1 | \chapter{Optical Elastic Model} |
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2 | |
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3 | \section{ The model description } |
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4 | |
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5 | This model is based on the optical approach in which a nucleus is considered to be a drop of |
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6 | an absorptive and refractive medium. Absorption results in the diffraction of a projectile |
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7 | hadron, while refraction (scattering) in the surface layer provides some smoothing of the |
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8 | diffraction picture. The differential cross section in the center mass system can be written |
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9 | \cite{bethe40,ap45,bs98}: |
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10 | \[ |
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11 | \frac{d\sigma_{el}}{d\Omega}= R^2F_d^2(k\,d\,\theta) |
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12 | \left\{\frac{J_1^2(k\,R\,\theta)}{\theta^2}+ |
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13 | \left[(\gamma\,k)^2+ |
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14 | (\delta\,k^2\,\theta)^2\right]J_0^2(k\,R\,\theta)\right\}, |
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15 | \quad d\Omega=2\pi\sin\theta\,d\theta, |
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16 | \] |
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17 | where $J_k$ is the Bessel function of k-th order ($k = 0,1$) and $\Omega$ and $\theta$ are |
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18 | the solid and polar scattering angles, respectively. |
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19 | $R$ and $d$ are nuclear geometrical parameters of the Woods-Saxon density form |
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20 | \[ |
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21 | \rho(r)=\rho_o\left\{1+\displaystyle\exp\left[\frac{(r-R)}{d}\right]\right\}^{-1}, |
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22 | \] |
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23 | in which $k$ is the projectile wave vector ($k=p/\hbar\,c$, $p$ is the projectile |
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24 | momentum multiplied by $c$), and $F_d$ is the dumping factor: |
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25 | \[ |
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26 | F_d(k\,d\,\theta)=\frac{\pi\,k\,d\,\theta}{\sinh(\pi\,k\,d\,\theta)}. |
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27 | \] |
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28 | $\gamma$ is the refraction (nuclear scattering) parameter, and $\delta$ is the |
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29 | spin-orbital interaction parameter. |
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30 | |
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31 | The model can be modified for charged particles by taking into account that a simple |
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32 | summation of nuclear and Coulomb cross sections does not work. However for small Coulomb |
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33 | scattering |
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34 | $\delta f_{Coulomb}(\theta)\ll f_{nuclear}(\theta)$ the amplitudes can be summarized: |
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35 | \[ |
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36 | \frac{d\sigma_{el}}{d\Omega} = | \delta f_{Coulomb}(\theta)+f_{nuclear}(\theta)|^2. |
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37 | \] |
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38 | This case is assumed in the diffuse optical model considered, where we should correct the |
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39 | nuclear refraction (scattering) term: |
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40 | \[ |
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41 | \gamma\,k \rightarrow \gamma\,k + \frac{n}{2kR}\left[\sin^2(\frac{\theta}{2}) + A_m \right]^{-1}, |
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42 | \quad n = \frac{\alpha Z_1 Z_2}{\beta} \ll kR, |
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43 | \] |
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44 | where $n$ is Zommerfeld parameter for the Coulomb field, $\beta=v/c$ is the particle velocity, |
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45 | and $Z_1$ and $Z_2$ are the hadron and nucleus charges, respectively. |
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46 | The parameter $A_m$ reflects Coulomb atomic shell screening and can be estimated as: |
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47 | \[ |
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48 | A_m = \frac{1.13 + 3.76n^2}{(1.77ka_oZ^{-1/3})^2}, |
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49 | \] |
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50 | where $a_o$ is the Bohr radius, and $Z$ is the atomic number. |
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51 | |
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52 | This method corresponds to elastic Coulomb-Wentzel scattering recently implemented |
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53 | in electromagnetic physics \cite{fer93}: |
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54 | \[ |
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55 | \frac{d\sigma_{el}^{cw}}{d\Omega}= \frac{n^2}{4k^2} |
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56 | \left[\sin^2(\frac{\theta}{2}) + A_m \right]^{-2}, \quad V(r)=\frac{e^2Z_1Z_2}{r^2}\exp(-r/R), |
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57 | \quad A_m = (2kR)^{-2}. |
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58 | \] |
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59 | The Coulomb-Wentzel cross-section reads: |
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60 | \[ |
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61 | \sigma_{el}^{cw}= \frac{n^2}{k^2}\frac{\pi}{A_m(1+A_m)}, \quad |
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62 | \sigma_{el}^{cw}(\theta_1,\theta_2)=2\pi\frac{n^2}{k^2}\frac{\cos\theta_1-\cos\theta_2} |
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63 | {(1-\cos\theta_1+2A_m)(1-\cos\theta_2+2A_m)}. |
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64 | \] |
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65 | This cross-section is much bigger than the integral nuclear cross section. However the |
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66 | main contribution comes from very small angles where the differential nuclear elastic |
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67 | cross section is approximately constant. In this region the Coulomb correction can be |
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68 | switched off. This avoids double counting of Coulomb scattering in electromagnetic and nuclear |
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69 | processes. |
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70 | |
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71 | \section{Validation} |
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72 | |
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73 | |
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74 | Figs. \ref{pPb1GeV}-\ref{pipC9p92GeVc} show the elastic differential cross sections of protons |
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75 | and pions with energies in the range 1 - 301 GeV on different targets: helium, carbon, silicon |
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76 | and lead. |
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77 | One can see that Coulomb corrections improve the description of differential cross sections |
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78 | in the regions of the minima. |
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79 | |
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80 | \begin{figure*} |
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81 | \centering \includegraphics[width=10cm,height=7cm]{hadronic/elastic/pPbsumT1GeV.eps} |
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82 | \vspace{-0.5cm} |
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83 | \caption{ Differential elastic scattering cross sections of protons with energy |
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84 | 1 GeV on lead. Curves show pure nuclear and Coulomb-corrected models. Points are |
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85 | experimental data. The Coulomb cross section dominates at small angles resulting |
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86 | in a large integral cross section: $\sigma_{el}^{nucl}=1210$ mb, with $\lambda = 25$ cm, |
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87 | $\sigma_{el}^{cw}(2^{0}, 16^{o})=642.65$ mb, with $\lambda = 47$ cm, |
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88 | $\sigma_{el}^{cw}(0,1^o)\sim \sigma_{el}^{cw} = 2.36423\cdot10^{9}$ mb, |
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89 | with $\lambda = 0.13$ micron.} |
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90 | \label{pPb1GeV} |
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91 | \end{figure*} |
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92 | |
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93 | \begin{figure*} |
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94 | \centering \includegraphics[width=10cm,height=7cm]{hadronic/elastic/pSisumT1GeV.eps} |
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95 | \vspace{-0.5cm} |
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96 | \caption{ Differential elastic scattering cross sections of protons with energy |
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97 | 1 GeV on silicon. Curves show pure nuclear and Coulomb-corrected models. Points are |
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98 | experimental data. The Coulomb cross section dominates at small angles resulting in |
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99 | a large integral cross-section: |
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100 | $\sigma_{el}^{cw}(0,1^o)\sim \sigma_{el}^{cw}=5.46779\cdot10^{8}$ mb, |
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101 | with $\lambda = 0.366068$ micron.} |
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102 | \label{pSi1GeV} |
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103 | \end{figure*} |
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104 | |
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105 | \begin{figure*} |
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106 | \centering \includegraphics[width=10cm,height=7cm]{hadronic/elastic/pPbsum9p92GeVc.eps} |
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107 | \vspace{-0.5cm} |
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108 | \caption{ Differential elastic scattering cross sections of protons with momentum |
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109 | 9.92 GeV/c on lead. Curves show pure nuclear and Coulomb-corrected models. Points are |
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110 | experimental data. The Coulomb cross section dominates at small angles resulting in a |
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111 | large integral cross section |
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112 | $\sigma_{el}^{cw}(0,4 \ mrad)\sim \sigma_{el}^{cw}=2.11854\cdot10^{9}$ mb, |
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113 | with $\lambda = 0.143101$ micron. |
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114 | $\sigma_{el}^{cw}(4 \ mrad,40 \ mrad)=919.547$ mb, |
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115 | with $\lambda = 329689$ micron (33 cm).} |
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116 | \label{pPb9p92GeVc} |
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117 | \end{figure*} |
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118 | |
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119 | \begin{figure*} |
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120 | \centering \includegraphics[width=10cm,height=7cm]{hadronic/elastic/pHeT45GeVsum.eps} |
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121 | \vspace{-0.5cm} |
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122 | \caption{ Differential elastic scattering cross sections of protons with energy |
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123 | 45 GeV on helium. Curves show pure nuclear and Coulomb-corrected models. Points are |
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124 | experimental data.} |
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125 | \label{pHe45GeV} |
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126 | \end{figure*} |
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127 | |
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128 | \begin{figure*} |
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129 | \centering \includegraphics[width=10cm,height=7cm]{hadronic/elastic/pHeT301GeVsum.eps} |
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130 | \vspace{-0.5cm} |
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131 | \caption{ Differential elastic scattering cross sections of protons with energy |
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132 | 301 GeV on helium. Curves show pure nuclear and Coulomb-corrected models. Points are |
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133 | experimental data.} |
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134 | \label{pHe301GeV} |
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135 | \end{figure*} |
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136 | |
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137 | \begin{figure*} |
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138 | \centering \includegraphics[width=10cm,height=7cm]{hadronic/elastic/pimPbsum9p92GeVc.eps} |
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139 | \vspace{-0.5cm} |
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140 | \caption{ Differential elastic scattering cross sections of $\pi^{-}$ with momentum |
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141 | 9.92 GeV/c on lead. Curves show pure nuclear and Coulomb-corrected models. Points are |
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142 | experimental data. } |
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143 | \label{pimPb9p92GeVc} |
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144 | \end{figure*} |
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145 | |
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146 | \begin{figure*} |
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147 | \centering \includegraphics[width=10cm,height=7cm]{hadronic/elastic/pipPbsum9p92GeVc.eps} |
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148 | \vspace{-0.5cm} |
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149 | \caption{ Differential elastic scattering cross sections of $\pi^{+}$ with momentum |
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150 | 9.92 GeV/c on lead. Curves show pure nuclear and Coulomb-corrected models. Points are |
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151 | experimental data.} |
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152 | \label{pipPb9p92GeVc} |
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153 | \end{figure*} |
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154 | |
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155 | \begin{figure*} |
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156 | \centering \includegraphics[width=10cm,height=7cm]{hadronic/elastic/pipCsum9p92GeVc.eps} |
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157 | \vspace{-0.5cm} |
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158 | \caption{ Differential elastic scattering cross sections of $\pi^{+}$ with momentum |
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159 | 9.92 GeV/c on carbon. Curves show pure nuclear and Coulomb-corrected models. Points are |
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160 | experimental data.} |
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161 | \label{pipC9p92GeVc} |
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162 | \end{figure*} |
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163 | |
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164 | \section{Implementation} |
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165 | |
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166 | This model is implemented in the class {\em G4DiffuseElastic}. It conforms to the general |
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167 | interface of hadronic models and performs initialization for different elements during |
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168 | tracking. |
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169 | |
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170 | |
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171 | \section{Status of this document} |
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172 | 18.11.07 created by V. Grichine \\ |
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173 | |
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174 | |
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175 | |
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176 | \begin{latexonly} |
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177 | |
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178 | \begin{thebibliography}{99} |
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179 | |
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180 | \bibitem{bethe40} P. Placzec, H.A. Bethe, {\em Phys. Rev.}, {\bf 57} (1940) 1075A. |
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181 | |
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182 | \bibitem{ap45} A. Akhiezer, I. Pomeranchuk, {\em J. Phys. (USSR)}, {\bf 9} (1945) 471; |
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183 | {\em Sov. Phys. USPEKHI}, {\bf 65} (1958) 593. |
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184 | |
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185 | \bibitem{bs98} Yu.A. Berezhnoy, V.A. Slipko, {\em Int. J. of Mod. Phys.}, {\bf E7} (1998) 723. |
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186 | |
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187 | \bibitem{fer93} J.M. Fernandez-Varea, R. Moyol, J. Baro, F. Salvat, |
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188 | {\em Nucl. Instr. and Math. in Phys. Res.}, {\bf B73} (1993) 447. |
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189 | |
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190 | |
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191 | \end{thebibliography} |
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192 | |
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193 | \end{latexonly} |
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194 | |
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195 | \begin{htmlonly} |
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196 | |
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197 | \section{Bibliography} |
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198 | |
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199 | \begin{enumerate} |
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200 | |
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201 | \item P. Placzec, H.A. Bethe, {\em Phys. Rev.}, {\bf 57} (1940) 1075A. |
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202 | |
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203 | |
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204 | \item A. Akhiezer, I. Pomeranchuk, {\em J. Phys. (USSR)}, {\bf 9} (1945) 471; |
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205 | {\em Sov. Phys. USPEKHI}, {\bf 65} (1958) 593. |
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206 | |
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207 | \item Yu.A. Berezhnoy, V.A. Slipko, {\em Int. J. of Mod. Phys.}, {\bf E7} (1998) 723. |
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208 | |
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209 | \item J.M. Fernandez-Varea, R. Moyol, J. Baro, F. Salvat, |
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210 | {\em Nucl. Instr. and Math. in Phys. Res.}, {\bf B73} (1993) 447. |
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211 | |
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212 | |
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213 | \end{enumerate} |
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214 | |
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215 | \end{htmlonly} |
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216 | |
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