\section[Positron - Electron Annihilation]{Positron - Electron Annihilation}\label{sec:em.annil}

\subsection{Introduction}
The process {\tt G4eplusAnnihilation} simulates the in-flight annihilation 
of a positron with an atomic electron.  As is usually done in shower 
programs \cite{egs4}, it is assumed here that the atomic electron is 
initially free and at rest.  Also, annihilation processes producing one, or 
three or more, photons are ignored because these processes are negligible 
compared to the annihilation into two photons \cite{egs4,messel}.

\subsection{Cross Section and Mean Free Path}
\subsubsection{Cross Section per Atom}
The annihilation in flight of a positron and electron is described by 
the cross section formula of Heitler \cite{heitler,egs4}: 

\begin{eqnarray}
\sigma(Z,E) & = & \frac{Z \pi r_e^2}{\gamma +1}
                  \left[\frac{\gamma^2 + 4 \gamma +1}{\gamma^2 -1}
                  \ln \left(\gamma +\sqrt{\gamma^2 -1} \right)  -\frac
                  {\gamma +3}{\sqrt{\gamma^2 -1}} \right]  \\
{\rm where}\nonumber\\
E      & = & \mbox{total energy of the incident positron}  \nonumber \\
\gamma & = & E/m c^2  \nonumber \\
r_e    & = & \mbox{classical electron radius}  \nonumber    
\end{eqnarray}

\subsubsection{Mean Free Path}
In a given material the mean free path, $\lambda$, for a positron to be 
annihilated with an electron is given by
\begin{equation}
\lambda(E) =
 \left( \sum_i n_{ati} \cdot \sigma(Z_i,E) \right)^{-1}
\end{equation}
where $n_{ati}$ is the number of atoms per volume of the $i^{th}$ element
composing the material.

\subsection {Sampling the final state}

The final state of the $e+e-$ annihilation process 
\[e^+ \; e^- \to \gamma_a \; \gamma_b \]
is simulated by first determining the kinematic limits of the photon energy
and then sampling the photon energy within those limits using the 
differential cross section.  Conservation of energy-momentum is then used to
determine the directions of the final state photons.
 
\subsubsection{Kinematic Limits}

If the incident $e^+$ has a kinetic energy $T$, then the total energy is
$E_e = T + mc^2$ and the momentum is $Pc = \sqrt{T(T+2mc^2)}$.
The total available energy is $E_{tot} = E_e + mc^2 = E_a + E_b$ and
momentum conservation requires $ \vec{P} = \vec{P}_{\gamma_a} + \vec{P}_{\gamma_b}$ .  The fraction of the total energy transferred to one photon 
(say $\gamma_a$) is
\begin{equation}
 \epsilon = \frac{E_a}{E_{tot}} \equiv \frac{E_a}{T+2mc^2} .
\end{equation}
The energy transfered to $\gamma_a$ is largest when $\gamma_a$ is emitted in
the direction of the incident $e^+$.  In that case 
$E_{a max} = (E_{tot}+Pc)/2$ .  The energy transfered to $\gamma_a$ is 
smallest when $\gamma_a$ is emitted in the opposite direction of the 
incident $e^+$.  Then $E_{a min} = (E_{tot}-Pc)/2$ .  Hence,
\begin {eqnarray}
  \epsilon_{max} &=& \frac{E_a \mathsf{max}}{E_{tot}} =
  \frac{1}{2} \left\lbrack 1+ \sqrt{\frac{\gamma - 1}{\gamma+1}} \right\rbrack \\ 
  \epsilon_{min} &=& \frac{E_a \mathsf{min}}{E_{tot}} =
  \frac{1}{2} \left\lbrack 1- \sqrt{\frac{\gamma - 1}{\gamma+1}} 
  \right\rbrack
\end {eqnarray}
\\
where $\qquad \gamma = (T + mc^2)/mc^2$ .  Therefore the range of $\epsilon$
is 
 $\quad \lbrack \epsilon_{min} \; ; \; \epsilon_{max} \rbrack $
 $\qquad ( \equiv \lbrack \epsilon_{min} \; ; \; 1-\epsilon_{min} \rbrack) $.
      
\subsubsection{Sampling the Gamma Energy}

A short overview of the sampling method is given in Chapter \ref{secmessel}.

The differential cross section of the two-photon positron-electron 
annihilation can be written as 
\cite{heitler,egs4}:
\begin{equation}
   \frac{d \sigma (Z, \epsilon)} {d \epsilon} =
   \frac{Z \pi r_e^2}{\gamma - 1} \ \frac{1}{\epsilon} \ 
   \left[
   1+\frac{2\gamma}{(\gamma+1)^2}-\epsilon-\frac{1}{(\gamma+1)^2}\frac{1}{\epsilon}
   \right]
\end{equation}
where
$Z$ is the atomic number of the material, $r_e$ the classical electron 
radius, and $\epsilon \in [ \epsilon_{min} \; ; \; \epsilon_{max} ]$ .
The differential cross section can be decomposed as
\begin{equation}
   \frac{d \sigma (Z, \epsilon)} {d \epsilon} =
   \frac{Z \pi r_e^2}{\gamma - 1} 
   \alpha f(\epsilon) g(\epsilon)
\end{equation}
where
\begin{eqnarray}
\alpha      &=& \ln (\epsilon_{max}/\epsilon_{min})  \nonumber \\  
f(\epsilon) &=& \frac{1}{\alpha \epsilon} \\
g(\epsilon) &=& \left[
    1+\frac{2\gamma}{(\gamma+1)^2}-\epsilon-\frac{1}{(\gamma+1)^2}\frac{1}{\epsilon}
    \right]  \equiv  
    1-\epsilon+\frac{2 \gamma \epsilon -1}{\epsilon (\gamma +1)^2}
\end{eqnarray}
Given two random numbers $r, r' \in [0,1]$, the photon energies are chosen
as follows:
\begin{enumerate}
\item 
  sample $\epsilon$ from $f(\epsilon):
  \epsilon =\epsilon_{min} \left( \frac{\epsilon_{max}}{\epsilon_{min}} \right)^r$
\item
  test the rejection function: if $g(\epsilon) \geq r'$ accept
  $\epsilon$, otherwise return to step 1.
\end{enumerate}
Then the photon energies are
 $E_a = \epsilon E_{tot} \qquad E_b = (1-\epsilon) E_{tot}$ .

\subsubsection{Computing the Final State Kinematics}
If $\theta$ is the angle between the incident $e^+$ and $\gamma_a$, then 
from energy-momentum conservation,
\begin{equation}
\cos \theta = \frac{1}{Pc} \left[ T+mc^2 \frac{2\epsilon -1}{\epsilon} \right]
 = \frac{\epsilon(\gamma +1) - 1}{\epsilon \sqrt{\gamma^2 -1}} .
\end{equation} \\
The azimuthal angle, $\phi$, is generated isotropically and the photon 
momentum vectors, $\vec{P_{\gamma_a}}$ and $\vec{P_{\gamma_b}}$, 
are computed from energy-momentum conservation and transformed into the lab
coordinate system. 

\subsubsection{Annihilation at Rest} 
The method {\tt AtRestDoIt} treats the special case when a positron comes 
to rest before annihilating.  It generates two photons, each with energy 
$k=mc^2$ and an isotropic angular distribution.

\subsection{Status of this document}
 09.10.98 created by M.Maire.     \\
 01.08.01 minor corrections (mma) \\
 09.01.02 MeanFreePath (mma) \\
 01.12.02 Re-written by D.H. Wright \\
   
\begin{latexonly}

\begin{thebibliography}{99}
\bibitem{egs4} R. Ford and W. Nelson.
   {\em SLAC-265, UC-32} (1985)      
\bibitem{messel} H. Messel and D. Crawford.
   {\em Electron-Photon shower distribution, Pergamon Press} (1970)
\bibitem{heitler} W. Heitler.
   {\em The Quantum Theory of Radiation, Clarendon Press, Oxford} (1954)
\end{thebibliography}

\end{latexonly}
   
\begin{htmlonly}

\subsection{Bibliography}

\begin{enumerate}
\item R. Ford and W. Nelson.
   {\em SLAC-265, UC-32} (1985)      
\item H. Messel and D. Crawford.
   {\em Electron-Photon shower distribution, Pergamon Press} (1970)
\item W. Heitler.
   {\em The Quantum Theory of Radiation, Clarendon Press, Oxford} (1954)
\end{enumerate}

\end{htmlonly}