[1208] | 1 | \subsection{Glauber model at high energies.} |
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| 2 | |
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| 3 | \hspace{1.0em}We can use Glauber approach \cite{Glauber55} to calculate |
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| 4 | the total, elastic and differential elastic hadron-nucleus and nucleus- |
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| 5 | nucleus cross sections at high (more than hundreds of MeV) energies. |
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| 6 | |
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| 7 | \subsubsection{The hadron--nucleus and nucleus--nucleus total and |
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| 8 | elastic cross sections.} |
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| 9 | \hspace{1.0em} |
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| 10 | The knowledge of the nuclear elastic scattering amplitude $F(\vec{q},s)$, |
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| 11 | where $s$ is the total hadron-nucleon or nucleon-nucleon c.m. energy squared |
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| 12 | and $\vec{q}$ is the momentum transfer vector, |
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| 13 | gives us a possibility to calculate the total cross section (the optical |
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| 14 | theorem) |
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| 15 | \begin{equation} |
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| 16 | \label{GM1} \sigma_{tot}(s) = \frac{4\pi}{k} Im F(0,s), |
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| 17 | \end{equation} |
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| 18 | where $k$ is a hadron or nucleon projectile momentum in the target nucleus |
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| 19 | rest frame. |
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| 20 | Using this amplitude we are also able to calculate the differential elastic |
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| 21 | cross section |
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| 22 | \begin{equation} |
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| 23 | \label{GM2} \frac{d\sigma_{el}(s)}{d\Omega} = |F(\vec{q},s)|^2 |
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| 24 | \end{equation} |
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| 25 | or |
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| 26 | \begin{equation} |
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| 27 | \label{GM3} \frac{d\sigma_{el}(s)}{dt} =\frac{\pi}{k^2} |F(\vec{q},s)|^2 |
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| 28 | \end{equation} |
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| 29 | and total elastic cross section |
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| 30 | \begin{equation} |
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| 31 | \label{GM4} \sigma_{el}(s)= \int d\Omega|F(\vec{q},s)|^2 = |
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| 32 | \frac{1}{k^2}\int dq|F(\vec{q},s)|^2. |
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| 33 | \end{equation} |
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| 34 | |
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| 35 | The elastic scattering amplitude can be expressed through the profile |
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| 36 | function |
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| 37 | \begin{equation} |
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| 38 | \label{GM5}\Gamma(\vec{B},s)= 1-S(\vec{B},s) |
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| 39 | \end{equation} |
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| 40 | as the following |
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| 41 | \begin{equation} |
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| 42 | \label{GM6} F(\vec{q},s)=\frac{ik}{2\pi}\int |
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| 43 | d^2\vec{B}\exp{[i\vec{q}\Gamma(\vec{B},s)]}, |
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| 44 | \end{equation} |
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| 45 | where $S(\vec{B},s)$ is the $S$-matrix and $\vec{B}$ |
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| 46 | is the impact parameter vector perpendicular to the |
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| 47 | incident momentum $\vec{k}$. |
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| 48 | |
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| 49 | The total and elastic cross sections can be obtained from |
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| 50 | the profile function $\Gamma(\vec{B},s)$: |
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| 51 | \begin{equation} |
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| 52 | \label{GM7} \sigma_{tot}(s)= 2\int d^2\vec{B}Re\Gamma(\vec{B},s) |
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| 53 | \end{equation} |
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| 54 | and |
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| 55 | \begin{equation} |
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| 56 | \label{GM8} \sigma_{el}(s)= \int d^2\vec{B}|\Gamma(\vec{B},s)|^2. |
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| 57 | \end{equation} |
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| 58 | |
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| 59 | Thus to calculate the total, elastic and differential cross sections we need |
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| 60 | to know $S$-matrix $S(\vec{B},s)$. |
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| 61 | |
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| 62 | \subsubsection{The hadron--nucleus and nucleus--nucleus |
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| 63 | $S$-matrix.} |
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| 64 | \hspace{1.0em} |
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| 65 | |
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| 66 | Let us consider the nucleus-nucleus scattering at |
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| 67 | given impact parameter $\vec{B}$ and at given squared total |
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| 68 | c.m. nucleon--nucleon energy $s$. |
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| 69 | |
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| 70 | In Glauber approach \cite{Glauber55} an |
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| 71 | elastic nucleus--nucleus interaction is a result of the |
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| 72 | interactions between nucleons from the projectile and target nuclei. |
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| 73 | Thus, the $S$-scattering matrix $S^{AB}(\vec B,s)$ for |
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| 74 | nucleus $A$ on nucleus $B$ |
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| 75 | collision in the impact parameter representation can |
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| 76 | be expressed as follows: |
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| 77 | \begin{equation} |
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| 78 | \label{GM9}S^{AB}( \vec |
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| 79 | B,s)=<\prod\limits_{i=1}^A\prod\limits_{j=1}^B |
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| 80 | [1 - \Gamma_{ij}(\vec{B}+ \vec b_i^A-\vec b_j^B,s)]> |
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| 81 | \end{equation} |
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| 82 | where $<...>$ means integration over the sets $\{\vec b_i^A\}$ and |
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| 83 | $\{\vec b_j^B\}$ with weight functions $T _A$$(\{\vec b^A\})$ and |
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| 84 | $T_B$$(\{\vec b^B\})$. These functions |
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| 85 | \begin{equation} |
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| 86 | \label{GM10} T_{A,B}(\vec b_i^{A,B})=\int |
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| 87 | \rho ((\vec b_i^{A,B}z_i)dz_i |
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| 88 | \end{equation} |
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| 89 | can be obtained from the nucleon |
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| 90 | densities $\rho ((\vec b_i^{A,B},z_i)$. The nucleon profile function is |
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| 91 | \begin{equation} |
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| 92 | \label{GM11} \Gamma_{ij}(\vec{B}+ \vec b_i^A-\vec b_j^B,s) = |
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| 93 | \frac{\sigma_{ij}(s)}{4\pi \beta_{ij}(s)} \exp{[-\frac{(\vec{B} + |
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| 94 | \vec b_i^A-\vec b_j^B)^2}{2\beta_{ij}(s)}]}. |
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| 95 | \end{equation} |
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| 96 | The last equation can be obtained in the case of nucleon-nucleon |
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| 97 | amplitude parametrization: |
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| 98 | \begin{equation} |
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| 99 | \label{GM12} f_{ij}(q,s) = \frac{ik \sigma_{ij}(s)}{4\pi} |
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| 100 | \exp{[-\frac{1}{2}\beta_{ij}(s) q^2]}. |
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| 101 | \end{equation} |
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| 102 | The equation $(\ref{GM9})$ is a result of the assumptions that |
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| 103 | the $AB$-scattering phase is sum of |
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| 104 | the nucleon--nucleon scattering phases and no correlations between nucleons |
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| 105 | inside a nucleus are taken into account. |
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| 106 | |
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| 107 | The hadron-nucleus $S$-matrix is calculated in similar way using |
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| 108 | Eq. $(\ref{GM9})$ for $i = 1$ and $\vec b_i = 0$. In this case |
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| 109 | we need to use the corresponding parameter $\sigma_{hN}(s)$ |
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| 110 | and $\beta_{hN}(s)$ |
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| 111 | in nucleon profile function. |
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| 112 | |
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| 113 | As we will show below |
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| 114 | the hadron-nucleon and nucleon--nucleon elastic scattering amplitudes at high |
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| 115 | energies |
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| 116 | can be expressed |
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| 117 | through the reggeon-nucleon vertex parameters and the parameters of the |
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| 118 | reggeon trajectory\cite{BT76}. |
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| 119 | |
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