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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