
\section[Single Scattering]{Single Scattering}

Single elastic scattering process is an alternative to the 
multiple scattering process. The advantage of the single scattering process is 
in possibility of usage of theory based cross sections, in contrary to
the Geant4 multiple scattering model \cite{singscat.urban}, which uses a number of 
phenomenological approximations on top of Lewis theory. 
The process $G4CoulombScattering$ was created 
for simulation of single scattering of muons, it also applicable  with some physical 
limitations to electrons, muons and ions.
Because each of elastic collisions are simulated the number of steps of charged
particles significantly increasing in comparison with the multiple scattering 
approach, correspondingly its CPU performance is pure. However, in low-density media 
(vacuum, low-density gas) multiple scattering may provide wrong results and
single scattering processes is more adequate. 

\subsection{Coulomb Scattering}

The single scattering model of Wentzel \cite{singscat.wentzel}
is used in many of multiple scattering models including Penelope code 
\cite{singscat.penelope}. The Wentzel for describing elastic
scattering of particles with charge $ze$ ($z=-1$ for electron) 
by atomic nucleus with atomic number $Z$
based on simplified scattering potential
\begin{equation}
V(r) =  \frac {zZe^2}{r}exp(-r/R),
\label{singscat.a}
\end{equation}
where the exponential factor tries to reproduce the effect of screening.
The parameter $R$ is a screening radius, which may be estimated from Thomas-Fermi model of 
the atom
\begin{equation}
R =  0.885 Z^{-1/3} r_B,
\label{singscat.a1}
\end{equation}
where $r_B$ is the Bohr radius. In the first Born approximation the
elastic scattering cross section $\sigma^(W)$ can be obtained as
\begin{equation}
\frac{d\sigma^{(W)}(\theta)}{d\Omega}= \frac{zZe^2}{(p\beta c)^2}\frac{1}{(2A + 1 - cos\theta)^2},
\label{singscat.a2}
\end{equation}
where $p$ is the momentum and $\beta$ is the velocity of the projectile particle.
The screening parameter $A$ according to Moliere and Bethe \cite{singscat.bethe}
\begin{equation}
A =  \left(\frac{\hslash}{2pR}\right)^2(1.13 + 3.76(\alpha Z/\beta)^2),
\label{singscat.a3}
\end{equation}
where $\alpha$ is a fine structure constant and the factor in brackets 
is used to take into account second order corrections to  
the first Born approximation.\\

The total elastic cross section $\sigma$ can be expressed via Wentzel cross section
(\ref{singscat.a2})
\begin{equation}
\frac{d\sigma(\theta)}{d\Omega}= \frac{d\sigma^{(W)}(\theta)}{d\Omega}\left(\frac{1}{(1 + \frac{(qR_N)^2}{12})^2} + \frac{1}{Z}\right),
\label{singscat.a4}
\end{equation}
where $q$ is momentum transfer to the nucleus, $R_N$ is nuclear radius. This term takes into
account nuclear size effect \cite{singscat.kokoulin}, the second term takes into account scattering off
electrons. The results of simulation with the single scattering model (Fig.\ref{plot:Alumin})
are competitive with the results of the multiple scattering.
\begin{figure}[htbp]
\center
\includegraphics[scale=0.4]{electromagnetic/standard/al.eps}
\caption{Scattering of muons off 1.5 mm aluminum foil: data \cite{singscat.attwood} - black squares; 
simulation - colored markers corresponding 
different options of multiple scattering and single scattering model; in the bottom
plot - relative difference between the simulation and the data in percents; 
hashed area demonstrates one standard 
deviation of the data.}
\label{plot:Alumin}
\end{figure}
\noindent
 
\subsection{Implementation Details}

The total cross section of the process is obtained as a result of integration 
of the differential cross section (\ref{singscat.a4}). The first term of this cross section
is integrated in the interval $(0,\pi)$. The second term in the smaller interval $(0,\theta_m)$,
where $\theta_m$ is the maximum scattering angle off electrons, which is determined using the cut value 
for the delta electron production. Before sampling of angular distribution the random
choice is performed between scattering off the nucleus and off electrons.

\subsection{Status of this document}
 06.12.07  created by V. Ivanchenko \\
 
\begin{latexonly}

\begin{thebibliography}{99}

\bibitem{singscat.urban} L.~Urban, A multiple scattering model,
   {\em CERN-OPEN-2006-077, Dec 2006. 18 pp.}
\bibitem{singscat.wentzel}G.~Wentzel,
   {\em Z. Phys. 40 (1927) 590.}
\bibitem{singscat.bethe}H.A.~Bethe,
   {\em Phys. Rev. 89 (1953) 1256.}
\bibitem{singscat.penelope}J.M.~Fernandez-Varea et al.
   {\em NIM B 73 (1993) 447.}
\bibitem{singscat.kokoulin} A.V.~Butkevich et al.,
   {\em NIM A 488 (2002) 282. }
\bibitem{singscat.attwood} D.~Attwood et al.
   {\em NIM B 251 (2006) 41.}
\end{thebibliography}

\end{latexonly}

\begin{htmlonly}

\subsection{Bibliography}

\begin{enumerate}

\item L.~Urban, A multiple scattering model,
   {\em CERN-OPEN-2006-077, Dec 2006. 18 pp.}
\item G.~Wentzel,
   {\em Z. Phys. 40 (1927) 590.}
\item H.A.~Bethe,
   {\em Phys. Rev. 89 (1953) 1256.}
\item J.M.~Fernandez-Varea et al.
   {\em NIM B 73 (1993) 447.}
\item A.V.~Butkevich et al.,
   {\em NIM A 488 (2002) 282. }
\item  D.~Attwood et al.
   {\em NIM B 251 (2006) 41.}
\end{enumerate}

\end{htmlonly}



