% ====================================================================== %\newcommand{\bvec}[1]{{\mathbf{#1}}} \newcommand{\bvec}[1]{{\boldsymbol{#1}}} %% use boldsymbol if amsmath is available! \section{Introduction}\label{sec:pol.intro} With the EM polarization extension it is possible to track polarized particles (leptons and photons). Special emphasis will be put in the proper treatment of polarized matter and its interaction with longitudinal polarized electrons/positrons or circularly polarized photons, which is for instance essential for the simulation of positron polarimetry. The implementation is base on Stokes vectors \cite{polIntro:McMaster:1961}. Further details can be found in \cite{polIntro:Laihem:thesis}. In its current state, the following polarization dependent processes are considered \begin{itemize} \item Bhabha/M{\o}ller scattering, \item Positron Annihilation, \item Compton scattering, \item Pair creation, \item Bremsstrahlung. \end{itemize} %\subsection{Existing codes for the simulation of polarized processes} Several simulation packages for the realistic description of the development of electromagnetic showers in matter have been developed. A prominent example of such codes is EGS (Electron Gamma Shower)\cite{polIntro:Nelson:1985ec}. For this simulation framework extensions with the treatment of polarized particles exist \cite{polIntro:Floettmann:thesis,polIntro:Namito:1993sv,polIntro:Liu:2000ey}; the most complete has been developed by K.~Fl{\"o}ttmann \cite{polIntro:Floettmann:thesis}. It is based on the matrix formalism \cite{polIntro:McMaster:1961}, which enables a very general treatment of polarization. However, the Fl{\"o}ttmann extension concentrates on evaluation of polarization transfer, i.e.\ the effects of polarization induced asymmetries are neglected, and interactions with polarized media are not considered. Another important simulation tool for detector studies is \textsc{Geant3} \cite{polIntro:Brun:1985ps}. Here also some effort has been made to include polarization \cite{polIntro:Alexander:2003fh,polIntro:Hoogduin:thesis}, but these extensions are not publicly available. %\section{Definitions} In general the implementation of polarization in this EM polarization library follows very closely the approach by K.~Fl{\"o}tt\-mann \cite{polIntro:Floettmann:thesis}. The basic principle is to associate a {\em Stokes vector} to each particle and track the mean polarization from one interaction to another. The basics for this approach is the matrix formalism as introduced in \cite{polIntro:McMaster:1961}. \subsection{Stokes vector} The {\em Stokes vector} \cite{polIntro:Stokes:1852,polIntro:McMaster:1961} is a rather simple object (in comparison to e.g.\ the spin density matrix), three real numbers are sufficient for the characterization of the polarization state of any single electron, positron or photon. Using {\em Stokes vectors} {\bf all} possible polarization states can be described, i.e.\ circular and linear polarized photons can be handled with the same formalism as longitudinal and transverse polarized electron/positrons. The {\em Stokes vector} can be used also for beams, in the sense that it defines a mean polarization. In the EM polarization library the Stokes vector is defined as follows: \begin{center} %\rotatebox{90}{ Method A} \renewcommand{\arraystretch}{1.15} \begin{tabular}{|c|c|c|} \hline & Photons & Electrons \\ \hline $\xi_1$ & linear polarization & polarization in x direction \\ $\xi_2$ & linear polarization but $\pi/4$ to right & polarization in y direction \\ $\xi_3$ & circular polarization & polarization in z direction \\ \hline \end{tabular} \end{center} This definition is assumed in the {\em particle reference frame}, i.e. with the momentum of the particle pointing to the z direction, cf.\ also next section about coordinate transformations. Correspondingly a 100\% longitudinally polarized electron or positron is characterized by \begin{equation} \bvec{\xi}=\mbox{$\scriptscriptstyle\left(\begin{array}{c}0\\0\\\pm1\end{array}\right)$}, \end{equation} where $\pm1$ corresponds to spin parallel (anti parallel) to particle's momentum. % Note that this definition is similar, but not identical to the definition used in McMaster \cite{polIntro:McMaster:1961}. Many scattering cross sections of polarized processes using Stokes vectors for the characterization of initial and final states are available in \cite{polIntro:McMaster:1961}. In general a differential cross section has the form \begin{equation} \frac{d\sigma(\bvec{\zeta}^{(1)},\bvec{\zeta}^{(2)},\bvec{\xi}^{(1)},\bvec{\xi}^{(2)})}{d\Omega}\;, \end{equation} i.e.\ it is a function of the polarization states of the initial particles $\bvec{\zeta}^{(1)}$ and $\bvec{\zeta}^{(2)}$, as well as of the polarization states of the final state particles $\bvec{\xi}^{(1)}$ and $\bvec{\xi}^{(2)}$ (in addition to the kinematic variables $E$, $\theta$, and $\phi$). Consequently, in a simulation we have to account for \begin{itemize} \item Asymmetries: \item[] Polarization of beam ($\bvec{\zeta}^{(1)}$) and target ($\bvec{\zeta}^{(2)}$) can induce azimuthal and polar asymmetries, and may also influence on the total cross section ({\tt Geant4: GetMeanFreePath()}). \item Polarization transfer / depolarization effects \item[] The dependence on the final state polarizations defines a possible transfer from initial polarization to final state particles. \end{itemize} \subsection{Transfer matrix} %For asymmetries one can extent the existing standard EM physics classes, %introducing the polarization of the initial states. On the other hand %for a general simulation of polarization transfer one has to work harder. Using the formalism of McMaster, differential cross section and polarization transfer from the initial state ($\bvec{\zeta}^{(1)}$) to one final state particle ($\bvec{\xi}^{(1)}$) are combined in an interaction matrix $T$: \begin{equation} \left(\begin{array}{c} O \\ \bvec{\xi}^{(1)} \end{array}\right) = T \, \left(\begin{array}{c} I \\ \bvec{\zeta}^{(1)} \end{array}\right)\;, \end{equation} where $I$ and $O$ are the incoming and outgoing currents, respectively. % In general the $4\times4$ matrix $T$ depends on the target polarization $\bvec{\zeta}^{(2)}$ (and of course on the kinematic variables $E$, $\theta$, $\phi$). Similarly one can define a matrix defining the polarization transfer to second final state particle like \begin{equation} \left(\begin{array}{c} O \\ \bvec{\xi}^{(2)} \end{array}\right) = T' \, \left(\begin{array}{c}I\\ \bvec{\zeta}^{(1)}\end{array}\right) \;. \end{equation} % %The components $I$ and $O$ refer to the incoming and outgoing %intensities, respectively. In this framework the transfer matrix $T$ is of the form \begin{equation} T = \left( \begin{array}{llll} S & A_1 & A_2 & A_3 \\ P_1 & M_{11} & M_{21} & M_{31} \\ P_2 & M_{12} & M_{22} & M_{32} \\ P_3 & M_{13} & M_{23} & M_{33} \\ \end{array} \right) \;. \end{equation} The matrix elements $T_{ij}$ can be identified as (unpolarized) differential cross section ($S$), polarized differential cross section ($A_j$), polarization transfer ($M_{ij}$), and (de)polarization ($P_i$). In the Fl{\"o}ttmann extension the elements $A_j$ and $P_i$ have been neglected, thus concentrating on polarization transfer only. Using the full matrix takes now all polarization effects into account. The transformation matrix, i.e.\ the dependence of the mean polarization of final state particles, can be derived from the asymmetry of the differential cross section w.r.t.\ this particular polarization. Where the asymmetry is defined as usual by \begin{equation} A = \frac{\sigma(+1)-\sigma(-1)}{\sigma(+1)+\sigma(-1)} \;. \end{equation} The mean final state polarizations can be determined coefficient by coefficient. % %For instance the components of the mean Stokes vector %% following eq.\ \eqref{eq:diffxsec} %$\bvec{\hat\xi}^{(1)}$ of the first final state particle is obtained %by %\begin{equation} % \hat\xi^{(2)}_1 = \frac{\sigma(\bvec{\zeta}^{(1)},\bvec{\zeta}^{(2)}, %\bvec{\xi}^{(1)}=\mbox{\small$\left(\!\!\begin{array}{c}+1\\0\\0\end{array}\!\!\right)$}, %\bvec{\xi}^{(2)}=\mbox{\small$\left(\!\!\begin{array}{c}0\\0\\0\end{array}\!\!\right)$}) %- %\sigma(\bvec{\zeta}^{(1)},\bvec{\zeta}^{(2)}, %\bvec{\xi}^{(1)}=\mbox{\small$\left(\!\!\begin{array}{c}-1\\0\\0\end{array}\!\!\right)$}, %\bvec{\xi}^{(2)}=\mbox{\small$\left(\!\!\begin{array}{c}0\\0\\0\end{array}\!\!\right)$}) %}{\sigma(\dots)+\sigma(\dots)} %\end{equation} % %\begin{equation} % \hat\xi^{(2)}_2 = \frac{\sigma(\bvec{\zeta}^{(1)},\bvec{\zeta}^{(2)}, %\bvec{\xi}^{(1)}=\mbox{\small$\left(\!\!\begin{array}{c}0\\+1\\0\end{array}\!\!\right)$}, %\bvec{\xi}^{(2)}=\mbox{\small$\left(\!\!\begin{array}{c}0\\0\\0\end{array}\!\!\right)$}) %- %\sigma(\bvec{\zeta}^{(1)},\bvec{\zeta}^{(2)}, %\bvec{\xi}^{(1)}=\mbox{\small$\left(\!\!\begin{array}{c}0\\-1\\0\end{array}\!\!\right)$}, %\bvec{\xi}^{(2)}=\mbox{\small$\left(\!\!\begin{array}{c}0\\0\\0\end{array}\!\!\right)$}) %}{\sigma(\dots)+\sigma(\dots)} %\end{equation} % %\begin{equation} % \hat\xi^{(2)}_3 = \frac{\sigma(\bvec{\zeta}^{(1)},\bvec{\zeta}^{(2)}, %\bvec{\xi}^{(1)}=\mbox{\small$\left(\!\!\begin{array}{c}0\\0\\+1\end{array}\!\!\right)$}, %\bvec{\xi}^{(2)}=\mbox{\small$\left(\!\!\begin{array}{c}0\\0\\0\end{array}\!\!\right)$}) %- %\sigma(\bvec{\zeta}^{(1)},\bvec{\zeta}^{(2)}, %\bvec{\xi}^{(1)}=\mbox{\small$\left(\!\!\begin{array}{c}0\\0\\-1\end{array}\!\!\right)$}, %\bvec{\xi}^{(2)}=\mbox{\small$\left(\!\!\begin{array}{c}0\\0\\0\end{array}\!\!\right)$}) %}{\sigma(\dots)+\sigma(\dots)} %\end{equation} % In general, the differential cross section is a linear function of the polarizations, i.e. \begin{eqnarray} \frac{d\sigma(\bvec{\zeta}^{(1)},\bvec{\zeta}^{(2)},\bvec{\xi}^{(1)},\bvec{\xi}^{(2)})}{d\Omega} &=& \Phi_{(\zeta^{(1)},\zeta^{(2)})} + \bvec{A}_{(\zeta^{(1)},\zeta^{(2)})} \cdot\bvec{\xi}^{(1)} + \bvec{B}_{(\zeta^{(1)},\zeta^{(2)})} \cdot\bvec{\xi}^{(2)} \nonumber\\ && \quad \quad \quad +\, {\bvec{\xi}^{(1)}}^T M_{(\zeta^{(1)},\zeta^{(2)})} \,\bvec{\xi}^{(2)} \end{eqnarray} In this form, the mean polarization of the final state can be read off easily, and one obtains \begin{eqnarray} \langle\bvec{\xi}^{(1)}\rangle &=& \frac{1}{\Phi_{(\zeta^{(1)},\zeta^{(2)})}} \bvec{A}_{(\zeta^{(1)},\zeta^{(2)})} \;\; \mbox{and} \\ \langle\bvec{\xi}^{(2)}\rangle &=& \frac{1}{\Phi_{(\zeta^{(1)},\zeta^{(2)})}} \bvec{B}_{(\zeta^{(1)},\zeta^{(2)})} \;. \end{eqnarray} Note, that the {\em mean} polarization states do not depend on the correlation matrix $M_{(\zeta^{(1)},\zeta^{(2)})}$. In order to account for correlation one has to generate {\em single} particle Stokes vector explicitly, i.e.\ on an event by event basis. However, this implementation generates {\em mean} polarization states, and neglects correlation effects. %\newpage \subsection{Coordinate transformations} \begin{figure}[h!] \centerline{\includegraphics[width=8.cm]{electromagnetic/standard/plots/frames.eps}} \caption{\label{pol.interframe} The {\em interaction frame} and the {\em particle frames} for the example of Compton scattering. The momenta of all participating particle lie in the $x$-$z$-plane, the scattering plane. The incoming photon gives the $z$ direction. The outgoing photon is defined as {\em particle 1} and gives the $x$-direction, perpendicular to the $z$-axis. The $y$-axis is then perpendicular to the scattering plane and completes the definition of a right handed coordinate system called {\em interaction frame}. The {\em particle frame} is defined by the Geant4 routine {\tt G4ThreeMomemtum::rotateUz()}.} \end{figure} Three different coordinate systems are used in the evaluation of polarization states: \begin{itemize} \item {\bf World frame} %\item[] The geometry of the target, and the momenta of all particles in Geant4 are noted in the world frame $X$, $Y$, $Z$ (the {\em global reference frame}, GRF). It is the basis of the calculation of any other coordinate system. \item {\bf Particle frame} %\item[] Each particle is carrying its own coordinate system. In this system the direction of motion coincides with the $z$-direction. Geant4 provides a transformation from any particle frame to the World frame by the method {\tt G4ThreeMomemtum::rotateUz()}. Thus, the $y$-axis of the {\em particle reference frame} (PRF) lies in the $X$-$Y$-plane of the world frame. The Stokes vector of any moving particle is defined w.r.t. the corresponding particle frame. Particles at rest (e.g.\ electrons of a media) use the world frame as particle frame. \item {\bf Interaction frame} %\item[] For the evaluation of the polarization transfer another coordinate system is used, defined by the scattering plane, cf.\ fig.\ \ref{pol.interframe}. There the $z$-axis is defined by the direction of motion of the incoming particle. The scattering plane is spanned by the $z$-axis and the $x$-axis, in a way, that the direction of {\em particle~1} has a positive $x$ component. The definition of {\em particle~1} depends on the process, for instance in Compton scattering, the outgoing photon is referred as {\em particle~1}\footnote{Note, for an incoming particle travelling on the $Z$-axis (of GRF), the $y$-axis of the PRF of both outgoing particles is parallel to the $y$-axis of the {\em interaction frame}.}. \end{itemize} All frames are right handed. \subsection{Polarized beam and material} Polarization of beam particles is well established. It can be used for simulating low-energy Compton scattering of linear polarized photons. The interpretation as Stokes vector allows now the usage in a more general framework. % The polarization state of a (initial) beam particle can be fixed using standard the ParticleGunMessenger class. For example, the class {\tt G4ParticleGun} provides the method {\tt SetParticlePolarization()}, which is usually accessable via \begin{verbatim} /gun/polarization \end{verbatim} in a macro file. In addition for the simulation of polarized media, a possibility to assign Stokes vectors to physical volumes is provided by a new class, the so-called {\em G4PolarizationManager}. %It also provides some helper routines for the evaluation of Stokes %vectors in different frames of reference. % The procedure to assign a polarization vector to a media, is done during the {\em detector construction}. There the {\em logical volumes} with certain polarization are made known to {\em polarization manager}. One example {\tt DetectorConstruction} might look like follows: \begin{verbatim} G4double Targetthickness = .010*mm; G4double Targetradius = 2.5*mm; G4Tubs *solidTarget = new G4Tubs("solidTarget", 0.0, Targetradius, Targetthickness/2, 0.0*deg, 360.0*deg ); G4LogicalVolume * logicalTarget = new G4LogicalVolume(solidTarget, iron, "logicalTarget", 0,0,0); G4VPhysicalVolume * physicalTarget = new G4PVPlacement(0,G4ThreeVector(0.*mm, 0.*mm, 0.*mm), logicalTarget, "physicalTarget", worldLogical, false, 0); G4PolarizationManager * polMgr = G4PolarizationManager::GetInstance(); polMgr->SetVolumePolarization(logicalTarget,G4ThreeVector(0.,0.,0.08)); \end{verbatim} Once a logical volume is known to the {\tt G4PolarizationManager}, its polarization vector can be accessed from a macro file by its name, e.g.\ the polarization of the logical volume called ``logicalTarget'' can be changed via \begin{verbatim} /polarization/volume/set logicalTarget 0. 0. -0.08 \end{verbatim} Note, the polarization of a material is stated in the world frame. \subsection{Status of this document} 20.11.06 created by A.Sch{\"a}licke\\ \begin{latexonly} \begin{thebibliography}{10} \bibitem{polIntro:McMaster:1961} W.~H.~McMaster, Rev.\ Mod.\ Phys.\ {\bf 33} (1961) 8; and references therein. \bibitem{polIntro:Laihem:thesis} K.~Laihem, PhD thesis, Measurement of the positron polarization at an helical undulator based positron source for the International Linear Collider ILC, Humboldt University Berlin, Germany, (2008). %%EGS \bibitem{polIntro:Nelson:1985ec} W.~R.~Nelson, H.~Hirayama, D.~W.~O.\ Rogers, %``The Egs4 Code System,'' SLAC-R-0265. \bibitem{polIntro:Floettmann:thesis} K.~Fl\"ottmann, PhD thesis, DESY Hamburg (1993); DESY-93-161. %kek extension \bibitem{polIntro:Namito:1993sv} Y.~Namito, S.~Ban, H.~Hirayama, %``Implementation of linearly polarized photon scattering into the EGS4 code,'' Nucl.\ Instrum.\ Meth.\ A {\bf 332} (1993) 277. \bibitem{polIntro:Liu:2000ey} J.~C.~Liu, T.~Kotseroglou, W.~R.~Nelson, D.~C.~Schultz, %``Polarization study for NLC positron source using EGS4,'' SLAC-PUB-8477. %Geant3 \bibitem{polIntro:Brun:1985ps} R.~Brun, M.~Caillat, M.~Maire, G.~N.~Patrick, L.~Urban, %``The Geant3 Electromagnetic Shower Program And A Comparison With The Egs3 %Code,'' CERN-DD/85/1. %% E166 \bibitem{polIntro:Alexander:2003fh} G.~Alexander {\it et al.}, %``Undulator-based production of polarized positrons: A proposal for % the 50-GeV beam in the FFTB,'' SLAC-TN-04-018, SLAC-PROPOSAL-E-166. \bibitem{polIntro:Hoogduin:thesis} J.~Hoogduin, PhD thesis, Rijksuniversiteit Groningen (1997). \bibitem{polIntro:Stokes:1852} G.~Stokes, Trans.\ Cambridge Phil.\ Soc.\ {\bf 9} (1852) 399. \end{thebibliography} \end{latexonly} \begin{htmlonly} \begin{enumerate}{10} \item W.~H.~McMaster, Rev.\ Mod.\ Phys.\ {\bf 33} (1961) 8; and references therein. \item K.~Laihem, PhD thesis, Measurement of the positron polarization at an helical undulator based positron source for the International Linear Collider ILC, Humboldt University Berlin, Germany, (2008). %%EGS \item W.~R.~Nelson, H.~Hirayama, D.~W.~O.\ Rogers, %``The Egs4 Code System,'' SLAC-R-0265. \item K.~Fl\"ottmann, PhD thesis, DESY Hamburg (1993); DESY-93-161. %kek extension \item Y.~Namito, S.~Ban, H.~Hirayama, %``Implementation of linearly polarized photon scattering into the EGS4 code,'' Nucl.\ Instrum.\ Meth.\ A {\bf 332} (1993) 277. \item J.~C.~Liu, T.~Kotseroglou, W.~R.~Nelson, D.~C.~Schultz, %``Polarization study for NLC positron source using EGS4,'' SLAC-PUB-8477. %Geant3 \item R.~Brun, M.~Caillat, M.~Maire, G.~N.~Patrick, L.~Urban, %``The Geant3 Electromagnetic Shower Program And A Comparison With The Egs3 %Code,'' CERN-DD/85/1. %% E166 \item G.~Alexander {\it et al.}, %``Undulator-based production of polarized positrons: A proposal for % the 50-GeV beam in the FFTB,'' SLAC-TN-04-018, SLAC-PROPOSAL-E-166. \item J.~Hoogduin, PhD thesis, Rijksuniversiteit Groningen (1997). \item G.~Stokes, Trans.\ Cambridge Phil.\ Soc.\ {\bf 9} (1852) 399. \end{enumerate} \end{htmlonly} % ====================================================================== \newcommand{\Mvariable}[1]{r_e} \newpage \section{Ionization}\label{sec:polarizedIonization} \subsection{Method} The class {\em G4ePolarizedIonization} provides continuous and discrete energy losses of polarized electrons and positrons in a material. It evaluates polarization transfer and -- if the material is polarized -- asymmetries in the explicit delta rays production. The implementation baseline follows the approach derived for the class {\em G4eIonization} described in sections \ref{en_loss} and \ref{sec:em.eion}. For continuous energy losses the effects of a polarized beam or target are negligible provided the separation cut $T_{\rm cut}$ is small, and are therefore not considered separately. On the other hand, in the explicit production of delta rays by M{\o}ller or Bhabha scattering, the effects of polarization on total cross section and mean free path, on distribution of final state particles and the average polarization of final state particles are taken into account. % ---------------------------------------------------------------------- \subsection{Total cross section and mean free path} Kinematics of Bhabha and M{\o}ller scattering is fixed by initial energy \begin{equation} \gamma=\frac{E_{k_1}}{m c^2}% =\frac{s}{2m^2}-1 \end{equation} and variable \begin{equation} \epsilon = \frac{E_{p_2}-m c^2}{E_{k_1}-m c^2}, \end{equation} which is the part of kinetic energy of initial particle carried out by scatter. Lower kinematic limit for $\epsilon$ is $0$, but in order to avoid divergencies in both total and differential cross sections one sets \begin{equation} \epsilon_{min}= x = \frac{T_{min}}{E_{k_1}-mc^2}, \end{equation} where $T_{min}$ has meaning of minimal kinetic energy of secondary electron. And, $\epsilon_{\rm max}=1(1/2)$ for Bhabha(M{\o}ller) scatterings. % ---------------------------------------------------------------------- \subsubsection{Total M{\o}ller cross section} The total cross section of the polarized M{\o}ller scattering can be expressed as follows \begin{equation}\label{totalMoller} \sigma^M_{pol}=\frac{2\pi\gamma^2 r_e^2}{(\gamma-1)^2(\gamma+1)}\left[ \sigma^M_0 + \zeta_3^{(1)}\zeta_3^{(2)}\sigma^M_L + \left(\zeta_1^{(1)}\zeta_1^{(2)}+\zeta_2^{(1)}\zeta_2^{(2)}\right)\sigma^M_T\right], \end{equation} where the $r_e$ is classical electron radius, and \begin{eqnarray} \sigma^M_0&=& - \frac{1}{1 - x} + \frac{1}{x} - \frac{{\left( \gamma - 1 \right)}^2}{{\gamma}^2} \left(\frac{1}{2} - x \right) + \frac{ 2 - 4\,\gamma }{2\,{\gamma}^2} \,\ln \left(\frac{1-x}{x}\right) \nonumber\\ \sigma^M_L&=& \frac{ \left( -3 + 2\,\gamma + {\gamma}^2 \right) \,\left( 1 - 2\,x \right) }{2\, {\gamma}^2} + \frac{2\,\gamma\,\left( -1 + 2\,\gamma \right)}{2\, {\gamma}^2} \,\ln \left(\frac{1-x}{x}\right) \nonumber\\ \sigma^M_T&=& \frac{2\,\left( \gamma - 1 \right) \,\left( 2\,x -1 \right)}{2\,{\gamma}^2} + \frac{ \left( 1 - 3\,\gamma \right) }{2\,{\gamma}^2} \,\ln \left(\frac{1-x}{x}\right) \label{mollertotal} \end{eqnarray} % ---------------------------------------------------------------------- \subsubsection{Total Bhabha cross section} The total cross section of the polarized Bhabha scattering can be expressed as follows \begin{equation}\label{totalBhabha} \sigma^B_{pol}=\frac{2\pi r_e^2}{\gamma-1} \left[ \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 \right], \end{equation} where \begin{eqnarray} \sigma^B_0&=& \frac{1 - x}{2\,\left( \gamma - 1 \right) \,x} + \frac{2\,\left( -1 + 3\,x - 6\,x^2 + 4\,x^3 \right) } {3\,{\left( 1 + \gamma \right) }^3} \nonumber\\ &+&\frac{-1 - 5\,x + 12\,x^2 - 10\,x^3 + 4\,x^4}{2\,\left( 1 + \gamma \right) \,x} + \frac{-3 - x + 8\,x^2 - 4\,x^3 - \ln (x)}{{\left( 1 + \gamma \right) }^2} \nonumber\\ &+&\frac{3 + 4\,x - 9\,x^2 + 3\,x^3 - x^4 + 6\,x\,\ln (x)}{3\,x} \nonumber\\ \sigma^B_L&=& \frac{2\,\left( 1 - 3\,x + 6\,x^2 - 4\,x^3 \right) }{3\,{\left( 1 + \gamma \right) }^3} + \frac{-14 + 15\,x - 3\,x^2 + 2\,x^3 - 9\,\ln (x)}{3\,\left( 1 + \gamma \right) } \nonumber\\ &+&\frac{5 + 3\,x - 12\,x^2 + 4\,x^3 + 3\,\ln (x)}{3\,{\left( 1 + \gamma \right) }^2} + \frac{7 - 9\,x + 3\,x^2 - x^3 + 6\,\ln (x)}{3} \nonumber\\ \sigma^B_T&=& \frac{2\,\left( -1 + 3\,x - 6\,x^2 + 4\,x^3 \right) }{3\,{\left( 1 + \gamma \right) }^3} + \frac{-7 - 3\,x + 18\,x^2 - 8\,x^3 - 3\,\ln (x)}{3\,{\left( 1 + \gamma \right) }^2} \nonumber\\ &+&\frac{5 + 3\,x - 12\,x^2 + 4\,x^3 + 9\,\ln (x)}{6\,\left( 1 + \gamma \right) } \end{eqnarray} % ---------------------------------------------------------------------- \subsubsection{Mean free path} With the help of the total polarized M{\o}ller cross section one can define a longitudinal asymmetry $A^M_L$ and the transverse asymmetry $A^M_T$, by \begin{tabular}{ccc} $ A^M_L = \displaystyle \frac{\sigma^M_L}{\sigma^M_0} \quad$ & and & $\quad A^M_T = \displaystyle \frac{\sigma^M_T}{\sigma^M_0}\;$. \end{tabular} Similarly, using the polarized Bhabha cross section one can introduce a longitudinal asymmetry $A^B_L$ and the transverse asymmetry $A^B_T$ via \begin{tabular}{ccc} $ A^B_L = \displaystyle \frac{\sigma^B_L}{\sigma^B_0} \quad$ & and & $\quad A^B_T = \displaystyle \frac{\sigma^B_T}{\sigma^B_0}\;$. \end{tabular} These asymmetries are depicted in figures \ref{pol.moller1} and \ref{pol.bhabha1} respectively. If both beam and target are polarized the mean free path as defined in section \ref{sec:em.eion} has to be modified. In the class {\em G4ePolarizedIonization} the polarized mean free path $\lambda^{\rm pol}$ is derived from the unpolarized mean free path $\lambda^{\rm unpol}$ via \begin{equation} \lambda^{\rm pol} = \frac{\lambda^{\rm unpol}}{1 + \zeta_3^{(1)}\zeta_3^{(2)}\, A_L + \left(\zeta_1^{(1)}\zeta_1^{(2)}+\zeta_2^{(1)}\zeta_2^{(2)}\right) \,A_T} \end{equation} % \begin{figure}[t] \begin{center} \includegraphics[width=6.5cm]{electromagnetic/standard/plots/MollerTA1.eps} \includegraphics[width=6.5cm]{electromagnetic/standard/plots/MollerTA2.eps} \end{center} \caption{\label{pol.moller1}M{\o}ller total cross section asymmetries depending on the total energy of the incoming electron, with a cut-off $T_{\rm cut}= 1 {\rm keV}$. Transverse asymmetry is plotted in blue, longitudinal asymmetry in red. Left part, between 0.5 MeV and 2 MeV, right part up to 10 MeV.} %\end{figure} % %\begin{figure}[t] \begin{center} \includegraphics[width=6.5cm]{electromagnetic/standard/plots/BhabhaTA1.eps} \includegraphics[width=6.5cm]{electromagnetic/standard/plots/BhabhaTA2.eps} \end{center} \caption{\label{pol.bhabha1}Bhabha total cross section asymmetries depending on the total energy of the incoming positron, with a cut-off $T_{\rm cut}= 1 {\rm keV}$. Transverse asymmetry is plotted in blue, longitudinal asymmetry in red. Left part, between 0.5 MeV and 2 MeV, right part up to 10 MeV.} \end{figure} % ---------------------------------------------------------------------- \subsection{Sampling the final state} \subsubsection{Differential cross section} The polarized differential cross section is rather complicated, the full result can be found in \cite{polIoni:Star:2006,polIoni:Ford:1957,polIoni:Stehle:1957}. In {\em G4PolarizedMollerCrossSection} the complete result is available taking all mass effects into account, only binding effects are neglected. Here we state only the ultra-relativistic approximation (URA), to show the general dependencies. \begin{eqnarray} &&\frac{d\sigma_{URA}^M}{d\epsilon d\varphi}= \frac{{{r_\epsilon}}^2}{ \gamma + 1} \times \nonumber\\ &&\Bigg[ \frac{{\left( 1 - \epsilon + \epsilon^2 \right) }^2}{4\,{\left( \epsilon - 1 \right) }^2\,\epsilon^2} + \zeta_3^{(1)}\zeta_3^{(2)}\frac{2 - \epsilon + \epsilon^2}{-4\,\epsilon ( 1 - \epsilon)} + \left(\zeta_2^{(1)}\zeta_2^{(2)} -\zeta_1^{(1)}\zeta_1^{(2)}\right)\frac{1}{4} \nonumber\\ &&+ \left(\xi_3^{(1)}\zeta_3^{(1)} - \xi_3^{(2)}\zeta_3^{(2)}\right) \frac{1 - \epsilon + 2\,\epsilon^2}{4\,\left( 1 - \epsilon \right) \,\epsilon^2} + \left(\xi_3^{(2)}\zeta_3^{(1)} - \xi_3^{(1)}\zeta_3^{(2)}\right) \frac{2 - 3\,\epsilon + 2\,\epsilon^2}{4\,{\left( 1 - \epsilon \right) }^2\,\epsilon} \Bigg] \nonumber\\ && \end{eqnarray} % The corresponding cross section for Bhabha cross section is implemented in {\em G4PolarizedBhabhaCrossSection}. In the ultra-relativistic approximation it reads \begin{eqnarray} &&\frac{d\sigma_{URA}^B}{d\epsilon d\varphi}= \frac{{{r_\epsilon}}^2}{ \gamma - 1} \times \nonumber\\ &&\Bigg[ \frac{{\left( 1 - \epsilon + \epsilon^2 \right) }^2}{4\,\epsilon^2} + \zeta_3^{(1)}\zeta_3^{(2)}\frac{\left( \epsilon - 1 \right) \,\left( 2 - \epsilon + \epsilon^2 \right) }{4\,\epsilon} +\left(\zeta_2^{(1)}\zeta_2^{(2)} -\zeta_1^{(1)}\zeta_1^{(2)}\right)\frac{(1-\epsilon)^2}{4} \nonumber\\ &&+ \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} + \left(\xi_3^{(2)}\zeta_3^{(1)} - \xi_3^{(1)}\zeta_3^{(2)}\right)\frac{ 2 - 3\,\epsilon + 2\,\epsilon^2}{4\epsilon} \Bigg] \nonumber\\ && \end{eqnarray} where \begin{tabular}[t]{l@{\ = \ }l} $r_e$ & classical electron radius \\ $\gamma$ & $E_{k_1}/m_e c^2$ \\ $\epsilon$ & ($E_{p_1}-m_e c^2)/(E_{k_1}-m_e c^2)$ \\ $E_{k_1}$ & energy of the incident electron/positron \\ $E_{p_1}$ & energy of the scattered electron/positron \\ $m_e c^2$ & electron mass \\ $\bvec{\zeta}^{(1)}$ & Stokes vector of the incoming electron/positron \\ $\bvec{\zeta}^{(2)}$ & Stokes vector of the target electron \\ $\bvec{\xi}^{(1)}$ & Stokes vector of the outgoing electron/positron \\ $\bvec{\xi}^{(2)}$ & Stokes vector of the outgoing (2nd) electron . \end{tabular} \subsubsection{Sampling} The delta ray is sampled according to methods discussed in Chapter 2. After exploitation of the symmetry in the M{\o}ller cross section under exchanging $\epsilon$ versus $(1-\epsilon)$, the differential cross section can be approximated by a simple function $f^M(\epsilon)$: \begin{equation} f^M(\epsilon) = \frac{1}{\epsilon^2} \frac{\epsilon_0}{1-2\epsilon_0} \end{equation} with the kinematic limits given by \begin{equation} \epsilon_0 = \frac{T_{\rm cut}}{E_{k_1}-m_e c^2} \le \epsilon \le \frac{1}{2} \end{equation} A similar function $f^B(\epsilon)$ can be found for Bhabha scattering: \begin{equation} f^B(\epsilon) = \frac{1}{\epsilon^2} \frac{\epsilon_0}{1-\epsilon_0} \end{equation} with the kinematic limits given by \begin{equation} \epsilon_0 = \frac{T_{\rm cut}}{E_{k_1}-m_e c^2} \le \epsilon \le 1 \end{equation} The kinematic of the delta ray production is constructed by the following steps: \begin{enumerate} \item $\epsilon$ is sampled from $f(\epsilon)$ \item calculate the differential cross section, depending on the initial polarizations $\bvec{\zeta}^{(1)}$ and $\bvec{\zeta}^{(2)}$. \item $\epsilon$ is accepted with the probability defined by ratio of the differential cross section over the approximation function. \item The $\varphi$ is diced uniformly. \item $\varphi$ is determined from the differential cross section, depending on the initial polarizations $\bvec{\zeta}^{(1)}$ and $\bvec{\zeta}^{(2)}$ \end{enumerate} Note, for initial states without transverse polarization components, the $\varphi$ distribution is always uniform. In figure \ref{pol.moller2} the asymmetries indicate the influence of polarization. In general the effect is largest around $\epsilon=\frac{1}{2}$. % %\begin{figure}[ht] %\includegraphics[scale=0.5]{electromagnetic/standard/plots/MollerXS.eps} %\caption{M{\o}ller differential cross section in arbitrary units. Black - unpolarized, Red - (+-),Blue (++). %This cross section is symmetric around point $\epsilon=1/2$. %} %\end{figure} %\begin{figure}[ht] %\includegraphics[scale=0.5]{electromagnetic/standard/plots/BhabhaXS.eps} %\caption{Bhabha differential cross section in arbitrary units. Black - unpolarized, Red - (+-),Blue (++)} %\end{figure} % \begin{figure}[ht] \begin{center} \includegraphics[width=6.5cm]{electromagnetic/standard/plots/MollerAsym.eps} \includegraphics[width=6.5cm]{electromagnetic/standard/plots/BhabhaAsym.eps} \end{center} %\caption{M{\o}ller differential cross section asymmetries in\%. %Red - ZZ, Gren - XX, Blue - YY, LightBlue -ZX} \caption{\label{pol.moller2}Differential cross section asymmetries in\% for M{\o}ller (left) and Bhabha (right) scattering ( red - $A_{ZZ}(\epsilon)$, green - $A_{XX}(\epsilon)$, blue - $A_{YY}(\epsilon)$, lightblue - $A_{ZX}(\epsilon)$)} \end{figure} After both $\phi$ and $\epsilon$ are known, the kinematic can be constructed fully. Using momentum conservation the momenta of the scattered incident particle and the ejected electron are constructed in global coordinate system. \subsubsection{Polarization transfer} After the kinematics is fixed the polarization properties of the outgoing particles are determined. Using the dependence of the differential cross section on the final state polarization a mean polarization is calculated according to method described in section \ref{sec:pol.intro}. The resulting polarization transfer functions $\xi^{(1,2)}_3(\epsilon)$ are depicted in figures \ref{pol.moller3} and \ref{pol.bhabha3}. \begin{figure}[ht] \includegraphics[width=6.5cm]{electromagnetic/standard/plots/MollerTransfer1.eps} \includegraphics[width=6.5cm]{electromagnetic/standard/plots/MollerTransfer2.eps} \caption{\label{pol.moller3}Polarization transfer functions in M{\o}ller scattering. Longitudinal polarization $\xi^{(2)}_3$ of electron with energy $E_{p_2}$ in blue; longitudinal polarization $\xi^{(1)}_3$ of second electron in red. Kinetic energy of incoming electron $T_{k_1} = 10 {\rm MeV}$}. \end{figure} \begin{figure}[ht] \includegraphics[width=6.5cm]{electromagnetic/standard/plots/BhabhaTransfer1.eps} \includegraphics[width=6.5cm]{electromagnetic/standard/plots/BhabhaTransfer2.eps} \caption{\label{pol.bhabha3}Polarization Transfer in Bhabha scattering. Longitudinal polarization $\xi^{(2)}_3$ of electron with energy $E_{p_2}$ in blue; longitudinal polarization $\xi^{(1)}_3$ of scattered positron. Kinetic energy of incoming positron $T_{k_1} = 10 {\rm MeV}$}. \end{figure} % ---------------------------------------------------------------------- \subsection{Status of this document} 20.11.06 created by P.Starovoitov\\ 21.02.07 minor update by A.Sch{\"a}licke\\ \begin{latexonly} \begin{thebibliography}{9} \bibitem{polIoni:Star:2006} P.~Starovoitov {\em et.al.}, in preparation. \bibitem{polIoni:Ford:1957} G.~W.~Ford, C.~J.~Mullin, Phys.~Rev.\ {\bf 108} (1957) 477. \bibitem{polIoni:Stehle:1957} P.~Stehle, Phys.~Rev.\ {\bf 110} (1958) 1458. \end{thebibliography} \end{latexonly} \begin{htmlonly} \subsection{Bibliography} \begin{enumerate} \item %{Star:2006} P.~Starovoitov {\em et.al.}, in preparation. \item %{Ford:1957} G.~W.~Ford, C.~J.~Mullin, Phys.~Rev.\ {\bf 108} (1957) 477. \item % {Stehle:1957} P.~Stehle, Phys.~Rev.\ {\bf 110} (1958) 1458. \end{enumerate} \end{htmlonly} \clearpage % ====================================================================== \section{Positron - Electron Annihilation} \subsection{Method} The class {\em G4eplusPolarizedAnnihilation} simulates annihilation of polarized positrons with electrons in a material. The implementation baseline follows the approach derived for the class {\em G4eplusAnnihilation} described in section \ref{sec:em.annil}. It evaluates polarization transfer and -- if the material is polarized -- asymmetries in the produced photons. Thus, it takes the effects of polarization on total cross section and mean free path, on distribution of final state photons into account. And calculates the average polarization of these generated photons. The material electrons are assumed to be free and at rest. \subsection{Total cross section and mean free path} Kinematics of annihilation process is fixed by initial energy \begin{equation} \gamma=\frac{E_{k_1}}{mc^2}%=\frac{s}{2(mc^2)^2}-1 \end{equation} and variable \begin{equation} \epsilon = \frac{E_{p_1}}{E_{k_1}+mc^2}, \end{equation} which is the part of total energy available in initial state carried out by first photon. This variable has the following kinematical limits \begin{equation} \frac{1}{2}\left(1-\sqrt{\frac{\gamma-1}{\gamma+1}}\right)\;<\; \epsilon \;<\;\frac{1}{2}\left(1+\sqrt{\frac{\gamma-1}{\gamma+1}}\right) \;. \end{equation} % ---------------------------------------------------------------------- \subsubsection{Total Cross Section} The total cross section of the annihilation of a polarized $e^+e^-$ pair into two photons could be expressed as follows \begin{equation}\label{totalAnnih} \sigma^A_{pol}=\frac{\pi r_e^2}{\gamma+1}\left[ \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], \end{equation} where \renewcommand{\Mvariable}[1]{\gamma} \begin{equation} \sigma^A_0= \frac{- \left( 3 + \Mvariable{gam} \right) \,{\sqrt{-1 + {\Mvariable{gam}}^2}} + \left( 1 + \Mvariable{gam}\,\left( 4 + \Mvariable{gam} \right) \right) \, \ln (\Mvariable{gam} + {\sqrt{-1 + {\Mvariable{gam}}^2}})}{4\, \left( {\Mvariable{gam}}^2 - 1 \right) } \end{equation} \begin{equation} \sigma^A_L= \frac{- {\sqrt{-1 + {\Mvariable{gam}}^2}}\, \left( 5 + \Mvariable{gam}\,\left( 4 + 3\,\Mvariable{gam} \right) \right) + \left( 3 + \Mvariable{gam}\,\left( 7 + \Mvariable{gam} + {\Mvariable{gam}}^2 \right) \right) \, \ln (\Mvariable{gam} + {\sqrt{{\Mvariable{gam}}^2-1 }})}{4\, {\left( \Mvariable{gam} -1\right) }^2\,\left( 1 + \Mvariable{gam} \right) } \end{equation} \begin{equation} \sigma^A_T= \frac{\left( 5 + \Mvariable{gam} \right) \,{\sqrt{-1 + {\Mvariable{gam}}^2}} - \left( 1 + 5\,\Mvariable{gam} \right) \,\ln (\Mvariable{gam} + {\sqrt{-1 + {\Mvariable{gam}}^2}})} {4\,{\left( -1 + \Mvariable{gam} \right) }^2\,\left( 1 + \Mvariable{gam} \right) } \end{equation} \subsubsection{Mean free path} With the help of the total polarized annihilation cross section one can define a longitudinal asymmetry $A^A_L$ and the transverse asymmetry $A^A_T$, by \begin{tabular}{ccc} $ A^A_L = \displaystyle \frac{\sigma^A_L}{\sigma^A_0} \quad$ & and & $\quad A^A_T = \displaystyle \frac{\sigma^A_T}{\sigma^A_0}\;$. \end{tabular} These asymmetries are depicted in figure \ref{pol.annihi1}. If both incident positron and target electron are polarized the mean free path as defined in section \ref{sec:em.annil} has to be modified. The polarized mean free path $\lambda^{\rm pol}$ is derived from the unpolarized mean free path $\lambda^{\rm unpol}$ via \begin{equation} \lambda^{\rm pol} = \frac{\lambda^{\rm unpol}}{1 + \zeta_3^{(1)}\zeta_3^{(2)}\, A_L + \left(\zeta_1^{(1)}\zeta_1^{(2)}+\zeta_2^{(1)}\zeta_2^{(2)}\right) \,A_T} \end{equation} \begin{figure}[ht] \begin{center} \includegraphics[width=6.5cm]{electromagnetic/standard/plots/AnnihTA1.eps} \includegraphics[width=6.5cm]{electromagnetic/standard/plots/AnnihTA2.eps} \end{center} \caption{\label{pol.annihi1}Annihilation total cross section asymmetries depending on the total energy of the incoming positron $E_{k_1}$. The transverse asymmetry is shown in blue, the longitudinal asymmetry in red. } \end{figure} \clearpage % ---------------------------------------------------------------------- \subsection{Sampling the final state} \subsubsection{Differential Cross Section} The fully polarized differential cross section is implemented in the class {\em G4PolarizedAnnihilationCrossSection}, which takes all mass effects into account, but binding effects are neglected \cite{polAnnihi:Star:2006,polAnnihi:Page:1957}. In the ultra-relativistic approximation (URA) and concentrating on longitudinal polarization states only the cross section is rather simple: \begin{eqnarray} \frac{d\sigma_{URA}^A}{d\epsilon d\varphi} & = & \frac{{{r_e}}^2}{ \gamma - 1} \times \Bigg( \frac{1 - 2\,\epsilon + 2\,\epsilon^2}{8\,\epsilon - 8\,\epsilon^2}\left(1 + \zeta_3^{(1)}\zeta_3^{(2)}\right) \nonumber\\ &&\quad\quad + \frac{ \left( 1 - 2\,\epsilon \right) \,\left( \zeta _{3}^{(1)} + \zeta _{3}^{(2)} \right) \, \left( \xi _{3}^{(1)} - \xi _{3}^{(2)} \right) }{8\,\left( \epsilon -1 \right) \,\epsilon} \Bigg) \end{eqnarray} % where \begin{tabular}[t]{l@{\ = \ }l} $r_e$ & classical electron radius \\ $\gamma$ & $E_{k_1}/m_e c^2$ \\ $E_{k_1}$ & energy of the incident positron \\ $m_e c^2$ & electron mass \\ $\bvec{\zeta}^{(1)}$ & Stokes vector of the incoming positron \\ $\bvec{\zeta}^{(2)}$ & Stokes vector of the target electron \\ $\bvec{\xi}^{(1)}$ & Stokes vector of the 1st photon \\ $\bvec{\xi}^{(2)}$ & Stokes vector of the 2nd photon . \end{tabular} % \begin{figure}[ht] \begin{center} \includegraphics[width=9.5cm]{electromagnetic/standard/plots/AnnihXS.eps} \end{center} \caption{Annihilation differential cross section in arbitrary units. Black line corresponds to unpolarized cross section; red line -- to the antiparallel spins of initial particles, and blue line -- to the parallel spins. Kinetic energy of the incoming positron $T_{k_1} = 10 {\rm MeV}$.} \end{figure} \subsubsection{Sampling} The photon energy is sampled according to methods discussed in Chapter 2. After exploitation of the symmetry in the Annihilation cross section under exchanging $\epsilon$ versus $(1-\epsilon)$, the differential cross section can be approximated by a simple function $f(\epsilon)$: \begin{equation} f(\epsilon) = \frac{1}{\epsilon} \ln^{-1}\left(\frac{\epsilon_{\rm max}}{\epsilon_{\rm min}}\right) \end{equation} with the kinematic limits given by \begin{eqnarray} \epsilon_{\rm min} &=& \frac{1}{2}\left(1-\sqrt{\frac{\gamma-1}{\gamma+1}}\right)\;, \nonumber\\ \epsilon_{\rm max} &=& \frac{1}{2}\left(1+\sqrt{\frac{\gamma-1}{\gamma+1}}\right) \;. \end{eqnarray} The kinematic of the two photon final state is constructed by the following steps: \begin{enumerate} \item $\epsilon$ is sampled from $f(\epsilon)$ \item calculate the differential cross section, depending on the initial polarizations $\bvec{\zeta}^{(1)}$ and $\bvec{\zeta}^{(2)}$. \item $\epsilon$ is accepted with the probability defined by the ratio of the differential cross section over the approximation function $f(\epsilon)$. \item The $\varphi$ is diced uniformly. \item $\varphi$ is determined from the differential cross section, depending on the initial polarizations $\bvec{\zeta}^{(1)}$ and $\bvec{\zeta}^{(2)}$. \end{enumerate} A short overview over the sampling method is given in Chapter 2. In figure \ref{pol.annihi2} the asymmetries indicate the influence of polarization for an 10MeV incoming positron. The actual behavior is very sensitive to the energy of the incoming positron. \begin{figure}[ht] \includegraphics[scale=0.5]{electromagnetic/standard/plots/AnnihAsym.eps} \caption{\label{pol.annihi2}Annihilation differential cross section asymmetries in\%. Red line corrsponds to $A_{ZZ}(\epsilon)$, green line -- $A_{XX}(\epsilon)$, blue line -- $A_{YY}(\epsilon)$, lightblue line -- $A_{ZX}(\epsilon)$). Kinetic energy of the incoming positron $T_{k_1} = 10 {\rm MeV}$.} \end{figure} \subsubsection{Polarization transfer} After the kinematics is fixed the polarization of the outgoing photon is determined. Using the dependence of the differential cross section on the final state polarizations a mean polarization is calculated for each photon according to method described in section \ref{sec:pol.intro}. The resulting polarization transfer functions $\xi^{(1,2)}(\epsilon)$ are depicted in figure \ref{pol.annihi3}. \begin{figure}[ht] \includegraphics[width=6.5cm]{electromagnetic/standard/plots/AnnihTransfer1.eps} \includegraphics[width=6.5cm]{electromagnetic/standard/plots/AnnihTransfer2.eps} \caption{\label{pol.annihi3} Polarization Transfer in annihilation process. Blue line corresponds to the circular polarization $\xi_3^{(1)}$ of the photon with energy $m(\gamma + 1)\epsilon$; red line -- circular polarization $\xi_3^{(2)}$ of the photon photon with energy $m(\gamma + 1)(1-\epsilon)$.} \end{figure} \subsection{Annihilation at Rest} The method \verb!AtRestDoIt! treats the special case where a positron comes to rest before annihilating. It generates two photons, each with energy $E_{p_{1/2}}=m c^2$ and an isotropic angular distribution. %Eventhough the asymmetry for annihilation at rest is 100\% (cf.\ %figure \ref{pol.annihi1}), there are always unpolarized electrons in %the a material. Starting with the differential cross section for annihilation with positron and electron spins opposed and parallel, respectively,\cite{polAnnihi:Page:1957} \begin{eqnarray} d\sigma_1 &=& \sim \frac{(1 - \beta^2) + \beta^2 (1 - \beta^2) (1 - \cos^2\theta)^2}{(1 - \beta^2\cos^2\theta)^2} d \cos\theta \\ d\sigma_2 &=& \sim \frac{\beta^2(1 - \cos^4\theta)}{(1 - \beta^2\cos^2\theta)^2} d \cos\theta \end{eqnarray} In the limit $\beta\to0$ the cross section $d\sigma_1$ becomes one, and the cross section $d\sigma_2$ vanishes. For the opposed spin state, the total angular momentum is zero and we have a uniform photon distribution. For the parallel case the total angular momentum is 1. Here the two photon final state is forbidden by angular momentum conservation, and it can be assumed that higher order processes (e.g.\ three photon final state) play a dominant role. However, in reality 100\% polarized electron targets do not exist, consequently there are always electrons with opposite spin, where the positron can annihilate with. % Leading again to a uniform distribution. Final state polarization does not play a role for the decay products of a spin zero state, and can be safely neglected. (Is set to zero) \subsection{Status of this document} 20.11.06 created by P.Starovoitov\\ 21.02.07 minor update by A.Sch{\"a}licke\\ \begin{latexonly} \begin{thebibliography}{9} \bibitem{polAnnihi:Star:2006} P.~Starovoitov {\em et.al.}, in preparation. \bibitem{polAnnihi:Page:1957} L.~A.~Page, %Polarization Effects in the Two-Quantum Annihilation of Positrons Phys.~Rev.\ {\bf 106} (1957) 394-398. \end{thebibliography} \end{latexonly} \begin{htmlonly} \subsection{Bibliography} \begin{enumerate} \item P.~Starovoitov {\em et.al.}, in preparation. \item L.~A.~Page, %Polarization Effects in the Two-Quantum Annihilation of Positrons Phys.~Rev.\ {\bf 106} (1957) 394-398. \end{enumerate} \end{htmlonly} % ====================================================================== \clearpage \section{Polarized Compton scattering} \subsection{Method} The class {\em G4PolarizedCompton} simulates Compton scattering of polarized photons with (possibly polarized) electrons in a material. The implementation follows the approach described for the class {\em G4ComptonScattering} introduced in section \ref{sec:em.compton}. Here the explicit production of a Compton scattered photon and the ejected electron is considered taking the effects of polarization on total cross section and mean free path as well as on the distribution of final state particles into account. Further the average polarizations of the scattered photon and electron are calculated. The material electrons are assumed to be free and at rest. \subsection{Total cross section and mean free path} Kinematics of the Compton process is fixed by the initial energy \begin{equation} X=\frac{E_{k_1}}{mc^2} \end{equation} and the variable \begin{equation} \epsilon = \frac{E_{p_1}}{E_{k_1}}, \end{equation} which is the part of total energy avaible in initial state carried out by scattered photon, and the scattering angle \begin{equation} \cos{\theta} = 1 - \frac{1}{X}\left(\frac{1}{\epsilon} - 1\right) \end{equation} The variable $\epsilon$ has the following limits: \begin{equation} \frac{1}{1+2X} \;<\; \epsilon \;<\;1 \end{equation} % ---------------------------------------------------------------------- \subsubsection{Total Cross Section} The total cross section of Compton scattering reads \begin{equation} \sigma^{C}_{pol}= %\frac{\pi \,{{r_e}}^2}{4\,X^2\,{\left( 1 + 2\,X \right) }^2} \frac{\pi \,{{r_e}}^2}{X^2\,{\left( 1 + 2\,X \right) }^2} \left[\sigma^{C}_0 + \zeta^{(1)}_3\zeta^{(2)}_3 \sigma^{C}_L\right] \end{equation} where \begin{equation} \sigma^{C}_0 = \frac{2\,X\,\left( 2 + X\,\left( 1 + X \right) \,\left( 8 + X \right) \right) - {\left( 1 + 2\,X \right) }^2\,\left( 2 + \left( 2 - X \right) \,X \right) \, \ln (1 + 2\,X)}{X} \end{equation} and \begin{equation} \sigma^{C}_L = 2\,X\,\left( 1 + X\,\left( 4 + 5\,X \right) \right) - \left( 1 + X \right) \,{\left( 1 + 2\,X \right) }^2\,\ln (1 + 2\,X) \end{equation} \begin{figure}[ht] \includegraphics[width=6.5cm]{electromagnetic/standard/plots/ComptonTA1.eps} \includegraphics[width=6.5cm]{electromagnetic/standard/plots/ComptonTA2.eps} \caption{\label{pol.compton1}Compton total cross section asymmetry depending on the energy of incoming photon. Left part, between $0$ and $\sim 1$ MeV, right part -- up to 10MeV. } \end{figure} \subsubsection{Mean free path} When simulating the Compton scattering of a photon with an atomic electron, an empirical cross section formula is used, which reproduces the cross section data down to 10 keV (see section \ref{sec:em.compton}). If both, beam and target, are polarized this mean free path has to be corrected. In the class {\em G4ComptonScattering} the polarized mean free path $\lambda^{\rm pol}$ is defined on the basis of the the unpolarized mean free path $\lambda^{\rm unpol}$ via \begin{equation} \lambda^{\rm pol} = \frac{\lambda^{\rm unpol}}{1 + \zeta_3^{(1)}\zeta_3^{(2)}\, A^C_L } \end{equation} where \begin{equation} A^C_L = \displaystyle \frac{\sigma^A_L}{\sigma^A_0} \end{equation} is the expected asymmetry from the the total polarized Compton cross section given above. This asymmetry is depicted in figure \ref{pol.compton1}. % ---------------------------------------------------------------------- \subsection{Sampling the final state} \subsubsection{Differential Compton Cross Section} In the ultra-relativistic approximation the dependence of the differential cross section on the longitudinal/circular degree of polarization is very simple. It reads \begin{eqnarray} &&\frac{d\sigma_{URA}^C}{de d\varphi}= %\frac{{{r_e}}^2 \,Z}{ 4X} \frac{{{r_e}}^2 }{ X} \Bigg( \frac{\epsilon^2 + 1}{2\,\epsilon} + \frac{ \epsilon^2 -1 }{2\,\epsilon} \left(\zeta_3^{(1)}\zeta_3^{(2)} + \zeta _{3}^{(2)}\,\xi _{3}^{(1)} - \zeta _{3}^{(1)}\,\xi _{3}^{(2)}\right) \nonumber\\ &&+\frac{\epsilon^2 + 1}{2\,\epsilon} \left( \zeta _{3}^{(1)}\,\xi _{3}^{(1)} - \zeta _{3}^{(2)} \,\xi _{3}^{(2)} \right) \Bigg) \end{eqnarray} where \begin{tabular}[t]{l@{\ = \ }l} $r_e$ & classical electron radius \\ $X$ & $E_{k_1}/m_e c^2$ \\ $E_{k_1}$ & energy of the incident photon \\ $m_e c^2$ & electron mass \\ \end{tabular} The fully polarized differential cross section is available in the class {\em G4PolarizedComptonCrossSection}. It takes all mass effects into account, but binding effects are neglected \cite{polCompt:Star:2006,polCompt:Lipps:1954}. The cross section dependence on $\epsilon$ for right handed circularly polarized photons and longitudinally polarized electrons is plotted in figure \ref{pol.compton2a} % \begin{figure} \includegraphics[scale=0.5]{electromagnetic/standard/plots/ComptonXS.eps} \caption{\label{pol.compton2a} Compton scattering differential cross section in arbitrary units. Black line corresponds to the unpolarized cross section; red line -- to the antiparallel spins of initial particles, and blue line -- to the parallel spins. Energy of the incoming photon $E_{k_1} = 10 {\rm MeV}$. } \end{figure} % \begin{figure} \includegraphics[scale=0.5]{electromagnetic/standard/plots/ComptonAsym.eps} \caption{\label{pol.compton2}Compton scattering differential cross section asymmetries in\%. Red line corresponds to the asymmetry due to circular photon and longitudinal electron initial state polarization, green line -- due to circular photon and transverse electron initial state polarization, blue line -- due to linear photon and transverse electron initial state polarization.} \end{figure} \subsubsection{Sampling} The photon energy is sampled according to methods discussed in Chapter 2. The differential cross section can be approximated by a simple function $\Phi(\epsilon)$: \begin{equation} \Phi(\epsilon) = \frac{1}{\epsilon} + \epsilon \end{equation} with the kinematic limits given by \begin{eqnarray} \epsilon_{\rm min} &=& \frac{1}{1+2X} \\ \epsilon_{\rm max} &=& 1 \end{eqnarray} The kinematic of the scattered photon is constructed by the following steps: \begin{enumerate} \item $\epsilon$ is sampled from $\Phi(\epsilon)$ \item calculate the differential cross section, depending on the initial polarizations $\bvec{\zeta}^{(1)}$ and $\bvec{\zeta}^{(2)}$, which the correct normalization. \item $\epsilon$ is accepted with the probability defined by ratio of the differential cross section over the approximation function. \item The $\varphi$ is diced uniformly. \item $\varphi$ is determined from the differential cross section, depending on the initial polarizations $\bvec{\zeta}^{(1)}$ and $\bvec{\zeta}^{(2)}$. \end{enumerate} In figure \ref{pol.compton2} the asymmetries indicate the influence of polarization for an 10MeV incoming positron. The actual behavior is very sensitive to energy of the incoming positron. \subsubsection{Polarization transfer} After the kinematics is fixed the polarization of the outgoing photon is determined. Using the dependence of the differential cross section on the final state polarizations a mean polarization is calculated for each photon according to the method described in section \ref{sec:pol.intro}. The resulting polarization transfer functions $\xi^{(1,2)}(\epsilon)$ are depicted in figure \ref{pol.compton3}. \begin{figure}[ht] \includegraphics[width=6.5cm]{electromagnetic/standard/plots/ComptonTransfer1.eps} \includegraphics[width=6.5cm]{electromagnetic/standard/plots/ComptonTransfer2.eps} \caption{\label{pol.compton3} Polarization Transfer in Compton scattering. Blue line corresponds to the longitudinal polarization $\xi_3^{(2)}$ of the electron, red line -- circular polarization $\xi_3^{(1)}$ of the photon.} \end{figure} \subsection{Status of this document} 20.11.06 created by P.Starovoitov\\ 21.02.07 corrected cross section and some minor update by A.Sch{\"a}licke\\ \begin{latexonly} \begin{thebibliography}{9} \bibitem{polCompt:Star:2006} P.~Starovoitov {\em et.al.}, in preparation. %\bibitem{polCompt:Stokes:1852} %G.~Stokes, Trans.\ Cambridge Phil.\ Soc.\ {\bf 9} (1852) 399. % %\bibitem{polCompt:McMaster:1961} %W.~H.~McMaster, Rev.\ Mod.\ Phys.\ {\bf 33} (1961) 8; and references therein. \bibitem{polCompt:Lipps:1954} F.W.~Lipps, H.A.~Tolhoek, %Polarization Phenomena of Electrons and Photons I, Physica {\bf 20} (1954) 85; F.W.~Lipps, H.A.~Tolhoek, %Polarization Phenomena of Electrons and Photons II, Physica {\bf 20} (1954) 395. \end{thebibliography} \end{latexonly} \begin{htmlonly} \subsection{Bibliography} \begin{enumerate} \item P.~Starovoitov {\em et.al.}, in preparation. \item F.W.~Lipps, H.A.~Tolhoek, %Polarization Phenomena of Electrons and Photons I, Physica {\bf 20} (1954) 85; F.W.~Lipps, H.A.~Tolhoek, %Polarization Phenomena of Electrons and Photons II, Physica {\bf 20} (1954) 395. \end{enumerate} \end{htmlonly} \newpage \section{Polarized Bremsstrahlung for electron and positron}\label{sec:pol.bremsstrahlung} \subsection{Method} The polarized version of Bremsstrahlung is based on the unpolarized cross section. Energy loss, mean free path, and distribution of explicitly generated final state particles are treated by the unpolarized version {\em G4eBremsstrahlung}. For details consult section \ref{sec:em.ebrem}. The remaining task is to attribute polarization vectors to the generated final state particles, which is discussed in the following. \subsection{Polarization in gamma conversion and brems\-strahlung} Gamma conversion and bremsstrahlung are cross-symmetric processes (i.e. the Feynman diagram for electron bremsstrahlung can be obtained from the gamma conversion diagram by flipping the incoming photon and outgoing positron lines) and their cross sections closely related. For both processes, the interaction occurs in the field of the nucleus and the total and differential cross section are polarization independent. Therefore, only the polarization transfer from the polarized incoming particle to the outgoing particles is taken into account. % \begin{figure}[htb] \begin{center} \includegraphics [scale=.33] {electromagnetic/standard/plots/Fyn_diag.eps} \caption {Feynman diagrams of Gamma conversion and bremsstrahlung processes.} \end{center} \end{figure} \noindent For both processes, the scattering can be formulated by: \begin{equation} \mathcal{K}_{1}(k_{1},\bvec{\zeta}^{(1)}) + \mathcal{N}_{1}(k_{\mathcal {N}_{1}}, \bvec{\zeta}^{(\mathcal {N}_{1})}) \longrightarrow \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})}) \end{equation} % Where $\mathcal{N}_{1}(k_{\mathcal {N}_{1}}, \bvec{\zeta}^{(\mathcal {N}_{1})})$ and $\mathcal{N}_{2}(p_{\mathcal{N}_{2}}, \bvec{\xi}^{(\mathcal{N}_{2})})$ are the initial and final state of the field of the nucleus respectively assumed to be unchanged, at rest and unpolarized. This leads to $k_{\mathcal {N}_{1}} = k_{\mathcal {N}_{2}} = 0$ and $\bvec{\zeta}^{(\mathcal {N}_{1})} = \bvec{\xi}^{(\mathcal{N}_{2})} = 0$ % Gamma conversion process \textbf{In the case of gamma conversion process}:\\ $\mathcal{K}_{1}(k_{1},\bvec{\zeta}^{(1)})$ is the incoming photon initial state with momentum $k_{1}$ and polarization state $\bvec{\zeta}^{(1)}$. \\ $\mathcal{P}_{1}(p_{1},\bvec{\xi}^{(1)})$ and $\mathcal{P}_{2}(p_{2},\bvec{\xi}^{(2)})$ are the two photons final states with momenta $p_{1}$ and $p_{2}$ and polarization states $\bvec{\xi}^{(1)}$ and $\bvec{\xi}^{(2)}$. % Bremsstrahlung process \textbf{In the case of bremsstrahlung process}:\\ $\mathcal{K}_{1}(k_{1},\bvec{\zeta}^{(1)})$ is the incoming lepton $e^{-}(e^{+})$ initial state with momentum $k_{1}$ and polarization state $\bvec{\zeta}^{(1)}$. \\ $\mathcal{P}_{1}(p_{1},\bvec{\xi}^{(1)})$ is the lepton $e^{-}(e^{+})$ final state with momentum $p_{1}$ and polarization state $\bvec{\xi}^{(1)}$. \\ $\mathcal{P}_{2}(p_{2},\bvec{\xi}^{(2)})$ is the bremsstrahlung photon in final state with momentum $p_{2}$ and polarization state $\bvec{\xi}^{(2)}$. \subsection[Polarization transfer to the photon]{Polarization transfer from the lepton $e^{-}(e^{+})$ to a photon} The polarization transfer from an electron (positron) to a photon in a brems\-strahlung process was first calculated by Olsen and Maximon \cite{polBrems:Olsen_Maximon} taking into account both Coulomb and screening effects. In the Stokes vector formalism, the $e^{-}(e^{+})$ polarization state can be transformed to a photon polarization finale state by means of interaction matrix $T_{\gamma}^{b}$. It defined via % \begin{equation} \left(\begin{array}{c} O \\ \bvec{\xi}^{(2)} \end{array}\right) = T_{\gamma}^{b} \, \left(\begin{array}{c} 1 \\ \bvec{\zeta}^{(1)} \end{array}\right)\;, \label{eq:brem_gamma} \end{equation} % and % \begin{equation} T_{\gamma}^{b}\approx \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ D & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & T & 0 & L \\ \end{array} \right)\;, \label{eq:matrix_brem_g} \end{equation} % where \begin{eqnarray} I &=& (\epsilon_{1}^{2}+\epsilon_{2}^{2})(3+2\Gamma)-2\epsilon_{1}\epsilon_{2}(1+4u^{2}\hat\xi^{2}\Gamma)\\ D &=& \left\lbrace 8\epsilon_{1}\epsilon_{2}u^{2}\hat\xi^{2}\Gamma \right\rbrace / I\\ T &=& \left\lbrace -4k\epsilon_{2}\hat\xi(1-2\hat\xi)u \Gamma \right\rbrace / I \\ L &=& k\lbrace(\epsilon_{1}+\epsilon_{2})(3+2\Gamma)-2\epsilon_{2}(1+4u^{2}\hat\xi^{2}\Gamma)\rbrace / I \label{eq:polbremdef} \end{eqnarray} % and % \begin{center} \begin{tabular}{ll} $\epsilon_{1}$ & Total energy of the incoming lepton $e^{+}(e^{-})$ in units $mc^{2}$\\ $\epsilon_{2}$ & Total energy of the outgoing lepton $e^{+}(e^{-})$ in units $mc^{2}$\\ $k$ &$=(\epsilon_{1}-\epsilon_{2})$, the energy of the bremsstrahlung photon in units of $mc^{2}$ \\ $\bvec{p}$ & Electron (positron) initial momentum in units $mc$\\ $\bvec{k}$ & Bremsstrahlung photon momentum in units $mc$\\ $\bvec{u}$ & Component of $\bvec{p}$ perpendicular to $\bvec{k}$ in units $mc$ and $u=\vert \bvec{u} \vert $\\ $\hat\xi$ & $ = 1/(1+u^{2})$ \end{tabular} \end{center} % Coulomb and screening effects are contained in \(\Gamma\), defined as follows \begin{eqnarray} \Gamma &=& \ln\left(\frac{1}{\delta}\right)-2-f(Z)+ \mathcal{F}\left(\frac{\hat\xi}{\delta}\right) \quad \mbox{for } \Delta \le 120 \\ \Gamma &=& \ln\left( \frac{111}{\hat\xi Z^{\frac{1}{3}}}\right)-2-f(z) \quad \mbox{for } \Delta \ge 120 \end{eqnarray} % with % \begin{eqnarray} \Delta &=& \frac{12 Z^{\frac{1}{3}}\epsilon_{1}\epsilon_{2} \hat\xi}{121 k} \quad \mbox{with $Z$ the atomic number and } \delta = \frac{k}{2\epsilon_{1}\epsilon{2}} \end{eqnarray} % % \noindent $f(Z)$ is the coulomb correction term derived by Davies, Bethe and Maximon \cite{polBrems:Davise}. $ \mathcal{F}({\hat\xi}/{\delta})$ contains the screening effects and is zero for $\Delta \le 0.5 $ (No screening effects). For $0.5 \le \Delta \le 120 $ (intermediate screening) it is a slowly decreasing function. The $\mathcal{F}({\hat\xi}/{\delta})$ values versus $\Delta$ are given in table \ref{koch} and used with a linear interpolation in between. The polarization vector of the incoming $e^{-}(e^{+})$ must be rotated into the frame defined by the scattering plane (x-z-plane) and the direction of the outgoing photon (z-axis). The resulting polarization vector of the bremsstrahlung photon is also given in this frame. \begin{table}[h] \caption{$ \mathcal{F}({\hat\xi}/{\delta})$ for intermediate values of the screening factor \cite{polBrems:koch}.} \label{koch} \begin{center} \begin{tabular}{|cc|cc|} \hline $\Delta$ &$ -\mathcal{F}\left({\hat\xi}/{\delta}\right)$ & $\Delta$& $ -\mathcal{F}\left({\hat\xi}/{\delta}\right)$\\ \hline 0.5 & 0.0145 & 40.0 & 2.00 \\ 1.0 & 0.0490 & 45.0 & 2.114\\ 2.0 & 0.1400 & 50.0 & 2.216\\ 4.0 & 0.3312 & 60.0 & 2.393\\ 8.0 & 0.6758 & 70.0 & 2.545\\ 15.0 & 1.126 & 80.0 & 2.676\\ 20.0 & 1.367 & 90.0 & 2.793\\ 25.0 & 1.564 & 100.0 & 2.897\\ 30.0 & 1.731 & 120.0 & 3.078\\ 35.0 & 1.875 & & \\ \hline \end{tabular} \end{center} \end{table} % Using Eq.\ (\ref{eq:brem_gamma}) and the transfer matrix given by Eq.\ (\ref{eq:matrix_brem_g}) the bremsstrahlung photon polarization state in the Stokes formalism \cite{polBrems:McMaster1, polBrems:McMaster2} is given by % \begin{equation} \xi^{(2)} = \left( \begin{array}{c} \xi_{1}^{(2)}\\ \xi_{2}^{(2)} \\ \xi_{3}^{(2)} \\ \end{array} \right) \approx \left( \begin{array}{c} D \\ 0 \\ \zeta_{1}^{(1)}L + \zeta_{2}^{(1)}T \\ \end{array} \right) \end{equation} \subsection[Polarization transfer to the lepton]{Remaining polarization of the lepton after emitting a bremsstrahlung photon} The \(e^{-}(e^{+})\) polarization final state after emitting a bremsstrahlung photon can be calculated using the interaction matrix \(T_{l}^{b}\) which describes the lepton depolarization. The polarization vector for the outgoing \(e^{-}(e^{+})\) is not given by Olsen and Maximon. However, their results can be used to calculate the following transfer matrix \cite{polBrems:klausFl,polBrems:hoogduin}. % \begin{equation} \left(\begin{array}{c} O \\ \bvec{\xi}^{(1)} \end{array}\right) = T_{l}^{b} \, \left(\begin{array}{c} 1 \\ \bvec{\zeta}^{(1)} \end{array}\right) \label{eq:brem_lepton} \end{equation} % \begin{equation} T_{l}^{b}\approx \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ D & M & 0 & E \\ 0 & 0 & M & 0 \\ 0 & F & 0 & M+P \\ \end{array} \right) \label{eq:matrix_brem_l} \end{equation} % where % \begin{eqnarray} I &=&(\epsilon_{1}^{2}+\epsilon_{2}^{2})(3+2\Gamma)-2\epsilon_{1}\epsilon_{2}(1+4u^{2}\hat\xi^{2}\Gamma)\\ F &=& \epsilon_{2} \left\lbrace 4k\hat\xi u (1-2\hat\xi)\Gamma\right\rbrace /I \\ E &=& \epsilon_{1} \left\lbrace 4k\hat\xi u (2\hat\xi-1)\Gamma \right\rbrace /I\\ M &=& \left\lbrace 4k\epsilon_{1}\epsilon_{2}(1+\Gamma - 2 u^{2}\hat\xi^{2} \Gamma)\right\rbrace / I \\ P &=& \left\lbrace k^{2} (1+8 \Gamma(\hat\xi - 0.5)^{2}\right\rbrace / I \end{eqnarray} % and % \begin{center} \begin{tabular}{ll} $\epsilon_{1}$ & Total energy of the incoming $e^{+}/e^{-}$ in units $mc^{2}$\\ $\epsilon_{2}$ & Total energy of the outgoing $e^{+}/e^{-}$ in units $mc^{2}$\\ $k$ & $=(\epsilon_{1}-\epsilon_{2})$, energy of the photon in units of $mc^{2}$\\ $\bvec{p}$ & Electron (positron) initial momentum in units $mc$\\ $\bvec{k}$ & Photon momentum in units $mc$\\ $\bvec{u}$ & Component of $\bvec{p}$ perpendicular to $\bvec{k}$ in units $mc$ and $u=\vert \bvec{u} \vert $ \end{tabular} \end{center} Using Eq.\ (\ref{eq:brem_lepton}) and the transfer matrix given by Eq.\ (\ref{eq:matrix_brem_l}) the \(e^{-}(e^{+})\) polarization state after emitting a bremsstrahlung photon is given in the Stokes formalism by % \begin{equation} \xi^{(1)} = \left( \begin{array}{c} \xi_{1}^{(1)}\\ \xi_{2}^{(1)} \\ \xi_{3}^{(1)} \\ \end{array} \right) \approx \left( \begin{array}{c} \zeta_{1}^{(1)} M + \zeta_{3}^{(1)} E \\ \zeta_{2}^{(1)} M \\ \zeta_{3}^{(1)}(M+P) + \zeta_{1}^{(1)} F \\ \end{array} \right) \;. \end{equation} \subsection{Status of this document} 20.11.06 created by K.Laihem\\ 21.02.07 minor update by A.Sch{\"a}licke\\ 27.11.08 correction in Eq.\ \eqref{eq:polbremdef} by A.Sch{\"a}licke \begin{latexonly} \begin{thebibliography}{7} \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. \bibitem{polBrems:McMaster1} W.H.~McMaster. Polarization and the Stokes parameters. American Journal of Physics, 22(6):351-362, 1954. \bibitem{polBrems:McMaster2}W.H.~McMaster. Matrix representation of polarization. Reviews of Modern Physics, 33(1):8-29, 1961. \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. \bibitem{polBrems:hoogduin}Hoogduin, Johannes Marinus, Electron, positron and photon polarimetry. PhD thesis, Rijksuniversiteit Groningen 1997. \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. \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. \bibitem{polBrems:Laihem:thesis} K.~Laihem, PhD thesis, Measurement of the positron polarization at an helical undulator based positron source for the International Linear Collider ILC, Humboldt University Berlin, Germany, (2008). \end{thebibliography} \end{latexonly} \begin{htmlonly} \subsection{Bibliography} \begin{enumerate} \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. \item W.H.~McMaster. Polarization and the Stokes parameters. American Journal of Physics, 22(6):351-362, 1954. \item W.H.~McMaster. Matrix representation of polarization. Reviews of Modern Physics, 33(1):8-29, 1961. \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. \item Hoogduin, Johannes Marinus. Electron, positron and photon polarimetry. PhD thesis, Rijksuniversiteit Groningen 1997. \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. \item H.W.~Koch and J.W.~Motz. Bremsstrahlung cross-section formulas and related data. Review Mod. Phys., 31(4):920-955, 1959. \item K.~Laihem, PhD thesis, Measurement of the positron polarization at an helical undulator based positron source for the International Linear Collider ILC, Humboldt University Berlin, Germany, (2008). \end{enumerate} \end{htmlonly} \newpage \section{Polarized Gamma conversion into an electron--positron pair}\label{sec:pol.conv} \subsection{Method} The polarized version of gamma conversion is based on the EM standard process {\em G4GammaConversion}. Mean free path and the distribution of explicitly generated final state particles are treated by this version. For details consult section \ref{sec:em.conv}. The remaining task is to attribute polarization vectors to the generated final state leptons, which is discussed in the following. \subsection[Polarization transfer]{Polarization transfer from the photon to the two leptons} Gamma conversion process is essentially the inverse process of Bremsstrahlung and the interaction matrix is obtained by inverting the rows and columns of the bremsstrahlung matrix and changing the sign of \(\epsilon_{2}\), cf.\ section \ref{sec:pol.bremsstrahlung}. It follows from the work by Olsen and Maximon \cite{polPair:Olsen_Maximon} that the polarization state \(\xi^{(1)}\) of an electron or positron after pair production is obtained by % \begin{equation} \left(\begin{array}{c} O \\ \bvec{\xi}^{(1)} \end{array}\right) = T_{l}^{p} \, \left(\begin{array}{c} 1 \\ \bvec{\zeta}^{(1)} \end{array}\right) \label{eq:conv_lepton} \end{equation} % and % \begin{equation} T_{l}^{p}\approx \left( \begin{array}{cccc} 1 & D & 0 & 0 \\ 0 & 0 & 0 & T \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & L \\ \end{array} \right) \;, \label{eq:matrix_conv} \end{equation} % where \begin{eqnarray} I &=& (\epsilon_{1}^{2}+\epsilon_{2}^{2})(3+2\Gamma)+2\epsilon_{1}\epsilon_{2}(1+4u^{2}\hat\xi^{2}\Gamma)\\ D &=& \left\lbrace -8\epsilon_{1}\epsilon_{2}u^{2}\hat\xi^{2}\Gamma \right\rbrace / I\\ T &=& \left\lbrace -4k\epsilon_{2}\hat\xi(1-2\hat\xi)u \Gamma \right\rbrace / I \\ L &=& k\lbrace(\epsilon_{1}-\epsilon_{2})(3+2\Gamma)+2\epsilon_{2}(1+4u^{2}\hat\xi^{2}\Gamma)\rbrace/ I \label{eq:polpairdef} \end{eqnarray} and \begin{center} \begin{tabular}{ll} $\epsilon_{1}$ & total energy of the first lepton $e^{+}(e^{-})$ in units $mc^{2}$\\ $\epsilon_{2}$ & total energy of the second lepton $e^{-}(e^{+})$ in units $mc^{2}$\\ $k=(\epsilon_{1}+\epsilon_{2})$ & energy of the incoming photon in units of $mc^{2}$\\ $\bvec{p}$ & electron=positron initial momentum in units $mc$\\ $\bvec{k}$ & photon momentum in units $mc$\\ $\bvec{u}$ & electron/positron initial momentum in units $mc$\\ $u$ & $=\vert \bvec{u} \vert $ \end{tabular} \end{center} % %Here, $\epsilon_{1}(\epsilon_{2})$ is the energy of the observed %electron or positron. The matrix (\ref{eq:matrix_conv}) for pair %production is the transpose of matrix (\ref{eq:matrix_brem_g}). Coulomb and screening effects are contained in \(\Gamma\), defined in section \ref{sec:pol.bremsstrahlung}. Using Eq.\ (\ref{eq:conv_lepton}) and the transfer matrix given by Eq.\ (\ref{eq:matrix_conv}) the polarization state of the produced $e^{-}(e^{+})$ is given in the Stokes formalism by: \begin{equation} \xi^{(1)} = \left( \begin{array}{c} \xi_{1}^{(1)}\\ \xi_{2}^{(1)} \\ \xi_{3}^{(1)} \\ \end{array} \right) \approx \left( \begin{array}{c} \zeta_{3}^{(1)} T \\ 0 \\ \zeta_{3}^{(1)} L \\ \end{array} \right) \end{equation} \subsection{Status of this document} 20.11.06 created by K.Laihem\\ 21.02.07 minor update by A.Sch{\"a}licke\\ 27.11.08 correction in Eq.\ \eqref{eq:polpairdef} by A.Sch{\"a}licke \begin{latexonly} \begin{thebibliography}{9} \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. \bibitem{polPair:Laihem:thesis} K.~Laihem, PhD thesis, Measurement of the positron polarization at an helical undulator based positron source for the International Linear Collider ILC, Humboldt University Berlin, Germany, (2008). \end{thebibliography} \end{latexonly} \begin{htmlonly} \subsection{Bibliography} \begin{enumerate} \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. \item K.~Laihem, PhD thesis, Measurement of the positron polarization at an helical undulator based positron source for the International Linear Collider ILC, Humboldt University Berlin, Germany, (2008). \end{enumerate} \end{htmlonly} \newpage \section{Polarized Photoelectric Effect} % \subsection{Method} % This section describes the basic formulas of polarization transfer in the photoelectric effect class ({\em G4PolarizedPhotoElectricEffect}). The photoelectric effect is the emission of electrons from matter upon the absorption of electromagnetic radiation, such as ultraviolet radiation or x-rays. The energy of the photon is completely absorbed by the electron and, if sufficient, the electron can escape from the material with a finite kinetic energy. A single photon can only eject a single electron, as the energy of one photon is only absorbed by one electron. The electrons that are emitted are often called photoelectrons. If the photon energy is higher than the binding energy the remaining energy is transferred to the electron as a kinetic energy \begin{equation} E_{kin}^{e^-} = k-B_{shell} \end{equation} % In Geant4 the photoelectric effect process is taken into account if: \begin{equation} k > B_{shell} \end{equation} % Where $k$ is the incoming photon energy and $B_{shell}$ the electron binding energy provided by the class {\it G4AtomicShells}. The polarized version of the photoelectric effect is based on the EM standard process {\em G4PhotoElectricEffect}. Mean free path and the distribution of explicitly generated final state particles are treated by this version. For details consult section \ref{sec:em.pee}. The remaining task is to attribute polarization vectors to the generated final state electron, which is discussed in the following. \subsection{Polarization transfer} % The polarization state of an incoming polarized photon is described by the Stokes vector $\vec{\zeta}^{(1)}$. % The polarization transfer to the photoelectron can be described in the Stokes formalism using the same approach as for the Bremsstrahlung and gamma conversion processes, cf.~\ref{sec:pol.bremsstrahlung} and \ref{sec:pol.conv}. The relation between the photoelectron's Stokes parameters and the incoming photon's Stokes parameters is described by the interaction matrix $T_{l}^{p}$ derived from H. Olsen \cite{polPEE:H.Olsen.Kgl} and reviewed by H.W McMaster \cite{polPEE:McMaster2}: \begin{equation} \left(\begin{array}{c} I^{\prime} \\ \vec{\xi}^{(1)} \end{array}\right) = T_{l}^{p} \, \left(\begin{array}{c} I_0 \\ \vec{\zeta}^{(1)} \end{array}\right) \label{eq:photo_lepton} \end{equation} % In general, for the photoelectric effect as a two-body scattering, the cross section should be correlated with the spin states of the incoming photon and the target electron. In our implementation the target electron is not polarized and only the polarization transfer from the photon to the photoelectron is taken into account. In this case the cross section of the process remains polarization independent. To compute the matrix elements we take advantage of the available kinematic variables provided by the generic {\it G4PhotoelectricEffect} class. To compute the photoelectron spin state (Stokes parameters), four main parameters are needed: \begin{itemize} \item The incoming photon Stokes vector $\vec{\zeta}^{(1)}$ \item The incoming photon's energy $k$. \item the photoelectron's kinetic energy $E_{kin}^{e^-}$ or the Lorentz factors $\beta$ and $\gamma$. \item The photoelectron's polar angle $\theta$ or $\cos\theta$. \end{itemize} % The interaction matrix derived by H. Olsen \cite{polPEE:H.Olsen.Kgl} is given by: % \begin{equation} T_{l}^{P}= %\frac{Z^{5}}{(137)^{4}}r_{0}^{2}\beta^{3}\frac{\epsilon}{k^{3}}\frac{\sin^{2}\theta}{(1-\beta \cos\theta)^{3}} \left( \begin{array}{cccc} 1+D & -D & 0 & 0 \\ 0 & 0 & 0 & B \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & A \\ \end{array} \right) \label{eq:matrix_photo} \end{equation} Where \begin{eqnarray} D &=& \frac{1}{k}\left[\frac{2}{k\epsilon(1-\beta \cos\theta)}-1 \right]\\ A &=& \frac{\epsilon}{\epsilon+1}\left[\frac{2}{k\epsilon}+\beta\cos\theta+\frac{2}{k\epsilon^2(1-\beta \cos\theta)}\right]\\ B &=& \frac{\epsilon}{\epsilon+1}\beta\sin\theta\left[\frac{2}{k\epsilon(1-\beta \cos\theta)}-1\right] \end{eqnarray} Using Eq.~(\ref{eq:photo_lepton}) and the transfer matrix given by Eq.~(\ref{eq:matrix_photo}) the polarization state of the produced $e^{-}$ is given in the Stokes formalism by: \begin{equation} \vec{\xi}^{(1)} = \left( \begin{array}{c} \xi_{1}^{(1)}\\ \xi_{2}^{(1)} \\ \xi_{3}^{(1)} \\ \end{array} \right) = \left( \begin{array}{c} \zeta_{3}^{(1)} B \\ 0 \\ \zeta_{3}^{(1)} A \\ \end{array} \right) \label{eq:final_stat} \end{equation} From equation (\ref{eq:final_stat}) one can see that a longitudinally (transversally) polarized photoelectron can only be produced if the incoming photon is circularly polarized. \subsection{Status of this document} 20.11.07 created by K.Laihem\\ 03.12.07 minor update by A.Sch{\"a}licke\\ \begin{latexonly} \begin{thebibliography}{9} %\bibitem{polBrems:McMaster1} W.H.~McMaster. Polarization and the Stokes parameters. American Journal of Physics, 22(6):351-362, 1954. \bibitem{polPEE:H.Olsen.Kgl} H. Olsen, Kgl.~N.~Videnskab. Selskabs Forh. 31, Nos 11, 11a (1958). \bibitem{polPEE:McMaster2}W.H.~McMaster. Matrix representation of polarization. Reviews of Modern Physics, 33(1):8-29, 1961. %\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. \bibitem{polPEE:Laihem:thesis} K.~Laihem, PhD thesis, Measurement of the positron polarization at an helical undulator based positron source for the International Linear Collider ILC, Humboldt University Berlin, Germany, (2008). \end{thebibliography} \end{latexonly} \begin{htmlonly} \subsection{Bibliography} \begin{enumerate} \item{polPEE:H.Olsen.Kgl} H. Olsen, Kgl.~N.~Videnskab. Selskabs Forh. 31, Nos 11, 11a (1958). \item{polPEE:McMaster2}W.H.~McMaster. Matrix representation of polarization. Reviews of Modern Physics, 33(1):8-29, 1961. \item K.~Laihem, PhD thesis, Measurement of the positron polarization at an helical undulator based positron source for the International Linear Collider ILC, Humboldt University Berlin, Germany, (2008). \end{enumerate} \end{htmlonly} % LocalWords: Bhabha