\chapter{ Hadron-nucleus Elastic Scattering at Medium and High Energy.}

\vspace{2ex}
\section{Method of Calculation}

The Glauber model \cite{helast.1} is used as an alternative method of
calculating differential cross sections for elastic and quasi-elastic 
hadron-nucleus scattering at high and intermediate energies. 

For high energies this includes corrections for inelastic screening and
for quasi-elastic scattering the exitation of a discrete level or a state in
the continuum is considered.
 
The usual expression for the Glauber model amplitude for multiple scattering 
was used

\begin{equation}
   F(q)=\frac{ik}{2\pi} \int d^2{b}e^{\vec{\mathstrut q}
   \cdot \vec{\mathstrut b}} M(\vec b) .
\label{helast.eq1}
\end{equation}
Here $M(\vec b)$ is the hadron-nucleus amplitude in the impact 
parameter representation
\begin{equation}
 M(\vec b) = 1-[1-e^{-A\int d^{3}r\Gamma(\vec{\mathstrut b}-
      \vec{\mathstrut s})\rho(\vec {r})}]^A,
 \label{helast.eq2}
\end{equation}
$k$ is the incident particle momentum,
$\vec q = \vec k' - \vec k$  is the momentum transfer, and
$\vec k'$ is the scattered particle momentum.
Note that $\left| \vec q \right| ^2 = -t$ - invariant momentum transfer
squared in the center of mass system.
$\Gamma(\vec {b})$ is the hadron-nucleon amplitude of 
elastic scattering in the impact-parameter representation

\begin{equation}
  \Gamma(\vec b)=\frac{\displaystyle 1}
                      {\displaystyle 2\pi ik^{hN}}
  \int d\vec {q} e^{-\vec {\mathstrut q} \cdot \vec{\mathstrut b}} 
  f(\vec {q}).
 \label{helast.eq3}
\end{equation}

The exponential parameterization of the hadron-nucleon amplitude is usually 
used:

\begin{equation}
  f(\vec {q})=\frac{\displaystyle ik^{hN}\sigma^{hN}}{2\pi}
   e^{-0.5q^2B}.
 \label{helast.eq4}
\end{equation}
Here  $\sigma^{hN}=\sigma_{tot}^{hN}(1-i\alpha)$, 
$\sigma_{tot}^{hN}$ is the total
cross section of a hadron-nucleon scattering, $B$ is the slope of the 
diffraction cone and $\alpha$ is the ratio of the real to imaginary parts of 
the amplitude at $q=0$.  The value $k^{hN}$ is the hadron momentum in the 
hadron-nucleon coordinate system.

The important difference of these calculations from the usual ones 
is that the two-gaussian form of the nuclear density was used

\begin{equation}
   \rho(r) = C (e^{-(r/R_{1})^2}-pe^{-(r/R_2)^2}),   
 \label{helast.eq5}
\end{equation}
where $R_1$, $R_2$ and $p$ are the fitting parameters and $C$ is a 
normalization constant.

This density representation allows the expressions for amplitude and 
differential cross section to be put into analytical form.  It was earlier 
used for light \cite{helast.2, helast.3} and medium \cite{helast.4} nuclei. 
Described below is an extension of this method to heavy nuclei.
The form \ref{helast.eq5} is not physical for a heavy nucleus, but 
nevertheless works rather well (see figures below).  The reason is that 
the nucleus absorbs the hadrons very strongly, especially at small impact 
parameters where the absorption is full. As a result only the peripherial part 
of the nucleus participates in elastic scattering.  Eq. \ref{helast.eq5}
therefore describes only the edge of a heavy nucleus.

 Substituting Eqs. \ref{helast.eq5} and \ref{helast.eq4} into Eqs.
\ref{helast.eq1}, \ref{helast.eq2} and \ref{helast.eq3} yields the following 
formula

\begin{eqnarray}
   F(q) = \frac{ik\pi}{2} 
   \sum\limits_{k=1}^A (-1)^k {A\choose k} 
   [\frac{\sigma^{hN}}{2\pi(R_1^3-pR_2^3)}]^k 
   \sum\limits_{m=0}^{k} (-1)^m {k \choose m}
   \left[ \frac{R_1^3}{R_1^2+2B}
   \right]^{k-m} \nonumber
\end{eqnarray}

\begin{eqnarray}
  \times \qquad {} \left[ 
  \frac{pR_2^3}{R_2^2+2B}
  \right]^{m} \left( 
  \frac{m}{R_2^2+2B}+\frac{k-m}{R_1^2+2B} \right)^{-1} \nonumber
\end{eqnarray}

\begin{eqnarray}
  \times \qquad {} \exp\left[ -\frac{-q^2}{4} 
  \left(
  \frac{m}{R_2^2+2B}+
  \frac{k-m}{R_1^2+2B} \right)^{-1}
  \right].
\label{helast.eq6}
\end{eqnarray}
% \end{eqnarray}

An analogous procedure can be used to get the inelastic screening corrections 
to the hadron-nucleus amplitude $\Delta M(\vec b)$ \cite{helast.5}.
In this case an intermediate inelastic diffractive state is created which 
rescatters on the nucleons of the nucleus and then returns into the initial 
hadron. Hence it is nessesary to integrate the production cross section over
the mass distribution of the exited system $\frac{d\sigma^{diff}}{dtdM_x^2}$. 
The expressions for the corresponding amplitude are quite long and so are not
presented here.  The corrections for the total cross-sections can be found in 
\cite{helast.5}.

The full amplitude is the sum $M(\vec b)+ \Delta M(\vec b)$.

The differential cross section is connected with the amplitude in the 
following way

\begin{equation}
  \frac{d\sigma}{d\Omega_{CM}}=\left| F(q) \right|^2, \qquad 
   \frac{d\sigma}{|dt|}=\frac{d\sigma}{dq_{CM}^2}=
  \frac{\pi}{k_{CM}^2} \left|F(q) \right|^2.
 \label{helast.eq7}
\end{equation}
                      
The main energy dependence of the hadron-nucleus elastic scattering 
cross section comes from the energy dependence of the parameters of 
hadron-nucleon scattering 
($\sigma_{tot}^{hN} \alpha$, $B$ and 
$\frac{d\sigma^{diff}}{dtdM_x^2}$). 
At interesting energies these parameters were fixed at their well-known 
values.  The fitting of the nuclear density parameters was performed 
over a wide range of atomic numbers $(A=4 - 208)$ using experimental data 
on proton-nuclei elastic scattering at a kinetic energy of $T_p=1 GeV$. 

The fitting was perfomed both for individual nuclei and for the entire set 
of nuclei at once. 
%%In the last event the following dependensies for nuclei
%%parameters $(for A=12 \div 208)$ were obtained

%\vspace{2ex}
%$R_1=4.18A^{0.302}$, $R_2=3.81(A-10)^{0.268}$, $p=0.95$.
%\hspace{3cm} (8)
%\vspace{2ex}

%%\vspace{2ex}
%%$R_1=4.45(A-1)^{0.309}$, $R_2=2.3A^{0.36}$, 

%%$p=0.176+0.0017A+8.7.10^{-6}A^2$.
%%\hspace{3cm} (8)
%%\vspace{2ex} 

It is necessary to note that for every nucleus 
an optimal set of density parameters exists and it
differs slightly from the one derived for the full set of nuclei.

A comparision of the phenomenological cross sections \cite{helast.6} with 
experiment is presented in Figs. \ref{helast.fig1} - \ref{helast.fig9}

% where the data of the proton-nucleus scattering 
% are imaged.
  
In this comparison, the individual nuclei parameters were used.
The experimental data were obtained in Gatchina (Russia) and in 
Saclay (France) \cite{helast.6}. 
The horizontal axis is the scattering angle in the center of mass
system $\Theta_{CM}$ and the vertical axis is
$\frac{\displaystyle d\sigma}{\displaystyle d\Omega_{CM}}$ 
in $\frac{\displaystyle mb}{\displaystyle Ster}$.
%The lower curve in the figures is the diferencial cross-section 
%of a coherent elastic scattering but upper one is the
%cross-section for noncoherent scattering.

Comparisions were also made for $p ^4He$ elastic scatering at 
$T_ = 1 GeV$[7], $45 GeV$ and $301 GeV$ [3]. The resulting
cross sections $\frac {d \sigma}{d\left| t \right|}$ are shown 
in the Figs. \ref{helast.fig10} - \ref{helast.fig12}.

In order to generate events the distribution function ${\cal F}$ of a 
corresponding process must be known. 
The differential cross section is proportional to the density distribution.
Therefore to get the distribution function it is sufficient to integrate the
differential cross section and normalize it:

\begin{equation}
    {\cal F}(q^2)=
 \frac
 {\displaystyle
   \int\limits_{0}^{q^2}d(q^2)
   \frac {\displaystyle d\sigma}{\displaystyle d(q^2)}
  }
 {\displaystyle
   \int\limits_{0}^{q_{max}^2}d(q^2)
   \frac {\displaystyle d\sigma}{\displaystyle d(q^2)} .
  }
\label{helast.eq8}
\end{equation}

Expressions \ref{helast.eq6} and \ref{helast.eq7} allow analytic integration
in Eq. \ref{helast.eq8} but the result is too long to be given here.

For light and medium nuclei the analytic expression is more convenient for 
calculations than the numerical integration of Eq. \ref{helast.eq8}, but for
heavy nuclei the latter is preferred due to the large number of terms in the 
analytic expression.

\section{Status of this document}
18.06.04 created by Nikolai Starkov \\
19.06.04 re-written for spelling and grammar by D.H. Wright \\


\begin{latexonly}

\begin{thebibliography}{599}

\bibitem{helast.1} R.J. Glauber, 
    in "High Energy Physics and Nuclear Structure",
    edited by S. Devons (Plenum Press, NY 1970).


\bibitem{helast.2} R. H. Bassel, W. Wilkin, Phys. Rev., 174, p. 1179, 1968; \\ 
T. T. Chou, Phys. Rev., 168, 1594, 1968; \\
M. A. Nasser, M. M. Gazzaly, J. V. Geaga et al., Nucl. Phys.,
 A312, pp. 209-216, 1978. 


\bibitem{helast.3} Bujak, P. Devensky, A. Kuznetsov et al.,
Phys. Rev., D23, N 9, pp. 1895-1910, 1981.


\bibitem{helast.4} V. L. Korotkikh, N. I. Starkov, Sov. Journ. of Nucl. Phys., 
v. 37, N 4, pp. 610-613, 1983; \\
 N. T. Ermekov, V. L. Korotkikh,
N. I. Starkov, Sov. Journ. of Nucl. Phys., 33, N 6, pp. 775-777,
1981.

 
\bibitem{helast.5} R.A. Nam, S. I. Nikol'skii, N. I. Starkov et al., 
Sov. Journ. of Nucl. Phys., v. 26, N 5,  pp. 550-555, 1977.


\bibitem{helast.6} G.D. Alkhazov et al., Phys. Rep., 1978, C42, N 2, pp. 89-144;
\bibitem{helast.7} J. V. Geaga, M. M. Gazzaly, G. J. Jgo et al., 
  Phys. Rev. Lett. 38, N 22, pp. 1265-1268; \\
 S. J. Wallace. Y. Alexander,  Phys. Rev. Lett. 38, N 22, pp. 1269-1272.

\end{thebibliography}

\end{latexonly}

\begin{htmlonly}

\section{Bibliography}

\begin{enumerate}
\item R.J. Glauber, 
    in "High Energy Physics and Nuclear Structure",
    edited by S. Devons (Plenum Press, NY 1970).


\item R. H. Bassel, W. Wilkin, Phys. Rev., 174, p. 1179, 1968; \\ 
T. T. Chou, Phys. Rev., 168, 1594, 1968; \\
M. A. Nasser, M. M. Gazzaly, J. V. Geaga et al., Nucl. Phys.,
 A312, pp. 209-216, 1978. 


\item Bujak, P. Devensky, A. Kuznetsov et al.,
Phys. Rev., D23, N 9, pp. 1895-1910, 1981.


\item V. L. Korotkikh, N. I. Starkov, Sov. Journ. of Nucl. Phys., 
v. 37, N 4, pp. 610-613, 1983; \\
 N. T. Ermekov, V. L. Korotkikh,
N. I. Starkov, Sov. Journ. of Nucl. Phys., 33, N 6, pp. 775-777,
1981.

 
\item R.A. Nam, S. I. Nikol'skii, N. I. Starkov et al., 
Sov. Journ. of Nucl. Phys., v. 26, N 5,  pp. 550-555, 1977.


\item G.D. Alkhazov et al., Phys. Rep., 1978, C42, N 2, pp. 89-144;
\bibitem{helast.7} J. V. Geaga, M. M. Gazzaly, G. J. Jgo et al., 
  Phys. Rev. Lett. 38, N 22, pp. 1265-1268; \\
 S. J. Wallace. Y. Alexander,  Phys. Rev. Lett. 38, N 22, pp. 1269-1272.

\end{enumerate}

\end{htmlonly}


\begin{figure}
  \includegraphics[scale=0.7]{hadronic/he_elastic/p_Be9_1.eps}
  \caption{Elastic proton scattering on $^9$Be at 1 GeV}
  \label{helast.fig1}
\end{figure}
\begin{figure}
  \includegraphics[scale=0.7]{hadronic/he_elastic/p_B11_1.eps}
  \caption{Elastic proton scattering on $^{11}$B at 1 GeV}
  \label{helast.fig2}
\end{figure}
\begin{figure}
  \includegraphics[scale=0.7]{hadronic/he_elastic/p_C12_1.eps}
  \caption{Elastic proton scattering on $^{12}$C at 1 GeV}
  \label{helast.fig3}
\end{figure}
\begin{figure}
  \includegraphics[scale=0.7]{hadronic/he_elastic/p_016_1.eps}
  \caption{Elastic proton scattering on $^{16}$O at 1 GeV}
  \label{helast.fig4}
\end{figure}
\begin{figure}
  \includegraphics[scale=0.7]{hadronic/he_elastic/p_Si28_1.eps}
  \caption{Elastic proton scattering on $^{28}$Si at 1 GeV}
  \label{helast.fig5}
\end{figure}
\begin{figure}
  \includegraphics[scale=0.7]{hadronic/he_elastic/p_Ca40_1.eps}
  \caption{Elastic proton scattering on $^{40}$Ca at 1 GeV}
  \label{helast.fig6}
\end{figure}
\begin{figure}
  \includegraphics[scale=0.7]{hadronic/he_elastic/p_Ni58_1.eps}
  \caption{Elastic proton scattering on $^{58}$Ni at 1 GeV}
  \label{helast.fig7}
\end{figure}
\begin{figure}
  \includegraphics[scale=0.7]{hadronic/he_elastic/p_Zr90_1.eps}
  \caption{Elastic proton scattering on $^{90}$Zr at 1 GeV}
 \label{helast.fig8}
\end{figure}
\begin{figure}
  \includegraphics[scale=0.7]{hadronic/he_elastic/p_Pb208_1.eps}
  \caption{Elastic proton scattering on $^{208}$Pb at 1 GeV}
  \label{helast.fig9}
\end{figure}

\begin{figure}
  \includegraphics[scale=0.7]{hadronic/he_elastic/p_He4_1.eps}
  \caption{Elastic proton scattering on $^4$He at 1 GeV}
  \label{helast.fig10}
\end{figure}
\begin{figure}
  \includegraphics[scale=0.7]{hadronic/he_elastic/p_He4_45.eps}
  \caption{Elastic proton scattering on $^4$He at 45 GeV}
  \label{helast.fig11}
\end{figure}
\begin{figure}
  \includegraphics[scale=0.7]{hadronic/he_elastic/p_He4_301.eps}
  \caption{Elastic proton scattering on $^4$He at 301 GeV}
  \label{helast.fig12}
\end{figure}
