\chapter{Optical Elastic Model}

\section{ The model description  }

This model is based on the optical approach in which a nucleus is considered to be a drop of 
an absorptive and refractive medium.  Absorption results in the diffraction of a projectile 
hadron, while refraction (scattering) in the surface layer provides some smoothing of the 
diffraction picture. The differential cross section in the center mass system can be written 
 \cite{bethe40,ap45,bs98}:
\[
\frac{d\sigma_{el}}{d\Omega}= R^2F_d^2(k\,d\,\theta)
\left\{\frac{J_1^2(k\,R\,\theta)}{\theta^2}+
\left[(\gamma\,k)^2+
(\delta\,k^2\,\theta)^2\right]J_0^2(k\,R\,\theta)\right\},
\quad d\Omega=2\pi\sin\theta\,d\theta,
\]
where $J_k$ is the Bessel function of k-th order ($k = 0,1$) and $\Omega$ and $\theta$ are 
the solid and polar scattering angles, respectively. 
$R$ and $d$ are nuclear geometrical parameters of the Woods-Saxon density form
\[
\rho(r)=\rho_o\left\{1+\displaystyle\exp\left[\frac{(r-R)}{d}\right]\right\}^{-1},
\]
in which $k$ is the projectile wave vector ($k=p/\hbar\,c$, $p$ is the projectile 
momentum multiplied by $c$), and $F_d$ is the dumping factor:
\[
F_d(k\,d\,\theta)=\frac{\pi\,k\,d\,\theta}{\sinh(\pi\,k\,d\,\theta)}.
\]
$\gamma$ is the refraction (nuclear scattering) parameter, and $\delta$ is the 
spin-orbital interaction parameter.

The model can be modified for charged particles by taking into account that a simple 
summation of nuclear and Coulomb cross sections does not work.  However for small Coulomb 
scattering  
$\delta f_{Coulomb}(\theta)\ll f_{nuclear}(\theta)$ the amplitudes can be summarized:
\[
\frac{d\sigma_{el}}{d\Omega} = | \delta f_{Coulomb}(\theta)+f_{nuclear}(\theta)|^2. 
\]
This case is assumed in the diffuse optical model considered, where we should correct the 
nuclear refraction (scattering) term:
\[
\gamma\,k \rightarrow \gamma\,k + \frac{n}{2kR}\left[\sin^2(\frac{\theta}{2}) + A_m \right]^{-1},
\quad n = \frac{\alpha Z_1  Z_2}{\beta} \ll kR,
\]
where $n$ is Zommerfeld parameter for the Coulomb field, $\beta=v/c$ is the particle velocity, 
and $Z_1$ and $Z_2$ are the hadron and nucleus charges, respectively. 
The parameter $A_m$ reflects Coulomb atomic shell screening and can be estimated as:
\[
A_m = \frac{1.13 + 3.76n^2}{(1.77ka_oZ^{-1/3})^2},
\]
where $a_o$ is the Bohr radius, and $Z$ is the atomic number.

This method corresponds to elastic Coulomb-Wentzel scattering recently implemented 
in electromagnetic physics \cite{fer93}:
\[
\frac{d\sigma_{el}^{cw}}{d\Omega}= \frac{n^2}{4k^2}
\left[\sin^2(\frac{\theta}{2}) + A_m \right]^{-2}, \quad V(r)=\frac{e^2Z_1Z_2}{r^2}\exp(-r/R),
\quad A_m = (2kR)^{-2}.
\]
The Coulomb-Wentzel cross-section reads:
\[
\sigma_{el}^{cw}= \frac{n^2}{k^2}\frac{\pi}{A_m(1+A_m)}, \quad 
\sigma_{el}^{cw}(\theta_1,\theta_2)=2\pi\frac{n^2}{k^2}\frac{\cos\theta_1-\cos\theta_2}
{(1-\cos\theta_1+2A_m)(1-\cos\theta_2+2A_m)}.
\]
This cross-section is much bigger than the integral nuclear cross section.  However the 
main contribution comes from very small angles where the differential nuclear elastic 
cross section is approximately constant. In this region the Coulomb correction can be 
switched off. This avoids double counting of Coulomb scattering in electromagnetic and nuclear 
processes.

\section{Validation}


Figs. \ref{pPb1GeV}-\ref{pipC9p92GeVc} show the elastic differential cross sections of protons 
and pions with energies in the range 1 - 301 GeV on different targets: helium, carbon, silicon 
and lead.
One can see that Coulomb corrections improve the description of differential cross sections 
in the regions of the minima.

\begin{figure*}
\centering \includegraphics[width=10cm,height=7cm]{hadronic/elastic/pPbsumT1GeV.eps}
\vspace{-0.5cm}
\caption{ Differential elastic scattering cross sections of protons with energy 
1 GeV on lead. Curves show pure nuclear and Coulomb-corrected models. Points are 
experimental data. The Coulomb cross section dominates at small angles resulting 
in a large integral cross section: $\sigma_{el}^{nucl}=1210$ mb, with $\lambda = 25$ cm, 
$\sigma_{el}^{cw}(2^{0}, 16^{o})=642.65$ mb, with $\lambda = 47$ cm, 
$\sigma_{el}^{cw}(0,1^o)\sim \sigma_{el}^{cw} = 2.36423\cdot10^{9}$ mb, 
with $\lambda = 0.13$ micron.}
\label{pPb1GeV}
\end{figure*}

\begin{figure*}
\centering \includegraphics[width=10cm,height=7cm]{hadronic/elastic/pSisumT1GeV.eps}
\vspace{-0.5cm}
\caption{ Differential elastic scattering cross sections of protons with energy 
1 GeV on silicon. Curves show pure nuclear and Coulomb-corrected models. Points are 
experimental data. The Coulomb cross section dominates at small angles resulting in 
a large integral cross-section: 
$\sigma_{el}^{cw}(0,1^o)\sim \sigma_{el}^{cw}=5.46779\cdot10^{8}$ mb, 
with $\lambda = 0.366068$ micron.}
\label{pSi1GeV}
\end{figure*}

\begin{figure*}
\centering \includegraphics[width=10cm,height=7cm]{hadronic/elastic/pPbsum9p92GeVc.eps}
\vspace{-0.5cm}
\caption{ Differential elastic scattering cross sections of protons with momentum 
9.92 GeV/c on lead. Curves show pure nuclear and Coulomb-corrected models. Points are 
experimental data. The Coulomb cross section dominates at small angles resulting in a
large integral cross section
$\sigma_{el}^{cw}(0,4 \ mrad)\sim \sigma_{el}^{cw}=2.11854\cdot10^{9}$ mb, 
with $\lambda = 0.143101$ micron. 
$\sigma_{el}^{cw}(4  \ mrad,40 \ mrad)=919.547$ mb, 
with $\lambda = 329689$ micron (33 cm).}
\label{pPb9p92GeVc}
\end{figure*}

\begin{figure*}
\centering \includegraphics[width=10cm,height=7cm]{hadronic/elastic/pHeT45GeVsum.eps}
\vspace{-0.5cm}
\caption{ Differential elastic scattering cross sections of protons with energy 
45 GeV on helium. Curves show pure nuclear and Coulomb-corrected models. Points are 
experimental data.}
\label{pHe45GeV}
\end{figure*}

\begin{figure*}
\centering \includegraphics[width=10cm,height=7cm]{hadronic/elastic/pHeT301GeVsum.eps}
\vspace{-0.5cm}
\caption{ Differential elastic scattering cross sections of protons with energy 
301 GeV on helium. Curves show pure nuclear and Coulomb-corrected models. Points are 
experimental data.}
\label{pHe301GeV}
\end{figure*}

\begin{figure*}
\centering \includegraphics[width=10cm,height=7cm]{hadronic/elastic/pimPbsum9p92GeVc.eps}
\vspace{-0.5cm}
\caption{ Differential elastic scattering cross sections of $\pi^{-}$ with momentum 
9.92 GeV/c on lead. Curves show pure nuclear and Coulomb-corrected models. Points are 
experimental data. }
\label{pimPb9p92GeVc}
\end{figure*}

\begin{figure*}
\centering \includegraphics[width=10cm,height=7cm]{hadronic/elastic/pipPbsum9p92GeVc.eps}
\vspace{-0.5cm}
\caption{ Differential elastic scattering cross sections of $\pi^{+}$ with momentum 
9.92 GeV/c on lead. Curves show pure nuclear and Coulomb-corrected models. Points are 
experimental data.}
\label{pipPb9p92GeVc}
\end{figure*}

\begin{figure*}
\centering \includegraphics[width=10cm,height=7cm]{hadronic/elastic/pipCsum9p92GeVc.eps}
\vspace{-0.5cm}
\caption{ Differential elastic scattering cross sections of $\pi^{+}$ with momentum 
9.92 GeV/c on carbon. Curves show pure nuclear and Coulomb-corrected models. Points are 
experimental data.}
\label{pipC9p92GeVc}
\end{figure*}

\section{Implementation} 

This model is implemented in the class {\em G4DiffuseElastic}. It conforms to the general 
interface of hadronic models and performs initialization for different elements during 
tracking.


\section{Status of this document}
18.11.07 created by V. Grichine \\



\begin{latexonly}

\begin{thebibliography}{99}

\bibitem{bethe40} P. Placzec, H.A. Bethe, {\em Phys. Rev.}, {\bf 57} (1940) 1075A.

\bibitem{ap45} A. Akhiezer, I. Pomeranchuk, {\em J. Phys. (USSR)}, {\bf 9} (1945) 471; 
{\em Sov. Phys. USPEKHI}, {\bf 65} (1958) 593.

\bibitem{bs98} Yu.A. Berezhnoy, V.A. Slipko, {\em Int. J. of Mod. Phys.}, {\bf E7} (1998) 723.

\bibitem{fer93} J.M. Fernandez-Varea, R. Moyol, J. Baro, F. Salvat, 
{\em Nucl. Instr. and Math. in Phys. Res.}, {\bf B73} (1993) 447.


\end{thebibliography}

\end{latexonly}

\begin{htmlonly}

\section{Bibliography}

\begin{enumerate}

\item P. Placzec, H.A. Bethe, {\em Phys. Rev.}, {\bf 57} (1940) 1075A.


\item A. Akhiezer, I. Pomeranchuk, {\em J. Phys. (USSR)}, {\bf 9} (1945) 471; 
{\em Sov. Phys. USPEKHI}, {\bf 65} (1958) 593.

\item Yu.A. Berezhnoy, V.A. Slipko, {\em Int. J. of Mod. Phys.}, {\bf E7} (1998) 723.

\item J.M. Fernandez-Varea, R. Moyol, J. Baro, F. Salvat, 
{\em Nucl. Instr. and Math. in Phys. Res.}, {\bf B73} (1993) 447.


\end{enumerate}

\end{htmlonly}

