[1208] | 1 | \chapter{ Hadron-nucleus Elastic Scattering at Medium and High Energy.} |
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| 2 | |
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| 3 | \vspace{2ex} |
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| 4 | \section{Method of Calculation} |
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| 5 | |
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| 6 | The Glauber model \cite{helast.1} is used as an alternative method of |
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| 7 | calculating differential cross sections for elastic and quasi-elastic |
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| 8 | hadron-nucleus scattering at high and intermediate energies. |
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| 9 | |
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| 10 | For high energies this includes corrections for inelastic screening and |
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| 11 | for quasi-elastic scattering the exitation of a discrete level or a state in |
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| 12 | the continuum is considered. |
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| 13 | |
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| 14 | The usual expression for the Glauber model amplitude for multiple scattering |
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| 15 | was used |
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| 16 | |
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| 17 | \begin{equation} |
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| 18 | F(q)=\frac{ik}{2\pi} \int d^2{b}e^{\vec{\mathstrut q} |
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| 19 | \cdot \vec{\mathstrut b}} M(\vec b) . |
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| 20 | \label{helast.eq1} |
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| 21 | \end{equation} |
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| 22 | Here $M(\vec b)$ is the hadron-nucleus amplitude in the impact |
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| 23 | parameter representation |
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| 24 | \begin{equation} |
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| 25 | M(\vec b) = 1-[1-e^{-A\int d^{3}r\Gamma(\vec{\mathstrut b}- |
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| 26 | \vec{\mathstrut s})\rho(\vec {r})}]^A, |
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| 27 | \label{helast.eq2} |
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| 28 | \end{equation} |
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| 29 | $k$ is the incident particle momentum, |
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| 30 | $\vec q = \vec k' - \vec k$ is the momentum transfer, and |
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| 31 | $\vec k'$ is the scattered particle momentum. |
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| 32 | Note that $\left| \vec q \right| ^2 = -t$ - invariant momentum transfer |
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| 33 | squared in the center of mass system. |
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| 34 | $\Gamma(\vec {b})$ is the hadron-nucleon amplitude of |
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| 35 | elastic scattering in the impact-parameter representation |
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| 36 | |
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| 37 | \begin{equation} |
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| 38 | \Gamma(\vec b)=\frac{\displaystyle 1} |
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| 39 | {\displaystyle 2\pi ik^{hN}} |
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| 40 | \int d\vec {q} e^{-\vec {\mathstrut q} \cdot \vec{\mathstrut b}} |
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| 41 | f(\vec {q}). |
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| 42 | \label{helast.eq3} |
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| 43 | \end{equation} |
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| 44 | |
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| 45 | The exponential parameterization of the hadron-nucleon amplitude is usually |
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| 46 | used: |
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| 47 | |
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| 48 | \begin{equation} |
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| 49 | f(\vec {q})=\frac{\displaystyle ik^{hN}\sigma^{hN}}{2\pi} |
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| 50 | e^{-0.5q^2B}. |
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| 51 | \label{helast.eq4} |
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| 52 | \end{equation} |
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| 53 | Here $\sigma^{hN}=\sigma_{tot}^{hN}(1-i\alpha)$, |
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| 54 | $\sigma_{tot}^{hN}$ is the total |
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| 55 | cross section of a hadron-nucleon scattering, $B$ is the slope of the |
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| 56 | diffraction cone and $\alpha$ is the ratio of the real to imaginary parts of |
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| 57 | the amplitude at $q=0$. The value $k^{hN}$ is the hadron momentum in the |
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| 58 | hadron-nucleon coordinate system. |
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| 59 | |
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| 60 | The important difference of these calculations from the usual ones |
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| 61 | is that the two-gaussian form of the nuclear density was used |
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| 62 | |
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| 63 | \begin{equation} |
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| 64 | \rho(r) = C (e^{-(r/R_{1})^2}-pe^{-(r/R_2)^2}), |
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| 65 | \label{helast.eq5} |
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| 66 | \end{equation} |
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| 67 | where $R_1$, $R_2$ and $p$ are the fitting parameters and $C$ is a |
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| 68 | normalization constant. |
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| 69 | |
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| 70 | This density representation allows the expressions for amplitude and |
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| 71 | differential cross section to be put into analytical form. It was earlier |
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| 72 | used for light \cite{helast.2, helast.3} and medium \cite{helast.4} nuclei. |
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| 73 | Described below is an extension of this method to heavy nuclei. |
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| 74 | The form \ref{helast.eq5} is not physical for a heavy nucleus, but |
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| 75 | nevertheless works rather well (see figures below). The reason is that |
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| 76 | the nucleus absorbs the hadrons very strongly, especially at small impact |
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| 77 | parameters where the absorption is full. As a result only the peripherial part |
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| 78 | of the nucleus participates in elastic scattering. Eq. \ref{helast.eq5} |
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| 79 | therefore describes only the edge of a heavy nucleus. |
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| 80 | |
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| 81 | Substituting Eqs. \ref{helast.eq5} and \ref{helast.eq4} into Eqs. |
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| 82 | \ref{helast.eq1}, \ref{helast.eq2} and \ref{helast.eq3} yields the following |
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| 83 | formula |
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| 84 | |
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| 85 | \begin{eqnarray} |
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| 86 | F(q) = \frac{ik\pi}{2} |
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| 87 | \sum\limits_{k=1}^A (-1)^k {A\choose k} |
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| 88 | [\frac{\sigma^{hN}}{2\pi(R_1^3-pR_2^3)}]^k |
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| 89 | \sum\limits_{m=0}^{k} (-1)^m {k \choose m} |
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| 90 | \left[ \frac{R_1^3}{R_1^2+2B} |
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| 91 | \right]^{k-m} \nonumber |
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| 92 | \end{eqnarray} |
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| 93 | |
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| 94 | \begin{eqnarray} |
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| 95 | \times \qquad {} \left[ |
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| 96 | \frac{pR_2^3}{R_2^2+2B} |
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| 97 | \right]^{m} \left( |
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| 98 | \frac{m}{R_2^2+2B}+\frac{k-m}{R_1^2+2B} \right)^{-1} \nonumber |
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| 99 | \end{eqnarray} |
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| 100 | |
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| 101 | \begin{eqnarray} |
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| 102 | \times \qquad {} \exp\left[ -\frac{-q^2}{4} |
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| 103 | \left( |
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| 104 | \frac{m}{R_2^2+2B}+ |
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| 105 | \frac{k-m}{R_1^2+2B} \right)^{-1} |
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| 106 | \right]. |
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| 107 | \label{helast.eq6} |
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| 108 | \end{eqnarray} |
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| 109 | % \end{eqnarray} |
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| 110 | |
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| 111 | An analogous procedure can be used to get the inelastic screening corrections |
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| 112 | to the hadron-nucleus amplitude $\Delta M(\vec b)$ \cite{helast.5}. |
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| 113 | In this case an intermediate inelastic diffractive state is created which |
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| 114 | rescatters on the nucleons of the nucleus and then returns into the initial |
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| 115 | hadron. Hence it is nessesary to integrate the production cross section over |
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| 116 | the mass distribution of the exited system $\frac{d\sigma^{diff}}{dtdM_x^2}$. |
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| 117 | The expressions for the corresponding amplitude are quite long and so are not |
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| 118 | presented here. The corrections for the total cross-sections can be found in |
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| 119 | \cite{helast.5}. |
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| 120 | |
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| 121 | The full amplitude is the sum $M(\vec b)+ \Delta M(\vec b)$. |
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| 122 | |
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| 123 | The differential cross section is connected with the amplitude in the |
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| 124 | following way |
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| 125 | |
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| 126 | \begin{equation} |
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| 127 | \frac{d\sigma}{d\Omega_{CM}}=\left| F(q) \right|^2, \qquad |
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| 128 | \frac{d\sigma}{|dt|}=\frac{d\sigma}{dq_{CM}^2}= |
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| 129 | \frac{\pi}{k_{CM}^2} \left|F(q) \right|^2. |
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| 130 | \label{helast.eq7} |
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| 131 | \end{equation} |
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| 132 | |
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| 133 | The main energy dependence of the hadron-nucleus elastic scattering |
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| 134 | cross section comes from the energy dependence of the parameters of |
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| 135 | hadron-nucleon scattering |
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| 136 | ($\sigma_{tot}^{hN} \alpha$, $B$ and |
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| 137 | $\frac{d\sigma^{diff}}{dtdM_x^2}$). |
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| 138 | At interesting energies these parameters were fixed at their well-known |
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| 139 | values. The fitting of the nuclear density parameters was performed |
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| 140 | over a wide range of atomic numbers $(A=4 - 208)$ using experimental data |
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| 141 | on proton-nuclei elastic scattering at a kinetic energy of $T_p=1 GeV$. |
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| 142 | |
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| 143 | The fitting was perfomed both for individual nuclei and for the entire set |
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| 144 | of nuclei at once. |
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| 145 | %%In the last event the following dependensies for nuclei |
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| 146 | %%parameters $(for A=12 \div 208)$ were obtained |
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| 147 | |
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| 148 | %\vspace{2ex} |
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| 149 | %$R_1=4.18A^{0.302}$, $R_2=3.81(A-10)^{0.268}$, $p=0.95$. |
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| 150 | %\hspace{3cm} (8) |
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| 151 | %\vspace{2ex} |
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| 152 | |
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| 153 | %%\vspace{2ex} |
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| 154 | %%$R_1=4.45(A-1)^{0.309}$, $R_2=2.3A^{0.36}$, |
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| 155 | |
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| 156 | %%$p=0.176+0.0017A+8.7.10^{-6}A^2$. |
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| 157 | %%\hspace{3cm} (8) |
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| 158 | %%\vspace{2ex} |
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| 159 | |
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| 160 | It is necessary to note that for every nucleus |
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| 161 | an optimal set of density parameters exists and it |
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| 162 | differs slightly from the one derived for the full set of nuclei. |
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| 163 | |
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| 164 | A comparision of the phenomenological cross sections \cite{helast.6} with |
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| 165 | experiment is presented in Figs. \ref{helast.fig1} - \ref{helast.fig9} |
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| 166 | |
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| 167 | % where the data of the proton-nucleus scattering |
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| 168 | % are imaged. |
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| 169 | |
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| 170 | In this comparison, the individual nuclei parameters were used. |
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| 171 | The experimental data were obtained in Gatchina (Russia) and in |
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| 172 | Saclay (France) \cite{helast.6}. |
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| 173 | The horizontal axis is the scattering angle in the center of mass |
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| 174 | system $\Theta_{CM}$ and the vertical axis is |
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| 175 | $\frac{\displaystyle d\sigma}{\displaystyle d\Omega_{CM}}$ |
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| 176 | in $\frac{\displaystyle mb}{\displaystyle Ster}$. |
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| 177 | %The lower curve in the figures is the diferencial cross-section |
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| 178 | %of a coherent elastic scattering but upper one is the |
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| 179 | %cross-section for noncoherent scattering. |
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| 180 | |
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| 181 | Comparisions were also made for $p ^4He$ elastic scatering at |
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| 182 | $T_ = 1 GeV$[7], $45 GeV$ and $301 GeV$ [3]. The resulting |
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| 183 | cross sections $\frac {d \sigma}{d\left| t \right|}$ are shown |
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| 184 | in the Figs. \ref{helast.fig10} - \ref{helast.fig12}. |
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| 185 | |
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| 186 | In order to generate events the distribution function ${\cal F}$ of a |
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| 187 | corresponding process must be known. |
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| 188 | The differential cross section is proportional to the density distribution. |
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| 189 | Therefore to get the distribution function it is sufficient to integrate the |
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| 190 | differential cross section and normalize it: |
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| 191 | |
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| 192 | \begin{equation} |
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| 193 | {\cal F}(q^2)= |
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| 194 | \frac |
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| 195 | {\displaystyle |
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| 196 | \int\limits_{0}^{q^2}d(q^2) |
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| 197 | \frac {\displaystyle d\sigma}{\displaystyle d(q^2)} |
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| 198 | } |
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| 199 | {\displaystyle |
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| 200 | \int\limits_{0}^{q_{max}^2}d(q^2) |
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| 201 | \frac {\displaystyle d\sigma}{\displaystyle d(q^2)} . |
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| 202 | } |
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| 203 | \label{helast.eq8} |
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| 204 | \end{equation} |
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| 205 | |
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| 206 | Expressions \ref{helast.eq6} and \ref{helast.eq7} allow analytic integration |
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| 207 | in Eq. \ref{helast.eq8} but the result is too long to be given here. |
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| 208 | |
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| 209 | For light and medium nuclei the analytic expression is more convenient for |
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| 210 | calculations than the numerical integration of Eq. \ref{helast.eq8}, but for |
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| 211 | heavy nuclei the latter is preferred due to the large number of terms in the |
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| 212 | analytic expression. |
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| 213 | |
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| 214 | \section{Status of this document} |
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| 215 | 18.06.04 created by Nikolai Starkov \\ |
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| 216 | 19.06.04 re-written for spelling and grammar by D.H. Wright \\ |
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| 217 | |
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| 218 | |
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| 219 | \begin{latexonly} |
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| 220 | |
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| 221 | \begin{thebibliography}{599} |
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| 222 | |
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| 223 | \bibitem{helast.1} R.J. Glauber, |
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| 224 | in "High Energy Physics and Nuclear Structure", |
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| 225 | edited by S. Devons (Plenum Press, NY 1970). |
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| 226 | |
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| 227 | |
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| 228 | \bibitem{helast.2} R. H. Bassel, W. Wilkin, Phys. Rev., 174, p. 1179, 1968; \\ |
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| 229 | T. T. Chou, Phys. Rev., 168, 1594, 1968; \\ |
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| 230 | M. A. Nasser, M. M. Gazzaly, J. V. Geaga et al., Nucl. Phys., |
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| 231 | A312, pp. 209-216, 1978. |
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| 232 | |
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| 233 | |
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| 234 | \bibitem{helast.3} Bujak, P. Devensky, A. Kuznetsov et al., |
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| 235 | Phys. Rev., D23, N 9, pp. 1895-1910, 1981. |
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| 236 | |
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| 237 | |
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| 238 | \bibitem{helast.4} V. L. Korotkikh, N. I. Starkov, Sov. Journ. of Nucl. Phys., |
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| 239 | v. 37, N 4, pp. 610-613, 1983; \\ |
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| 240 | N. T. Ermekov, V. L. Korotkikh, |
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| 241 | N. I. Starkov, Sov. Journ. of Nucl. Phys., 33, N 6, pp. 775-777, |
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| 242 | 1981. |
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| 243 | |
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| 244 | |
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| 245 | \bibitem{helast.5} R.A. Nam, S. I. Nikol'skii, N. I. Starkov et al., |
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| 246 | Sov. Journ. of Nucl. Phys., v. 26, N 5, pp. 550-555, 1977. |
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| 247 | |
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| 248 | |
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| 249 | \bibitem{helast.6} G.D. Alkhazov et al., Phys. Rep., 1978, C42, N 2, pp. 89-144; |
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| 250 | \bibitem{helast.7} J. V. Geaga, M. M. Gazzaly, G. J. Jgo et al., |
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| 251 | Phys. Rev. Lett. 38, N 22, pp. 1265-1268; \\ |
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| 252 | S. J. Wallace. Y. Alexander, Phys. Rev. Lett. 38, N 22, pp. 1269-1272. |
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| 253 | |
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| 254 | \end{thebibliography} |
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| 255 | |
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| 256 | \end{latexonly} |
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| 257 | |
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| 258 | \begin{htmlonly} |
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| 259 | |
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| 260 | \section{Bibliography} |
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| 261 | |
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| 262 | \begin{enumerate} |
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| 263 | \item R.J. Glauber, |
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| 264 | in "High Energy Physics and Nuclear Structure", |
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| 265 | edited by S. Devons (Plenum Press, NY 1970). |
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| 266 | |
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| 267 | |
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| 268 | \item R. H. Bassel, W. Wilkin, Phys. Rev., 174, p. 1179, 1968; \\ |
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| 269 | T. T. Chou, Phys. Rev., 168, 1594, 1968; \\ |
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| 270 | M. A. Nasser, M. M. Gazzaly, J. V. Geaga et al., Nucl. Phys., |
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| 271 | A312, pp. 209-216, 1978. |
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| 272 | |
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| 273 | |
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| 274 | \item Bujak, P. Devensky, A. Kuznetsov et al., |
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| 275 | Phys. Rev., D23, N 9, pp. 1895-1910, 1981. |
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| 276 | |
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| 277 | |
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| 278 | \item V. L. Korotkikh, N. I. Starkov, Sov. Journ. of Nucl. Phys., |
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| 279 | v. 37, N 4, pp. 610-613, 1983; \\ |
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| 280 | N. T. Ermekov, V. L. Korotkikh, |
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| 281 | N. I. Starkov, Sov. Journ. of Nucl. Phys., 33, N 6, pp. 775-777, |
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| 282 | 1981. |
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| 283 | |
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| 284 | |
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| 285 | \item R.A. Nam, S. I. Nikol'skii, N. I. Starkov et al., |
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| 286 | Sov. Journ. of Nucl. Phys., v. 26, N 5, pp. 550-555, 1977. |
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| 287 | |
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| 288 | |
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| 289 | \item G.D. Alkhazov et al., Phys. Rep., 1978, C42, N 2, pp. 89-144; |
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| 290 | \bibitem{helast.7} J. V. Geaga, M. M. Gazzaly, G. J. Jgo et al., |
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| 291 | Phys. Rev. Lett. 38, N 22, pp. 1265-1268; \\ |
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| 292 | S. J. Wallace. Y. Alexander, Phys. Rev. Lett. 38, N 22, pp. 1269-1272. |
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| 293 | |
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| 294 | \end{enumerate} |
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| 295 | |
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| 296 | \end{htmlonly} |
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| 297 | |
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| 298 | |
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| 299 | \begin{figure} |
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| 300 | \includegraphics[scale=0.7]{hadronic/he_elastic/p_Be9_1.eps} |
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| 301 | \caption{Elastic proton scattering on $^9$Be at 1 GeV} |
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| 302 | \label{helast.fig1} |
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| 303 | \end{figure} |
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| 304 | \begin{figure} |
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| 305 | \includegraphics[scale=0.7]{hadronic/he_elastic/p_B11_1.eps} |
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| 306 | \caption{Elastic proton scattering on $^{11}$B at 1 GeV} |
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| 307 | \label{helast.fig2} |
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| 308 | \end{figure} |
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| 309 | \begin{figure} |
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| 310 | \includegraphics[scale=0.7]{hadronic/he_elastic/p_C12_1.eps} |
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| 311 | \caption{Elastic proton scattering on $^{12}$C at 1 GeV} |
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| 312 | \label{helast.fig3} |
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| 313 | \end{figure} |
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| 314 | \begin{figure} |
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| 315 | \includegraphics[scale=0.7]{hadronic/he_elastic/p_016_1.eps} |
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| 316 | \caption{Elastic proton scattering on $^{16}$O at 1 GeV} |
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| 317 | \label{helast.fig4} |
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| 318 | \end{figure} |
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| 319 | \begin{figure} |
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| 320 | \includegraphics[scale=0.7]{hadronic/he_elastic/p_Si28_1.eps} |
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| 321 | \caption{Elastic proton scattering on $^{28}$Si at 1 GeV} |
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| 322 | \label{helast.fig5} |
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| 323 | \end{figure} |
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| 324 | \begin{figure} |
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| 325 | \includegraphics[scale=0.7]{hadronic/he_elastic/p_Ca40_1.eps} |
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| 326 | \caption{Elastic proton scattering on $^{40}$Ca at 1 GeV} |
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| 327 | \label{helast.fig6} |
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| 328 | \end{figure} |
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| 329 | \begin{figure} |
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| 330 | \includegraphics[scale=0.7]{hadronic/he_elastic/p_Ni58_1.eps} |
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| 331 | \caption{Elastic proton scattering on $^{58}$Ni at 1 GeV} |
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| 332 | \label{helast.fig7} |
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| 333 | \end{figure} |
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| 334 | \begin{figure} |
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| 335 | \includegraphics[scale=0.7]{hadronic/he_elastic/p_Zr90_1.eps} |
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| 336 | \caption{Elastic proton scattering on $^{90}$Zr at 1 GeV} |
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| 337 | \label{helast.fig8} |
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| 338 | \end{figure} |
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| 339 | \begin{figure} |
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| 340 | \includegraphics[scale=0.7]{hadronic/he_elastic/p_Pb208_1.eps} |
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| 341 | \caption{Elastic proton scattering on $^{208}$Pb at 1 GeV} |
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| 342 | \label{helast.fig9} |
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| 343 | \end{figure} |
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| 344 | |
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| 345 | \begin{figure} |
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| 346 | \includegraphics[scale=0.7]{hadronic/he_elastic/p_He4_1.eps} |
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| 347 | \caption{Elastic proton scattering on $^4$He at 1 GeV} |
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| 348 | \label{helast.fig10} |
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| 349 | \end{figure} |
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| 350 | \begin{figure} |
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| 351 | \includegraphics[scale=0.7]{hadronic/he_elastic/p_He4_45.eps} |
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| 352 | \caption{Elastic proton scattering on $^4$He at 45 GeV} |
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| 353 | \label{helast.fig11} |
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| 354 | \end{figure} |
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| 355 | \begin{figure} |
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| 356 | \includegraphics[scale=0.7]{hadronic/he_elastic/p_He4_301.eps} |
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| 357 | \caption{Elastic proton scattering on $^4$He at 301 GeV} |
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| 358 | \label{helast.fig12} |
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| 359 | \end{figure} |
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