
\section[Gamma Conversion into an Electron - Positron Pair]
 {Gamma Conversion into an Electron - Positron Pair}\label{sec:em.conv}
 
\subsection{Cross Section and Mean Free Path}

\subsubsection{Cross Section per Atom}

The total cross-section per atom for the conversion of a gamma into an 
$(e^+,e^-)$ pair has been parameterized as 
\begin{equation} \label{conv1}
\sigma(Z,E_\gamma) = Z(Z+1) \: \left[ F_1(X) + F_2(X) \: Z + \frac{F_3(X)}{Z}
                               \right],
\end{equation}
where $E_{\gamma}$ is the incident gamma energy and $X = \ln (E_{\gamma}/m_{e}c^2)$ .  The functions $F_n$ are given by
\begin{eqnarray}
F_1(X) & = & a_0 + a_1 X + a_2 X^2 + a_3 X^3 + a_4 X^4 + a_5 X^5  \\
F_2(X) & = & b_0 + b_1 X + b_2 X^2 + b_3 X^3 + b_4 X^4 + b_5 X^5  \nonumber \\
F_3(X) & = & c_0 + c_1 X + c_2 X^2 + c_3 X^3 + c_4 X^4 + c_5 X^5 ,  \nonumber
\end{eqnarray} 
with the parameters $a_i, b_i, c_i$ taken from a least-squares fit to the 
data \cite{conv.hubb}.  Their values can be found in the function which 
computes formula \ref{conv1}.
\\ 
This parameterization describes the data in the range 
\begin{eqnarray*}
 1\leq Z\leq 100
\end{eqnarray*}
and
\begin{eqnarray*}
E_\gamma \in [1.5 \mbox{ MeV} , 100 \mbox{ GeV}] . 
\end{eqnarray*}
The accuracy of the fit was estimated to be 
$\frac{\Delta\ \sigma}{\sigma}\leq 5\% $ with a mean value of
$\approx 2.2\%$.  Above 100 GeV the cross section is constant.  Below 
$E_{low} = 1.5 \mbox{ MeV}$ the extrapolation
\begin{equation}
\sigma(E) = \sigma(E_{low}) \cdot \left( \frac{E-2m_e c^2}{E_{low}-2m_e c^2}
                                  \right)^2
\end{equation}
is used.

\subsubsection{Mean Free Path}
In a given material the mean free path, $\lambda$, for a photon to convert into
an $(e^+,e^-)$ pair is
\begin{equation}
\lambda(E_{\gamma}) =
 \left( \sum_i n_{ati} \cdot \sigma (Z_i,E_{\gamma}) \right)^{-1}
\end{equation}
where $n_{ati}$ is the number of atoms per volume of the $i^{th}$ element of
the material.

\subsection{Final State}
\subsubsection{Choosing an Element}
The differential cross section depends on the atomic number $Z$ of the 
material in which the interaction occurs.  In a compound material the element
$i$ in which the interaction occurs is chosen randomly according to the 
probability
\begin{equation}
  Prob(Z_i,E_{\gamma}) = 
                      \frac{n_{ati} \sigma(Z_i,E_{\gamma})}
                      {\sum_i [ n_{ati} \cdot \sigma_i (E_{\gamma})]} .
\end{equation}

\subsubsection{Corrected Bethe-Heitler Cross Section}
As written in \cite{conv.heit}, the Bethe-Heitler formula corrected for various
effects is 
\begin{eqnarray} \label{conv.eq1}
\frac{d \sigma(Z,\epsilon)}{d \epsilon} & = & \alpha r_e^2 Z [Z + \xi(Z)]
\left \{ [\epsilon^2 + ( 1 -\epsilon)^2] 
\left[ \Phi_1(\delta(\epsilon)) - \frac{F(Z)}{2} \right] \right. \nonumber \\ 
& & + \left. \frac{2}{3}\epsilon (1-\epsilon) 
\left[ \Phi_2(\delta(\epsilon)) - \frac{F(Z)}{2} \right] \right \} 
\end{eqnarray}
where $\alpha$ is the fine-structure constant and $r_e$ the classical electron
radius.  Here $\epsilon = E/E_\gamma$, $E_\gamma$ is the energy of the photon 
and $E$ is the total energy carried by one particle of the $(e^+,e^-)$ pair.
The kinematical limits of $\epsilon$ are therefore
\begin{equation}
\frac{m_e c^2}{E_\gamma} = \epsilon_0 \leq \epsilon \leq 1-\epsilon_0 .
\end{equation}

\paragraph*{Screening Effect}
The {\it screening variable}, $\delta$, is a function of $\epsilon$ 
\begin{equation}
\delta(\epsilon)=\frac{136}{Z^{1/3}} \ \frac{\epsilon_0}{\epsilon(1-\epsilon)},
\end{equation}
and measures the 'impact parameter' of the projectile.  Two screening 
functions are introduced in the Bethe-Heitler formula :
\begin{eqnarray}
\mbox{for } \delta \leq 1 & \Phi_1 (\delta) = 
                          & 20.867 - 3.242 \delta + 0.625 \delta^2 \\
                          & \Phi_2 (\delta) =
                          & 20.209 - 1.930 \delta - 0.086 \delta^2 \nonumber \\
\mbox{for } \delta > 1    & \Phi_1 (\delta) =
                          & \Phi_2 (\delta) = 21.12 - 4.184 \ln(\delta+0.952).
                            \nonumber 
\end{eqnarray}
Because the formula \ref{conv.eq1} is symmetric under the exchange $\epsilon
 \leftrightarrow (1-\epsilon)$, the range of $\epsilon$ can be restricted to 
\begin{equation}
\epsilon \in [ \epsilon_0 , 1/2] .
\end{equation}

\paragraph*{Born Approximation}
The Bethe-Heitler formula is calculated with plane waves, but Coulomb waves 
should be used instead.  To correct for this, a 
{\it Coulomb correction function} is introduced in the Bethe-Heitler formula :
\begin{eqnarray}
\mbox{for $E_\gamma$ } <    50 \mbox{ MeV :} & F(z) = & 8/3 \ln Z \\
\mbox{for $E_\gamma$ } \geq 50 \mbox{ MeV :} & F(z) = & 8/3 \ln Z + 8 f_c (Z)
                            \nonumber 
\end{eqnarray}
with
\begin{eqnarray} 
f_c(Z) &=& (\alpha Z)^2 \left[ \frac{1}{1+(\alpha Z)^2} \right.
 \\ & & \left.
+ 0.20206 - 0.0369 (\alpha Z)^2 + 0.0083 (\alpha Z)^4 - 0.0020 (\alpha Z)^6 
+ \cdots \right]. \nonumber
\end{eqnarray}
It should be mentioned that, after these additions, the cross section
becomes negative if
\begin{equation} 
\delta > \delta_{max} (\epsilon_1) = \exp \left[ \frac{42.24 - F(Z)}{8.368}
                                          \right] - 0.952  .
\end{equation}
This gives an additional constraint on $\epsilon$ :
\begin{equation}
\delta \leq \delta_{max} \Longrightarrow
\epsilon \geq \epsilon_1 = \frac{1}{2} -
 \frac{1}{2} \sqrt{1-\frac{\delta_{min}}{\delta_{max}}}
\end{equation}
where
\begin{equation}
\delta_{min} = \delta \left(\epsilon = \frac{1}{2} \right) 
             = \frac{136}{Z^{1/3}} \ 4 \epsilon_0
\end{equation}
has been introduced.  Finally the range of $\epsilon$ becomes
\begin{equation}
\epsilon \in [ \epsilon_{min}=\max (\epsilon_0,\epsilon_1),\ 1/2] .
\end{equation}
\vspace{0mm}
\begin{center}
\includegraphics*
[width=\textwidth,height=0.4\textheight,draft=false]
                             {electromagnetic/standard/conv.eps}
\end{center}
\paragraph*{Gamma Conversion in the Electron Field}
The electron cloud gives an additional contribution to pair creation,
proportional to $Z$ (instead of $Z^2$).  This is taken into account through
the expression
\begin{equation}
\xi(Z) = \frac{\ln(1440/Z^{2/3})}{\ln(183/Z^{1/3}) - f_c(Z)} .
\end{equation}

\paragraph*{Factorization of the Cross Section}
$\epsilon$ is sampled using the techniques of 'composition+rejection', as 
treated in \cite{conv.ford,conv.butch,conv.messel}.  First, two auxiliary 
screening functions should be introduced:
\begin{eqnarray}
F_1(\delta) &=& 3 \Phi_1(\delta) - \Phi_2(\delta) - F(Z)  \nonumber \\
F_2(\delta) &=& \frac{3}{2} \Phi_1(\delta) - \frac{1}{2} \Phi_2(\delta) - F(Z)
\end{eqnarray}
It can be seen that $F_1(\delta)$ and $F_2(\delta)$ are decreasing
functions of $\delta$, $\forall \delta \in [\delta_{min} , \delta_{max}]$.
They reach their maximum for $\delta_{min} = \delta(\epsilon = 1/2)$ :
\begin{eqnarray}
F_{10} = \max F_1(\delta) = F_1(\delta_{min})  \nonumber \\
F_{20} = \max F_2(\delta) = F_2(\delta_{min}) . \,
\end{eqnarray}
After some algebraic manipulations the formula \ref{conv.eq1} can be written :
\begin{eqnarray} \label{conv.eq2}
\frac{d \sigma(Z,\epsilon)}{d \epsilon} &=& \alpha r_e^2 Z [Z + \xi(Z)]
\frac{2}{9} \left[ \frac{1}{2} - \epsilon_{min} \right] \nonumber \\
 & & \times \left[
N_1 \ f_1(\epsilon) \ g_1(\epsilon) + N_2 \ f_2(\epsilon) \ g_2(\epsilon)
\right] ,
\end{eqnarray}
where
\begin{eqnarray*}
N_1 = \left[ \frac{1}{2} - \epsilon_{min} \right]^2 F_{10} \hspace{5mm} &
f_1(\epsilon) = \frac{3}{\left[ \frac{1}{2} - \epsilon_{min} \right]^3}
                \left[ \frac{1}{2} - \epsilon \right]^2    &
\hspace{5mm} g_1(\epsilon) = \frac{F_1(\epsilon)}{F_{10}}
\\                              
N_2 = \frac{3}{2} F_{20} \hspace{5mm} &
f_2(\epsilon) = \mbox{const}
              = \frac{1}{\left[ \frac{1}{2} - \epsilon_{min} \right]} &
\hspace{5mm} g_2(\epsilon) = \frac{F_2(\epsilon)}{F_{20}} .
\end{eqnarray*}
$f_1(\epsilon)$ and $f_2(\epsilon)$ are probability density functions on the
interval $\epsilon \in [\epsilon_{min} , 1/2]$ such that 
$$ \int_{\epsilon_{min}}^{1/2} f_i(\epsilon) \, d\epsilon = 1 $$ ,
and $g_1(\epsilon)$ and $g_2(\epsilon)$ are valid rejection functions:
$0 < g_i (\epsilon) \leq 1$ .

\paragraph*{Sampling the Energy}
Given a triplet of uniformly distributed random numbers $(r_a, r_b, r_c)$ :
\begin{enumerate}
\item use $r_a$ to choose which decomposition term in \ref{conv.eq2} to use:
\begin{equation}
 \mbox{if } r_a < N_1/(N_1+N_2) \rightarrow f_1(\epsilon)\ g_1(\epsilon) \\
 \mbox{  otherwise } \rightarrow f_2(\epsilon)\ g_2(\epsilon)
\end{equation}
\item sample $\epsilon$ from $f_1(\epsilon)$ or $f_2(\epsilon)$ with $r_b$ :
\begin{equation}
\epsilon = \frac{1}{2} - \left(\frac{1}{2} - \epsilon_{min} \right) r_b^{1/3}
\hspace{5mm} \mbox{or} \hspace{5mm}
\epsilon = \epsilon_{min} + \left(\frac{1}{2} - \epsilon_{min} \right) r_b
\end{equation}
\item reject $\epsilon$ if $g_1(\epsilon) \mbox{or }  g_2(\epsilon) < r_c$   
\end{enumerate}
{\sc note} :
below $E_{\gamma} = 2$ MeV it is enough to sample $\epsilon$ uniformly on
$[\epsilon_0,\ 1/2]$, without rejection.
\paragraph*{Charge}
The charge of each particle of the pair is fixed randomly.

\subsubsection{Polar Angle of the Electron or Positron}
The polar angle of the electron (or positron) is defined with respect to the
direction of the parent photon.  The energy-angle distribution given by 
Tsai \cite{conv.tsai} is quite complicated to sample and can be approximated
by a density function suggested by Urban \cite{conv.urban} :
\begin{equation} \label{conv.eq3}
\forall u \in [0,\ \infty [  \hspace{3mm}
f(u) = \frac{9a^2}{9+d} \left[ u \exp (-a u) + d\ u \exp (-3a u) \right] 
\end{equation}
with
\begin{equation}
a=\frac{5}{8} \hspace{5mm} d=27 \hspace{1cm} \mbox{and } 
  \theta_\pm = \frac{mc^2}{E_\pm} \ u .
\end{equation}
A sampling of the distribution \ref{conv.eq3} requires a triplet of random 
numbers such that
\begin{equation}
\mbox{if } r_1 < \frac{9}{9+d} \rightarrow u = \frac{-\ln(r_2 r_3)}{a}
          \hspace{5mm} \mbox{otherwise } u = \frac{-\ln(r_2 r_3)}{3a} .
\end{equation}
The azimuthal angle $\phi$ is generated isotropically.
\paragraph*{Final State}
The $e^+$ and $e^-$ momenta are assumed to be coplanar with the parent photon. 
This information, together with energy conservation, is used to calculate the 
momentum vectors of the $(e^+,e^-)$ pair and to rotate them to the global
reference system.

\subsection{Status of this document}
12.01.02 created by M.Maire. \\
21.03.02 corrections in angular distribution (mma) \\
22.04.02 re-worded by D.H. Wright \\ 

\begin{latexonly}

\begin{thebibliography}{99}
\bibitem{conv.hubb} J.H.Hubbell, H.A.Gimm, I.Overbo 
 {\it Jou. Phys. Chem. Ref. Data 9:1023} (1980)
\bibitem{conv.heit} W. Heitler
 {\it The Quantum Theory of Radiation, Oxford University Press} (1957)  
\bibitem{conv.ford} R. Ford and W. Nelson.
 {\it SLAC-210, UC-32} (1978)    
\bibitem{conv.butch} J.C. Butcher and H. Messel.
 {\it Nucl. Phys. 20} 15 (1960)   
\bibitem{conv.messel} H. Messel and D. Crawford.
 {\it Electron-Photon shower distribution, Pergamon Press} (1970)
\bibitem{conv.tsai}
  Y. S. Tsai, {\em Rev. Mod. Phys. 46} 815 (1974),
  Y. S. Tsai, {\em Rev. Mod. Phys. 49} 421 (1977)
\bibitem{conv.urban} L.Urban in {\sc Geant3} writeup, section PHYS-211.
 {\it Cern Program Library} (1993) 
\end{thebibliography}

\end{latexonly}

\begin{htmlonly}

\subsection{Bibliography}

\begin{enumerate}
\item J.H.Hubbell, H.A.Gimm, I.Overbo 
 {\it Jou. Phys. Chem. Ref. Data 9:1023} (1980)
\item W. Heitler
 {\it The Quantum Theory of Radiation, Oxford University Press} (1957)  
\item R. Ford and W. Nelson.
 {\it SLAC-210, UC-32} (1978)    
\item J.C. Butcher and H. Messel.
 {\it Nucl. Phys. 20} 15 (1960)   
\item H. Messel and D. Crawford.
 {\it Electron-Photon shower distribution, Pergamon Press} (1970)
\item Y. S. Tsai, {\em Rev. Mod. Phys. 46} 815 (1974),
  Y. S. Tsai, {\em Rev. Mod. Phys. 49} 421 (1977)
\item L.Urban in {\sc Geant3} writeup, section PHYS-211.
 {\it Cern Program Library} (1993) 
\end{enumerate}

\end{htmlonly}
