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, Humboldt University Berlin, Germany, (2007). |
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402 | |
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403 | %%EGS |
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404 | \bibitem{polIntro:Nelson:1985ec} |
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405 | W.~R.~Nelson, H.~Hirayama, D.~W.~O.\ Rogers, |
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406 | %``The Egs4 Code System,'' |
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407 | SLAC-R-0265. |
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408 | |
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409 | \bibitem{polIntro:Floettmann:thesis} |
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410 | K.~Fl\"ottmann, PhD thesis, DESY Hamburg (1993); DESY-93-161. |
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411 | |
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412 | %kek extension |
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413 | \bibitem{polIntro:Namito:1993sv} |
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414 | Y.~Namito, S.~Ban, H.~Hirayama, |
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415 | %``Implementation of linearly polarized photon scattering into the EGS4 code,'' |
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416 | Nucl.\ Instrum.\ Meth.\ A {\bf 332} (1993) 277. |
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417 | |
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418 | \bibitem{polIntro:Liu:2000ey} |
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419 | J.~C.~Liu, T.~Kotseroglou, W.~R.~Nelson, D.~C.~Schultz, |
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420 | %``Polarization study for NLC positron source using EGS4,'' |
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421 | SLAC-PUB-8477. |
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422 | %Geant3 |
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423 | \bibitem{polIntro:Brun:1985ps} |
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424 | R.~Brun, M.~Caillat, M.~Maire, G.~N.~Patrick, L.~Urban, |
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425 | %``The Geant3 Electromagnetic Shower Program And A Comparison With The Egs3 |
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426 | %Code,'' |
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427 | CERN-DD/85/1. |
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428 | |
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429 | %% E166 |
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430 | \bibitem{polIntro:Alexander:2003fh} |
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431 | G.~Alexander {\it et al.}, |
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432 | %``Undulator-based production of polarized positrons: A proposal for |
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433 | % the 50-GeV beam in the FFTB,'' |
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434 | SLAC-TN-04-018, SLAC-PROPOSAL-E-166. |
---|
435 | |
---|
436 | \bibitem{polIntro:Hoogduin:thesis} |
---|
437 | J.~Hoogduin, PhD thesis, Rijksuniversiteit Groningen (1997). |
---|
438 | |
---|
439 | \bibitem{polIntro:Stokes:1852} |
---|
440 | G.~Stokes, |
---|
441 | Trans.\ Cambridge Phil.\ Soc.\ {\bf 9} (1852) 399. |
---|
442 | |
---|
443 | |
---|
444 | \end{thebibliography} |
---|
445 | |
---|
446 | \end{latexonly} |
---|
447 | |
---|
448 | \begin{htmlonly} |
---|
449 | |
---|
450 | \begin{enumerate}{10} |
---|
451 | \item |
---|
452 | W.~H.~McMaster, Rev.\ Mod.\ Phys.\ {\bf 33} (1961) 8; and references therein. |
---|
453 | |
---|
454 | \item |
---|
455 | K.~Laihem, PhD thesis, Humboldt University Berlin, Germany, (2007). |
---|
456 | |
---|
457 | %%EGS |
---|
458 | \item |
---|
459 | W.~R.~Nelson, H.~Hirayama, D.~W.~O.\ Rogers, |
---|
460 | %``The Egs4 Code System,'' |
---|
461 | SLAC-R-0265. |
---|
462 | |
---|
463 | \item |
---|
464 | K.~Fl\"ottmann, PhD thesis, DESY Hamburg (1993); DESY-93-161. |
---|
465 | |
---|
466 | %kek extension |
---|
467 | \item |
---|
468 | Y.~Namito, S.~Ban, H.~Hirayama, |
---|
469 | %``Implementation of linearly polarized photon scattering into the EGS4 code,'' |
---|
470 | Nucl.\ Instrum.\ Meth.\ A {\bf 332} (1993) 277. |
---|
471 | |
---|
472 | \item |
---|
473 | J.~C.~Liu, T.~Kotseroglou, W.~R.~Nelson, D.~C.~Schultz, |
---|
474 | %``Polarization study for NLC positron source using EGS4,'' |
---|
475 | SLAC-PUB-8477. |
---|
476 | %Geant3 |
---|
477 | \item |
---|
478 | R.~Brun, M.~Caillat, M.~Maire, G.~N.~Patrick, L.~Urban, |
---|
479 | %``The Geant3 Electromagnetic Shower Program And A Comparison With The Egs3 |
---|
480 | %Code,'' |
---|
481 | CERN-DD/85/1. |
---|
482 | |
---|
483 | %% E166 |
---|
484 | \item |
---|
485 | G.~Alexander {\it et al.}, |
---|
486 | %``Undulator-based production of polarized positrons: A proposal for |
---|
487 | % the 50-GeV beam in the FFTB,'' |
---|
488 | SLAC-TN-04-018, SLAC-PROPOSAL-E-166. |
---|
489 | |
---|
490 | \item |
---|
491 | J.~Hoogduin, PhD thesis, Rijksuniversiteit Groningen (1997). |
---|
492 | |
---|
493 | \item |
---|
494 | G.~Stokes, |
---|
495 | Trans.\ Cambridge Phil.\ Soc.\ {\bf 9} (1852) 399. |
---|
496 | |
---|
497 | \end{enumerate} |
---|
498 | |
---|
499 | \end{htmlonly} |
---|
500 | |
---|
501 | |
---|
502 | |
---|
503 | |
---|
504 | % ====================================================================== |
---|
505 | \newcommand{\Mvariable}[1]{r_e} |
---|
506 | |
---|
507 | \newpage |
---|
508 | \section{Ionization}\label{sec:polarizedIonization} |
---|
509 | \subsection{Method} |
---|
510 | The class {\em G4ePolarizedIonization} provides continuous and |
---|
511 | discrete energy losses of polarized electrons and positrons in a |
---|
512 | material. It evaluates polarization transfer and -- if the material |
---|
513 | is polarized -- asymmetries in the explicit delta rays production. |
---|
514 | The implementation baseline follows the approach derived for the |
---|
515 | class {\em G4eIonization} described in sections |
---|
516 | \ref{en_loss} and \ref{sec:em.eion}. |
---|
517 | For continuous energy losses the effects of a polarized beam or |
---|
518 | target are negligible provided the separation cut $T_{\rm cut}$ is |
---|
519 | small, and are therefore not considered separately. On the other |
---|
520 | hand, in the explicit production of delta rays by M{\o}ller or |
---|
521 | Bhabha scattering, the effects of polarization on total cross |
---|
522 | section and mean free path, on distribution of final state particles |
---|
523 | and the average polarization of final state particles are taken into |
---|
524 | account. |
---|
525 | |
---|
526 | % ---------------------------------------------------------------------- |
---|
527 | |
---|
528 | \subsection{Total cross section and mean free path} |
---|
529 | |
---|
530 | Kinematics of Bhabha and M{\o}ller scattering is fixed by initial |
---|
531 | energy |
---|
532 | \begin{equation} |
---|
533 | \gamma=\frac{E_{k_1}}{m c^2}% =\frac{s}{2m^2}-1 |
---|
534 | \end{equation} |
---|
535 | and variable |
---|
536 | \begin{equation} |
---|
537 | \epsilon = \frac{E_{p_2}-m c^2}{E_{k_1}-m c^2}, |
---|
538 | \end{equation} |
---|
539 | which is the part of kinetic energy of initial particle carried out by |
---|
540 | scatter. Lower kinematic limit for $\epsilon$ is $0$, but in order |
---|
541 | to avoid divergencies in both total and differential cross sections |
---|
542 | one sets |
---|
543 | \begin{equation} |
---|
544 | \epsilon_{min}= x = \frac{T_{min}}{E_{k_1}-mc^2}, |
---|
545 | \end{equation} |
---|
546 | where $T_{min}$ has meaning of minimal kinetic energy of secondary |
---|
547 | electron. And, $\epsilon_{\rm max}=1(1/2)$ for Bhabha(M{\o}ller) |
---|
548 | scatterings. |
---|
549 | |
---|
550 | % ---------------------------------------------------------------------- |
---|
551 | \subsubsection{Total M{\o}ller cross section} |
---|
552 | |
---|
553 | The total cross section of the polarized M{\o}ller scattering can be expressed as follows |
---|
554 | \begin{equation}\label{totalMoller} |
---|
555 | \sigma^M_{pol}=\frac{2\pi\gamma^2 r_e^2}{(\gamma-1)^2(\gamma+1)}\left[ |
---|
556 | \sigma^M_0 + \zeta_3^{(1)}\zeta_3^{(2)}\sigma^M_L |
---|
557 | + \left(\zeta_1^{(1)}\zeta_1^{(2)}+\zeta_2^{(1)}\zeta_2^{(2)}\right)\sigma^M_T\right], |
---|
558 | \end{equation} |
---|
559 | where the $r_e$ is classical electron radius, and |
---|
560 | \begin{eqnarray} |
---|
561 | \sigma^M_0&=& |
---|
562 | - \frac{1}{1 - x} + \frac{1}{x} |
---|
563 | - \frac{{\left( \gamma - 1 \right)}^2}{{\gamma}^2} |
---|
564 | \left(\frac{1}{2} - x \right) |
---|
565 | + \frac{ 2 - 4\,\gamma }{2\,{\gamma}^2} |
---|
566 | \,\ln \left(\frac{1-x}{x}\right) |
---|
567 | \nonumber\\ |
---|
568 | \sigma^M_L&=& |
---|
569 | \frac{ \left( -3 + 2\,\gamma + {\gamma}^2 \right) |
---|
570 | \,\left( 1 - 2\,x \right) }{2\, {\gamma}^2} |
---|
571 | + \frac{2\,\gamma\,\left( -1 + 2\,\gamma \right)}{2\, |
---|
572 | {\gamma}^2} \,\ln \left(\frac{1-x}{x}\right) |
---|
573 | \nonumber\\ |
---|
574 | \sigma^M_T&=& |
---|
575 | \frac{2\,\left( \gamma - 1 \right) \,\left( 2\,x -1 \right)}{2\,{\gamma}^2} |
---|
576 | + \frac{ |
---|
577 | \left( 1 - 3\,\gamma \right) }{2\,{\gamma}^2} \,\ln \left(\frac{1-x}{x}\right) |
---|
578 | \label{mollertotal} |
---|
579 | \end{eqnarray} |
---|
580 | |
---|
581 | % ---------------------------------------------------------------------- |
---|
582 | \subsubsection{Total Bhabha cross section} |
---|
583 | |
---|
584 | The total cross section of the polarized Bhabha scattering can be expressed as follows |
---|
585 | \begin{equation}\label{totalBhabha} |
---|
586 | \sigma^B_{pol}=\frac{2\pi r_e^2}{\gamma-1} |
---|
587 | \left[ |
---|
588 | \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 |
---|
589 | \right], |
---|
590 | \end{equation} |
---|
591 | where |
---|
592 | \begin{eqnarray} |
---|
593 | \sigma^B_0&=& |
---|
594 | \frac{1 - x}{2\,\left( \gamma - 1 \right) \,x} + |
---|
595 | \frac{2\,\left( -1 + 3\,x - 6\,x^2 + 4\,x^3 \right) } |
---|
596 | {3\,{\left( 1 + \gamma \right) }^3} |
---|
597 | \nonumber\\ |
---|
598 | &+&\frac{-1 - 5\,x + 12\,x^2 - 10\,x^3 + 4\,x^4}{2\,\left( 1 + \gamma \right) \,x} |
---|
599 | + \frac{-3 - x + 8\,x^2 - 4\,x^3 - \ln (x)}{{\left( 1 + \gamma \right) }^2} |
---|
600 | \nonumber\\ |
---|
601 | &+&\frac{3 + 4\,x - 9\,x^2 + 3\,x^3 - x^4 + 6\,x\,\ln (x)}{3\,x} |
---|
602 | \nonumber\\ |
---|
603 | \sigma^B_L&=& |
---|
604 | \frac{2\,\left( 1 - 3\,x + 6\,x^2 - 4\,x^3 \right) }{3\,{\left( 1 + \gamma \right) }^3} + |
---|
605 | \frac{-14 + 15\,x - 3\,x^2 + 2\,x^3 - 9\,\ln (x)}{3\,\left( 1 + \gamma \right) } |
---|
606 | \nonumber\\ |
---|
607 | &+&\frac{5 + 3\,x - 12\,x^2 + 4\,x^3 + 3\,\ln (x)}{3\,{\left( 1 + \gamma \right) }^2} + |
---|
608 | \frac{7 - 9\,x + 3\,x^2 - x^3 + 6\,\ln (x)}{3} |
---|
609 | \nonumber\\ |
---|
610 | \sigma^B_T&=& |
---|
611 | \frac{2\,\left( -1 + 3\,x - 6\,x^2 + 4\,x^3 \right) }{3\,{\left( 1 + \gamma \right) }^3} + |
---|
612 | \frac{-7 - 3\,x + 18\,x^2 - 8\,x^3 - 3\,\ln (x)}{3\,{\left( 1 + \gamma \right) }^2} |
---|
613 | \nonumber\\ |
---|
614 | &+&\frac{5 + 3\,x - 12\,x^2 + 4\,x^3 + 9\,\ln (x)}{6\,\left( 1 + \gamma \right) } |
---|
615 | \end{eqnarray} |
---|
616 | |
---|
617 | % ---------------------------------------------------------------------- |
---|
618 | \subsubsection{Mean free path} |
---|
619 | |
---|
620 | With the help of the total polarized M{\o}ller cross section |
---|
621 | one can define a longitudinal asymmetry $A^M_L$ and the transverse |
---|
622 | asymmetry $A^M_T$, by |
---|
623 | |
---|
624 | \begin{tabular}{ccc} |
---|
625 | $ A^M_L = \displaystyle \frac{\sigma^M_L}{\sigma^M_0} \quad$ & and & |
---|
626 | $\quad A^M_T = \displaystyle \frac{\sigma^M_T}{\sigma^M_0}\;$. |
---|
627 | \end{tabular} |
---|
628 | |
---|
629 | Similarly, using the polarized Bhabha cross section one can introduce a |
---|
630 | longitudinal asymmetry $A^B_L$ and the transverse asymmetry $A^B_T$ |
---|
631 | via |
---|
632 | |
---|
633 | \begin{tabular}{ccc} |
---|
634 | $ A^B_L = \displaystyle \frac{\sigma^B_L}{\sigma^B_0} \quad$ & and & |
---|
635 | $\quad A^B_T = \displaystyle \frac{\sigma^B_T}{\sigma^B_0}\;$. |
---|
636 | \end{tabular} |
---|
637 | |
---|
638 | These asymmetries are depicted in figures \ref{pol.moller1} and |
---|
639 | \ref{pol.bhabha1} respectively. |
---|
640 | |
---|
641 | If both beam and target are polarized the mean free path as defined in |
---|
642 | section \ref{sec:em.eion} has to be modified. In the class {\em |
---|
643 | G4ePolarizedIonization} the polarized mean free path $\lambda^{\rm |
---|
644 | pol}$ is derived from the unpolarized mean free path $\lambda^{\rm |
---|
645 | unpol}$ via |
---|
646 | \begin{equation} |
---|
647 | \lambda^{\rm pol} = \frac{\lambda^{\rm unpol}}{1 + |
---|
648 | \zeta_3^{(1)}\zeta_3^{(2)}\, A_L + |
---|
649 | \left(\zeta_1^{(1)}\zeta_1^{(2)}+\zeta_2^{(1)}\zeta_2^{(2)}\right) \,A_T} |
---|
650 | \end{equation} |
---|
651 | |
---|
652 | % |
---|
653 | \begin{figure}[t] |
---|
654 | \begin{center} |
---|
655 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/MollerTA1.eps} |
---|
656 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/MollerTA2.eps} |
---|
657 | \end{center} |
---|
658 | \caption{\label{pol.moller1}M{\o}ller total cross section |
---|
659 | asymmetries depending on the total energy of the incoming |
---|
660 | electron, with a cut-off $T_{\rm cut}= 1 {\rm keV}$. Transverse |
---|
661 | asymmetry is plotted in blue, longitudinal asymmetry in red. Left |
---|
662 | part, between 0.5 MeV and 2 MeV, right part up to 10 MeV.} |
---|
663 | %\end{figure} |
---|
664 | % |
---|
665 | %\begin{figure}[t] |
---|
666 | \begin{center} |
---|
667 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/BhabhaTA1.eps} |
---|
668 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/BhabhaTA2.eps} |
---|
669 | \end{center} |
---|
670 | \caption{\label{pol.bhabha1}Bhabha total cross section |
---|
671 | asymmetries depending on the total energy of the incoming |
---|
672 | positron, with a cut-off $T_{\rm cut}= 1 {\rm keV}$. Transverse |
---|
673 | asymmetry is plotted in blue, longitudinal asymmetry in red. Left |
---|
674 | part, between 0.5 MeV and 2 MeV, right part up to 10 MeV.} |
---|
675 | \end{figure} |
---|
676 | |
---|
677 | |
---|
678 | |
---|
679 | |
---|
680 | % ---------------------------------------------------------------------- |
---|
681 | \subsection{Sampling the final state} |
---|
682 | |
---|
683 | \subsubsection{Differential cross section} |
---|
684 | |
---|
685 | The polarized differential cross section is rather complicated, |
---|
686 | the full result can be found in \cite{polIoni:Star:2006,polIoni:Ford:1957,polIoni:Stehle:1957}. |
---|
687 | In {\em G4PolarizedMollerCrossSection} the complete result is |
---|
688 | available taking all mass effects into account, only binding effects |
---|
689 | are neglected. |
---|
690 | Here we state only the ultra-relativistic approximation (URA), to show |
---|
691 | the general dependencies. |
---|
692 | \begin{eqnarray} |
---|
693 | &&\frac{d\sigma_{URA}^M}{d\epsilon d\varphi}= |
---|
694 | \frac{{{r_\epsilon}}^2}{ \gamma + 1} \times |
---|
695 | \nonumber\\ |
---|
696 | &&\Bigg[ |
---|
697 | \frac{{\left( 1 - \epsilon + \epsilon^2 \right) }^2}{4\,{\left( \epsilon - 1 \right) }^2\,\epsilon^2} + |
---|
698 | \zeta_3^{(1)}\zeta_3^{(2)}\frac{2 - \epsilon + |
---|
699 | \epsilon^2}{-4\,\epsilon ( 1 - \epsilon)} + |
---|
700 | \left(\zeta_2^{(1)}\zeta_2^{(2)} -\zeta_1^{(1)}\zeta_1^{(2)}\right)\frac{1}{4} |
---|
701 | \nonumber\\ |
---|
702 | &&+ |
---|
703 | \left(\xi_3^{(1)}\zeta_3^{(1)} - \xi_3^{(2)}\zeta_3^{(2)}\right) |
---|
704 | \frac{1 - \epsilon + 2\,\epsilon^2}{4\,\left( 1 - \epsilon \right) \,\epsilon^2} |
---|
705 | + \left(\xi_3^{(2)}\zeta_3^{(1)} - \xi_3^{(1)}\zeta_3^{(2)}\right) |
---|
706 | \frac{2 - 3\,\epsilon + 2\,\epsilon^2}{4\,{\left( 1 - \epsilon \right) }^2\,\epsilon} |
---|
707 | \Bigg] \nonumber\\ |
---|
708 | && |
---|
709 | \end{eqnarray} |
---|
710 | % |
---|
711 | The corresponding cross section for Bhabha cross section is |
---|
712 | implemented in {\em G4PolarizedBhabhaCrossSection}. In the |
---|
713 | ultra-relativistic approximation it reads |
---|
714 | \begin{eqnarray} |
---|
715 | &&\frac{d\sigma_{URA}^B}{d\epsilon d\varphi}= |
---|
716 | \frac{{{r_\epsilon}}^2}{ \gamma - 1} \times |
---|
717 | \nonumber\\ |
---|
718 | &&\Bigg[ |
---|
719 | \frac{{\left( 1 - \epsilon + \epsilon^2 \right) }^2}{4\,\epsilon^2} + |
---|
720 | \zeta_3^{(1)}\zeta_3^{(2)}\frac{\left( \epsilon - 1 \right) \,\left( 2 - \epsilon + \epsilon^2 \right) }{4\,\epsilon} |
---|
721 | +\left(\zeta_2^{(1)}\zeta_2^{(2)} -\zeta_1^{(1)}\zeta_1^{(2)}\right)\frac{(1-\epsilon)^2}{4} |
---|
722 | \nonumber\\ |
---|
723 | &&+ |
---|
724 | \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} |
---|
725 | + \left(\xi_3^{(2)}\zeta_3^{(1)} - \xi_3^{(1)}\zeta_3^{(2)}\right)\frac{ 2 - 3\,\epsilon + 2\,\epsilon^2}{4\epsilon} |
---|
726 | \Bigg] \nonumber\\ |
---|
727 | && |
---|
728 | \end{eqnarray} |
---|
729 | where |
---|
730 | \begin{tabular}[t]{l@{\ = \ }l} |
---|
731 | $r_e$ & classical electron radius \\ |
---|
732 | $\gamma$ & $E_{k_1}/m_e c^2$ \\ |
---|
733 | $\epsilon$ & ($E_{p_1}-m_e c^2)/(E_{k_1}-m_e c^2)$ \\ |
---|
734 | $E_{k_1}$ & energy of the incident electron/positron \\ |
---|
735 | $E_{p_1}$ & energy of the scattered electron/positron \\ |
---|
736 | $m_e c^2$ & electron mass \\ |
---|
737 | $\bvec{\zeta}^{(1)}$ & Stokes vector of the incoming electron/positron \\ |
---|
738 | $\bvec{\zeta}^{(2)}$ & Stokes vector of the target electron \\ |
---|
739 | $\bvec{\xi}^{(1)}$ & Stokes vector of the outgoing electron/positron \\ |
---|
740 | $\bvec{\xi}^{(2)}$ & Stokes vector of the outgoing (2nd) electron . |
---|
741 | \end{tabular} |
---|
742 | |
---|
743 | \subsubsection{Sampling} |
---|
744 | |
---|
745 | The delta ray is sampled according to methods discussed in Chapter |
---|
746 | 2. After exploitation of the symmetry in the M{\o}ller cross section |
---|
747 | under exchanging $\epsilon$ versus $(1-\epsilon)$, the differential |
---|
748 | cross section can be approximated by a simple function $f^M(\epsilon)$: |
---|
749 | \begin{equation} |
---|
750 | f^M(\epsilon) = \frac{1}{\epsilon^2} \frac{\epsilon_0}{1-2\epsilon_0} |
---|
751 | \end{equation} |
---|
752 | with the kinematic limits given by |
---|
753 | \begin{equation} |
---|
754 | \epsilon_0 = \frac{T_{\rm cut}}{E_{k_1}-m_e c^2} \le \epsilon \le |
---|
755 | \frac{1}{2} |
---|
756 | \end{equation} |
---|
757 | A similar function $f^B(\epsilon)$ can be found for Bhabha scattering: |
---|
758 | \begin{equation} |
---|
759 | f^B(\epsilon) = \frac{1}{\epsilon^2} \frac{\epsilon_0}{1-\epsilon_0} |
---|
760 | \end{equation} |
---|
761 | with the kinematic limits given by |
---|
762 | \begin{equation} |
---|
763 | \epsilon_0 = \frac{T_{\rm cut}}{E_{k_1}-m_e c^2} \le \epsilon \le 1 |
---|
764 | \end{equation} |
---|
765 | |
---|
766 | The kinematic of the delta ray production is constructed by the |
---|
767 | following steps: |
---|
768 | \begin{enumerate} |
---|
769 | \item $\epsilon$ is sampled from $f(\epsilon)$ |
---|
770 | \item calculate the differential cross section, depending on the |
---|
771 | initial polarizations $\bvec{\zeta}^{(1)}$ and $\bvec{\zeta}^{(2)}$. |
---|
772 | \item $\epsilon$ is accepted with the probability defined by ratio |
---|
773 | of the differential cross section over the approximation |
---|
774 | function. |
---|
775 | \item The $\varphi$ is diced uniformly. |
---|
776 | \item $\varphi$ is determined from the differential cross section, |
---|
777 | depending on the initial polarizations $\bvec{\zeta}^{(1)}$ and $\bvec{\zeta}^{(2)}$ |
---|
778 | \end{enumerate} |
---|
779 | Note, for initial states without transverse polarization components, the |
---|
780 | $\varphi$ distribution is always uniform. |
---|
781 | In figure \ref{pol.moller2} the asymmetries indicate the influence of |
---|
782 | polarization. In general the effect is largest around |
---|
783 | $\epsilon=\frac{1}{2}$. |
---|
784 | % |
---|
785 | %\begin{figure}[ht] |
---|
786 | %\includegraphics[scale=0.5]{electromagnetic/standard/plots/MollerXS.eps} |
---|
787 | %\caption{M{\o}ller differential cross section in arbitrary units. Black - unpolarized, Red - (+-),Blue (++). |
---|
788 | %This cross section is symmetric around point $\epsilon=1/2$. |
---|
789 | %} |
---|
790 | %\end{figure} |
---|
791 | %\begin{figure}[ht] |
---|
792 | %\includegraphics[scale=0.5]{electromagnetic/standard/plots/BhabhaXS.eps} |
---|
793 | %\caption{Bhabha differential cross section in arbitrary units. Black - unpolarized, Red - (+-),Blue (++)} |
---|
794 | %\end{figure} |
---|
795 | % |
---|
796 | \begin{figure}[ht] |
---|
797 | \begin{center} |
---|
798 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/MollerAsym.eps} |
---|
799 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/BhabhaAsym.eps} |
---|
800 | \end{center} |
---|
801 | %\caption{M{\o}ller differential cross section asymmetries in\%. |
---|
802 | %Red - ZZ, Gren - XX, Blue - YY, LightBlue -ZX} |
---|
803 | \caption{\label{pol.moller2}Differential cross section asymmetries in\% for M{\o}ller |
---|
804 | (left) and Bhabha (right) scattering ( red - $A_{ZZ}(\epsilon)$, |
---|
805 | green - $A_{XX}(\epsilon)$, blue - $A_{YY}(\epsilon)$, lightblue - $A_{ZX}(\epsilon)$)} |
---|
806 | \end{figure} |
---|
807 | |
---|
808 | After both $\phi$ and $\epsilon$ are known, the kinematic can be |
---|
809 | constructed fully. Using momentum conservation the momenta of the |
---|
810 | scattered incident particle and the ejected electron are constructed |
---|
811 | in global coordinate system. |
---|
812 | |
---|
813 | \subsubsection{Polarization transfer} |
---|
814 | |
---|
815 | After the kinematics is fixed the polarization properties of the |
---|
816 | outgoing particles are determined. Using the dependence of |
---|
817 | the differential cross section on the final state polarization a mean |
---|
818 | polarization is calculated according to method described in section |
---|
819 | \ref{sec:pol.intro}. |
---|
820 | |
---|
821 | The resulting polarization transfer functions $\xi^{(1,2)}_3(\epsilon)$ |
---|
822 | are depicted in figures \ref{pol.moller3} and \ref{pol.bhabha3}. |
---|
823 | |
---|
824 | \begin{figure}[ht] |
---|
825 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/MollerTransfer1.eps} |
---|
826 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/MollerTransfer2.eps} |
---|
827 | \caption{\label{pol.moller3}Polarization transfer functions in |
---|
828 | M{\o}ller scattering. Longitudinal polarization |
---|
829 | $\xi^{(2)}_3$ of electron with energy $E_{p_2}$ in blue; longitudinal |
---|
830 | polarization $\xi^{(1)}_3$ of second electron in red. Kinetic energy of incoming electron $T_{k_1} = 10 {\rm MeV}$}. |
---|
831 | \end{figure} |
---|
832 | |
---|
833 | \begin{figure}[ht] |
---|
834 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/BhabhaTransfer1.eps} |
---|
835 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/BhabhaTransfer2.eps} |
---|
836 | \caption{\label{pol.bhabha3}Polarization Transfer in Bhabha scattering. |
---|
837 | Longitudinal polarization |
---|
838 | $\xi^{(2)}_3$ of electron with energy $E_{p_2}$ in blue; longitudinal |
---|
839 | polarization $\xi^{(1)}_3$ of scattered positron. Kinetic energy of incoming positron $T_{k_1} = 10 {\rm MeV}$}. |
---|
840 | \end{figure} |
---|
841 | |
---|
842 | % ---------------------------------------------------------------------- |
---|
843 | \subsection{Status of this document} |
---|
844 | 20.11.06 created by P.Starovoitov\\ |
---|
845 | 21.02.07 minor update by A.Sch{\"a}licke\\ |
---|
846 | |
---|
847 | \begin{latexonly} |
---|
848 | |
---|
849 | \begin{thebibliography}{9} |
---|
850 | \bibitem{polIoni:Star:2006} P.~Starovoitov {\em et.al.}, in preparation. |
---|
851 | \bibitem{polIoni:Ford:1957} |
---|
852 | G.~W.~Ford, C.~J.~Mullin, |
---|
853 | Phys.~Rev.\ {\bf 108} (1957) 477. |
---|
854 | \bibitem{polIoni:Stehle:1957} |
---|
855 | P.~Stehle, |
---|
856 | Phys.~Rev.\ {\bf 110} (1958) 1458. |
---|
857 | |
---|
858 | \end{thebibliography} |
---|
859 | |
---|
860 | \end{latexonly} |
---|
861 | |
---|
862 | \begin{htmlonly} |
---|
863 | |
---|
864 | \subsection{Bibliography} |
---|
865 | \begin{enumerate} |
---|
866 | \item %{Star:2006} |
---|
867 | P.~Starovoitov {\em et.al.}, in preparation. |
---|
868 | \item %{Ford:1957} |
---|
869 | G.~W.~Ford, C.~J.~Mullin, |
---|
870 | Phys.~Rev.\ {\bf 108} (1957) 477. |
---|
871 | \item % {Stehle:1957} |
---|
872 | P.~Stehle, |
---|
873 | Phys.~Rev.\ {\bf 110} (1958) 1458. |
---|
874 | \end{enumerate} |
---|
875 | |
---|
876 | \end{htmlonly} |
---|
877 | |
---|
878 | |
---|
879 | \clearpage |
---|
880 | % ====================================================================== |
---|
881 | \section{Positron - Electron Annihilation} |
---|
882 | \subsection{Method} |
---|
883 | The class {\em G4eplusPolarizedAnnihilation} simulates |
---|
884 | annihilation of polarized positrons with electrons in a material. |
---|
885 | The implementation baseline follows the approach derived for the class |
---|
886 | {\em G4eplusAnnihilation} described in section |
---|
887 | \ref{sec:em.annil}. |
---|
888 | It evaluates polarization transfer and -- if the material is polarized -- |
---|
889 | asymmetries in the produced photons. Thus, it takes the effects of |
---|
890 | polarization on total cross section and mean free path, on |
---|
891 | distribution of final state photons into account. And |
---|
892 | calculates the average polarization of these generated photons. |
---|
893 | The material electrons are assumed to be free and at rest. |
---|
894 | |
---|
895 | \subsection{Total cross section and mean free path} |
---|
896 | Kinematics of annihilation process is fixed by initial energy |
---|
897 | \begin{equation} |
---|
898 | \gamma=\frac{E_{k_1}}{mc^2}%=\frac{s}{2(mc^2)^2}-1 |
---|
899 | \end{equation} |
---|
900 | and variable |
---|
901 | \begin{equation} |
---|
902 | \epsilon = \frac{E_{p_1}}{E_{k_1}+mc^2}, |
---|
903 | \end{equation} |
---|
904 | which is the part of total energy available in initial state carried out by first photon. |
---|
905 | This variable has the following kinematical limits |
---|
906 | \begin{equation} |
---|
907 | \frac{1}{2}\left(1-\sqrt{\frac{\gamma-1}{\gamma+1}}\right)\;<\; |
---|
908 | \epsilon |
---|
909 | \;<\;\frac{1}{2}\left(1+\sqrt{\frac{\gamma-1}{\gamma+1}}\right) |
---|
910 | \;. |
---|
911 | \end{equation} |
---|
912 | |
---|
913 | % ---------------------------------------------------------------------- |
---|
914 | \subsubsection{Total Cross Section} |
---|
915 | The total cross section of the annihilation of a polarized $e^+e^-$ |
---|
916 | pair into two photons could be expressed as follows |
---|
917 | \begin{equation}\label{totalAnnih} |
---|
918 | \sigma^A_{pol}=\frac{\pi r_e^2}{\gamma+1}\left[ |
---|
919 | \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], |
---|
920 | \end{equation} |
---|
921 | where |
---|
922 | \renewcommand{\Mvariable}[1]{\gamma} |
---|
923 | \begin{equation} |
---|
924 | \sigma^A_0= |
---|
925 | \frac{- \left( 3 + \Mvariable{gam} \right) \,{\sqrt{-1 + {\Mvariable{gam}}^2}} + |
---|
926 | \left( 1 + \Mvariable{gam}\,\left( 4 + \Mvariable{gam} \right) \right) \, |
---|
927 | \ln (\Mvariable{gam} + {\sqrt{-1 + {\Mvariable{gam}}^2}})}{4\, |
---|
928 | \left( {\Mvariable{gam}}^2 - 1 \right) } |
---|
929 | \end{equation} |
---|
930 | \begin{equation} |
---|
931 | \sigma^A_L= |
---|
932 | \frac{- {\sqrt{-1 + {\Mvariable{gam}}^2}}\, |
---|
933 | \left( 5 + \Mvariable{gam}\,\left( 4 + 3\,\Mvariable{gam} \right) \right) + |
---|
934 | \left( 3 + \Mvariable{gam}\,\left( 7 + \Mvariable{gam} + {\Mvariable{gam}}^2 \right) \right) \, |
---|
935 | \ln (\Mvariable{gam} + {\sqrt{{\Mvariable{gam}}^2-1 }})}{4\, |
---|
936 | {\left( \Mvariable{gam} -1\right) }^2\,\left( 1 + \Mvariable{gam} \right) } |
---|
937 | \end{equation} |
---|
938 | \begin{equation} |
---|
939 | \sigma^A_T= |
---|
940 | \frac{\left( 5 + \Mvariable{gam} \right) \,{\sqrt{-1 + {\Mvariable{gam}}^2}} - |
---|
941 | \left( 1 + 5\,\Mvariable{gam} \right) \,\ln (\Mvariable{gam} + {\sqrt{-1 + {\Mvariable{gam}}^2}})} |
---|
942 | {4\,{\left( -1 + \Mvariable{gam} \right) }^2\,\left( 1 + \Mvariable{gam} \right) } |
---|
943 | \end{equation} |
---|
944 | |
---|
945 | |
---|
946 | \subsubsection{Mean free path} |
---|
947 | |
---|
948 | With the help of the total polarized annihilation cross section |
---|
949 | one can define a longitudinal asymmetry $A^A_L$ and the transverse |
---|
950 | asymmetry $A^A_T$, by |
---|
951 | |
---|
952 | \begin{tabular}{ccc} |
---|
953 | $ A^A_L = \displaystyle \frac{\sigma^A_L}{\sigma^A_0} \quad$ & and & |
---|
954 | $\quad A^A_T = \displaystyle \frac{\sigma^A_T}{\sigma^A_0}\;$. |
---|
955 | \end{tabular} |
---|
956 | |
---|
957 | These asymmetries are depicted in figure \ref{pol.annihi1}. |
---|
958 | |
---|
959 | If both incident positron and target electron are polarized the mean |
---|
960 | free path as defined in section \ref{sec:em.annil} has to be |
---|
961 | modified. The polarized mean free path $\lambda^{\rm pol}$ is derived |
---|
962 | from the unpolarized mean free path $\lambda^{\rm unpol}$ via |
---|
963 | \begin{equation} |
---|
964 | \lambda^{\rm pol} = \frac{\lambda^{\rm unpol}}{1 + |
---|
965 | \zeta_3^{(1)}\zeta_3^{(2)}\, A_L + |
---|
966 | \left(\zeta_1^{(1)}\zeta_1^{(2)}+\zeta_2^{(1)}\zeta_2^{(2)}\right) \,A_T} |
---|
967 | \end{equation} |
---|
968 | |
---|
969 | \begin{figure}[ht] |
---|
970 | \begin{center} |
---|
971 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/AnnihTA1.eps} |
---|
972 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/AnnihTA2.eps} |
---|
973 | \end{center} |
---|
974 | \caption{\label{pol.annihi1}Annihilation total cross section asymmetries depending on the |
---|
975 | total energy of the incoming positron $E_{k_1}$. The transverse asymmetry |
---|
976 | is shown in blue, the longitudinal asymmetry in red. } |
---|
977 | \end{figure} |
---|
978 | |
---|
979 | \clearpage |
---|
980 | |
---|
981 | % ---------------------------------------------------------------------- |
---|
982 | \subsection{Sampling the final state} |
---|
983 | \subsubsection{Differential Cross Section} |
---|
984 | The fully polarized differential cross section is implemented in the |
---|
985 | class {\em G4PolarizedAnnihilationCrossSection}, which takes all mass |
---|
986 | effects into account, but binding effects are neglected \cite{polAnnihi:Star:2006,polAnnihi:Page:1957}. |
---|
987 | In the ultra-relativistic approximation (URA) and concentrating on |
---|
988 | longitudinal polarization states only the cross section is |
---|
989 | rather simple: |
---|
990 | \begin{eqnarray} |
---|
991 | \frac{d\sigma_{URA}^A}{d\epsilon d\varphi} & = & |
---|
992 | \frac{{{r_e}}^2}{ \gamma - 1} \times |
---|
993 | \Bigg( |
---|
994 | \frac{1 - 2\,\epsilon + 2\,\epsilon^2}{8\,\epsilon - 8\,\epsilon^2}\left(1 + \zeta_3^{(1)}\zeta_3^{(2)}\right) |
---|
995 | \nonumber\\ |
---|
996 | &&\quad\quad |
---|
997 | + \frac{ \left( 1 - 2\,\epsilon \right) \,\left( \zeta _{3}^{(1)} + \zeta _{3}^{(2)} \right) \, |
---|
998 | \left( \xi _{3}^{(1)} - \xi _{3}^{(2)} \right) }{8\,\left( \epsilon -1 \right) \,\epsilon} |
---|
999 | \Bigg) |
---|
1000 | \end{eqnarray} |
---|
1001 | % |
---|
1002 | where |
---|
1003 | \begin{tabular}[t]{l@{\ = \ }l} |
---|
1004 | $r_e$ & classical electron radius \\ |
---|
1005 | $\gamma$ & $E_{k_1}/m_e c^2$ \\ |
---|
1006 | $E_{k_1}$ & energy of the incident positron \\ |
---|
1007 | $m_e c^2$ & electron mass \\ |
---|
1008 | $\bvec{\zeta}^{(1)}$ & Stokes vector of the incoming positron \\ |
---|
1009 | $\bvec{\zeta}^{(2)}$ & Stokes vector of the target electron \\ |
---|
1010 | $\bvec{\xi}^{(1)}$ & Stokes vector of the 1st photon \\ |
---|
1011 | $\bvec{\xi}^{(2)}$ & Stokes vector of the 2nd photon . |
---|
1012 | \end{tabular} |
---|
1013 | % |
---|
1014 | \begin{figure}[ht] |
---|
1015 | \begin{center} |
---|
1016 | \includegraphics[width=9.5cm]{electromagnetic/standard/plots/AnnihXS.eps} |
---|
1017 | \end{center} |
---|
1018 | \caption{Annihilation differential cross section in arbitrary |
---|
1019 | units. Black line corresponds to unpolarized cross section; |
---|
1020 | red line -- to the antiparallel spins of initial particles, and blue line -- to the parallel spins. |
---|
1021 | Kinetic energy of the incoming positron $T_{k_1} = 10 {\rm MeV}$.} |
---|
1022 | \end{figure} |
---|
1023 | |
---|
1024 | \subsubsection{Sampling} |
---|
1025 | |
---|
1026 | The photon energy is sampled according to methods discussed in Chapter |
---|
1027 | 2. After exploitation of the symmetry in the Annihilation cross section |
---|
1028 | under exchanging $\epsilon$ versus $(1-\epsilon)$, the differential |
---|
1029 | cross section can be approximated by a simple function $f(\epsilon)$: |
---|
1030 | \begin{equation} |
---|
1031 | f(\epsilon) = \frac{1}{\epsilon} |
---|
1032 | \ln^{-1}\left(\frac{\epsilon_{\rm max}}{\epsilon_{\rm min}}\right) |
---|
1033 | \end{equation} |
---|
1034 | with the kinematic limits given by |
---|
1035 | \begin{eqnarray} |
---|
1036 | \epsilon_{\rm min} &=& |
---|
1037 | \frac{1}{2}\left(1-\sqrt{\frac{\gamma-1}{\gamma+1}}\right)\;, \nonumber\\ |
---|
1038 | \epsilon_{\rm max} &=& |
---|
1039 | \frac{1}{2}\left(1+\sqrt{\frac{\gamma-1}{\gamma+1}}\right) |
---|
1040 | \;. |
---|
1041 | \end{eqnarray} |
---|
1042 | |
---|
1043 | The kinematic of the two photon final state is constructed by the |
---|
1044 | following steps: |
---|
1045 | \begin{enumerate} |
---|
1046 | \item $\epsilon$ is sampled from $f(\epsilon)$ |
---|
1047 | \item calculate the differential cross section, depending on the |
---|
1048 | initial polarizations $\bvec{\zeta}^{(1)}$ and $\bvec{\zeta}^{(2)}$. |
---|
1049 | \item $\epsilon$ is accepted with the probability defined by the ratio |
---|
1050 | of the differential cross section over the approximation |
---|
1051 | function $f(\epsilon)$. |
---|
1052 | \item The $\varphi$ is diced uniformly. |
---|
1053 | \item $\varphi$ is determined from the differential cross section, |
---|
1054 | depending on the initial polarizations $\bvec{\zeta}^{(1)}$ and $\bvec{\zeta}^{(2)}$. |
---|
1055 | \end{enumerate} |
---|
1056 | A short overview over the sampling method is given in Chapter 2. |
---|
1057 | In figure \ref{pol.annihi2} the asymmetries indicate the influence of |
---|
1058 | polarization for an 10MeV incoming positron. The actual behavior is |
---|
1059 | very sensitive to the energy of the incoming positron. |
---|
1060 | |
---|
1061 | |
---|
1062 | \begin{figure}[ht] |
---|
1063 | \includegraphics[scale=0.5]{electromagnetic/standard/plots/AnnihAsym.eps} |
---|
1064 | \caption{\label{pol.annihi2}Annihilation differential cross section |
---|
1065 | asymmetries in\%. |
---|
1066 | Red line corrsponds to $A_{ZZ}(\epsilon)$, green line -- $A_{XX}(\epsilon)$, |
---|
1067 | blue line -- $A_{YY}(\epsilon)$, lightblue line -- $A_{ZX}(\epsilon)$). |
---|
1068 | Kinetic energy of the incoming positron $T_{k_1} = 10 {\rm MeV}$.} |
---|
1069 | \end{figure} |
---|
1070 | |
---|
1071 | \subsubsection{Polarization transfer} |
---|
1072 | |
---|
1073 | After the kinematics is fixed the polarization of the |
---|
1074 | outgoing photon is determined. Using the dependence of |
---|
1075 | the differential cross section on the final state polarizations a mean |
---|
1076 | polarization is calculated for each photon according to method |
---|
1077 | described in section \ref{sec:pol.intro}. |
---|
1078 | |
---|
1079 | The resulting polarization transfer functions $\xi^{(1,2)}(\epsilon)$ |
---|
1080 | are depicted in figure \ref{pol.annihi3}. |
---|
1081 | |
---|
1082 | \begin{figure}[ht] |
---|
1083 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/AnnihTransfer1.eps} |
---|
1084 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/AnnihTransfer2.eps} |
---|
1085 | \caption{\label{pol.annihi3} |
---|
1086 | Polarization Transfer in annihilation process. |
---|
1087 | Blue line corresponds to the circular polarization $\xi_3^{(1)}$ of the photon with energy $m(\gamma + 1)\epsilon$; |
---|
1088 | red line -- circular polarization $\xi_3^{(2)}$ of the photon photon with energy $m(\gamma + 1)(1-\epsilon)$.} |
---|
1089 | \end{figure} |
---|
1090 | |
---|
1091 | \subsection{Annihilation at Rest} |
---|
1092 | |
---|
1093 | The method \verb!AtRestDoIt! treats the special case where a positron |
---|
1094 | comes to rest before annihilating. It generates two photons, each with |
---|
1095 | energy $E_{p_{1/2}}=m c^2$ and an isotropic angular distribution. |
---|
1096 | %Eventhough the asymmetry for annihilation at rest is 100\% (cf.\ |
---|
1097 | %figure \ref{pol.annihi1}), there are always unpolarized electrons in |
---|
1098 | %the a material. |
---|
1099 | Starting with the differential cross section for annihilation with |
---|
1100 | positron and electron spins opposed and parallel, |
---|
1101 | respectively,\cite{polAnnihi:Page:1957} |
---|
1102 | \begin{eqnarray} |
---|
1103 | d\sigma_1 &=& \sim \frac{(1 - \beta^2) + \beta^2 (1 - \beta^2) (1 - |
---|
1104 | \cos^2\theta)^2}{(1 - \beta^2\cos^2\theta)^2} d \cos\theta \\ |
---|
1105 | d\sigma_2 &=& \sim \frac{\beta^2(1 - |
---|
1106 | \cos^4\theta)}{(1 - \beta^2\cos^2\theta)^2} d \cos\theta \\ |
---|
1107 | \end{eqnarray} |
---|
1108 | In the limit $\beta\to0$ the cross section $d\sigma_1$ becomes one, |
---|
1109 | and the cross section $d\sigma_2$ vanishes. For the opposed spin |
---|
1110 | state, the total angular |
---|
1111 | momentum is zero and we have a uniform photon distribution. For the |
---|
1112 | parallel case the total angular momentum is 1. Here the two photon |
---|
1113 | final state is forbidden by angular momentum conservation, and it can |
---|
1114 | be assumed that higher order processes (e.g.\ three photon final |
---|
1115 | state) play a dominant role. However, in reality 100\% polarized |
---|
1116 | electron targets do not exist, consequently there are always electrons |
---|
1117 | with opposite spin, where the positron can annihilate with. |
---|
1118 | % Leading again to a uniform distribution. |
---|
1119 | Final state polarization does not play a role for the decay products |
---|
1120 | of a spin zero state, and can be safely neglected. (Is set to zero) |
---|
1121 | |
---|
1122 | \subsection{Status of this document} |
---|
1123 | 20.11.06 created by P.Starovoitov\\ |
---|
1124 | 21.02.07 minor update by A.Sch{\"a}licke\\ |
---|
1125 | |
---|
1126 | \begin{latexonly} |
---|
1127 | |
---|
1128 | \begin{thebibliography}{9} |
---|
1129 | \bibitem{polAnnihi:Star:2006} P.~Starovoitov {\em et.al.}, in preparation. |
---|
1130 | \bibitem{polAnnihi:Page:1957} |
---|
1131 | L.~A.~Page, |
---|
1132 | %Polarization Effects in the Two-Quantum Annihilation of Positrons |
---|
1133 | Phys.~Rev.\ {\bf 106} (1957) 394-398. |
---|
1134 | \end{thebibliography} |
---|
1135 | |
---|
1136 | \end{latexonly} |
---|
1137 | |
---|
1138 | \begin{htmlonly} |
---|
1139 | |
---|
1140 | \subsection{Bibliography} |
---|
1141 | \begin{enumerate} |
---|
1142 | \item P.~Starovoitov {\em et.al.}, in preparation. |
---|
1143 | \item L.~A.~Page, |
---|
1144 | %Polarization Effects in the Two-Quantum Annihilation of Positrons |
---|
1145 | Phys.~Rev.\ {\bf 106} (1957) 394-398. |
---|
1146 | \end{enumerate} |
---|
1147 | |
---|
1148 | \end{htmlonly} |
---|
1149 | |
---|
1150 | % ====================================================================== |
---|
1151 | \clearpage |
---|
1152 | \section{Polarized Compton scattering} |
---|
1153 | \subsection{Method} |
---|
1154 | The class {\em G4PolarizedCompton} simulates |
---|
1155 | Compton scattering of polarized photons with (possibly polarized) |
---|
1156 | electrons in a material. The implementation follows the approach |
---|
1157 | described for the class {\em G4ComptonScattering} introduced |
---|
1158 | in section \ref{sec:em.compton}. |
---|
1159 | Here the explicit production of a Compton scattered photon and the |
---|
1160 | ejected electron is considered taking the effects of polarization on |
---|
1161 | total cross section and mean free path as well as on the distribution |
---|
1162 | of final state particles into account. Further the average |
---|
1163 | polarizations of the scattered photon and electron are calculated. |
---|
1164 | The material electrons are assumed to be free and at rest. |
---|
1165 | |
---|
1166 | \subsection{Total cross section and mean free path} |
---|
1167 | |
---|
1168 | Kinematics of the Compton process is fixed by the initial energy |
---|
1169 | \begin{equation} |
---|
1170 | X=\frac{E_{k_1}}{mc^2} |
---|
1171 | \end{equation} |
---|
1172 | and the variable |
---|
1173 | \begin{equation} |
---|
1174 | \epsilon = \frac{E_{p_1}}{E_{k_1}}, |
---|
1175 | \end{equation} |
---|
1176 | which is the part of total energy avaible in initial state carried out |
---|
1177 | by scattered photon, and the scattering angle |
---|
1178 | \begin{equation} |
---|
1179 | \cos{\theta} = 1 - \frac{1}{X}\left(\frac{1}{\epsilon} - 1\right) |
---|
1180 | \end{equation} |
---|
1181 | The variable $\epsilon$ has the following limits: |
---|
1182 | \begin{equation} |
---|
1183 | \frac{1}{1+2X} \;<\; \epsilon \;<\;1 |
---|
1184 | \end{equation} |
---|
1185 | |
---|
1186 | |
---|
1187 | % ---------------------------------------------------------------------- |
---|
1188 | \subsubsection{Total Cross Section} |
---|
1189 | The total cross section of Compton scattering reads |
---|
1190 | \begin{equation} |
---|
1191 | \sigma^{C}_{pol}= |
---|
1192 | %\frac{\pi \,{{r_e}}^2}{4\,X^2\,{\left( 1 + 2\,X \right) }^2} |
---|
1193 | \frac{\pi \,{{r_e}}^2}{X^2\,{\left( 1 + 2\,X \right) }^2} |
---|
1194 | \left[\sigma^{C}_0 + \zeta^{(1)}_3\zeta^{(2)}_3 \sigma^{C}_L\right] |
---|
1195 | \end{equation} |
---|
1196 | where |
---|
1197 | \begin{equation} |
---|
1198 | \sigma^{C}_0 = \frac{2\,X\,\left( 2 + X\,\left( 1 + X \right) \,\left( 8 + X \right) \right) - |
---|
1199 | {\left( 1 + 2\,X \right) }^2\,\left( 2 + \left( 2 - X \right) \,X \right) \, |
---|
1200 | \ln (1 + 2\,X)}{X} |
---|
1201 | \end{equation} |
---|
1202 | and |
---|
1203 | \begin{equation} |
---|
1204 | \sigma^{C}_L = 2\,X\,\left( 1 + X\,\left( 4 + 5\,X \right) \right) - |
---|
1205 | \left( 1 + X \right) \,{\left( 1 + 2\,X \right) }^2\,\ln (1 + 2\,X) |
---|
1206 | \end{equation} |
---|
1207 | |
---|
1208 | \begin{figure}[ht] |
---|
1209 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/ComptonTA1.eps} |
---|
1210 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/ComptonTA2.eps} |
---|
1211 | \caption{\label{pol.compton1}Compton total cross section asymmetry depending on the energy of incoming photon. |
---|
1212 | Left part, between $0$ and $\sim 1$ MeV, right part -- up to 10MeV. } |
---|
1213 | \end{figure} |
---|
1214 | |
---|
1215 | |
---|
1216 | \subsubsection{Mean free path} |
---|
1217 | When simulating the Compton scattering of a photon with an atomic |
---|
1218 | electron, an empirical cross section formula is used, which reproduces |
---|
1219 | the cross section data down to 10 keV (see section |
---|
1220 | \ref{sec:em.compton}). If both, beam and target, are polarized this |
---|
1221 | mean free path has to be corrected. |
---|
1222 | |
---|
1223 | In the class {\em G4ComptonScattering} the polarized mean free path |
---|
1224 | $\lambda^{\rm pol}$ is defined on the basis of the the unpolarized |
---|
1225 | mean free path $\lambda^{\rm unpol}$ via |
---|
1226 | \begin{equation} |
---|
1227 | \lambda^{\rm pol} = \frac{\lambda^{\rm unpol}}{1 + |
---|
1228 | \zeta_3^{(1)}\zeta_3^{(2)}\, A^C_L } |
---|
1229 | \end{equation} |
---|
1230 | where |
---|
1231 | \begin{equation} |
---|
1232 | A^C_L = \displaystyle \frac{\sigma^A_L}{\sigma^A_0} |
---|
1233 | \end{equation} |
---|
1234 | is the expected asymmetry from the the total polarized Compton |
---|
1235 | cross section given above. |
---|
1236 | This asymmetry is depicted in figure \ref{pol.compton1}. |
---|
1237 | |
---|
1238 | |
---|
1239 | % ---------------------------------------------------------------------- |
---|
1240 | \subsection{Sampling the final state} |
---|
1241 | \subsubsection{Differential Compton Cross Section} |
---|
1242 | |
---|
1243 | In the ultra-relativistic approximation the dependence of the |
---|
1244 | differential cross section on the longitudinal/circular degree of |
---|
1245 | polarization is very simple. It reads |
---|
1246 | \begin{eqnarray} |
---|
1247 | &&\frac{d\sigma_{URA}^C}{de d\varphi}= |
---|
1248 | %\frac{{{r_e}}^2 \,Z}{ 4X} |
---|
1249 | \frac{{{r_e}}^2 }{ X} |
---|
1250 | \Bigg( |
---|
1251 | \frac{\epsilon^2 + 1}{2\,\epsilon} + |
---|
1252 | \frac{ \epsilon^2 -1 }{2\,\epsilon} \left(\zeta_3^{(1)}\zeta_3^{(2)} + |
---|
1253 | \zeta _{3}^{(2)}\,\xi _{3}^{(1)} - \zeta _{3}^{(1)}\,\xi _{3}^{(2)}\right) |
---|
1254 | \nonumber\\ |
---|
1255 | &&+\frac{\epsilon^2 + 1}{2\,\epsilon} \left( \zeta _{3}^{(1)}\,\xi _{3}^{(1)} - \zeta _{3}^{(2)} \,\xi _{3}^{(2)} \right) |
---|
1256 | \Bigg) |
---|
1257 | \end{eqnarray} |
---|
1258 | where |
---|
1259 | \begin{tabular}[t]{l@{\ = \ }l} |
---|
1260 | $r_e$ & classical electron radius \\ |
---|
1261 | $X$ & $E_{k_1}/m_e c^2$ \\ |
---|
1262 | $E_{k_1}$ & energy of the incident photon \\ |
---|
1263 | $m_e c^2$ & electron mass \\ |
---|
1264 | \end{tabular} |
---|
1265 | |
---|
1266 | The fully polarized differential cross section is available in the class {\em |
---|
1267 | G4PolarizedComptonCrossSection}. It takes all mass effects into |
---|
1268 | account, but binding effects are neglected \cite{polCompt:Star:2006,polCompt:Lipps:1954}. |
---|
1269 | The cross section dependence on $\epsilon$ for right handed circularly polarized |
---|
1270 | photons and longitudinally polarized electrons is plotted in figure \ref{pol.compton2a} |
---|
1271 | % |
---|
1272 | \begin{figure} |
---|
1273 | \includegraphics[scale=0.5]{electromagnetic/standard/plots/ComptonXS.eps} |
---|
1274 | \caption{\label{pol.compton2a} |
---|
1275 | Compton scattering differential cross section in arbitrary |
---|
1276 | units. Black line corresponds to the unpolarized cross section; |
---|
1277 | red line -- to the antiparallel spins of initial particles, and blue line -- to the parallel spins. |
---|
1278 | Energy of the incoming photon $E_{k_1} = 10 {\rm MeV}$. |
---|
1279 | } |
---|
1280 | \end{figure} |
---|
1281 | % |
---|
1282 | \begin{figure} |
---|
1283 | \includegraphics[scale=0.5]{electromagnetic/standard/plots/ComptonAsym.eps} |
---|
1284 | \caption{\label{pol.compton2}Compton scattering differential cross section asymmetries in\%. |
---|
1285 | Red line corresponds to the asymmetry due to circular photon and longitudinal electron initial state polarization, |
---|
1286 | green line -- due to circular photon and transverse electron initial state polarization, |
---|
1287 | blue line -- due to linear photon and transverse electron initial state polarization.} |
---|
1288 | \end{figure} |
---|
1289 | |
---|
1290 | |
---|
1291 | \subsubsection{Sampling} |
---|
1292 | |
---|
1293 | The photon energy is sampled according to methods discussed in Chapter |
---|
1294 | 2. The differential cross section can be approximated by a simple |
---|
1295 | function $\Phi(\epsilon)$: |
---|
1296 | \begin{equation} |
---|
1297 | \Phi(\epsilon) = \frac{1}{\epsilon} + \epsilon |
---|
1298 | \end{equation} |
---|
1299 | with the kinematic limits given by |
---|
1300 | \begin{eqnarray} |
---|
1301 | \epsilon_{\rm min} &=& \frac{1}{1+2X} \\ |
---|
1302 | \epsilon_{\rm max} &=& 1 |
---|
1303 | \end{eqnarray} |
---|
1304 | |
---|
1305 | |
---|
1306 | |
---|
1307 | |
---|
1308 | The kinematic of the scattered photon is constructed by the |
---|
1309 | following steps: |
---|
1310 | \begin{enumerate} |
---|
1311 | \item $\epsilon$ is sampled from $\Phi(\epsilon)$ |
---|
1312 | \item calculate the differential cross section, depending on the |
---|
1313 | initial polarizations $\bvec{\zeta}^{(1)}$ and $\bvec{\zeta}^{(2)}$, which |
---|
1314 | the correct normalization. |
---|
1315 | \item $\epsilon$ is accepted with the probability defined by ratio |
---|
1316 | of the differential cross section over the approximation |
---|
1317 | function. |
---|
1318 | \item The $\varphi$ is diced uniformly. |
---|
1319 | \item $\varphi$ is determined from the differential cross section, |
---|
1320 | depending on the initial polarizations $\bvec{\zeta}^{(1)}$ and $\bvec{\zeta}^{(2)}$. |
---|
1321 | \end{enumerate} |
---|
1322 | In figure \ref{pol.compton2} the asymmetries indicate the influence of |
---|
1323 | polarization for an 10MeV incoming positron. The actual behavior is |
---|
1324 | very sensitive to energy of the incoming positron. |
---|
1325 | |
---|
1326 | \subsubsection{Polarization transfer} |
---|
1327 | |
---|
1328 | After the kinematics is fixed the polarization of the |
---|
1329 | outgoing photon is determined. Using the dependence of |
---|
1330 | the differential cross section on the final state polarizations a mean |
---|
1331 | polarization is calculated for each photon according to the method |
---|
1332 | described in section \ref{sec:pol.intro}. |
---|
1333 | |
---|
1334 | The resulting polarization transfer functions $\xi^{(1,2)}(\epsilon)$ |
---|
1335 | are depicted in figure \ref{pol.compton3}. |
---|
1336 | |
---|
1337 | \begin{figure}[ht] |
---|
1338 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/ComptonTransfer1.eps} |
---|
1339 | \includegraphics[width=6.5cm]{electromagnetic/standard/plots/ComptonTransfer2.eps} |
---|
1340 | \caption{\label{pol.compton3} Polarization Transfer in Compton scattering. |
---|
1341 | Blue line corresponds to the longitudinal polarization $\xi_3^{(2)}$ of the electron, |
---|
1342 | red line -- circular polarization $\xi_3^{(1)}$ of the photon.} |
---|
1343 | \end{figure} |
---|
1344 | |
---|
1345 | \subsection{Status of this document} |
---|
1346 | 20.11.06 created by P.Starovoitov\\ |
---|
1347 | 21.02.07 corrected cross section and some minor update by A.Sch{\"a}licke\\ |
---|
1348 | |
---|
1349 | \begin{latexonly} |
---|
1350 | |
---|
1351 | \begin{thebibliography}{9} |
---|
1352 | \bibitem{polCompt:Star:2006} P.~Starovoitov {\em et.al.}, in preparation. |
---|
1353 | %\bibitem{polCompt:Stokes:1852} |
---|
1354 | %G.~Stokes, Trans.\ Cambridge Phil.\ Soc.\ {\bf 9} (1852) 399. |
---|
1355 | % |
---|
1356 | %\bibitem{polCompt:McMaster:1961} |
---|
1357 | %W.~H.~McMaster, Rev.\ Mod.\ Phys.\ {\bf 33} (1961) 8; and references therein. |
---|
1358 | \bibitem{polCompt:Lipps:1954} |
---|
1359 | F.W.~Lipps, H.A.~Tolhoek, |
---|
1360 | %Polarization Phenomena of Electrons and Photons I, |
---|
1361 | Physica {\bf 20} (1954) 85; |
---|
1362 | F.W.~Lipps, H.A.~Tolhoek, |
---|
1363 | %Polarization Phenomena of Electrons and Photons II, |
---|
1364 | Physica {\bf 20} (1954) 395. |
---|
1365 | |
---|
1366 | \end{thebibliography} |
---|
1367 | |
---|
1368 | \end{latexonly} |
---|
1369 | |
---|
1370 | \begin{htmlonly} |
---|
1371 | |
---|
1372 | \subsection{Bibliography} |
---|
1373 | \begin{enumerate} |
---|
1374 | \item P.~Starovoitov {\em et.al.}, in preparation. |
---|
1375 | \item |
---|
1376 | F.W.~Lipps, H.A.~Tolhoek, |
---|
1377 | %Polarization Phenomena of Electrons and Photons I, |
---|
1378 | Physica {\bf 20} (1954) 85; |
---|
1379 | F.W.~Lipps, H.A.~Tolhoek, |
---|
1380 | %Polarization Phenomena of Electrons and Photons II, |
---|
1381 | Physica {\bf 20} (1954) 395. |
---|
1382 | \end{enumerate} |
---|
1383 | |
---|
1384 | \end{htmlonly} |
---|
1385 | |
---|
1386 | |
---|
1387 | \newpage |
---|
1388 | \section{Polarized Bremsstrahlung for electron and positron}\label{sec:pol.bremsstrahlung} |
---|
1389 | \subsection{Method} |
---|
1390 | |
---|
1391 | The polarized version of Bremsstrahlung is based on the unpolarized |
---|
1392 | cross section. Energy loss, mean free path, and distribution of |
---|
1393 | explicitly generated final state particles are treated by the |
---|
1394 | unpolarized version {\em G4eBremsstrahlung}. For details consult |
---|
1395 | section \ref{sec:em.ebrem}. |
---|
1396 | |
---|
1397 | The remaining task is to attribute polarization vectors to the |
---|
1398 | generated final state particles, which is discussed in the following. |
---|
1399 | |
---|
1400 | \subsection{Polarization in gamma conversion and brems\-strahlung} |
---|
1401 | |
---|
1402 | Gamma conversion and bremsstrahlung are cross-symmetric processes |
---|
1403 | (i.e. the Feynman diagram for electron bremsstrahlung can be obtained |
---|
1404 | from the gamma conversion diagram by flipping the incoming photon and |
---|
1405 | outgoing positron lines) and their cross sections closely related. For |
---|
1406 | both processes, the interaction occurs in the field of the nucleus and |
---|
1407 | the total and differential cross section are polarization |
---|
1408 | independent. Therefore, only the polarization transfer from the |
---|
1409 | polarized incoming particle to the outgoing particles is taken into |
---|
1410 | account. |
---|
1411 | % |
---|
1412 | \begin{figure}[htb] |
---|
1413 | \begin{center} |
---|
1414 | \includegraphics [scale=.33] {electromagnetic/standard/plots/Fyn_diag.eps} |
---|
1415 | \caption {Feynman diagrams of Gamma conversion and bremsstrahlung processes.} |
---|
1416 | \end{center} |
---|
1417 | \end{figure} |
---|
1418 | |
---|
1419 | |
---|
1420 | \noindent |
---|
1421 | For both processes, the scattering can be formulated by: |
---|
1422 | \begin{equation} |
---|
1423 | \mathcal{K}_{1}(k_{1},\bvec{\zeta}^{(1)}) + \mathcal{N}_{1}(k_{\mathcal |
---|
1424 | {N}_{1}}, \bvec{\zeta}^{(\mathcal {N}_{1})}) |
---|
1425 | \longrightarrow |
---|
1426 | \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})}) |
---|
1427 | \end{equation} |
---|
1428 | % |
---|
1429 | Where $\mathcal{N}_{1}(k_{\mathcal {N}_{1}}, \bvec{\zeta}^{(\mathcal |
---|
1430 | {N}_{1})})$ and $\mathcal{N}_{2}(p_{\mathcal{N}_{2}}, |
---|
1431 | \bvec{\xi}^{(\mathcal{N}_{2})})$ are the initial and final state of the |
---|
1432 | field of the nucleus respectively assumed to be unchanged, at rest and |
---|
1433 | unpolarized. This leads to $k_{\mathcal {N}_{1}} = k_{\mathcal |
---|
1434 | {N}_{2}} = 0$ and $\bvec{\zeta}^{(\mathcal {N}_{1})} = |
---|
1435 | \bvec{\xi}^{(\mathcal{N}_{2})} = 0$ |
---|
1436 | |
---|
1437 | % Gamma conversion process |
---|
1438 | \textbf{In the case of gamma conversion process}:\\ |
---|
1439 | $\mathcal{K}_{1}(k_{1},\bvec{\zeta}^{(1)})$ is the incoming photon initial |
---|
1440 | state with momentum $k_{1}$ and polarization state $\bvec{\zeta}^{(1)}$. \\ |
---|
1441 | $\mathcal{P}_{1}(p_{1},\bvec{\xi}^{(1)})$ and |
---|
1442 | $\mathcal{P}_{2}(p_{2},\bvec{\xi}^{(2)})$ are the two photons final states with |
---|
1443 | momenta $p_{1}$ and $p_{2}$ and polarization states $\bvec{\xi}^{(1)}$ and $\bvec{\xi}^{(2)}$. |
---|
1444 | |
---|
1445 | % Bremsstrahlung process |
---|
1446 | \textbf{In the case of bremsstrahlung process}:\\ |
---|
1447 | $\mathcal{K}_{1}(k_{1},\bvec{\zeta}^{(1)})$ is the incoming lepton |
---|
1448 | $e^{-}(e^{+})$ initial state with momentum $k_{1}$ and polarization |
---|
1449 | state $\bvec{\zeta}^{(1)}$. \\ |
---|
1450 | $\mathcal{P}_{1}(p_{1},\bvec{\xi}^{(1)})$ is the lepton $e^{-}(e^{+})$ final |
---|
1451 | state with momentum $p_{1}$ and polarization state $\bvec{\xi}^{(1)}$. \\ |
---|
1452 | $\mathcal{P}_{2}(p_{2},\bvec{\xi}^{(2)})$ is the bremsstrahlung photon in |
---|
1453 | final state with momentum $p_{2}$ and polarization state $\bvec{\xi}^{(2)}$. |
---|
1454 | |
---|
1455 | \subsection[Polarization transfer to the photon]{Polarization transfer from the lepton $e^{-}(e^{+})$ to a photon} |
---|
1456 | The polarization transfer from an electron (positron) to a photon in a |
---|
1457 | brems\-strahlung process was first calculated by Olsen and Maximon |
---|
1458 | \cite{polBrems:Olsen_Maximon} taking into account both Coulomb and screening |
---|
1459 | effects. In the Stokes vector formalism, the $e^{-}(e^{+})$ |
---|
1460 | polarization state can be transformed to a photon polarization finale |
---|
1461 | state by means of interaction matrix $T_{\gamma}^{b}$. It defined via |
---|
1462 | % |
---|
1463 | \begin{equation} |
---|
1464 | \left(\begin{array}{c} |
---|
1465 | O \\ |
---|
1466 | \bvec{\xi}^{(2)} |
---|
1467 | \end{array}\right) |
---|
1468 | = T_{\gamma}^{b} \, |
---|
1469 | \left(\begin{array}{c} |
---|
1470 | 1 \\ |
---|
1471 | \bvec{\zeta}^{(1)} |
---|
1472 | \end{array}\right)\;, |
---|
1473 | \label{eq:brem_gamma} |
---|
1474 | \end{equation} |
---|
1475 | % |
---|
1476 | and |
---|
1477 | % |
---|
1478 | \begin{equation} |
---|
1479 | T_{\gamma}^{b}\approx |
---|
1480 | \left( |
---|
1481 | \begin{array}{cccc} |
---|
1482 | 1 & 0 & 0 & 0 \\ |
---|
1483 | D & 0 & 0 & 0 \\ |
---|
1484 | 0 & 0 & 0 & 0 \\ |
---|
1485 | 0 & T & 0 & L \\ |
---|
1486 | \end{array} |
---|
1487 | \right)\;, |
---|
1488 | \label{eq:matrix_brem_g} |
---|
1489 | \end{equation} |
---|
1490 | % |
---|
1491 | where |
---|
1492 | \begin{eqnarray} |
---|
1493 | I &=& (\epsilon_{1}^{2}+\epsilon_{2}^{2})(3+2\Gamma)-2\epsilon_{1}\epsilon_{2}(1+4u^{2}\hat\xi^{2}\Gamma)\\ |
---|
1494 | D &=& \left\lbrace 8\epsilon_{1}\epsilon_{2}u^{2}\hat\xi^{2}\Gamma \right\rbrace / I\\ |
---|
1495 | T &=& \left\lbrace -4k\epsilon_{2}\hat\xi(1-2\hat\xi)u \Gamma \right\rbrace / I \\ |
---|
1496 | L &=& |
---|
1497 | k\lbrace(\epsilon_{1}^{2}+\epsilon_{2}^{2})(3+2\Gamma)-2\epsilon_{2}(1+4u^{2}\hat\xi^{2}\Gamma)\rbrace |
---|
1498 | / I |
---|
1499 | \end{eqnarray} |
---|
1500 | % |
---|
1501 | and |
---|
1502 | % |
---|
1503 | \begin{center} |
---|
1504 | \begin{tabular}{ll} |
---|
1505 | $\epsilon_{1}$ & Total energy of the incoming lepton $e^{+}(e^{-})$ in units $mc^{2}$\\ |
---|
1506 | $\epsilon_{2}$ & Total energy of the outgoing lepton $e^{+}(e^{-})$ in units $mc^{2}$\\ |
---|
1507 | $k$ &$=(\epsilon_{1}-\epsilon_{2})$, the energy of the bremsstrahlung photon in units of $mc^{2}$ |
---|
1508 | \\ |
---|
1509 | $\bvec{p}$ & Electron (positron) initial momentum in units $mc$\\ |
---|
1510 | $\bvec{k}$ & Bremsstrahlung photon momentum in units $mc$\\ |
---|
1511 | $\bvec{u}$ & Component of $\bvec{p}$ |
---|
1512 | perpendicular to $\bvec{k}$ in units $mc$ and $u=\vert \bvec{u} \vert $\\ |
---|
1513 | $\hat\xi$ & $ = 1/(1+u^{2})$ |
---|
1514 | \end{tabular} |
---|
1515 | \end{center} |
---|
1516 | % |
---|
1517 | Coulomb and screening effects are contained in \(\Gamma\), defined as |
---|
1518 | follows |
---|
1519 | \begin{eqnarray} |
---|
1520 | \Gamma &=& \ln\left(\frac{1}{\delta}\right)-2-f(Z)+ |
---|
1521 | \mathcal{F}\left(\frac{\hat\xi}{\delta}\right) \quad \mbox{for } \Delta \le 120 \\ |
---|
1522 | \Gamma &=& \ln\left( \frac{111}{\hat\xi Z^{\frac{1}{3}}}\right)-2-f(z) |
---|
1523 | \quad \mbox{for } \Delta \ge 120 |
---|
1524 | \end{eqnarray} |
---|
1525 | % |
---|
1526 | with |
---|
1527 | % |
---|
1528 | \begin{eqnarray} |
---|
1529 | \Delta &=& \frac{12 Z^{\frac{1}{3}}\epsilon_{1}\epsilon_{2} \hat\xi}{121 |
---|
1530 | k} \quad \mbox{with $Z$ the atomic number and } \delta = |
---|
1531 | \frac{k}{2\epsilon_{1}\epsilon{2}} |
---|
1532 | \end{eqnarray} |
---|
1533 | % |
---|
1534 | % |
---|
1535 | \noindent |
---|
1536 | $f(Z)$ is the coulomb correction term derived by Davies, Bethe |
---|
1537 | and Maximon \cite{polBrems:Davise}. |
---|
1538 | $ \mathcal{F}({\hat\xi}/{\delta})$ contains the screening effects |
---|
1539 | and is zero for $\Delta \le 0.5 $ (No screening effects). For $0.5 \le |
---|
1540 | \Delta \le 120 $ (intermediate screening) it is a slowly decreasing |
---|
1541 | function. The $\mathcal{F}({\hat\xi}/{\delta})$ values versus |
---|
1542 | $\Delta$ are given in table \ref{koch} and used with a linear |
---|
1543 | interpolation in between. |
---|
1544 | |
---|
1545 | The polarization vector of the incoming $e^{-}(e^{+})$ must be rotated |
---|
1546 | into the frame defined by the scattering plane (x-z-plane) and the |
---|
1547 | direction of the outgoing photon (z-axis). The resulting polarization |
---|
1548 | vector of the bremsstrahlung photon is also given in this frame. |
---|
1549 | \begin{table}[h] |
---|
1550 | \caption{$ \mathcal{F}({\hat\xi}/{\delta})$ for intermediate values of the screening factor \cite{polBrems:koch}.} |
---|
1551 | \label{koch} |
---|
1552 | \begin{center} |
---|
1553 | \begin{tabular}{|cc|cc|} |
---|
1554 | \hline |
---|
1555 | $\Delta$ &$ -\mathcal{F}\left({\hat\xi}/{\delta}\right)$ & $\Delta$& $ -\mathcal{F}\left({\hat\xi}/{\delta}\right)$\\ |
---|
1556 | \hline |
---|
1557 | 0.5 & 0.0145 & 40.0 & 2.00 \\ |
---|
1558 | 1.0 & 0.0490 & 45.0 & 2.114\\ |
---|
1559 | 2.0 & 0.1400 & 50.0 & 2.216\\ |
---|
1560 | 4.0 & 0.3312 & 60.0 & 2.393\\ |
---|
1561 | 8.0 & 0.6758 & 70.0 & 2.545\\ |
---|
1562 | 15.0 & 1.126 & 80.0 & 2.676\\ |
---|
1563 | 20.0 & 1.367 & 90.0 & 2.793\\ |
---|
1564 | 25.0 & 1.564 & 100.0 & 2.897\\ |
---|
1565 | 30.0 & 1.731 & 120.0 & 3.078\\ |
---|
1566 | 35.0 & 1.875 & & \\ |
---|
1567 | \hline |
---|
1568 | \end{tabular} |
---|
1569 | \end{center} |
---|
1570 | \end{table} |
---|
1571 | % |
---|
1572 | Using Eq.\ (\ref{eq:brem_gamma}) and the transfer matrix given by |
---|
1573 | Eq.\ (\ref{eq:matrix_brem_g}) the bremsstrahlung photon polarization |
---|
1574 | state in the Stokes formalism \cite{polBrems:McMaster1, polBrems:McMaster2} is given by |
---|
1575 | % |
---|
1576 | \begin{equation} |
---|
1577 | \xi^{(2)} = \left( |
---|
1578 | \begin{array}{c} |
---|
1579 | \xi_{1}^{(2)}\\ |
---|
1580 | \xi_{2}^{(2)} \\ |
---|
1581 | \xi_{3}^{(2)} \\ |
---|
1582 | \end{array} |
---|
1583 | \right) |
---|
1584 | \approx |
---|
1585 | \left( |
---|
1586 | \begin{array}{c} |
---|
1587 | D \\ |
---|
1588 | 0 \\ |
---|
1589 | \zeta_{1}^{(1)}L + \zeta_{2}^{(1)}T \\ |
---|
1590 | \end{array} |
---|
1591 | \right) |
---|
1592 | \end{equation} |
---|
1593 | |
---|
1594 | \subsection[Polarization transfer to the lepton]{Remaining polarization of the lepton after emitting a bremsstrahlung photon} |
---|
1595 | The \(e^{-}(e^{+})\) polarization final state after emitting a |
---|
1596 | bremsstrahlung photon can be calculated using the interaction matrix |
---|
1597 | \(T_{l}^{b}\) which describes the lepton depolarization. The |
---|
1598 | polarization vector for the outgoing \(e^{-}(e^{+})\) is not given by |
---|
1599 | Olsen and Maximon. However, their results can be used to calculate the |
---|
1600 | following transfer matrix \cite{polBrems:klausFl,polBrems:hoogduin}. |
---|
1601 | % |
---|
1602 | \begin{equation} |
---|
1603 | \left(\begin{array}{c} |
---|
1604 | O \\ |
---|
1605 | \bvec{\xi}^{(1)} |
---|
1606 | \end{array}\right) |
---|
1607 | = T_{l}^{b} \, |
---|
1608 | \left(\begin{array}{c} |
---|
1609 | 1 \\ |
---|
1610 | \bvec{\zeta}^{(1)} |
---|
1611 | \end{array}\right) |
---|
1612 | \label{eq:brem_lepton} |
---|
1613 | \end{equation} |
---|
1614 | % |
---|
1615 | \begin{equation} |
---|
1616 | T_{l}^{b}\approx |
---|
1617 | \left( |
---|
1618 | \begin{array}{cccc} |
---|
1619 | 1 & 0 & 0 & 0 \\ |
---|
1620 | D & M & 0 & E \\ |
---|
1621 | 0 & 0 & M & 0 \\ |
---|
1622 | 0 & F & 0 & M+P \\ |
---|
1623 | \end{array} |
---|
1624 | \right) |
---|
1625 | \label{eq:matrix_brem_l} |
---|
1626 | \end{equation} |
---|
1627 | % |
---|
1628 | where |
---|
1629 | % |
---|
1630 | \begin{eqnarray} |
---|
1631 | I &=&(\epsilon_{1}^{2}+\epsilon_{2}^{2})(3+2\Gamma)-2\epsilon_{1}\epsilon_{2}(1+4u^{2}\hat\xi^{2}\Gamma)\\ |
---|
1632 | F &=& \epsilon_{2} \left\lbrace 4k\hat\xi u (1-2\hat\xi)\Gamma\right\rbrace /I \\ |
---|
1633 | E &=& \epsilon_{1} \left\lbrace 4k\hat\xi u (2\hat\xi-1)\Gamma \right\rbrace /I\\ |
---|
1634 | M &=& \left\lbrace 4k\epsilon_{1}\epsilon_{2}(1+\Gamma - 2 u^{2}\hat\xi^{2} \Gamma)\right\rbrace / I \\ |
---|
1635 | P &=& \left\lbrace k^{2} (1+8 \Gamma(\hat\xi - 0.5)^{2}\right\rbrace / I |
---|
1636 | \end{eqnarray} |
---|
1637 | % |
---|
1638 | and |
---|
1639 | % |
---|
1640 | \begin{center} |
---|
1641 | \begin{tabular}{ll} |
---|
1642 | $\epsilon_{1}$ & Total energy of the incoming $e^{+}/e^{-}$ in units $mc^{2}$\\ |
---|
1643 | $\epsilon_{2}$ & Total energy of the outgoing $e^{+}/e^{-}$ in units $mc^{2}$\\ |
---|
1644 | $k$ & $=(\epsilon_{1}-\epsilon_{2})$, energy of the photon in units of $mc^{2}$\\ |
---|
1645 | $\bvec{p}$ & Electron (positron) initial momentum in units $mc$\\ |
---|
1646 | $\bvec{k}$ & Photon momentum in units $mc$\\ |
---|
1647 | $\bvec{u}$ & Component of $\bvec{p}$ |
---|
1648 | perpendicular to $\bvec{k}$ in units $mc$ and $u=\vert \bvec{u} \vert $ |
---|
1649 | \end{tabular} |
---|
1650 | \end{center} |
---|
1651 | |
---|
1652 | Using Eq.\ (\ref{eq:brem_lepton}) and the transfer matrix given by |
---|
1653 | Eq.\ (\ref{eq:matrix_brem_l}) the \(e^{-}(e^{+})\) polarization state |
---|
1654 | after emitting a bremsstrahlung photon is given in the Stokes |
---|
1655 | formalism by |
---|
1656 | % |
---|
1657 | \begin{equation} |
---|
1658 | \xi^{(1)} = \left( |
---|
1659 | \begin{array}{c} |
---|
1660 | \xi_{1}^{(1)}\\ |
---|
1661 | \xi_{2}^{(1)} \\ |
---|
1662 | \xi_{3}^{(1)} \\ |
---|
1663 | \end{array} |
---|
1664 | \right) |
---|
1665 | \approx |
---|
1666 | \left( |
---|
1667 | \begin{array}{c} |
---|
1668 | \zeta_{1}^{(1)} M + \zeta_{3}^{(1)} E \\ |
---|
1669 | \zeta_{2}^{(1)} M \\ |
---|
1670 | \zeta_{3}^{(1)}(M+P) + \zeta_{1}^{(1)} F \\ |
---|
1671 | \end{array} |
---|
1672 | \right) |
---|
1673 | \;. |
---|
1674 | \end{equation} |
---|
1675 | |
---|
1676 | \subsection{Status of this document} |
---|
1677 | 20.11.06 created by K.Laihem\\ |
---|
1678 | 21.02.07 minor update by A.Sch{\"a}licke\\ |
---|
1679 | |
---|
1680 | \begin{latexonly} |
---|
1681 | |
---|
1682 | \begin{thebibliography}{7} |
---|
1683 | |
---|
1684 | \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. |
---|
1685 | |
---|
1686 | \bibitem{polBrems:McMaster1} W.H.~McMaster. Polarization and the Stokes parameters. American Journal of Physics, 22(6):351-362, 1954. |
---|
1687 | |
---|
1688 | \bibitem{polBrems:McMaster2}W.H.~McMaster. Matrix representation of polarization. Reviews of Modern Physics, 33(1):8-29, 1961. |
---|
1689 | |
---|
1690 | \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. |
---|
1691 | |
---|
1692 | \bibitem{polBrems:hoogduin}Hoogduin, Johannes Marinus. Electron, positron and photon polarimetry. PhD thesis, Rijksuniversiteit Groningen 1997. |
---|
1693 | |
---|
1694 | \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. |
---|
1695 | |
---|
1696 | \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. |
---|
1697 | |
---|
1698 | \bibitem{polBrems:Laihem:thesis} |
---|
1699 | K.~Laihem, PhD thesis, Humboldt University Berlin, Germany, (2007). |
---|
1700 | |
---|
1701 | \end{thebibliography} |
---|
1702 | \end{latexonly} |
---|
1703 | |
---|
1704 | \begin{htmlonly} |
---|
1705 | \begin{thebibliography}{9} |
---|
1706 | \begin{enumerate} |
---|
1707 | |
---|
1708 | \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. |
---|
1709 | |
---|
1710 | \item W.H.~McMaster. Polarization and the Stokes parameters. American Journal of Physics, 22(6):351-362, 1954. |
---|
1711 | |
---|
1712 | \item W.H.~McMaster. Matrix representation of polarization. Reviews of Modern Physics, 33(1):8-29, 1961. |
---|
1713 | |
---|
1714 | \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. |
---|
1715 | |
---|
1716 | \item Hoogduin, Johannes Marinus. Electron, positron and photon polarimetry. PhD thesis, Rijksuniversiteit Groningen 1997. |
---|
1717 | |
---|
1718 | \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. |
---|
1719 | |
---|
1720 | \item H.W.~Koch and J.W.~Motz. Bremsstrahlung cross-section formulas and related data. Review Mod. Phys., 31(4):920-955, 1959. |
---|
1721 | |
---|
1722 | \item K.~Laihem, PhD thesis, Humboldt University Berlin, Germany, (2007). |
---|
1723 | |
---|
1724 | \end{enumerate} |
---|
1725 | \end{htmlonly} |
---|
1726 | |
---|
1727 | \newpage |
---|
1728 | \section{Polarized Gamma conversion into an electron--positron pair} |
---|
1729 | \subsection{Method} |
---|
1730 | |
---|
1731 | The polarized version of gamma conversion is based on the EM standard |
---|
1732 | process {\em G4GammaConversion}. Mean free path and the distribution |
---|
1733 | of explicitly generated final state particles are treated by this |
---|
1734 | version. For details consult |
---|
1735 | section \ref{sec:em.conv}. |
---|
1736 | |
---|
1737 | The remaining task is to attribute polarization vectors to the |
---|
1738 | generated final state leptons, which is discussed in the following. |
---|
1739 | |
---|
1740 | |
---|
1741 | \subsection[Polarization transfer]{Polarization transfer from the photon to the two leptons} |
---|
1742 | Gamma conversion process is essentially the inverse process of |
---|
1743 | Bremsstrahlung and the interaction matrix is obtained by inverting the |
---|
1744 | rows and columns of the bremsstrahlung matrix and changing the sign of |
---|
1745 | \(\epsilon_{2}\), cf.\ section \ref{sec:pol.bremsstrahlung}. It |
---|
1746 | follows from the work by Olsen and Maximon |
---|
1747 | \cite{polPair:Olsen_Maximon} that the polarization state \(\xi^{(1)}\) of an |
---|
1748 | electron or positron after pair production is obtained by |
---|
1749 | % |
---|
1750 | \begin{equation} |
---|
1751 | \left(\begin{array}{c} |
---|
1752 | O \\ |
---|
1753 | \bvec{\xi}^{(1)} |
---|
1754 | \end{array}\right) |
---|
1755 | = T_{l}^{p} \, |
---|
1756 | \left(\begin{array}{c} |
---|
1757 | 1 \\ |
---|
1758 | \bvec{\zeta}^{(1)} |
---|
1759 | \end{array}\right) |
---|
1760 | \label{eq:conv_lepton} |
---|
1761 | \end{equation} |
---|
1762 | % |
---|
1763 | and |
---|
1764 | % |
---|
1765 | \begin{equation} |
---|
1766 | T_{l}^{p}\approx |
---|
1767 | \left( |
---|
1768 | \begin{array}{cccc} |
---|
1769 | 1 & D & 0 & 0 \\ |
---|
1770 | 0 & 0 & 0 & T \\ |
---|
1771 | 0 & 0 & 0 & 0 \\ |
---|
1772 | 0 & 0 & 0 & L \\ |
---|
1773 | \end{array} |
---|
1774 | \right) |
---|
1775 | \;, |
---|
1776 | \label{eq:matrix_conv} |
---|
1777 | \end{equation} |
---|
1778 | % |
---|
1779 | where |
---|
1780 | \begin{eqnarray} |
---|
1781 | I &=& (\epsilon_{1}^{2}+\epsilon_{2}^{2})(3+2\Gamma)+2\epsilon_{1}\epsilon_{2}(1+4u^{2}\hat\xi^{2}\Gamma)\\ |
---|
1782 | D &=& \left\lbrace -8\epsilon_{1}\epsilon_{2}u^{2}\hat\xi^{2}\Gamma \right\rbrace / I\\ |
---|
1783 | T &=& \left\lbrace -4k\epsilon_{2}\hat\xi(1-2\hat\xi)u \Gamma \right\rbrace / I \\ |
---|
1784 | L &=& |
---|
1785 | k\lbrace(\epsilon_{1}^{2}+\epsilon_{2}^{2})(3+2\Gamma)-2\epsilon_{2}(1+4u^{2}\hat\xi^{2}\Gamma)\rbrace/ I |
---|
1786 | \end{eqnarray} |
---|
1787 | and |
---|
1788 | \begin{center} |
---|
1789 | \begin{tabular}{ll} |
---|
1790 | $\epsilon_{1}$ & total energy of the first lepton $e^{+}(e^{-})$ in units $mc^{2}$\\ |
---|
1791 | $\epsilon_{2}$ & total energy of the second lepton $e^{-}(e^{+})$ in units $mc^{2}$\\ |
---|
1792 | $k=(\epsilon_{1}+\epsilon_{2})$ & energy of the incoming photon in units of $mc^{2}$\\ |
---|
1793 | $\bvec{p}$ & electron=positron initial momentum in units $mc$\\ |
---|
1794 | $\bvec{k}$ & photon momentum in units $mc$\\ |
---|
1795 | $\bvec{u}$ & electron/positron initial momentum in units $mc$\\ |
---|
1796 | $u$ & $=\vert \bvec{u} \vert $ |
---|
1797 | \end{tabular} |
---|
1798 | \end{center} |
---|
1799 | % |
---|
1800 | %Here, $\epsilon_{1}(\epsilon_{2})$ is the energy of the observed |
---|
1801 | %electron or positron. The matrix (\ref{eq:matrix_conv}) for pair |
---|
1802 | %production is the transpose of matrix (\ref{eq:matrix_brem_g}). |
---|
1803 | Coulomb and screening effects are contained in \(\Gamma\), defined in |
---|
1804 | section \ref{sec:pol.bremsstrahlung}. |
---|
1805 | |
---|
1806 | |
---|
1807 | Using Eq.\ (\ref{eq:conv_lepton}) and the transfer matrix given by |
---|
1808 | Eq.\ (\ref{eq:matrix_conv}) the polarization state of |
---|
1809 | the produced $e^{-}(e^{+})$ is given in the Stokes formalism by: |
---|
1810 | |
---|
1811 | \begin{equation} |
---|
1812 | \xi^{(1)} = \left( |
---|
1813 | \begin{array}{c} |
---|
1814 | \xi_{1}^{(1)}\\ |
---|
1815 | \xi_{2}^{(1)} \\ |
---|
1816 | \xi_{3}^{(1)} \\ |
---|
1817 | \end{array} |
---|
1818 | \right) |
---|
1819 | \approx |
---|
1820 | \left( |
---|
1821 | \begin{array}{c} |
---|
1822 | \zeta_{3}^{(1)} T \\ |
---|
1823 | 0 \\ |
---|
1824 | \zeta_{3}^{(1)} L \\ |
---|
1825 | \end{array} |
---|
1826 | \right) |
---|
1827 | \end{equation} |
---|
1828 | |
---|
1829 | |
---|
1830 | \subsection{Status of this document} |
---|
1831 | 20.11.06 created by K.Laihem\\ |
---|
1832 | 21.02.07 minor update by A.Sch{\"a}licke\\ |
---|
1833 | |
---|
1834 | \begin{latexonly} |
---|
1835 | |
---|
1836 | \begin{thebibliography}{9} |
---|
1837 | |
---|
1838 | \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. |
---|
1839 | |
---|
1840 | \bibitem{polPair:Laihem:thesis} |
---|
1841 | K.~Laihem, PhD thesis, Humboldt University Berlin, Germany, (2007). |
---|
1842 | |
---|
1843 | \end{thebibliography} |
---|
1844 | |
---|
1845 | \end{latexonly} |
---|
1846 | |
---|
1847 | \begin{htmlonly} |
---|
1848 | \begin{enumerate} |
---|
1849 | |
---|
1850 | \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. |
---|
1851 | |
---|
1852 | \item K.~Laihem, PhD thesis, Humboldt University Berlin, Germany, (2007). |
---|
1853 | |
---|
1854 | \end{enumerate} |
---|
1855 | |
---|
1856 | \end{htmlonly} |
---|
1857 | |
---|
1858 | |
---|
1859 | |
---|
1860 | % LocalWords: Bhabha |
---|