source: trunk/documents/UserDoc/DocBookUsersGuides/PhysicsReferenceManual/latex/electromagnetic/standard/polarisation.tex@ 1222

% ======================================================================

%\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 <Sx> <Sy> <Sz>
\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