
| 460 | Collider ILC, Humboldt University Berlin, Germany, (2008). |
---|
| 461 | |
---|
| 462 | |
---|
| 463 | %%EGS |
---|
| 464 | \item |
---|
| 465 | W.~R.~Nelson, H.~Hirayama, D.~W.~O.\ Rogers, |
---|
| 466 | %``The Egs4 Code System,'' |
---|
| 467 | SLAC-R-0265. |
---|
| 468 | |
---|
| 469 | \item |
---|
| 470 | K.~Fl\"ottmann, PhD thesis, DESY Hamburg (1993); DESY-93-161. |
---|
| 471 | |
---|
| 472 | %kek extension |
---|
| 473 | \item |
---|
| 474 | Y.~Namito, S.~Ban, H.~Hirayama, |
---|
| 475 | %``Implementation of linearly polarized photon scattering into the EGS4 code,'' |
---|
| 476 | Nucl.\ Instrum.\ Meth.\ A {\bf 332} (1993) 277. |
---|
| 477 | |
---|
| 478 | \item |
---|
| 479 | J.~C.~Liu, T.~Kotseroglou, W.~R.~Nelson, D.~C.~Schultz, |
---|
| 480 | %``Polarization study for NLC positron source using EGS4,'' |
---|
| 481 | SLAC-PUB-8477. |
---|
| 482 | %Geant3 |
---|
| 483 | \item |
---|
| 484 | R.~Brun, M.~Caillat, M.~Maire, G.~N.~Patrick, L.~Urban, |
---|
| 485 | %``The Geant3 Electromagnetic Shower Program And A Comparison With The Egs3 |
---|
| 486 | %Code,'' |
---|
| 487 | CERN-DD/85/1. |
---|
| 488 | |
---|
| 489 | %% E166 |
---|
| 490 | \item |
---|
| 491 | G.~Alexander {\it et al.}, |
---|
| 492 | %``Undulator-based production of polarized positrons: A proposal for |
---|
| 493 | % the 50-GeV beam in the FFTB,'' |
---|
| 494 | SLAC-TN-04-018, SLAC-PROPOSAL-E-166. |
---|
| 495 | |
---|
| 496 | \item |
---|
| 497 | J.~Hoogduin, PhD thesis, Rijksuniversiteit Groningen (1997). |
---|
| 498 | |
---|
| 499 | \item |
---|
| 500 | G.~Stokes, |
---|
| 501 | Trans.\ Cambridge Phil.\ Soc.\ {\bf 9} (1852) 399. |
---|
| 502 | |
---|
| 503 | \end{enumerate} |
---|
| 504 | |
---|
| 505 | \end{htmlonly} |
---|
| 506 | |
---|
| 507 | |
---|
| 508 | |
---|
| 509 | |
---|
| 510 | % ====================================================================== |
---|
| 511 | \newcommand{\Mvariable}[1]{r_e} |
---|
| 512 | |
---|
| 513 | \newpage |
---|
| 514 | \section{Ionization}\label{sec:polarizedIonization} |
---|
| 515 | \subsection{Method} |
---|
| 516 | The class {\em G4ePolarizedIonization} provides continuous and |
---|
| 517 | discrete energy losses of polarized electrons and positrons in a |
---|
| 518 | material. It evaluates polarization transfer and -- if the material |
---|
| 519 | is polarized -- asymmetries in the explicit delta rays production. |
---|
| 520 | The implementation baseline follows the approach derived for the |
---|
| 521 | class {\em G4eIonization} described in sections |
---|
| 522 | \ref{en_loss} and \ref{sec:em.eion}. |
---|
| 523 | For continuous energy losses the effects of a polarized beam or |
---|
| 524 | target are negligible provided the separation cut $T_{\rm cut}$ is |
---|
| 525 | small, and are therefore not considered separately. On the other |
---|
| 526 | hand, in the explicit production of delta rays by M{\o}ller or |
---|
| 527 | Bhabha scattering, the effects of polarization on total cross |
---|
| 528 | section and mean free path, on distribution of final state particles |
---|
| 529 | and the average polarization of final state particles are taken into |
---|
| 530 | account. |
---|
| 531 | |
---|
| 532 | % ---------------------------------------------------------------------- |
---|
| 533 | |
---|
| 534 | \subsection{Total cross section and mean free path} |
---|
| 535 | |
---|
| 536 | Kinematics of Bhabha and M{\o}ller scattering is fixed by initial |
---|
| 537 | energy |
---|
| 538 | \begin{equation} |
---|
| 539 | \gamma=\frac{E_{k_1}}{m c^2}% =\frac{s}{2m^2}-1 |
---|
| 540 | \end{equation} |
---|
| 541 | and variable |
---|
| 542 | \begin{equation} |
---|
| 543 | \epsilon = \frac{E_{p_2}-m c^2}{E_{k_1}-m c^2}, |
---|
| 544 | \end{equation} |
---|
| 545 | which is the part of kinetic energy of initial particle carried out by |
---|
| 546 | scatter. Lower kinematic limit for $\epsilon$ is $0$, but in order |
---|
| 547 | to avoid divergencies in both total and differential cross sections |
---|
| 548 | one sets |
---|
| 549 | \begin{equation} |
---|
| 550 | \epsilon_{min}= x = \frac{T_{min}}{E_{k_1}-mc^2}, |
---|
| 551 | \end{equation} |
---|
| 552 | where $T_{min}$ has meaning of minimal kinetic energy of secondary |
---|
| 553 | electron. And, $\epsilon_{\rm max}=1(1/2)$ for Bhabha(M{\o}ller) |
---|
| 554 | scatterings. |
---|
| 555 | |
---|
| 556 | % ---------------------------------------------------------------------- |
---|
| 557 | \subsubsection{Total M{\o}ller cross section} |
---|
| 558 | |
---|
| 559 | The total cross section of the polarized M{\o}ller scattering can be expressed as follows |
---|
| 560 | \begin{equation}\label{totalMoller} |
---|
| 561 | \sigma^M_{pol}=\frac{2\pi\gamma^2 r_e^2}{(\gamma-1)^2(\gamma+1)}\left[ |
---|
| 562 | \sigma^M_0 + \zeta_3^{(1)}\zeta_3^{(2)}\sigma^M_L |
---|
| 563 | + \left(\zeta_1^{(1)}\zeta_1^{(2)}+\zeta_2^{(1)}\zeta_2^{(2)}\right)\sigma^M_T\right], |
---|
| 564 | \end{equation} |
---|
| 565 | where the $r_e$ is classical electron radius, and |
---|
| 566 | \begin{eqnarray} |
---|
| 567 | \sigma^M_0&=& |
---|
| 568 | - \frac{1}{1 - x} + \frac{1}{x} |
---|
| 569 | - \frac{{\left( \gamma - 1 \right)}^2}{{\gamma}^2} |
---|
| 570 | \left(\frac{1}{2} - x \right) |
---|
| 571 | + \frac{ 2 - 4\,\gamma }{2\,{\gamma}^2} |
---|
| 572 | \,\ln \left(\frac{1-x}{x}\right) |
---|
| 573 | \nonumber\\ |
---|
| 574 | \sigma^M_L&=& |
---|
| 575 | \frac{ \left( -3 + 2\,\gamma + {\gamma}^2 \right) |
---|
| 576 | \,\left( 1 - 2\,x \right) }{2\, {\gamma}^2} |
---|
| 577 | + \frac{2\,\gamma\,\left( -1 + 2\,\gamma \right)}{2\, |
---|
| 578 | {\gamma}^2} \,\ln \left(\frac{1-x}{x}\right) |
---|
| 579 | \nonumber\\ |
---|
| 580 | \sigma^M_T&=& |
---|
| 581 | \frac{2\,\left( \gamma - 1 \right) \,\left( 2\,x -1 \right)}{2\,{\gamma}^2} |
---|
| 582 | + \frac{ |
---|
| 583 | \left( 1 - 3\,\gamma \right) }{2\,{\gamma}^2} \,\ln \left(\frac{1-x}{x}\right) |
---|
| 584 | \label{mollertotal} |
---|
| 585 | \end{eqnarray} |
---|
| 586 | |
---|
| 587 | % ---------------------------------------------------------------------- |
---|
| 588 | \subsubsection{Total Bhabha cross section} |
---|
| 589 | |
---|
| 590 | The total cross section of the polarized Bhabha scattering can be expressed as follows |
---|
| 591 | \begin{equation}\label{totalBhabha} |
---|
| 592 | \sigma^B_{pol}=\frac{2\pi r_e^2}{\gamma-1} |
---|
| 593 | \left[ |
---|
| 594 | \sigma^B_0 + \zeta_3^{(1)}\zeta_3^{(2)}\sigma^B_L + \left(\zeta_1^{(1)}\zeta_1^{(2)} + \zeta_2^{(1)}\zeta_2^{(2)}\right)\sigma^B_T |
---|
| 595 | \right], |
---|
| 596 | \end{equation} |
---|
| 597 | where |
---|
| 598 | \begin{eqnarray} |
---|
| 599 | \sigma^B_0&=& |
---|
| 600 | \frac{1 - x}{2\,\left( \gamma - 1 \right) \,x} + |
---|
| 601 | \frac{2\,\left( -1 + 3\,x - 6\,x^2 + 4\,x^3 \right) } |
---|
| 602 | {3\,{\left( 1 + \gamma \right) }^3} |
---|
| 603 | \nonumber\\ |
---|
| 604 | &+&\frac{-1 - 5\,x + 12\,x^2 - 10\,x^3 + 4\,x^4}{2\,\left( 1 + \gamma \right) \,x} |
---|
| 605 | + \frac{-3 - x + 8\,x^2 - 4\,x^3 - \ln (x)}{{\left( 1 + \gamma \right) }^2} |
---|
| 606 | \nonumber\\ |
---|
| 607 | &+&\frac{3 + 4\,x - 9\,x^2 + 3\,x^3 - x^4 + 6\,x\,\ln (x)}{3\,x} |
---|
| 608 | \nonumber\\ |
---|
| 609 | \sigma^B_L&=& |
---|
| 610 | \frac{2\,\left( 1 - 3\,x + 6\,x^2 - 4\,x^3 \right) }{3\,{\left( 1 + \gamma \right) }^3} + |
---|
| 611 | \frac{-14 + 15\,x - 3\,x^2 + 2\,x^3 - 9\,\ln (x)}{3\,\left( 1 + \gamma \right) } |
---|
| 612 | \nonumber\\ |
---|
| 613 | &+&\frac{5 + 3\,x - 12\,x^2 + 4\,x^3 + 3\,\ln (x)}{3\,{\left( 1 + \gamma \right) }^2} + |
---|
| 614 | \frac{7 - 9\,x + 3\,x^2 - x^3 + 6\,\ln (x)}{3} |
---|
| 615 | \nonumber\\ |
---|
| 616 | \sigma^B_T&=& |
---|
| 617 | \frac{2\,\left( -1 + 3\,x - 6\,x^2 + 4\,x^3 \right) }{3\,{\left( 1 + \gamma \right) }^3} + |
---|
| 618 | \frac{-7 - 3\,x + 18\,x^2 - 8\,x^3 - 3\,\ln (x)}{3\,{\left( 1 + \gamma \right) }^2} |
---|
| 619 | \nonumber\\ |
---|
| 620 | &+&\frac{5 + 3\,x - 12\,x^2 + 4\,x^3 + 9\,\ln (x)}{6\,\left( 1 + \gamma \right) } |
---|
| 621 | \end{eqnarray} |
---|
| 622 | |
---|
| 623 | % ---------------------------------------------------------------------- |
---|
| 624 | \subsubsection{Mean free path} |
---|
| 625 | |
---|
| 626 | With the help of the total polarized M{\o}ller cross section |
---|
| 627 | one can define a longitudinal asymmetry $A^M_L$ and the transverse |
---|
| 628 | asymmetry $A^M_T$, by |
---|
| 629 | |
---|
| 630 | \begin{tabular}{ccc} |
---|
| 631 | $ A^M_L = \displaystyle \frac{\sigma^M_L}{\sigma^M_0} \quad$ & and & |
---|
| 632 | $\quad A^M_T = \displaystyle \frac{\sigma^M_T}{\sigma^M_0}\;$. |
---|
| 633 | \end{tabular} |
---|
| 634 | |
---|
| 635 | Similarly, using the polarized Bhabha cross section one can introduce a |
---|
| 636 | longitudinal asymmetry $A^B_L$ and the transverse asymmetry $A^B_T$ |
---|
| 637 | via |
---|
| 638 | |
---|
| 639 | \begin{tabular}{ccc} |
---|
| 640 | $ A^B_L = \displaystyle \frac{\sigma^B_L}{\sigma^B_0} \quad$ & and & |
---|
| 641 | $\quad A^B_T = \displaystyle \frac{\sigma^B_T}{\sigma^B_0}\;$. |
---|
| 642 | \end{tabular} |
---|
| 643 | |
---|
| 644 | These asymmetries are depicted in figures \ref{pol.moller1} and |
---|
| 645 | \ref{pol.bhabha1} respectively. |
---|
| 646 | |
---|
| 647 | If both beam and target are polarized the mean free path as defined in |
---|
| 648 | section \ref{sec:em.eion} has to be modified. In the class {\em |
---|
| 649 | G4ePolarizedIonization} the polarized mean free path $\lambda^{\rm |
---|
| 650 | pol}$ is derived from the unpolarized mean free path $\lambda^{\rm |
---|
| 651 | unpol}$ via |
---|
| 652 | \begin{equation} |
---|
| 653 | \lambda^{\rm pol} = \frac{\lambda^{\rm unpol}}{1 + |
---|
| 654 | \zeta_3^{(1)}\zeta_3^{(2)}\, A_L + |
---|
| 655 | \left(\zeta_1^{(1)}\zeta_1^{(2)}+\zeta_2^{(1)}\zeta_2^{(2)}\right) \,A_T} |
---|
| 656 | \end{equation} |
---|
| 657 | |
---|
| 658 | % |
---|
| 659 | \begin{figure}[t] |
---|
| 660 | \begin{center} |
---|
| 661 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/MollerTA1.eps} |
---|
| 662 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/MollerTA2.eps} |
---|
| 663 | \end{center} |
---|
| 664 | \caption{\label{pol.moller1}M{\o}ller total cross section |
---|
| 665 | asymmetries depending on the total energy of the incoming |
---|
| 666 | electron, with a cut-off $T_{\rm cut}= 1 {\rm keV}$. Transverse |
---|
| 667 | asymmetry is plotted in blue, longitudinal asymmetry in red. Left |
---|
| 668 | part, between 0.5 MeV and 2 MeV, right part up to 10 MeV.} |
---|
| 669 | %\end{figure} |
---|
| 670 | % |
---|
| 671 | %\begin{figure}[t] |
---|
| 672 | \begin{center} |
---|
| 673 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/BhabhaTA1.eps} |
---|
| 674 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/BhabhaTA2.eps} |
---|
| 675 | \end{center} |
---|
| 676 | \caption{\label{pol.bhabha1}Bhabha total cross section |
---|
| 677 | asymmetries depending on the total energy of the incoming |
---|
| 678 | positron, with a cut-off $T_{\rm cut}= 1 {\rm keV}$. Transverse |
---|
| 679 | asymmetry is plotted in blue, longitudinal asymmetry in red. Left |
---|
| 680 | part, between 0.5 MeV and 2 MeV, right part up to 10 MeV.} |
---|
| 681 | \end{figure} |
---|
| 682 | |
---|
| 683 | |
---|
| 684 | |
---|
| 685 | |
---|
| 686 | % ---------------------------------------------------------------------- |
---|
| 687 | \subsection{Sampling the final state} |
---|
| 688 | |
---|
| 689 | \subsubsection{Differential cross section} |
---|
| 690 | |
---|
| 691 | The polarized differential cross section is rather complicated, |
---|
| 692 | the full result can be found in \cite{polIoni:Star:2006,polIoni:Ford:1957,polIoni:Stehle:1957}. |
---|
| 693 | In {\em G4PolarizedMollerCrossSection} the complete result is |
---|
| 694 | available taking all mass effects into account, only binding effects |
---|
| 695 | are neglected. |
---|
| 696 | Here we state only the ultra-relativistic approximation (URA), to show |
---|
| 697 | the general dependencies. |
---|
| 698 | \begin{eqnarray} |
---|
| 699 | &&\frac{d\sigma_{URA}^M}{d\epsilon d\varphi}= |
---|
| 700 | \frac{{{r_\epsilon}}^2}{ \gamma + 1} \times |
---|
| 701 | \nonumber\\ |
---|
| 702 | &&\Bigg[ |
---|
| 703 | \frac{{\left( 1 - \epsilon + \epsilon^2 \right) }^2}{4\,{\left( \epsilon - 1 \right) }^2\,\epsilon^2} + |
---|
| 704 | \zeta_3^{(1)}\zeta_3^{(2)}\frac{2 - \epsilon + |
---|
| 705 | \epsilon^2}{-4\,\epsilon ( 1 - \epsilon)} + |
---|
| 706 | \left(\zeta_2^{(1)}\zeta_2^{(2)} -\zeta_1^{(1)}\zeta_1^{(2)}\right)\frac{1}{4} |
---|
| 707 | \nonumber\\ |
---|
| 708 | &&+ |
---|
| 709 | \left(\xi_3^{(1)}\zeta_3^{(1)} - \xi_3^{(2)}\zeta_3^{(2)}\right) |
---|
| 710 | \frac{1 - \epsilon + 2\,\epsilon^2}{4\,\left( 1 - \epsilon \right) \,\epsilon^2} |
---|
| 711 | + \left(\xi_3^{(2)}\zeta_3^{(1)} - \xi_3^{(1)}\zeta_3^{(2)}\right) |
---|
| 712 | \frac{2 - 3\,\epsilon + 2\,\epsilon^2}{4\,{\left( 1 - \epsilon \right) }^2\,\epsilon} |
---|
| 713 | \Bigg] \nonumber\\ |
---|
| 714 | && |
---|
| 715 | \end{eqnarray} |
---|
| 716 | % |
---|
| 717 | The corresponding cross section for Bhabha cross section is |
---|
| 718 | implemented in {\em G4PolarizedBhabhaCrossSection}. In the |
---|
| 719 | ultra-relativistic approximation it reads |
---|
| 720 | \begin{eqnarray} |
---|
| 721 | &&\frac{d\sigma_{URA}^B}{d\epsilon d\varphi}= |
---|
| 722 | \frac{{{r_\epsilon}}^2}{ \gamma - 1} \times |
---|
| 723 | \nonumber\\ |
---|
| 724 | &&\Bigg[ |
---|
| 725 | \frac{{\left( 1 - \epsilon + \epsilon^2 \right) }^2}{4\,\epsilon^2} + |
---|
| 726 | \zeta_3^{(1)}\zeta_3^{(2)}\frac{\left( \epsilon - 1 \right) \,\left( 2 - \epsilon + \epsilon^2 \right) }{4\,\epsilon} |
---|
| 727 | +\left(\zeta_2^{(1)}\zeta_2^{(2)} -\zeta_1^{(1)}\zeta_1^{(2)}\right)\frac{(1-\epsilon)^2}{4} |
---|
| 728 | \nonumber\\ |
---|
| 729 | &&+ |
---|
| 730 | \left(\xi_3^{(1)}\zeta_3^{(1)} - \xi_3^{(2)}\zeta_3^{(2)}\right)\frac{1 - 2\,\epsilon + 3\,\epsilon^2 - 2\,\epsilon^3}{4\,\epsilon^2} |
---|
| 731 | + \left(\xi_3^{(2)}\zeta_3^{(1)} - \xi_3^{(1)}\zeta_3^{(2)}\right)\frac{ 2 - 3\,\epsilon + 2\,\epsilon^2}{4\epsilon} |
---|
| 732 | \Bigg] \nonumber\\ |
---|
| 733 | && |
---|
| 734 | \end{eqnarray} |
---|
| 735 | where |
---|
| 736 | \begin{tabular}[t]{l@{\ = \ }l} |
---|
| 737 | $r_e$ & classical electron radius \\ |
---|
| 738 | $\gamma$ & $E_{k_1}/m_e c^2$ \\ |
---|
| 739 | $\epsilon$ & ($E_{p_1}-m_e c^2)/(E_{k_1}-m_e c^2)$ \\ |
---|
| 740 | $E_{k_1}$ & energy of the incident electron/positron \\ |
---|
| 741 | $E_{p_1}$ & energy of the scattered electron/positron \\ |
---|
| 742 | $m_e c^2$ & electron mass \\ |
---|
| 743 | $\bvec{\zeta}^{(1)}$ & Stokes vector of the incoming electron/positron \\ |
---|
| 744 | $\bvec{\zeta}^{(2)}$ & Stokes vector of the target electron \\ |
---|
| 745 | $\bvec{\xi}^{(1)}$ & Stokes vector of the outgoing electron/positron \\ |
---|
| 746 | $\bvec{\xi}^{(2)}$ & Stokes vector of the outgoing (2nd) electron . |
---|
| 747 | \end{tabular} |
---|
| 748 | |
---|
| 749 | \subsubsection{Sampling} |
---|
| 750 | |
---|
| 751 | The delta ray is sampled according to methods discussed in Chapter |
---|
| 752 | 2. After exploitation of the symmetry in the M{\o}ller cross section |
---|
| 753 | under exchanging $\epsilon$ versus $(1-\epsilon)$, the differential |
---|
| 754 | cross section can be approximated by a simple function $f^M(\epsilon)$: |
---|
| 755 | \begin{equation} |
---|
| 756 | f^M(\epsilon) = \frac{1}{\epsilon^2} \frac{\epsilon_0}{1-2\epsilon_0} |
---|
| 757 | \end{equation} |
---|
| 758 | with the kinematic limits given by |
---|
| 759 | \begin{equation} |
---|
| 760 | \epsilon_0 = \frac{T_{\rm cut}}{E_{k_1}-m_e c^2} \le \epsilon \le |
---|
| 761 | \frac{1}{2} |
---|
| 762 | \end{equation} |
---|
| 763 | A similar function $f^B(\epsilon)$ can be found for Bhabha scattering: |
---|
| 764 | \begin{equation} |
---|
| 765 | f^B(\epsilon) = \frac{1}{\epsilon^2} \frac{\epsilon_0}{1-\epsilon_0} |
---|
| 766 | \end{equation} |
---|
| 767 | with the kinematic limits given by |
---|
| 768 | \begin{equation} |
---|
| 769 | \epsilon_0 = \frac{T_{\rm cut}}{E_{k_1}-m_e c^2} \le \epsilon \le 1 |
---|
| 770 | \end{equation} |
---|
| 771 | |
---|
| 772 | The kinematic of the delta ray production is constructed by the |
---|
| 773 | following steps: |
---|
| 774 | \begin{enumerate} |
---|
| 775 | \item $\epsilon$ is sampled from $f(\epsilon)$ |
---|
| 776 | \item calculate the differential cross section, depending on the |
---|
| 777 | initial polarizations $\bvec{\zeta}^{(1)}$ and $\bvec{\zeta}^{(2)}$. |
---|
| 778 | \item $\epsilon$ is accepted with the probability defined by ratio |
---|
| 779 | of the differential cross section over the approximation |
---|
| 780 | function. |
---|
| 781 | \item The $\varphi$ is diced uniformly. |
---|
| 782 | \item $\varphi$ is determined from the differential cross section, |
---|
| 783 | depending on the initial polarizations $\bvec{\zeta}^{(1)}$ and $\bvec{\zeta}^{(2)}$ |
---|
| 784 | \end{enumerate} |
---|
| 785 | Note, for initial states without transverse polarization components, the |
---|
| 786 | $\varphi$ distribution is always uniform. |
---|
| 787 | In figure \ref{pol.moller2} the asymmetries indicate the influence of |
---|
| 788 | polarization. In general the effect is largest around |
---|
| 789 | $\epsilon=\frac{1}{2}$. |
---|
| 790 | % |
---|
| 791 | %\begin{figure}[ht] |
---|
| 792 | %\includegraphics[scale=0.5]{electromagnetic/standard/plots/MollerXS.eps} |
---|
| 793 | %\caption{M{\o}ller differential cross section in arbitrary units. Black - unpolarized, Red - (+-),Blue (++). |
---|
| 794 | %This cross section is symmetric around point $\epsilon=1/2$. |
---|
| 795 | %} |
---|
| 796 | %\end{figure} |
---|
| 797 | %\begin{figure}[ht] |
---|
| 798 | %\includegraphics[scale=0.5]{electromagnetic/standard/plots/BhabhaXS.eps} |
---|
| 799 | %\caption{Bhabha differential cross section in arbitrary units. Black - unpolarized, Red - (+-),Blue (++)} |
---|
| 800 | %\end{figure} |
---|
| 801 | % |
---|
| 802 | \begin{figure}[ht] |
---|
| 803 | \begin{center} |
---|
| 804 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/MollerAsym.eps} |
---|
| 805 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/BhabhaAsym.eps} |
---|
| 806 | \end{center} |
---|
| 807 | %\caption{M{\o}ller differential cross section asymmetries in\%. |
---|
| 808 | %Red - ZZ, Gren - XX, Blue - YY, LightBlue -ZX} |
---|
| 809 | \caption{\label{pol.moller2}Differential cross section asymmetries in\% for M{\o}ller |
---|
| 810 | (left) and Bhabha (right) scattering ( red - $A_{ZZ}(\epsilon)$, |
---|
| 811 | green - $A_{XX}(\epsilon)$, blue - $A_{YY}(\epsilon)$, lightblue - $A_{ZX}(\epsilon)$)} |
---|
| 812 | \end{figure} |
---|
| 813 | |
---|
| 814 | After both $\phi$ and $\epsilon$ are known, the kinematic can be |
---|
| 815 | constructed fully. Using momentum conservation the momenta of the |
---|
| 816 | scattered incident particle and the ejected electron are constructed |
---|
| 817 | in global coordinate system. |
---|
| 818 | |
---|
| 819 | \subsubsection{Polarization transfer} |
---|
| 820 | |
---|
| 821 | After the kinematics is fixed the polarization properties of the |
---|
| 822 | outgoing particles are determined. Using the dependence of |
---|
| 823 | the differential cross section on the final state polarization a mean |
---|
| 824 | polarization is calculated according to method described in section |
---|
| 825 | \ref{sec:pol.intro}. |
---|
| 826 | |
---|
| 827 | The resulting polarization transfer functions $\xi^{(1,2)}_3(\epsilon)$ |
---|
| 828 | are depicted in figures \ref{pol.moller3} and \ref{pol.bhabha3}. |
---|
| 829 | |
---|
| 830 | \begin{figure}[ht] |
---|
| 831 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/MollerTransfer1.eps} |
---|
| 832 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/MollerTransfer2.eps} |
---|
| 833 | \caption{\label{pol.moller3}Polarization transfer functions in |
---|
| 834 | M{\o}ller scattering. Longitudinal polarization |
---|
| 835 | $\xi^{(2)}_3$ of electron with energy $E_{p_2}$ in blue; longitudinal |
---|
| 836 | polarization $\xi^{(1)}_3$ of second electron in red. Kinetic energy of incoming electron $T_{k_1} = 10 {\rm MeV}$}. |
---|
| 837 | \end{figure} |
---|
| 838 | |
---|
| 839 | \begin{figure}[ht] |
---|
| 840 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/BhabhaTransfer1.eps} |
---|
| 841 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/BhabhaTransfer2.eps} |
---|
| 842 | \caption{\label{pol.bhabha3}Polarization Transfer in Bhabha scattering. |
---|
| 843 | Longitudinal polarization |
---|
| 844 | $\xi^{(2)}_3$ of electron with energy $E_{p_2}$ in blue; longitudinal |
---|
| 845 | polarization $\xi^{(1)}_3$ of scattered positron. Kinetic energy of incoming positron $T_{k_1} = 10 {\rm MeV}$}. |
---|
| 846 | \end{figure} |
---|
| 847 | |
---|
| 848 | % ---------------------------------------------------------------------- |
---|
| 849 | \subsection{Status of this document} |
---|
| 850 | 20.11.06 created by P.Starovoitov\\ |
---|
| 851 | 21.02.07 minor update by A.Sch{\"a}licke\\ |
---|
| 852 | |
---|
| 853 | \begin{latexonly} |
---|
| 854 | |
---|
| 855 | \begin{thebibliography}{9} |
---|
| 856 | \bibitem{polIoni:Star:2006} P.~Starovoitov {\em et.al.}, in preparation. |
---|
| 857 | \bibitem{polIoni:Ford:1957} |
---|
| 858 | G.~W.~Ford, C.~J.~Mullin, |
---|
| 859 | Phys.~Rev.\ {\bf 108} (1957) 477. |
---|
| 860 | \bibitem{polIoni:Stehle:1957} |
---|
| 861 | P.~Stehle, |
---|
| 862 | Phys.~Rev.\ {\bf 110} (1958) 1458. |
---|
| 863 | |
---|
| 864 | \end{thebibliography} |
---|
| 865 | |
---|
| 866 | \end{latexonly} |
---|
| 867 | |
---|
| 868 | \begin{htmlonly} |
---|
| 869 | |
---|
| 870 | \subsection{Bibliography} |
---|
| 871 | \begin{enumerate} |
---|
| 872 | \item %{Star:2006} |
---|
| 873 | P.~Starovoitov {\em et.al.}, in preparation. |
---|
| 874 | \item %{Ford:1957} |
---|
| 875 | G.~W.~Ford, C.~J.~Mullin, |
---|
| 876 | Phys.~Rev.\ {\bf 108} (1957) 477. |
---|
| 877 | \item % {Stehle:1957} |
---|
| 878 | P.~Stehle, |
---|
| 879 | Phys.~Rev.\ {\bf 110} (1958) 1458. |
---|
| 880 | \end{enumerate} |
---|
| 881 | |
---|
| 882 | \end{htmlonly} |
---|
| 883 | |
---|
| 884 | |
---|
| 885 | \clearpage |
---|
| 886 | % ====================================================================== |
---|
| 887 | \section{Positron - Electron Annihilation} |
---|
| 888 | \subsection{Method} |
---|
| 889 | The class {\em G4eplusPolarizedAnnihilation} simulates |
---|
| 890 | annihilation of polarized positrons with electrons in a material. |
---|
| 891 | The implementation baseline follows the approach derived for the class |
---|
| 892 | {\em G4eplusAnnihilation} described in section |
---|
| 893 | \ref{sec:em.annil}. |
---|
| 894 | It evaluates polarization transfer and -- if the material is polarized -- |
---|
| 895 | asymmetries in the produced photons. Thus, it takes the effects of |
---|
| 896 | polarization on total cross section and mean free path, on |
---|
| 897 | distribution of final state photons into account. And |
---|
| 898 | calculates the average polarization of these generated photons. |
---|
| 899 | The material electrons are assumed to be free and at rest. |
---|
| 900 | |
---|
| 901 | \subsection{Total cross section and mean free path} |
---|
| 902 | Kinematics of annihilation process is fixed by initial energy |
---|
| 903 | \begin{equation} |
---|
| 904 | \gamma=\frac{E_{k_1}}{mc^2}%=\frac{s}{2(mc^2)^2}-1 |
---|
| 905 | \end{equation} |
---|
| 906 | and variable |
---|
| 907 | \begin{equation} |
---|
| 908 | \epsilon = \frac{E_{p_1}}{E_{k_1}+mc^2}, |
---|
| 909 | \end{equation} |
---|
| 910 | which is the part of total energy available in initial state carried out by first photon. |
---|
| 911 | This variable has the following kinematical limits |
---|
| 912 | \begin{equation} |
---|
| 913 | \frac{1}{2}\left(1-\sqrt{\frac{\gamma-1}{\gamma+1}}\right)\;<\; |
---|
| 914 | \epsilon |
---|
| 915 | \;<\;\frac{1}{2}\left(1+\sqrt{\frac{\gamma-1}{\gamma+1}}\right) |
---|
| 916 | \;. |
---|
| 917 | \end{equation} |
---|
| 918 | |
---|
| 919 | % ---------------------------------------------------------------------- |
---|
| 920 | \subsubsection{Total Cross Section} |
---|
| 921 | The total cross section of the annihilation of a polarized $e^+e^-$ |
---|
| 922 | pair into two photons could be expressed as follows |
---|
| 923 | \begin{equation}\label{totalAnnih} |
---|
| 924 | \sigma^A_{pol}=\frac{\pi r_e^2}{\gamma+1}\left[ |
---|
| 925 | \sigma^A_0 + \zeta_3^{(1)}\zeta_3^{(2)}\sigma^A_L + \left(\zeta_1^{(1)}\zeta_1^{(2)}+\zeta_2^{(1)}\zeta_2^{(2)}\right)\sigma^A_T\right], |
---|
| 926 | \end{equation} |
---|
| 927 | where |
---|
| 928 | \renewcommand{\Mvariable}[1]{\gamma} |
---|
| 929 | \begin{equation} |
---|
| 930 | \sigma^A_0= |
---|
| 931 | \frac{- \left( 3 + \Mvariable{gam} \right) \,{\sqrt{-1 + {\Mvariable{gam}}^2}} + |
---|
| 932 | \left( 1 + \Mvariable{gam}\,\left( 4 + \Mvariable{gam} \right) \right) \, |
---|
| 933 | \ln (\Mvariable{gam} + {\sqrt{-1 + {\Mvariable{gam}}^2}})}{4\, |
---|
| 934 | \left( {\Mvariable{gam}}^2 - 1 \right) } |
---|
| 935 | \end{equation} |
---|
| 936 | \begin{equation} |
---|
| 937 | \sigma^A_L= |
---|
| 938 | \frac{- {\sqrt{-1 + {\Mvariable{gam}}^2}}\, |
---|
| 939 | \left( 5 + \Mvariable{gam}\,\left( 4 + 3\,\Mvariable{gam} \right) \right) + |
---|
| 940 | \left( 3 + \Mvariable{gam}\,\left( 7 + \Mvariable{gam} + {\Mvariable{gam}}^2 \right) \right) \, |
---|
| 941 | \ln (\Mvariable{gam} + {\sqrt{{\Mvariable{gam}}^2-1 }})}{4\, |
---|
| 942 | {\left( \Mvariable{gam} -1\right) }^2\,\left( 1 + \Mvariable{gam} \right) } |
---|
| 943 | \end{equation} |
---|
| 944 | \begin{equation} |
---|
| 945 | \sigma^A_T= |
---|
| 946 | \frac{\left( 5 + \Mvariable{gam} \right) \,{\sqrt{-1 + {\Mvariable{gam}}^2}} - |
---|
| 947 | \left( 1 + 5\,\Mvariable{gam} \right) \,\ln (\Mvariable{gam} + {\sqrt{-1 + {\Mvariable{gam}}^2}})} |
---|
| 948 | {4\,{\left( -1 + \Mvariable{gam} \right) }^2\,\left( 1 + \Mvariable{gam} \right) } |
---|
| 949 | \end{equation} |
---|
| 950 | |
---|
| 951 | |
---|
| 952 | \subsubsection{Mean free path} |
---|
| 953 | |
---|
| 954 | With the help of the total polarized annihilation cross section |
---|
| 955 | one can define a longitudinal asymmetry $A^A_L$ and the transverse |
---|
| 956 | asymmetry $A^A_T$, by |
---|
| 957 | |
---|
| 958 | \begin{tabular}{ccc} |
---|
| 959 | $ A^A_L = \displaystyle \frac{\sigma^A_L}{\sigma^A_0} \quad$ & and & |
---|
| 960 | $\quad A^A_T = \displaystyle \frac{\sigma^A_T}{\sigma^A_0}\;$. |
---|
| 961 | \end{tabular} |
---|
| 962 | |
---|
| 963 | These asymmetries are depicted in figure \ref{pol.annihi1}. |
---|
| 964 | |
---|
| 965 | If both incident positron and target electron are polarized the mean |
---|
| 966 | free path as defined in section \ref{sec:em.annil} has to be |
---|
| 967 | modified. The polarized mean free path $\lambda^{\rm pol}$ is derived |
---|
| 968 | from the unpolarized mean free path $\lambda^{\rm unpol}$ via |
---|
| 969 | \begin{equation} |
---|
| 970 | \lambda^{\rm pol} = \frac{\lambda^{\rm unpol}}{1 + |
---|
| 971 | \zeta_3^{(1)}\zeta_3^{(2)}\, A_L + |
---|
| 972 | \left(\zeta_1^{(1)}\zeta_1^{(2)}+\zeta_2^{(1)}\zeta_2^{(2)}\right) \,A_T} |
---|
| 973 | \end{equation} |
---|
| 974 | |
---|
| 975 | \begin{figure}[ht] |
---|
| 976 | \begin{center} |
---|
| 977 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/AnnihTA1.eps} |
---|
| 978 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/AnnihTA2.eps} |
---|
| 979 | \end{center} |
---|
| 980 | \caption{\label{pol.annihi1}Annihilation total cross section asymmetries depending on the |
---|
| 981 | total energy of the incoming positron $E_{k_1}$. The transverse asymmetry |
---|
| 982 | is shown in blue, the longitudinal asymmetry in red. } |
---|
| 983 | \end{figure} |
---|
| 984 | |
---|
| 985 | \clearpage |
---|
| 986 | |
---|
| 987 | % ---------------------------------------------------------------------- |
---|
| 988 | \subsection{Sampling the final state} |
---|
| 989 | \subsubsection{Differential Cross Section} |
---|
| 990 | The fully polarized differential cross section is implemented in the |
---|
| 991 | class {\em G4PolarizedAnnihilationCrossSection}, which takes all mass |
---|
| 992 | effects into account, but binding effects are neglected \cite{polAnnihi:Star:2006,polAnnihi:Page:1957}. |
---|
| 993 | In the ultra-relativistic approximation (URA) and concentrating on |
---|
| 994 | longitudinal polarization states only the cross section is |
---|
| 995 | rather simple: |
---|
| 996 | \begin{eqnarray} |
---|
| 997 | \frac{d\sigma_{URA}^A}{d\epsilon d\varphi} & = & |
---|
| 998 | \frac{{{r_e}}^2}{ \gamma - 1} \times |
---|
| 999 | \Bigg( |
---|
| 1000 | \frac{1 - 2\,\epsilon + 2\,\epsilon^2}{8\,\epsilon - 8\,\epsilon^2}\left(1 + \zeta_3^{(1)}\zeta_3^{(2)}\right) |
---|
| 1001 | \nonumber\\ |
---|
| 1002 | &&\quad\quad |
---|
| 1003 | + \frac{ \left( 1 - 2\,\epsilon \right) \,\left( \zeta _{3}^{(1)} + \zeta _{3}^{(2)} \right) \, |
---|
| 1004 | \left( \xi _{3}^{(1)} - \xi _{3}^{(2)} \right) }{8\,\left( \epsilon -1 \right) \,\epsilon} |
---|
| 1005 | \Bigg) |
---|
| 1006 | \end{eqnarray} |
---|
| 1007 | % |
---|
| 1008 | where |
---|
| 1009 | \begin{tabular}[t]{l@{\ = \ }l} |
---|
| 1010 | $r_e$ & classical electron radius \\ |
---|
| 1011 | $\gamma$ & $E_{k_1}/m_e c^2$ \\ |
---|
| 1012 | $E_{k_1}$ & energy of the incident positron \\ |
---|
| 1013 | $m_e c^2$ & electron mass \\ |
---|
| 1014 | $\bvec{\zeta}^{(1)}$ & Stokes vector of the incoming positron \\ |
---|
| 1015 | $\bvec{\zeta}^{(2)}$ & Stokes vector of the target electron \\ |
---|
| 1016 | $\bvec{\xi}^{(1)}$ & Stokes vector of the 1st photon \\ |
---|
| 1017 | $\bvec{\xi}^{(2)}$ & Stokes vector of the 2nd photon . |
---|
| 1018 | \end{tabular} |
---|
| 1019 | % |
---|
| 1020 | \begin{figure}[ht] |
---|
| 1021 | \begin{center} |
---|
| 1022 | \includegraphics[width=9.5cm]{electromagnetic/standard/plots/AnnihXS.eps} |
---|
| 1023 | \end{center} |
---|
| 1024 | \caption{Annihilation differential cross section in arbitrary |
---|
| 1025 | units. Black line corresponds to unpolarized cross section; |
---|
| 1026 | red line -- to the antiparallel spins of initial particles, and blue line -- to the parallel spins. |
---|
| 1027 | Kinetic energy of the incoming positron $T_{k_1} = 10 {\rm MeV}$.} |
---|
| 1028 | \end{figure} |
---|
| 1029 | |
---|
| 1030 | \subsubsection{Sampling} |
---|
| 1031 | |
---|
| 1032 | The photon energy is sampled according to methods discussed in Chapter |
---|
| 1033 | 2. After exploitation of the symmetry in the Annihilation cross section |
---|
| 1034 | under exchanging $\epsilon$ versus $(1-\epsilon)$, the differential |
---|
| 1035 | cross section can be approximated by a simple function $f(\epsilon)$: |
---|
| 1036 | \begin{equation} |
---|
| 1037 | f(\epsilon) = \frac{1}{\epsilon} |
---|
| 1038 | \ln^{-1}\left(\frac{\epsilon_{\rm max}}{\epsilon_{\rm min}}\right) |
---|
| 1039 | \end{equation} |
---|
| 1040 | with the kinematic limits given by |
---|
| 1041 | \begin{eqnarray} |
---|
| 1042 | \epsilon_{\rm min} &=& |
---|
| 1043 | \frac{1}{2}\left(1-\sqrt{\frac{\gamma-1}{\gamma+1}}\right)\;, \nonumber\\ |
---|
| 1044 | \epsilon_{\rm max} &=& |
---|
| 1045 | \frac{1}{2}\left(1+\sqrt{\frac{\gamma-1}{\gamma+1}}\right) |
---|
| 1046 | \;. |
---|
| 1047 | \end{eqnarray} |
---|
| 1048 | |
---|
| 1049 | The kinematic of the two photon final state is constructed by the |
---|
| 1050 | following steps: |
---|
| 1051 | \begin{enumerate} |
---|
| 1052 | \item $\epsilon$ is sampled from $f(\epsilon)$ |
---|
| 1053 | \item calculate the differential cross section, depending on the |
---|
| 1054 | initial polarizations $\bvec{\zeta}^{(1)}$ and $\bvec{\zeta}^{(2)}$. |
---|
| 1055 | \item $\epsilon$ is accepted with the probability defined by the ratio |
---|
| 1056 | of the differential cross section over the approximation |
---|
| 1057 | function $f(\epsilon)$. |
---|
| 1058 | \item The $\varphi$ is diced uniformly. |
---|
| 1059 | \item $\varphi$ is determined from the differential cross section, |
---|
| 1060 | depending on the initial polarizations $\bvec{\zeta}^{(1)}$ and $\bvec{\zeta}^{(2)}$. |
---|
| 1061 | \end{enumerate} |
---|
| 1062 | A short overview over the sampling method is given in Chapter 2. |
---|
| 1063 | In figure \ref{pol.annihi2} the asymmetries indicate the influence of |
---|
| 1064 | polarization for an 10MeV incoming positron. The actual behavior is |
---|
| 1065 | very sensitive to the energy of the incoming positron. |
---|
| 1066 | |
---|
| 1067 | |
---|
| 1068 | \begin{figure}[ht] |
---|
| 1069 | \includegraphics[scale=0.5]{electromagnetic/standard/plots/AnnihAsym.eps} |
---|
| 1070 | \caption{\label{pol.annihi2}Annihilation differential cross section |
---|
| 1071 | asymmetries in\%. |
---|
| 1072 | Red line corrsponds to $A_{ZZ}(\epsilon)$, green line -- $A_{XX}(\epsilon)$, |
---|
| 1073 | blue line -- $A_{YY}(\epsilon)$, lightblue line -- $A_{ZX}(\epsilon)$). |
---|
| 1074 | Kinetic energy of the incoming positron $T_{k_1} = 10 {\rm MeV}$.} |
---|
| 1075 | \end{figure} |
---|
| 1076 | |
---|
| 1077 | \subsubsection{Polarization transfer} |
---|
| 1078 | |
---|
| 1079 | After the kinematics is fixed the polarization of the |
---|
| 1080 | outgoing photon is determined. Using the dependence of |
---|
| 1081 | the differential cross section on the final state polarizations a mean |
---|
| 1082 | polarization is calculated for each photon according to method |
---|
| 1083 | described in section \ref{sec:pol.intro}. |
---|
| 1084 | |
---|
| 1085 | The resulting polarization transfer functions $\xi^{(1,2)}(\epsilon)$ |
---|
| 1086 | are depicted in figure \ref{pol.annihi3}. |
---|
| 1087 | |
---|
| 1088 | \begin{figure}[ht] |
---|
| 1089 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/AnnihTransfer1.eps} |
---|
| 1090 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/AnnihTransfer2.eps} |
---|
| 1091 | \caption{\label{pol.annihi3} |
---|
| 1092 | Polarization Transfer in annihilation process. |
---|
| 1093 | Blue line corresponds to the circular polarization $\xi_3^{(1)}$ of the photon with energy $m(\gamma + 1)\epsilon$; |
---|
| 1094 | red line -- circular polarization $\xi_3^{(2)}$ of the photon photon with energy $m(\gamma + 1)(1-\epsilon)$.} |
---|
| 1095 | \end{figure} |
---|
| 1096 | |
---|
| 1097 | \subsection{Annihilation at Rest} |
---|
| 1098 | |
---|
| 1099 | The method \verb!AtRestDoIt! treats the special case where a positron |
---|
| 1100 | comes to rest before annihilating. It generates two photons, each with |
---|
| 1101 | energy $E_{p_{1/2}}=m c^2$ and an isotropic angular distribution. |
---|
| 1102 | %Eventhough the asymmetry for annihilation at rest is 100\% (cf.\ |
---|
| 1103 | %figure \ref{pol.annihi1}), there are always unpolarized electrons in |
---|
| 1104 | %the a material. |
---|
| 1105 | Starting with the differential cross section for annihilation with |
---|
| 1106 | positron and electron spins opposed and parallel, |
---|
| 1107 | respectively,\cite{polAnnihi:Page:1957} |
---|
| 1108 | \begin{eqnarray} |
---|
| 1109 | d\sigma_1 &=& \sim \frac{(1 - \beta^2) + \beta^2 (1 - \beta^2) (1 - |
---|
| 1110 | \cos^2\theta)^2}{(1 - \beta^2\cos^2\theta)^2} d \cos\theta \\ |
---|
| 1111 | d\sigma_2 &=& \sim \frac{\beta^2(1 - |
---|
| 1112 | \cos^4\theta)}{(1 - \beta^2\cos^2\theta)^2} d \cos\theta |
---|
| 1113 | \end{eqnarray} |
---|
| 1114 | In the limit $\beta\to0$ the cross section $d\sigma_1$ becomes one, |
---|
| 1115 | and the cross section $d\sigma_2$ vanishes. For the opposed spin |
---|
| 1116 | state, the total angular |
---|
| 1117 | momentum is zero and we have a uniform photon distribution. For the |
---|
| 1118 | parallel case the total angular momentum is 1. Here the two photon |
---|
| 1119 | final state is forbidden by angular momentum conservation, and it can |
---|
| 1120 | be assumed that higher order processes (e.g.\ three photon final |
---|
| 1121 | state) play a dominant role. However, in reality 100\% polarized |
---|
| 1122 | electron targets do not exist, consequently there are always electrons |
---|
| 1123 | with opposite spin, where the positron can annihilate with. |
---|
| 1124 | % Leading again to a uniform distribution. |
---|
| 1125 | Final state polarization does not play a role for the decay products |
---|
| 1126 | of a spin zero state, and can be safely neglected. (Is set to zero) |
---|
| 1127 | |
---|
| 1128 | \subsection{Status of this document} |
---|
| 1129 | 20.11.06 created by P.Starovoitov\\ |
---|
| 1130 | 21.02.07 minor update by A.Sch{\"a}licke\\ |
---|
| 1131 | |
---|
| 1132 | \begin{latexonly} |
---|
| 1133 | |
---|
| 1134 | \begin{thebibliography}{9} |
---|
| 1135 | \bibitem{polAnnihi:Star:2006} P.~Starovoitov {\em et.al.}, in preparation. |
---|
| 1136 | \bibitem{polAnnihi:Page:1957} |
---|
| 1137 | L.~A.~Page, |
---|
| 1138 | %Polarization Effects in the Two-Quantum Annihilation of Positrons |
---|
| 1139 | Phys.~Rev.\ {\bf 106} (1957) 394-398. |
---|
| 1140 | \end{thebibliography} |
---|
| 1141 | |
---|
| 1142 | \end{latexonly} |
---|
| 1143 | |
---|
| 1144 | \begin{htmlonly} |
---|
| 1145 | |
---|
| 1146 | \subsection{Bibliography} |
---|
| 1147 | \begin{enumerate} |
---|
| 1148 | \item P.~Starovoitov {\em et.al.}, in preparation. |
---|
| 1149 | \item L.~A.~Page, |
---|
| 1150 | %Polarization Effects in the Two-Quantum Annihilation of Positrons |
---|
| 1151 | Phys.~Rev.\ {\bf 106} (1957) 394-398. |
---|
| 1152 | \end{enumerate} |
---|
| 1153 | |
---|
| 1154 | \end{htmlonly} |
---|
| 1155 | |
---|
| 1156 | % ====================================================================== |
---|
| 1157 | \clearpage |
---|
| 1158 | \section{Polarized Compton scattering} |
---|
| 1159 | \subsection{Method} |
---|
| 1160 | The class {\em G4PolarizedCompton} simulates |
---|
| 1161 | Compton scattering of polarized photons with (possibly polarized) |
---|
| 1162 | electrons in a material. The implementation follows the approach |
---|
| 1163 | described for the class {\em G4ComptonScattering} introduced |
---|
| 1164 | in section \ref{sec:em.compton}. |
---|
| 1165 | Here the explicit production of a Compton scattered photon and the |
---|
| 1166 | ejected electron is considered taking the effects of polarization on |
---|
| 1167 | total cross section and mean free path as well as on the distribution |
---|
| 1168 | of final state particles into account. Further the average |
---|
| 1169 | polarizations of the scattered photon and electron are calculated. |
---|
| 1170 | The material electrons are assumed to be free and at rest. |
---|
| 1171 | |
---|
| 1172 | \subsection{Total cross section and mean free path} |
---|
| 1173 | |
---|
| 1174 | Kinematics of the Compton process is fixed by the initial energy |
---|
| 1175 | \begin{equation} |
---|
| 1176 | X=\frac{E_{k_1}}{mc^2} |
---|
| 1177 | \end{equation} |
---|
| 1178 | and the variable |
---|
| 1179 | \begin{equation} |
---|
| 1180 | \epsilon = \frac{E_{p_1}}{E_{k_1}}, |
---|
| 1181 | \end{equation} |
---|
| 1182 | which is the part of total energy avaible in initial state carried out |
---|
| 1183 | by scattered photon, and the scattering angle |
---|
| 1184 | \begin{equation} |
---|
| 1185 | \cos{\theta} = 1 - \frac{1}{X}\left(\frac{1}{\epsilon} - 1\right) |
---|
| 1186 | \end{equation} |
---|
| 1187 | The variable $\epsilon$ has the following limits: |
---|
| 1188 | \begin{equation} |
---|
| 1189 | \frac{1}{1+2X} \;<\; \epsilon \;<\;1 |
---|
| 1190 | \end{equation} |
---|
| 1191 | |
---|
| 1192 | |
---|
| 1193 | % ---------------------------------------------------------------------- |
---|
| 1194 | \subsubsection{Total Cross Section} |
---|
| 1195 | The total cross section of Compton scattering reads |
---|
| 1196 | \begin{equation} |
---|
| 1197 | \sigma^{C}_{pol}= |
---|
| 1198 | %\frac{\pi \,{{r_e}}^2}{4\,X^2\,{\left( 1 + 2\,X \right) }^2} |
---|
| 1199 | \frac{\pi \,{{r_e}}^2}{X^2\,{\left( 1 + 2\,X \right) }^2} |
---|
| 1200 | \left[\sigma^{C}_0 + \zeta^{(1)}_3\zeta^{(2)}_3 \sigma^{C}_L\right] |
---|
| 1201 | \end{equation} |
---|
| 1202 | where |
---|
| 1203 | \begin{equation} |
---|
| 1204 | \sigma^{C}_0 = \frac{2\,X\,\left( 2 + X\,\left( 1 + X \right) \,\left( 8 + X \right) \right) - |
---|
| 1205 | {\left( 1 + 2\,X \right) }^2\,\left( 2 + \left( 2 - X \right) \,X \right) \, |
---|
| 1206 | \ln (1 + 2\,X)}{X} |
---|
| 1207 | \end{equation} |
---|
| 1208 | and |
---|
| 1209 | \begin{equation} |
---|
| 1210 | \sigma^{C}_L = 2\,X\,\left( 1 + X\,\left( 4 + 5\,X \right) \right) - |
---|
| 1211 | \left( 1 + X \right) \,{\left( 1 + 2\,X \right) }^2\,\ln (1 + 2\,X) |
---|
| 1212 | \end{equation} |
---|
| 1213 | |
---|
| 1214 | \begin{figure}[ht] |
---|
| 1215 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/ComptonTA1.eps} |
---|
| 1216 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/ComptonTA2.eps} |
---|
| 1217 | \caption{\label{pol.compton1}Compton total cross section asymmetry depending on the energy of incoming photon. |
---|
| 1218 | Left part, between $0$ and $\sim 1$ MeV, right part -- up to 10MeV. } |
---|
| 1219 | \end{figure} |
---|
| 1220 | |
---|
| 1221 | |
---|
| 1222 | \subsubsection{Mean free path} |
---|
| 1223 | When simulating the Compton scattering of a photon with an atomic |
---|
| 1224 | electron, an empirical cross section formula is used, which reproduces |
---|
| 1225 | the cross section data down to 10 keV (see section |
---|
| 1226 | \ref{sec:em.compton}). If both, beam and target, are polarized this |
---|
| 1227 | mean free path has to be corrected. |
---|
| 1228 | |
---|
| 1229 | In the class {\em G4ComptonScattering} the polarized mean free path |
---|
| 1230 | $\lambda^{\rm pol}$ is defined on the basis of the the unpolarized |
---|
| 1231 | mean free path $\lambda^{\rm unpol}$ via |
---|
| 1232 | \begin{equation} |
---|
| 1233 | \lambda^{\rm pol} = \frac{\lambda^{\rm unpol}}{1 + |
---|
| 1234 | \zeta_3^{(1)}\zeta_3^{(2)}\, A^C_L } |
---|
| 1235 | \end{equation} |
---|
| 1236 | where |
---|
| 1237 | \begin{equation} |
---|
| 1238 | A^C_L = \displaystyle \frac{\sigma^A_L}{\sigma^A_0} |
---|
| 1239 | \end{equation} |
---|
| 1240 | is the expected asymmetry from the the total polarized Compton |
---|
| 1241 | cross section given above. |
---|
| 1242 | This asymmetry is depicted in figure \ref{pol.compton1}. |
---|
| 1243 | |
---|
| 1244 | |
---|
| 1245 | % ---------------------------------------------------------------------- |
---|
| 1246 | \subsection{Sampling the final state} |
---|
| 1247 | \subsubsection{Differential Compton Cross Section} |
---|
| 1248 | |
---|
| 1249 | In the ultra-relativistic approximation the dependence of the |
---|
| 1250 | differential cross section on the longitudinal/circular degree of |
---|
| 1251 | polarization is very simple. It reads |
---|
| 1252 | \begin{eqnarray} |
---|
| 1253 | &&\frac{d\sigma_{URA}^C}{de d\varphi}= |
---|
| 1254 | %\frac{{{r_e}}^2 \,Z}{ 4X} |
---|
| 1255 | \frac{{{r_e}}^2 }{ X} |
---|
| 1256 | \Bigg( |
---|
| 1257 | \frac{\epsilon^2 + 1}{2\,\epsilon} + |
---|
| 1258 | \frac{ \epsilon^2 -1 }{2\,\epsilon} \left(\zeta_3^{(1)}\zeta_3^{(2)} + |
---|
| 1259 | \zeta _{3}^{(2)}\,\xi _{3}^{(1)} - \zeta _{3}^{(1)}\,\xi _{3}^{(2)}\right) |
---|
| 1260 | \nonumber\\ |
---|
| 1261 | &&+\frac{\epsilon^2 + 1}{2\,\epsilon} \left( \zeta _{3}^{(1)}\,\xi _{3}^{(1)} - \zeta _{3}^{(2)} \,\xi _{3}^{(2)} \right) |
---|
| 1262 | \Bigg) |
---|
| 1263 | \end{eqnarray} |
---|
| 1264 | where |
---|
| 1265 | \begin{tabular}[t]{l@{\ = \ }l} |
---|
| 1266 | $r_e$ & classical electron radius \\ |
---|
| 1267 | $X$ & $E_{k_1}/m_e c^2$ \\ |
---|
| 1268 | $E_{k_1}$ & energy of the incident photon \\ |
---|
| 1269 | $m_e c^2$ & electron mass \\ |
---|
| 1270 | \end{tabular} |
---|
| 1271 | |
---|
| 1272 | The fully polarized differential cross section is available in the class {\em |
---|
| 1273 | G4PolarizedComptonCrossSection}. It takes all mass effects into |
---|
| 1274 | account, but binding effects are neglected \cite{polCompt:Star:2006,polCompt:Lipps:1954}. |
---|
| 1275 | The cross section dependence on $\epsilon$ for right handed circularly polarized |
---|
| 1276 | photons and longitudinally polarized electrons is plotted in figure \ref{pol.compton2a} |
---|
| 1277 | % |
---|
| 1278 | \begin{figure} |
---|
| 1279 | \includegraphics[scale=0.5]{electromagnetic/standard/plots/ComptonXS.eps} |
---|
| 1280 | \caption{\label{pol.compton2a} |
---|
| 1281 | Compton scattering differential cross section in arbitrary |
---|
| 1282 | units. Black line corresponds to the unpolarized cross section; |
---|
| 1283 | red line -- to the antiparallel spins of initial particles, and blue line -- to the parallel spins. |
---|
| 1284 | Energy of the incoming photon $E_{k_1} = 10 {\rm MeV}$. |
---|
| 1285 | } |
---|
| 1286 | \end{figure} |
---|
| 1287 | % |
---|
| 1288 | \begin{figure} |
---|
| 1289 | \includegraphics[scale=0.5]{electromagnetic/standard/plots/ComptonAsym.eps} |
---|
| 1290 | \caption{\label{pol.compton2}Compton scattering differential cross section asymmetries in\%. |
---|
| 1291 | Red line corresponds to the asymmetry due to circular photon and longitudinal electron initial state polarization, |
---|
| 1292 | green line -- due to circular photon and transverse electron initial state polarization, |
---|
| 1293 | blue line -- due to linear photon and transverse electron initial state polarization.} |
---|
| 1294 | \end{figure} |
---|
| 1295 | |
---|
| 1296 | |
---|
| 1297 | \subsubsection{Sampling} |
---|
| 1298 | |
---|
| 1299 | The photon energy is sampled according to methods discussed in Chapter |
---|
| 1300 | 2. The differential cross section can be approximated by a simple |
---|
| 1301 | function $\Phi(\epsilon)$: |
---|
| 1302 | \begin{equation} |
---|
| 1303 | \Phi(\epsilon) = \frac{1}{\epsilon} + \epsilon |
---|
| 1304 | \end{equation} |
---|
| 1305 | with the kinematic limits given by |
---|
| 1306 | \begin{eqnarray} |
---|
| 1307 | \epsilon_{\rm min} &=& \frac{1}{1+2X} \\ |
---|
| 1308 | \epsilon_{\rm max} &=& 1 |
---|
| 1309 | \end{eqnarray} |
---|
| 1310 | |
---|
| 1311 | |
---|
| 1312 | |
---|
| 1313 | |
---|
| 1314 | The kinematic of the scattered photon is constructed by the |
---|
| 1315 | following steps: |
---|
| 1316 | \begin{enumerate} |
---|
| 1317 | \item $\epsilon$ is sampled from $\Phi(\epsilon)$ |
---|
| 1318 | \item calculate the differential cross section, depending on the |
---|
| 1319 | initial polarizations $\bvec{\zeta}^{(1)}$ and $\bvec{\zeta}^{(2)}$, which |
---|
| 1320 | the correct normalization. |
---|
| 1321 | \item $\epsilon$ is accepted with the probability defined by ratio |
---|
| 1322 | of the differential cross section over the approximation |
---|
| 1323 | function. |
---|
| 1324 | \item The $\varphi$ is diced uniformly. |
---|
| 1325 | \item $\varphi$ is determined from the differential cross section, |
---|
| 1326 | depending on the initial polarizations $\bvec{\zeta}^{(1)}$ and $\bvec{\zeta}^{(2)}$. |
---|
| 1327 | \end{enumerate} |
---|
| 1328 | In figure \ref{pol.compton2} the asymmetries indicate the influence of |
---|
| 1329 | polarization for an 10MeV incoming positron. The actual behavior is |
---|
| 1330 | very sensitive to energy of the incoming positron. |
---|
| 1331 | |
---|
| 1332 | \subsubsection{Polarization transfer} |
---|
| 1333 | |
---|
| 1334 | After the kinematics is fixed the polarization of the |
---|
| 1335 | outgoing photon is determined. Using the dependence of |
---|
| 1336 | the differential cross section on the final state polarizations a mean |
---|
| 1337 | polarization is calculated for each photon according to the method |
---|
| 1338 | described in section \ref{sec:pol.intro}. |
---|
| 1339 | |
---|
| 1340 | The resulting polarization transfer functions $\xi^{(1,2)}(\epsilon)$ |
---|
| 1341 | are depicted in figure \ref{pol.compton3}. |
---|
| 1342 | |
---|
| 1343 | \begin{figure}[ht] |
---|
| 1344 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/ComptonTransfer1.eps} |
---|
| 1345 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/ComptonTransfer2.eps} |
---|
| 1346 | \caption{\label{pol.compton3} Polarization Transfer in Compton scattering. |
---|
| 1347 | Blue line corresponds to the longitudinal polarization $\xi_3^{(2)}$ of the electron, |
---|
| 1348 | red line -- circular polarization $\xi_3^{(1)}$ of the photon.} |
---|
| 1349 | \end{figure} |
---|
| 1350 | |
---|
| 1351 | \subsection{Status of this document} |
---|
| 1352 | 20.11.06 created by P.Starovoitov\\ |
---|
| 1353 | 21.02.07 corrected cross section and some minor update by A.Sch{\"a}licke\\ |
---|
| 1354 | |
---|
| 1355 | \begin{latexonly} |
---|
| 1356 | |
---|
| 1357 | \begin{thebibliography}{9} |
---|
| 1358 | \bibitem{polCompt:Star:2006} P.~Starovoitov {\em et.al.}, in preparation. |
---|
| 1359 | %\bibitem{polCompt:Stokes:1852} |
---|
| 1360 | %G.~Stokes, Trans.\ Cambridge Phil.\ Soc.\ {\bf 9} (1852) 399. |
---|
| 1361 | % |
---|
| 1362 | %\bibitem{polCompt:McMaster:1961} |
---|
| 1363 | %W.~H.~McMaster, Rev.\ Mod.\ Phys.\ {\bf 33} (1961) 8; and references therein. |
---|
| 1364 | \bibitem{polCompt:Lipps:1954} |
---|
| 1365 | F.W.~Lipps, H.A.~Tolhoek, |
---|
| 1366 | %Polarization Phenomena of Electrons and Photons I, |
---|
| 1367 | Physica {\bf 20} (1954) 85; |
---|
| 1368 | F.W.~Lipps, H.A.~Tolhoek, |
---|
| 1369 | %Polarization Phenomena of Electrons and Photons II, |
---|
| 1370 | Physica {\bf 20} (1954) 395. |
---|
| 1371 | |
---|
| 1372 | \end{thebibliography} |
---|
| 1373 | |
---|
| 1374 | \end{latexonly} |
---|
| 1375 | |
---|
| 1376 | \begin{htmlonly} |
---|
| 1377 | |
---|
| 1378 | \subsection{Bibliography} |
---|
| 1379 | \begin{enumerate} |
---|
| 1380 | \item P.~Starovoitov {\em et.al.}, in preparation. |
---|
| 1381 | \item |
---|
| 1382 | F.W.~Lipps, H.A.~Tolhoek, |
---|
| 1383 | %Polarization Phenomena of Electrons and Photons I, |
---|
| 1384 | Physica {\bf 20} (1954) 85; |
---|
| 1385 | F.W.~Lipps, H.A.~Tolhoek, |
---|
| 1386 | %Polarization Phenomena of Electrons and Photons II, |
---|
| 1387 | Physica {\bf 20} (1954) 395. |
---|
| 1388 | \end{enumerate} |
---|
| 1389 | |
---|
| 1390 | \end{htmlonly} |
---|
| 1391 | |
---|
| 1392 | |
---|
| 1393 | \newpage |
---|
| 1394 | \section{Polarized Bremsstrahlung for electron and positron}\label{sec:pol.bremsstrahlung} |
---|
| 1395 | \subsection{Method} |
---|
| 1396 | |
---|
| 1397 | The polarized version of Bremsstrahlung is based on the unpolarized |
---|
| 1398 | cross section. Energy loss, mean free path, and distribution of |
---|
| 1399 | explicitly generated final state particles are treated by the |
---|
| 1400 | unpolarized version {\em G4eBremsstrahlung}. For details consult |
---|
| 1401 | section \ref{sec:em.ebrem}. |
---|
| 1402 | |
---|
| 1403 | The remaining task is to attribute polarization vectors to the |
---|
| 1404 | generated final state particles, which is discussed in the following. |
---|
| 1405 | |
---|
| 1406 | \subsection{Polarization in gamma conversion and brems\-strahlung} |
---|
| 1407 | |
---|
| 1408 | Gamma conversion and bremsstrahlung are cross-symmetric processes |
---|
| 1409 | (i.e. the Feynman diagram for electron bremsstrahlung can be obtained |
---|
| 1410 | from the gamma conversion diagram by flipping the incoming photon and |
---|
| 1411 | outgoing positron lines) and their cross sections closely related. For |
---|
| 1412 | both processes, the interaction occurs in the field of the nucleus and |
---|
| 1413 | the total and differential cross section are polarization |
---|
| 1414 | independent. Therefore, only the polarization transfer from the |
---|
| 1415 | polarized incoming particle to the outgoing particles is taken into |
---|
| 1416 | account. |
---|
| 1417 | % |
---|
| 1418 | \begin{figure}[htb] |
---|
| 1419 | \begin{center} |
---|
| 1420 | \includegraphics [scale=.33] {electromagnetic/standard/plots/Fyn_diag.eps} |
---|
| 1421 | \caption {Feynman diagrams of Gamma conversion and bremsstrahlung processes.} |
---|
| 1422 | \end{center} |
---|
| 1423 | \end{figure} |
---|
| 1424 | |
---|
| 1425 | |
---|
| 1426 | \noindent |
---|
| 1427 | For both processes, the scattering can be formulated by: |
---|
| 1428 | \begin{equation} |
---|
| 1429 | \mathcal{K}_{1}(k_{1},\bvec{\zeta}^{(1)}) + \mathcal{N}_{1}(k_{\mathcal |
---|
| 1430 | {N}_{1}}, \bvec{\zeta}^{(\mathcal {N}_{1})}) |
---|
| 1431 | \longrightarrow |
---|
| 1432 | \mathcal{P}_{1}(p_{1},\bvec{\xi}^{(1)}) + \mathcal{P}_{2}(p_{2},\bvec{\xi}^{(2)}) + \mathcal{N}_{2}(p_{\mathcal{N}_{2}}, \bvec{\xi}^{(\mathcal{N}_{2})}) |
---|
| 1433 | \end{equation} |
---|
| 1434 | % |
---|
| 1435 | Where $\mathcal{N}_{1}(k_{\mathcal {N}_{1}}, \bvec{\zeta}^{(\mathcal |
---|
| 1436 | {N}_{1})})$ and $\mathcal{N}_{2}(p_{\mathcal{N}_{2}}, |
---|
| 1437 | \bvec{\xi}^{(\mathcal{N}_{2})})$ are the initial and final state of the |
---|
| 1438 | field of the nucleus respectively assumed to be unchanged, at rest and |
---|
| 1439 | unpolarized. This leads to $k_{\mathcal {N}_{1}} = k_{\mathcal |
---|
| 1440 | {N}_{2}} = 0$ and $\bvec{\zeta}^{(\mathcal {N}_{1})} = |
---|
| 1441 | \bvec{\xi}^{(\mathcal{N}_{2})} = 0$ |
---|
| 1442 | |
---|
| 1443 | % Gamma conversion process |
---|
| 1444 | \textbf{In the case of gamma conversion process}:\\ |
---|
| 1445 | $\mathcal{K}_{1}(k_{1},\bvec{\zeta}^{(1)})$ is the incoming photon initial |
---|
| 1446 | state with momentum $k_{1}$ and polarization state $\bvec{\zeta}^{(1)}$. \\ |
---|
| 1447 | $\mathcal{P}_{1}(p_{1},\bvec{\xi}^{(1)})$ and |
---|
| 1448 | $\mathcal{P}_{2}(p_{2},\bvec{\xi}^{(2)})$ are the two photons final states with |
---|
| 1449 | momenta $p_{1}$ and $p_{2}$ and polarization states $\bvec{\xi}^{(1)}$ and $\bvec{\xi}^{(2)}$. |
---|
| 1450 | |
---|
| 1451 | % Bremsstrahlung process |
---|
| 1452 | \textbf{In the case of bremsstrahlung process}:\\ |
---|
| 1453 | $\mathcal{K}_{1}(k_{1},\bvec{\zeta}^{(1)})$ is the incoming lepton |
---|
| 1454 | $e^{-}(e^{+})$ initial state with momentum $k_{1}$ and polarization |
---|
| 1455 | state $\bvec{\zeta}^{(1)}$. \\ |
---|
| 1456 | $\mathcal{P}_{1}(p_{1},\bvec{\xi}^{(1)})$ is the lepton $e^{-}(e^{+})$ final |
---|
| 1457 | state with momentum $p_{1}$ and polarization state $\bvec{\xi}^{(1)}$. \\ |
---|
| 1458 | $\mathcal{P}_{2}(p_{2},\bvec{\xi}^{(2)})$ is the bremsstrahlung photon in |
---|
| 1459 | final state with momentum $p_{2}$ and polarization state $\bvec{\xi}^{(2)}$. |
---|
| 1460 | |
---|
| 1461 | \subsection[Polarization transfer to the photon]{Polarization transfer from the lepton $e^{-}(e^{+})$ to a photon} |
---|
| 1462 | The polarization transfer from an electron (positron) to a photon in a |
---|
| 1463 | brems\-strahlung process was first calculated by Olsen and Maximon |
---|
| 1464 | \cite{polBrems:Olsen_Maximon} taking into account both Coulomb and screening |
---|
| 1465 | effects. In the Stokes vector formalism, the $e^{-}(e^{+})$ |
---|
| 1466 | polarization state can be transformed to a photon polarization finale |
---|
| 1467 | state by means of interaction matrix $T_{\gamma}^{b}$. It defined via |
---|
| 1468 | % |
---|
| 1469 | \begin{equation} |
---|
| 1470 | \left(\begin{array}{c} |
---|
| 1471 | O \\ |
---|
| 1472 | \bvec{\xi}^{(2)} |
---|
| 1473 | \end{array}\right) |
---|
| 1474 | = T_{\gamma}^{b} \, |
---|
| 1475 | \left(\begin{array}{c} |
---|
| 1476 | 1 \\ |
---|
| 1477 | \bvec{\zeta}^{(1)} |
---|
| 1478 | \end{array}\right)\;, |
---|
| 1479 | \label{eq:brem_gamma} |
---|
| 1480 | \end{equation} |
---|
| 1481 | % |
---|
| 1482 | and |
---|
| 1483 | % |
---|
| 1484 | \begin{equation} |
---|
| 1485 | T_{\gamma}^{b}\approx |
---|
| 1486 | \left( |
---|
| 1487 | \begin{array}{cccc} |
---|
| 1488 | 1 & 0 & 0 & 0 \\ |
---|
| 1489 | D & 0 & 0 & 0 \\ |
---|
| 1490 | 0 & 0 & 0 & 0 \\ |
---|
| 1491 | 0 & T & 0 & L \\ |
---|
| 1492 | \end{array} |
---|
| 1493 | \right)\;, |
---|
| 1494 | \label{eq:matrix_brem_g} |
---|
| 1495 | \end{equation} |
---|
| 1496 | % |
---|
| 1497 | where |
---|
| 1498 | \begin{eqnarray} |
---|
| 1499 | I &=& (\epsilon_{1}^{2}+\epsilon_{2}^{2})(3+2\Gamma)-2\epsilon_{1}\epsilon_{2}(1+4u^{2}\hat\xi^{2}\Gamma)\\ |
---|
| 1500 | D &=& \left\lbrace 8\epsilon_{1}\epsilon_{2}u^{2}\hat\xi^{2}\Gamma \right\rbrace / I\\ |
---|
| 1501 | T &=& \left\lbrace -4k\epsilon_{2}\hat\xi(1-2\hat\xi)u \Gamma \right\rbrace / I \\ |
---|
| 1502 | L &=& |
---|
| 1503 | k\lbrace(\epsilon_{1}+\epsilon_{2})(3+2\Gamma)-2\epsilon_{2}(1+4u^{2}\hat\xi^{2}\Gamma)\rbrace |
---|
| 1504 | / I |
---|
| 1505 | \label{eq:polbremdef} |
---|
| 1506 | \end{eqnarray} |
---|
| 1507 | % |
---|
| 1508 | and |
---|
| 1509 | % |
---|
| 1510 | \begin{center} |
---|
| 1511 | \begin{tabular}{ll} |
---|
| 1512 | $\epsilon_{1}$ & Total energy of the incoming lepton $e^{+}(e^{-})$ in units $mc^{2}$\\ |
---|
| 1513 | $\epsilon_{2}$ & Total energy of the outgoing lepton $e^{+}(e^{-})$ in units $mc^{2}$\\ |
---|
| 1514 | $k$ &$=(\epsilon_{1}-\epsilon_{2})$, the energy of the bremsstrahlung photon in units of $mc^{2}$ |
---|
| 1515 | \\ |
---|
| 1516 | $\bvec{p}$ & Electron (positron) initial momentum in units $mc$\\ |
---|
| 1517 | $\bvec{k}$ & Bremsstrahlung photon momentum in units $mc$\\ |
---|
| 1518 | $\bvec{u}$ & Component of $\bvec{p}$ |
---|
| 1519 | perpendicular to $\bvec{k}$ in units $mc$ and $u=\vert \bvec{u} \vert $\\ |
---|
| 1520 | $\hat\xi$ & $ = 1/(1+u^{2})$ |
---|
| 1521 | \end{tabular} |
---|
| 1522 | \end{center} |
---|
| 1523 | % |
---|
| 1524 | Coulomb and screening effects are contained in \(\Gamma\), defined as |
---|
| 1525 | follows |
---|
| 1526 | \begin{eqnarray} |
---|
| 1527 | \Gamma &=& \ln\left(\frac{1}{\delta}\right)-2-f(Z)+ |
---|
| 1528 | \mathcal{F}\left(\frac{\hat\xi}{\delta}\right) \quad \mbox{for } \Delta \le 120 \\ |
---|
| 1529 | \Gamma &=& \ln\left( \frac{111}{\hat\xi Z^{\frac{1}{3}}}\right)-2-f(z) |
---|
| 1530 | \quad \mbox{for } \Delta \ge 120 |
---|
| 1531 | \end{eqnarray} |
---|
| 1532 | % |
---|
| 1533 | with |
---|
| 1534 | % |
---|
| 1535 | \begin{eqnarray} |
---|
| 1536 | \Delta &=& \frac{12 Z^{\frac{1}{3}}\epsilon_{1}\epsilon_{2} \hat\xi}{121 |
---|
| 1537 | k} \quad \mbox{with $Z$ the atomic number and } \delta = |
---|
| 1538 | \frac{k}{2\epsilon_{1}\epsilon{2}} |
---|
| 1539 | \end{eqnarray} |
---|
| 1540 | % |
---|
| 1541 | % |
---|
| 1542 | \noindent |
---|
| 1543 | $f(Z)$ is the coulomb correction term derived by Davies, Bethe |
---|
| 1544 | and Maximon \cite{polBrems:Davise}. |
---|
| 1545 | $ \mathcal{F}({\hat\xi}/{\delta})$ contains the screening effects |
---|
| 1546 | and is zero for $\Delta \le 0.5 $ (No screening effects). For $0.5 \le |
---|
| 1547 | \Delta \le 120 $ (intermediate screening) it is a slowly decreasing |
---|
| 1548 | function. The $\mathcal{F}({\hat\xi}/{\delta})$ values versus |
---|
| 1549 | $\Delta$ are given in table \ref{koch} and used with a linear |
---|
| 1550 | interpolation in between. |
---|
| 1551 | |
---|
| 1552 | The polarization vector of the incoming $e^{-}(e^{+})$ must be rotated |
---|
| 1553 | into the frame defined by the scattering plane (x-z-plane) and the |
---|
| 1554 | direction of the outgoing photon (z-axis). The resulting polarization |
---|
| 1555 | vector of the bremsstrahlung photon is also given in this frame. |
---|
| 1556 | \begin{table}[h] |
---|
| 1557 | \caption{$ \mathcal{F}({\hat\xi}/{\delta})$ for intermediate values of the screening factor \cite{polBrems:koch}.} |
---|
| 1558 | \label{koch} |
---|
| 1559 | \begin{center} |
---|
| 1560 | \begin{tabular}{|cc|cc|} |
---|
| 1561 | \hline |
---|
| 1562 | $\Delta$ &$ -\mathcal{F}\left({\hat\xi}/{\delta}\right)$ & $\Delta$& $ -\mathcal{F}\left({\hat\xi}/{\delta}\right)$\\ |
---|
| 1563 | \hline |
---|
| 1564 | 0.5 & 0.0145 & 40.0 & 2.00 \\ |
---|
| 1565 | 1.0 & 0.0490 & 45.0 & 2.114\\ |
---|
| 1566 | 2.0 & 0.1400 & 50.0 & 2.216\\ |
---|
| 1567 | 4.0 & 0.3312 & 60.0 & 2.393\\ |
---|
| 1568 | 8.0 & 0.6758 & 70.0 & 2.545\\ |
---|
| 1569 | 15.0 & 1.126 & 80.0 & 2.676\\ |
---|
| 1570 | 20.0 & 1.367 & 90.0 & 2.793\\ |
---|
| 1571 | 25.0 & 1.564 & 100.0 & 2.897\\ |
---|
| 1572 | 30.0 & 1.731 & 120.0 & 3.078\\ |
---|
| 1573 | 35.0 & 1.875 & & \\ |
---|
| 1574 | \hline |
---|
| 1575 | \end{tabular} |
---|
| 1576 | \end{center} |
---|
| 1577 | \end{table} |
---|
| 1578 | % |
---|
| 1579 | Using Eq.\ (\ref{eq:brem_gamma}) and the transfer matrix given by |
---|
| 1580 | Eq.\ (\ref{eq:matrix_brem_g}) the bremsstrahlung photon polarization |
---|
| 1581 | state in the Stokes formalism \cite{polBrems:McMaster1, polBrems:McMaster2} is given by |
---|
| 1582 | % |
---|
| 1583 | \begin{equation} |
---|
| 1584 | \xi^{(2)} = \left( |
---|
| 1585 | \begin{array}{c} |
---|
| 1586 | \xi_{1}^{(2)}\\ |
---|
| 1587 | \xi_{2}^{(2)} \\ |
---|
| 1588 | \xi_{3}^{(2)} \\ |
---|
| 1589 | \end{array} |
---|
| 1590 | \right) |
---|
| 1591 | \approx |
---|
| 1592 | \left( |
---|
| 1593 | \begin{array}{c} |
---|
| 1594 | D \\ |
---|
| 1595 | 0 \\ |
---|
| 1596 | \zeta_{1}^{(1)}L + \zeta_{2}^{(1)}T \\ |
---|
| 1597 | \end{array} |
---|
| 1598 | \right) |
---|
| 1599 | \end{equation} |
---|
| 1600 | |
---|
| 1601 | \subsection[Polarization transfer to the lepton]{Remaining polarization of the lepton after emitting a bremsstrahlung photon} |
---|
| 1602 | The \(e^{-}(e^{+})\) polarization final state after emitting a |
---|
| 1603 | bremsstrahlung photon can be calculated using the interaction matrix |
---|
| 1604 | \(T_{l}^{b}\) which describes the lepton depolarization. The |
---|
| 1605 | polarization vector for the outgoing \(e^{-}(e^{+})\) is not given by |
---|
| 1606 | Olsen and Maximon. However, their results can be used to calculate the |
---|
| 1607 | following transfer matrix \cite{polBrems:klausFl,polBrems:hoogduin}. |
---|
| 1608 | % |
---|
| 1609 | \begin{equation} |
---|
| 1610 | \left(\begin{array}{c} |
---|
| 1611 | O \\ |
---|
| 1612 | \bvec{\xi}^{(1)} |
---|
| 1613 | \end{array}\right) |
---|
| 1614 | = T_{l}^{b} \, |
---|
| 1615 | \left(\begin{array}{c} |
---|
| 1616 | 1 \\ |
---|
| 1617 | \bvec{\zeta}^{(1)} |
---|
| 1618 | \end{array}\right) |
---|
| 1619 | \label{eq:brem_lepton} |
---|
| 1620 | \end{equation} |
---|
| 1621 | % |
---|
| 1622 | \begin{equation} |
---|
| 1623 | T_{l}^{b}\approx |
---|
| 1624 | \left( |
---|
| 1625 | \begin{array}{cccc} |
---|
| 1626 | 1 & 0 & 0 & 0 \\ |
---|
| 1627 | D & M & 0 & E \\ |
---|
| 1628 | 0 & 0 & M & 0 \\ |
---|
| 1629 | 0 & F & 0 & M+P \\ |
---|
| 1630 | \end{array} |
---|
| 1631 | \right) |
---|
| 1632 | \label{eq:matrix_brem_l} |
---|
| 1633 | \end{equation} |
---|
| 1634 | % |
---|
| 1635 | where |
---|
| 1636 | % |
---|
| 1637 | \begin{eqnarray} |
---|
| 1638 | I &=&(\epsilon_{1}^{2}+\epsilon_{2}^{2})(3+2\Gamma)-2\epsilon_{1}\epsilon_{2}(1+4u^{2}\hat\xi^{2}\Gamma)\\ |
---|
| 1639 | F &=& \epsilon_{2} \left\lbrace 4k\hat\xi u (1-2\hat\xi)\Gamma\right\rbrace /I \\ |
---|
| 1640 | E &=& \epsilon_{1} \left\lbrace 4k\hat\xi u (2\hat\xi-1)\Gamma \right\rbrace /I\\ |
---|
| 1641 | M &=& \left\lbrace 4k\epsilon_{1}\epsilon_{2}(1+\Gamma - 2 u^{2}\hat\xi^{2} \Gamma)\right\rbrace / I \\ |
---|
| 1642 | P &=& \left\lbrace k^{2} (1+8 \Gamma(\hat\xi - 0.5)^{2}\right\rbrace / I |
---|
| 1643 | \end{eqnarray} |
---|
| 1644 | % |
---|
| 1645 | and |
---|
| 1646 | % |
---|
| 1647 | \begin{center} |
---|
| 1648 | \begin{tabular}{ll} |
---|
| 1649 | $\epsilon_{1}$ & Total energy of the incoming $e^{+}/e^{-}$ in units $mc^{2}$\\ |
---|
| 1650 | $\epsilon_{2}$ & Total energy of the outgoing $e^{+}/e^{-}$ in units $mc^{2}$\\ |
---|
| 1651 | $k$ & $=(\epsilon_{1}-\epsilon_{2})$, energy of the photon in units of $mc^{2}$\\ |
---|
| 1652 | $\bvec{p}$ & Electron (positron) initial momentum in units $mc$\\ |
---|
| 1653 | $\bvec{k}$ & Photon momentum in units $mc$\\ |
---|
| 1654 | $\bvec{u}$ & Component of $\bvec{p}$ |
---|
| 1655 | perpendicular to $\bvec{k}$ in units $mc$ and $u=\vert \bvec{u} \vert $ |
---|
| 1656 | \end{tabular} |
---|
| 1657 | \end{center} |
---|
| 1658 | |
---|
| 1659 | Using Eq.\ (\ref{eq:brem_lepton}) and the transfer matrix given by |
---|
| 1660 | Eq.\ (\ref{eq:matrix_brem_l}) the \(e^{-}(e^{+})\) polarization state |
---|
| 1661 | after emitting a bremsstrahlung photon is given in the Stokes |
---|
| 1662 | formalism by |
---|
| 1663 | % |
---|
| 1664 | \begin{equation} |
---|
| 1665 | \xi^{(1)} = \left( |
---|
| 1666 | \begin{array}{c} |
---|
| 1667 | \xi_{1}^{(1)}\\ |
---|
| 1668 | \xi_{2}^{(1)} \\ |
---|
| 1669 | \xi_{3}^{(1)} \\ |
---|
| 1670 | \end{array} |
---|
| 1671 | \right) |
---|
| 1672 | \approx |
---|
| 1673 | \left( |
---|
| 1674 | \begin{array}{c} |
---|
| 1675 | \zeta_{1}^{(1)} M + \zeta_{3}^{(1)} E \\ |
---|
| 1676 | \zeta_{2}^{(1)} M \\ |
---|
| 1677 | \zeta_{3}^{(1)}(M+P) + \zeta_{1}^{(1)} F \\ |
---|
| 1678 | \end{array} |
---|
| 1679 | \right) |
---|
| 1680 | \;. |
---|
| 1681 | \end{equation} |
---|
| 1682 | |
---|
| 1683 | \subsection{Status of this document} |
---|
| 1684 | 20.11.06 created by K.Laihem\\ |
---|
| 1685 | 21.02.07 minor update by A.Sch{\"a}licke\\ |
---|
| 1686 | 27.11.08 correction in Eq.\ \eqref{eq:polbremdef} by A.Sch{\"a}licke |
---|
| 1687 | |
---|
| 1688 | \begin{latexonly} |
---|
| 1689 | |
---|
| 1690 | \begin{thebibliography}{7} |
---|
| 1691 | |
---|
| 1692 | \bibitem{polBrems:Olsen_Maximon} H.~Olsen and L.C.~Maximon. Photon and electron polarization in high- energy bremsstrahlung and pair production with screening. Physical Review, 114:887-904, 1959. |
---|
| 1693 | |
---|
| 1694 | \bibitem{polBrems:McMaster1} W.H.~McMaster. Polarization and the Stokes parameters. American Journal of Physics, 22(6):351-362, 1954. |
---|
| 1695 | |
---|
| 1696 | \bibitem{polBrems:McMaster2}W.H.~McMaster. Matrix representation of polarization. Reviews of Modern Physics, 33(1):8-29, 1961. |
---|
| 1697 | |
---|
| 1698 | \bibitem{polBrems:klausFl}K.~Fl{\"o}ttmann. Investigations toward the development of polarized and unpolarized high intensity positron sources for linear colliders. PhD thesis, Universitat Hamburg, 1993. |
---|
| 1699 | |
---|
| 1700 | \bibitem{polBrems:hoogduin}Hoogduin, Johannes Marinus, Electron, positron and photon polarimetry. PhD thesis, Rijksuniversiteit Groningen 1997. |
---|
| 1701 | |
---|
| 1702 | \bibitem{polBrems:Davise}H.~Davies, H.A.~Bethe and L.C.~Maximon, Theory of Bremsstrahlung and Pair Production. II. Integral Cross Section for Pair Production, Physical Review, 93(4):788-795, 1954. |
---|
| 1703 | |
---|
| 1704 | \bibitem{polBrems:koch}H.W.~Koch and J.W.~Motz, Bremsstrahlung cross-section formulas and related data. Review Mod. Phys., 31(4):920-955, 1959. |
---|
| 1705 | |
---|
| 1706 | \bibitem{polBrems:Laihem:thesis} |
---|
| 1707 | K.~Laihem, PhD thesis, Measurement of the positron polarization at an |
---|
| 1708 | helical undulator based positron source for the International Linear |
---|
| 1709 | Collider ILC, Humboldt University Berlin, Germany, (2008). |
---|
| 1710 | |
---|
| 1711 | \end{thebibliography} |
---|
| 1712 | \end{latexonly} |
---|
| 1713 | |
---|
| 1714 | \begin{htmlonly} |
---|
| 1715 | |
---|
| 1716 | \subsection{Bibliography} |
---|
| 1717 | \begin{enumerate} |
---|
| 1718 | |
---|
| 1719 | \item H.~Olsen and L.C.~Maximon. Photon and electron polarization in high- energy bremsstrahlung and pair production with screening. Physical Review, 114:887-904, 1959. |
---|
| 1720 | |
---|
| 1721 | \item W.H.~McMaster. Polarization and the Stokes parameters. American Journal of Physics, 22(6):351-362, 1954. |
---|
| 1722 | |
---|
| 1723 | \item W.H.~McMaster. Matrix representation of polarization. Reviews of Modern Physics, 33(1):8-29, 1961. |
---|
| 1724 | |
---|
| 1725 | \item K.~Fl{\"o}ttmann. Investigations toward the development of polarized and unpolarized high intensity positron sources for linear colliders. PhD thesis, Universitat Hamburg, 1993. |
---|
| 1726 | |
---|
| 1727 | \item Hoogduin, Johannes Marinus. Electron, positron and photon polarimetry. PhD thesis, Rijksuniversiteit Groningen 1997. |
---|
| 1728 | |
---|
| 1729 | \item H.~Davies, H.A.~Bethe and L.C.~Maximon. Theory of Bremsstrahlung and Pair Production. II. Integral Cross Section for Pair Production. Physical Review, 93(4):788-795, 1954. |
---|
| 1730 | |
---|
| 1731 | \item H.W.~Koch and J.W.~Motz. Bremsstrahlung cross-section formulas and related data. Review Mod. Phys., 31(4):920-955, 1959. |
---|
| 1732 | |
---|
| 1733 | \item |
---|
| 1734 | K.~Laihem, PhD thesis, Measurement of the positron polarization at an |
---|
| 1735 | helical undulator based positron source for the International Linear |
---|
| 1736 | Collider ILC, Humboldt University Berlin, Germany, (2008). |
---|
| 1737 | |
---|
| 1738 | \end{enumerate} |
---|
| 1739 | |
---|
| 1740 | \end{htmlonly} |
---|
| 1741 | |
---|
| 1742 | \newpage |
---|
| 1743 | \section{Polarized Gamma conversion into an electron--positron pair}\label{sec:pol.conv} |
---|
| 1744 | \subsection{Method} |
---|
| 1745 | |
---|
| 1746 | The polarized version of gamma conversion is based on the EM standard |
---|
| 1747 | process {\em G4GammaConversion}. Mean free path and the distribution |
---|
| 1748 | of explicitly generated final state particles are treated by this |
---|
| 1749 | version. For details consult |
---|
| 1750 | section \ref{sec:em.conv}. |
---|
| 1751 | |
---|
| 1752 | The remaining task is to attribute polarization vectors to the |
---|
| 1753 | generated final state leptons, which is discussed in the following. |
---|
| 1754 | |
---|
| 1755 | |
---|
| 1756 | \subsection[Polarization transfer]{Polarization transfer from the photon to the two leptons} |
---|
| 1757 | Gamma conversion process is essentially the inverse process of |
---|
| 1758 | Bremsstrahlung and the interaction matrix is obtained by inverting the |
---|
| 1759 | rows and columns of the bremsstrahlung matrix and changing the sign of |
---|
| 1760 | \(\epsilon_{2}\), cf.\ section \ref{sec:pol.bremsstrahlung}. It |
---|
| 1761 | follows from the work by Olsen and Maximon |
---|
| 1762 | \cite{polPair:Olsen_Maximon} that the polarization state \(\xi^{(1)}\) of an |
---|
| 1763 | electron or positron after pair production is obtained by |
---|
| 1764 | % |
---|
| 1765 | \begin{equation} |
---|
| 1766 | \left(\begin{array}{c} |
---|
| 1767 | O \\ |
---|
| 1768 | \bvec{\xi}^{(1)} |
---|
| 1769 | \end{array}\right) |
---|
| 1770 | = T_{l}^{p} \, |
---|
| 1771 | \left(\begin{array}{c} |
---|
| 1772 | 1 \\ |
---|
| 1773 | \bvec{\zeta}^{(1)} |
---|
| 1774 | \end{array}\right) |
---|
| 1775 | \label{eq:conv_lepton} |
---|
| 1776 | \end{equation} |
---|
| 1777 | % |
---|
| 1778 | and |
---|
| 1779 | % |
---|
| 1780 | \begin{equation} |
---|
| 1781 | T_{l}^{p}\approx |
---|
| 1782 | \left( |
---|
| 1783 | \begin{array}{cccc} |
---|
| 1784 | 1 & D & 0 & 0 \\ |
---|
| 1785 | 0 & 0 & 0 & T \\ |
---|
| 1786 | 0 & 0 & 0 & 0 \\ |
---|
| 1787 | 0 & 0 & 0 & L \\ |
---|
| 1788 | \end{array} |
---|
| 1789 | \right) |
---|
| 1790 | \;, |
---|
| 1791 | \label{eq:matrix_conv} |
---|
| 1792 | \end{equation} |
---|
| 1793 | % |
---|
| 1794 | where |
---|
| 1795 | \begin{eqnarray} |
---|
| 1796 | I &=& (\epsilon_{1}^{2}+\epsilon_{2}^{2})(3+2\Gamma)+2\epsilon_{1}\epsilon_{2}(1+4u^{2}\hat\xi^{2}\Gamma)\\ |
---|
| 1797 | D &=& \left\lbrace -8\epsilon_{1}\epsilon_{2}u^{2}\hat\xi^{2}\Gamma \right\rbrace / I\\ |
---|
| 1798 | T &=& \left\lbrace -4k\epsilon_{2}\hat\xi(1-2\hat\xi)u \Gamma \right\rbrace / I \\ |
---|
| 1799 | L &=& |
---|
| 1800 | k\lbrace(\epsilon_{1}-\epsilon_{2})(3+2\Gamma)+2\epsilon_{2}(1+4u^{2}\hat\xi^{2}\Gamma)\rbrace/ |
---|
| 1801 | I |
---|
| 1802 | \label{eq:polpairdef} |
---|
| 1803 | \end{eqnarray} |
---|
| 1804 | and |
---|
| 1805 | \begin{center} |
---|
| 1806 | \begin{tabular}{ll} |
---|
| 1807 | $\epsilon_{1}$ & total energy of the first lepton $e^{+}(e^{-})$ in units $mc^{2}$\\ |
---|
| 1808 | $\epsilon_{2}$ & total energy of the second lepton $e^{-}(e^{+})$ in units $mc^{2}$\\ |
---|
| 1809 | $k=(\epsilon_{1}+\epsilon_{2})$ & energy of the incoming photon in units of $mc^{2}$\\ |
---|
| 1810 | $\bvec{p}$ & electron=positron initial momentum in units $mc$\\ |
---|
| 1811 | $\bvec{k}$ & photon momentum in units $mc$\\ |
---|
| 1812 | $\bvec{u}$ & electron/positron initial momentum in units $mc$\\ |
---|
| 1813 | $u$ & $=\vert \bvec{u} \vert $ |
---|
| 1814 | \end{tabular} |
---|
| 1815 | \end{center} |
---|
| 1816 | % |
---|
| 1817 | %Here, $\epsilon_{1}(\epsilon_{2})$ is the energy of the observed |
---|
| 1818 | %electron or positron. The matrix (\ref{eq:matrix_conv}) for pair |
---|
| 1819 | %production is the transpose of matrix (\ref{eq:matrix_brem_g}). |
---|
| 1820 | Coulomb and screening effects are contained in \(\Gamma\), defined in |
---|
| 1821 | section \ref{sec:pol.bremsstrahlung}. |
---|
| 1822 | |
---|
| 1823 | |
---|
| 1824 | Using Eq.\ (\ref{eq:conv_lepton}) and the transfer matrix given by |
---|
| 1825 | Eq.\ (\ref{eq:matrix_conv}) the polarization state of |
---|
| 1826 | the produced $e^{-}(e^{+})$ is given in the Stokes formalism by: |
---|
| 1827 | |
---|
| 1828 | \begin{equation} |
---|
| 1829 | \xi^{(1)} = \left( |
---|
| 1830 | \begin{array}{c} |
---|
| 1831 | \xi_{1}^{(1)}\\ |
---|
| 1832 | \xi_{2}^{(1)} \\ |
---|
| 1833 | \xi_{3}^{(1)} \\ |
---|
| 1834 | \end{array} |
---|
| 1835 | \right) |
---|
| 1836 | \approx |
---|
| 1837 | \left( |
---|
| 1838 | \begin{array}{c} |
---|
| 1839 | \zeta_{3}^{(1)} T \\ |
---|
| 1840 | 0 \\ |
---|
| 1841 | \zeta_{3}^{(1)} L \\ |
---|
| 1842 | \end{array} |
---|
| 1843 | \right) |
---|
| 1844 | \end{equation} |
---|
| 1845 | |
---|
| 1846 | |
---|
| 1847 | \subsection{Status of this document} |
---|
| 1848 | 20.11.06 created by K.Laihem\\ |
---|
| 1849 | 21.02.07 minor update by A.Sch{\"a}licke\\ |
---|
| 1850 | 27.11.08 correction in Eq.\ \eqref{eq:polpairdef} by A.Sch{\"a}licke |
---|
| 1851 | |
---|
| 1852 | |
---|
| 1853 | \begin{latexonly} |
---|
| 1854 | |
---|
| 1855 | \begin{thebibliography}{9} |
---|
| 1856 | |
---|
| 1857 | \bibitem{polPair:Olsen_Maximon} H.~Olsen and L.C.~Maximon. Photon and electron polarization in high- energy bremsstrahlung and pair production with screening. Physical Review, 114:887-904, 1959. |
---|
| 1858 | |
---|
| 1859 | \bibitem{polPair:Laihem:thesis} |
---|
| 1860 | K.~Laihem, PhD thesis, Measurement of the positron polarization at an |
---|
| 1861 | helical undulator based positron source for the International Linear |
---|
| 1862 | Collider ILC, Humboldt University Berlin, Germany, (2008). |
---|
| 1863 | |
---|
| 1864 | |
---|
| 1865 | \end{thebibliography} |
---|
| 1866 | |
---|
| 1867 | \end{latexonly} |
---|
| 1868 | |
---|
| 1869 | \begin{htmlonly} |
---|
| 1870 | |
---|
| 1871 | \subsection{Bibliography} |
---|
| 1872 | \begin{enumerate} |
---|
| 1873 | |
---|
| 1874 | \item H.~Olsen and L.C.~Maximon. Photon and electron polarization in high- energy bremsstrahlung and pair production with screening. Physical Review, 114:887-904, 1959. |
---|
| 1875 | |
---|
| 1876 | \item K.~Laihem, PhD thesis, Measurement of the positron polarization at an |
---|
| 1877 | helical undulator based positron source for the International Linear |
---|
| 1878 | Collider ILC, Humboldt University Berlin, Germany, (2008). |
---|
| 1879 | |
---|
| 1880 | |
---|
| 1881 | \end{enumerate} |
---|
| 1882 | |
---|
| 1883 | \end{htmlonly} |
---|
| 1884 | |
---|
| 1885 | \newpage |
---|
| 1886 | \section{Polarized Photoelectric Effect} |
---|
| 1887 | % |
---|
| 1888 | \subsection{Method} |
---|
| 1889 | % |
---|
| 1890 | This section describes the basic formulas of polarization transfer in |
---|
| 1891 | the photoelectric effect class ({\em G4PolarizedPhotoElectricEffect}). |
---|
| 1892 | The photoelectric effect is the emission of electrons from matter upon |
---|
| 1893 | the absorption of electromagnetic radiation, such as ultraviolet |
---|
| 1894 | radiation or x-rays. The energy of the photon is completely absorbed |
---|
| 1895 | by the electron and, if sufficient, the electron can escape from the |
---|
| 1896 | material with a finite kinetic energy. A single photon can only eject |
---|
| 1897 | a single electron, as the energy of one photon is only absorbed by one |
---|
| 1898 | electron. The electrons that are emitted are often called |
---|
| 1899 | photoelectrons. If the photon energy is higher than the binding energy |
---|
| 1900 | the remaining energy is transferred to the electron as a kinetic |
---|
| 1901 | energy |
---|
| 1902 | \begin{equation} |
---|
| 1903 | E_{kin}^{e^-} = k-B_{shell} |
---|
| 1904 | \end{equation} |
---|
| 1905 | % |
---|
| 1906 | In Geant4 the photoelectric effect process is taken into account if: |
---|
| 1907 | \begin{equation} |
---|
| 1908 | k > B_{shell} |
---|
| 1909 | \end{equation} |
---|
| 1910 | % |
---|
| 1911 | Where $k$ is the incoming photon energy and $B_{shell}$ the electron |
---|
| 1912 | binding energy provided by the class {\it G4AtomicShells}. |
---|
| 1913 | |
---|
| 1914 | The polarized version of the photoelectric effect is based on the EM |
---|
| 1915 | standard process {\em G4PhotoElectricEffect}. Mean free path and the |
---|
| 1916 | distribution of explicitly generated final state particles are |
---|
| 1917 | treated by this version. For details consult |
---|
| 1918 | section \ref{sec:em.pee}. |
---|
| 1919 | |
---|
| 1920 | The remaining task is to attribute polarization vectors to the |
---|
| 1921 | generated final state electron, which is discussed in the following. |
---|
| 1922 | |
---|
| 1923 | |
---|
| 1924 | \subsection{Polarization transfer} |
---|
| 1925 | % |
---|
| 1926 | The polarization state of an incoming polarized photon |
---|
| 1927 | is described by the Stokes vector $\vec{\zeta}^{(1)}$. |
---|
| 1928 | % |
---|
| 1929 | The polarization transfer to the photoelectron |
---|
| 1930 | can be described in the Stokes formalism using the same approach as |
---|
| 1931 | for the Bremsstrahlung and gamma conversion processes, |
---|
| 1932 | cf.~\ref{sec:pol.bremsstrahlung} and \ref{sec:pol.conv}. The relation |
---|
| 1933 | between the photoelectron's Stokes parameters and the incoming |
---|
| 1934 | photon's Stokes parameters is described by the interaction matrix |
---|
| 1935 | $T_{l}^{p}$ derived from H. Olsen \cite{polPEE:H.Olsen.Kgl} and reviewed by |
---|
| 1936 | H.W McMaster \cite{polPEE:McMaster2}: |
---|
| 1937 | \begin{equation} |
---|
| 1938 | \left(\begin{array}{c} |
---|
| 1939 | I^{\prime} \\ |
---|
| 1940 | \vec{\xi}^{(1)} |
---|
| 1941 | \end{array}\right) |
---|
| 1942 | = T_{l}^{p} \, |
---|
| 1943 | \left(\begin{array}{c} |
---|
| 1944 | I_0 \\ |
---|
| 1945 | \vec{\zeta}^{(1)} |
---|
| 1946 | \end{array}\right) |
---|
| 1947 | \label{eq:photo_lepton} |
---|
| 1948 | \end{equation} |
---|
| 1949 | % |
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| 1950 | In general, for the photoelectric effect as a two-body scattering, the |
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| 1951 | cross section should be correlated with the spin states of the |
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| 1952 | incoming photon and the target electron. In our implementation the |
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| 1953 | target electron is not polarized and only the polarization transfer |
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| 1954 | from the photon to the photoelectron is taken into account. In this |
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| 1955 | case the cross section of the process remains polarization |
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| 1956 | independent. To compute the matrix elements we take advantage of the |
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| 1957 | available kinematic variables provided by the generic {\it |
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| 1958 | G4PhotoelectricEffect} class. To compute the photoelectron spin |
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| 1959 | state (Stokes parameters), four main parameters are needed: |
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| 1960 | \begin{itemize} |
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| 1961 | \item The incoming photon Stokes vector $\vec{\zeta}^{(1)}$ |
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| 1962 | \item The incoming photon's energy $k$. |
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| 1963 | \item the photoelectron's kinetic energy $E_{kin}^{e^-}$ or the |
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| 1964 | Lorentz factors $\beta$ and $\gamma$. |
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| 1965 | \item The photoelectron's polar angle $\theta$ or $\cos\theta$. |
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| 1966 | \end{itemize} |
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| 1967 | % |
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| 1968 | The interaction matrix derived by H. Olsen \cite{polPEE:H.Olsen.Kgl} is given |
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| 1969 | by: |
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| 1970 | % |
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| 1971 | \begin{equation} |
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| 1972 | T_{l}^{P}= %\frac{Z^{5}}{(137)^{4}}r_{0}^{2}\beta^{3}\frac{\epsilon}{k^{3}}\frac{\sin^{2}\theta}{(1-\beta \cos\theta)^{3}} |
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| 1973 | \left( |
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| 1974 | \begin{array}{cccc} |
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| 1975 | 1+D & -D & 0 & 0 \\ |
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| 1976 | 0 & 0 & 0 & B \\ |
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| 1977 | 0 & 0 & 0 & 0 \\ |
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| 1978 | 0 & 0 & 0 & A \\ |
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| 1979 | \end{array} |
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| 1980 | \right) |
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| 1981 | \label{eq:matrix_photo} |
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| 1982 | \end{equation} |
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| 1983 | |
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| 1984 | |
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| 1985 | Where |
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| 1986 | \begin{eqnarray} |
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| 1987 | D &=& \frac{1}{k}\left[\frac{2}{k\epsilon(1-\beta \cos\theta)}-1 \right]\\ |
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| 1988 | A &=& \frac{\epsilon}{\epsilon+1}\left[\frac{2}{k\epsilon}+\beta\cos\theta+\frac{2}{k\epsilon^2(1-\beta \cos\theta)}\right]\\ |
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| 1989 | B &=& \frac{\epsilon}{\epsilon+1}\beta\sin\theta\left[\frac{2}{k\epsilon(1-\beta \cos\theta)}-1\right] |
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| 1990 | \end{eqnarray} |
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| 1991 | |
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| 1992 | Using Eq.~(\ref{eq:photo_lepton}) and the transfer matrix given by |
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| 1993 | Eq.~(\ref{eq:matrix_photo}) the polarization state of |
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| 1994 | the produced $e^{-}$ is given in the Stokes formalism by: |
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| 1995 | |
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| 1996 | \begin{equation} |
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| 1997 | \vec{\xi}^{(1)} = \left( |
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| 1998 | \begin{array}{c} |
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| 1999 | \xi_{1}^{(1)}\\ |
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| 2000 | \xi_{2}^{(1)} \\ |
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| 2001 | \xi_{3}^{(1)} \\ |
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| 2002 | \end{array} |
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| 2003 | \right) |
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| 2004 | = |
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| 2005 | \left( |
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| 2006 | \begin{array}{c} |
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| 2007 | \zeta_{3}^{(1)} B \\ |
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| 2008 | 0 \\ |
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| 2009 | \zeta_{3}^{(1)} A \\ |
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| 2010 | \end{array} |
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| 2011 | \right) |
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| 2012 | \label{eq:final_stat} |
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| 2013 | \end{equation} |
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| 2014 | |
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| 2015 | From equation (\ref{eq:final_stat}) one can see that a longitudinally |
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| 2016 | (transversally) polarized photoelectron can only be produced if the |
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| 2017 | incoming photon is circularly polarized. |
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| 2018 | |
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| 2019 | \subsection{Status of this document} |
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| 2020 | 20.11.07 created by K.Laihem\\ |
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| 2021 | 03.12.07 minor update by A.Sch{\"a}licke\\ |
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| 2022 | |
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| 2023 | \begin{latexonly} |
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| 2024 | |
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| 2025 | \begin{thebibliography}{9} |
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| 2026 | |
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| 2027 | %\bibitem{polBrems:McMaster1} W.H.~McMaster. Polarization and the Stokes parameters. American Journal of Physics, 22(6):351-362, 1954. |
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| 2028 | |
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| 2029 | \bibitem{polPEE:H.Olsen.Kgl} H. Olsen, Kgl.~N.~Videnskab. Selskabs Forh. 31, Nos 11, 11a (1958). |
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| 2030 | |
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| 2031 | \bibitem{polPEE:McMaster2}W.H.~McMaster. Matrix representation of polarization. Reviews of Modern Physics, 33(1):8-29, 1961. |
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| 2032 | |
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| 2033 | |
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| 2034 | %\bibitem{polPair:Olsen_Maximon} H.~Olsen and L.C.~Maximon. Photon and electron polarization in high- energy bremsstrahlung and pair production with screening. Physical Review, 114:887-904, 1959. |
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| 2035 | |
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| 2036 | \bibitem{polPEE:Laihem:thesis} |
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| 2037 | K.~Laihem, PhD thesis, Measurement of the positron polarization at an |
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| 2038 | helical undulator based positron source for the International Linear |
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| 2039 | Collider ILC, Humboldt University Berlin, Germany, (2008). |
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| 2040 | |
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| 2041 | |
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| 2042 | \end{thebibliography} |
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| 2043 | |
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| 2044 | \end{latexonly} |
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| 2045 | |
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| 2046 | \begin{htmlonly} |
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| 2047 | |
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| 2048 | \subsection{Bibliography} |
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| 2049 | \begin{enumerate} |
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| 2050 | \item{polPEE:H.Olsen.Kgl} H. Olsen, Kgl.~N.~Videnskab. Selskabs Forh. 31, Nos 11, 11a (1958). |
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| 2051 | |
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| 2052 | \item{polPEE:McMaster2}W.H.~McMaster. Matrix representation of polarization. Reviews of Modern Physics, 33(1):8-29, 1961. |
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| 2053 | |
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| 2054 | |
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| 2055 | \item K.~Laihem, PhD thesis, Measurement of the positron polarization at an |
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| 2056 | helical undulator based positron source for the International Linear |
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| 2057 | Collider ILC, Humboldt University Berlin, Germany, (2008). |
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| 2058 | |
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| 2059 | \end{enumerate} |
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| 2060 | |
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| 2061 | \end{htmlonly} |
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| 2062 | |
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| 2063 | % LocalWords: Bhabha |
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