1 | |
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2 | \chapter{Total Reaction Cross Section in Nucleus-nucleus Reactions} |
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3 | \noindent |
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4 | The transportation of heavy ions in matter is a subject of much interest in |
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5 | several fields of science. An important input for simulations of this process |
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6 | is the total reaction cross section, which is defined as the total |
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7 | ($\sigma_{T}$) minus the elastic ($\sigma_{el}$) cross section for |
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8 | nucleus-nucleus reactions: |
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9 | \begin{eqnarray*} |
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10 | \sigma_{R} = \sigma_{T} - \sigma_{el} . |
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11 | \end{eqnarray*} |
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12 | The total reaction cross section has been studied both theoretically and |
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13 | experimentally and several empirical parameterizations of it have been |
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14 | developed. In Geant4 the total reaction cross sections are calculated using |
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15 | four such parameterizations: the Sihver\cite{nnc.Sihver93}, |
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16 | Kox\cite{nnc.Kox87}, Shen\cite{nnc.Shen89} and Tripathi\cite{nnc.Tripathi97} |
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17 | formulae. Each of these is discussed in order below. |
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18 | |
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19 | \section{Sihver Formula} |
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20 | \noindent |
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21 | Of the four parameterizations, the Sihver formula has the simplest form: |
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22 | \begin{equation} |
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23 | \sigma_{R} = \pi r^{2}_{0}[A^{1/3}_{p} + A^{1/3}_{t} - b_{0} [A^{-1/3}_{p} + A^{-1/3}_{t}] ]^{2} |
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24 | \end{equation} |
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25 | where A$_{p}$ and A$_{t}$ are the mass numbers of the projectile and target |
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26 | nuclei, and |
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27 | \begin{eqnarray*} |
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28 | b_{0}=1.581-0.876(A^{-1/3}_{p} + A^{-1/3}_{t}) , |
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29 | \end{eqnarray*} |
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30 | \begin{eqnarray*} |
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31 | r_{0}=1.36fm. |
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32 | \end{eqnarray*} |
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33 | It consists of a nuclear geometrical term $(A^{1/3}_p + A^{1/3}_t)$ and an |
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34 | overlap or transparency parameter ($b_0$) for nucleons in the nucleus. The |
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35 | cross section is independent of energy and can be used for incident energies |
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36 | greater than 100 MeV/nucleon. |
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37 | |
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38 | \section{Kox and Shen Formulae} |
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39 | \noindent |
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40 | Both the Kox and Shen formulae are based on the strong absorption model. They |
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41 | express the total reaction cross section in terms of the interaction radius |
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42 | $R$, the nucleus-nucleus interaction barrier $B$, and the center-of-mass energy |
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43 | of the colliding system $E_{CM}$: |
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44 | \begin{equation} |
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45 | \sigma_{R} = \pi R^{2}[1-\frac{B}{E_{CM}}]. |
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46 | \end{equation} |
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47 | |
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48 | \noindent |
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49 | {\bf Kox formula:} Here $B$ is the Coulomb barrier ($B_c$) of the |
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50 | projectile-target system and is given by |
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51 | \begin{eqnarray*} |
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52 | B_{c}=\frac{Z_{t}Z_{p}e^{2}}{r_{C}(A^{1/3}_{t}+A^{1/3}_{p})}, |
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53 | \end{eqnarray*} |
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54 | where $r_{C}$ = 1.3 fm, $e$ is the electron charge and $Z_t$, $Z_p$ are the |
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55 | atomic numbers of the target and projectile nuclei. $R$ is the interaction |
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56 | radius $R_{int}$ which in the Kox formula is divided into volume and surface |
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57 | terms: |
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58 | \begin{eqnarray*} |
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59 | R_{int}=R_{vol}+R_{surf} . |
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60 | \end{eqnarray*} |
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61 | $R_{vol}$ and $R_{surf}$ correspond to the energy-independent and |
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62 | energy-dependent components of the reactions, respectively. Collisions which |
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63 | have relatively small impact parameters are independent of both energy and mass |
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64 | number. These core collisions are parameterized by $R_{vol}$. Therefore |
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65 | $R_{vol}$ can depend only on the volume of the projectile and target nuclei: |
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66 | \begin{eqnarray*} |
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67 | R_{vol}=r_{0}(A^{1/3}_{t}+A^{1/3}_{p}) . |
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68 | \end{eqnarray*} |
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69 | |
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70 | The second term of the interaction radius is a nuclear surface contribution and |
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71 | is parameterized by |
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72 | \begin{eqnarray*} |
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73 | R_{surf}=r_{0}[a\frac{A^{1/3}_{t}A^{1/3}_{p}}{A^{1/3}_{t}+A^{1/3}_{p}}-c]+D. |
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74 | \end{eqnarray*} |
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75 | |
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76 | The first term in brackets is the mass asymmetry which is related to the volume |
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77 | overlap of the projectile and target. The second term $c$ is an |
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78 | energy-dependent parameter which takes into account increasing surface |
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79 | transparency as the projectile energy increases. $D$ is the neutron-excess |
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80 | which becomes important in collisions of heavy or neutron-rich targets. It is |
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81 | given by |
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82 | \begin{eqnarray*} |
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83 | D=\frac{5(A_{t}-Z_{t})Z_{p}}{A_{p}A_{r}}. |
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84 | \end{eqnarray*} |
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85 | The surface component ($R_{surf}$) of the interaction radius is actually not |
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86 | part of the simple framework of the strong absorption model, but a better |
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87 | reproduction of experimental results is possible when it is used. |
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88 | |
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89 | The parameters $r_0$, $a$ and $c$ are obtained using a $\chi^{2}$ minimizing |
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90 | procedure with the experimental data. In this procedure the parameters $r_{0}$ |
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91 | and $a$ were fixed while $c$ was allowed to vary only with the beam energy per |
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92 | nucleon. The best $\chi^{2}$ fit is provided by $r_{0}$ = 1.1 fm and |
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93 | $a = 1.85$ with the corresponding values of $c$ listed in Table III and shown |
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94 | in Fig.~12 of Ref.~\cite{nnc.Kox87} as a function of beam energy per nucleon. |
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95 | This reference presents the values of $c$ only in chart and figure form, which |
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96 | is not suitable for Monte Carlo calculations. Therefore a simple analytical |
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97 | function is used to calculate $c$ in Geant4. The function is: |
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98 | \begin{eqnarray*} |
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99 | c=-\frac{10}{x^{5}}+2.0 \mbox{ } \rm{for} \mbox{ } x \ge 1.5 |
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100 | \end{eqnarray*} |
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101 | \begin{eqnarray*} |
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102 | c=(-\frac{10}{1.5^{5}}+2.0)\times(\frac{x}{1.5})^{3} \mbox{ } \rm{for} \mbox{ } x < 1.5 , |
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103 | \end{eqnarray*} |
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104 | \begin{eqnarray*} |
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105 | x=log(KE) , |
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106 | \end{eqnarray*} |
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107 | where $KE$ is the projectile kinetic energy in units of MeV/nucleon in the |
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108 | laboratory system. \\ |
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109 | |
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110 | \noindent |
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111 | {\bf Shen formula:} as mentioned earlier, this formula is also based on the |
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112 | strong absorption model, therefore it has a structure similar to the Kox |
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113 | formula: |
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114 | \begin{equation} |
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115 | \sigma_{R} = 10\pi R^{2}[1-\frac{B}{E_{CM}}]. |
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116 | \end{equation} |
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117 | However, different parameterized forms for $R$ and $B$ are applied. The |
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118 | interaction radius $R$ is given by |
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119 | \begin{eqnarray*} |
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120 | R=r_{0}[A^{1/3}_{t}+A^{1/3}_{p}+1.85\frac{A^{1/3}_{t}A^{1/3}_{p}}{A^{1/3}_{t}+A^{1/3}_{p}}-C'(KE)] \\ |
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121 | +\alpha\frac{5(A_{t}-Z_{t})Z_{p}}{A_{p}A_{r}}+\beta E^{-1/3}_{CM}\frac{A^{1/3}_{t}A^{1/3}_{p}}{A^{1/3}_{t}+A^{1/3}_{p}} |
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122 | \end{eqnarray*} |
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123 | where $\alpha$, $\beta$ and $r_0$ are |
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124 | \begin{eqnarray*} |
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125 | \alpha = 1 fm |
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126 | \end{eqnarray*} |
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127 | \begin{eqnarray*} |
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128 | \beta = 0.176MeV^{1/3} \cdot fm |
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129 | \end{eqnarray*} |
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130 | \begin{eqnarray*} |
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131 | r_{0}= 1.1 fm |
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132 | \end{eqnarray*} |
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133 | In Ref.~\cite{nnc.Shen89} as well, no functional form for $C'(KE)$ is given. |
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134 | Hence the same simple analytical function is used by Geant4 to derive $c$ |
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135 | values. |
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136 | |
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137 | The second term $B$ is called the nuclear-nuclear interaction barrier in the |
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138 | Shen formula and is given by |
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139 | \begin{eqnarray*} |
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140 | B=\frac{1.44Z_{t}Z_{p}}{r}-b\frac{R_{t}R_{p}}{R_{t}+R_{p}} (MeV) |
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141 | \end{eqnarray*} |
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142 | where $r$, $b$, $R_t$ and $R_p$ are given by |
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143 | \begin{eqnarray*} |
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144 | r=R_{t}+R_{p}+3.2fm |
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145 | \end{eqnarray*} |
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146 | \begin{eqnarray*} |
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147 | b=1MeV\cdot fm^{-1} |
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148 | \end{eqnarray*} |
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149 | \begin{eqnarray*} |
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150 | R_{i}=1.12A^{1/3}_{i} -0.94A^{-1/3}_{i} ~ (i=t,p) |
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151 | \end{eqnarray*} |
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152 | The difference between the Kox and Shen formulae appears at energies below |
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153 | 30 MeV/nucleon. In this region the Shen formula shows better agreement with the |
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154 | experimental data in most cases. |
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155 | |
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156 | \section{Tripathi formula} |
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157 | \noindent |
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158 | Because the Tripathi formula is also based on the strong absorption model its |
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159 | form is similar to the Kox and Shen formulae: |
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160 | \begin{equation} |
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161 | \sigma_{R} = \pi r_0^2 (A^{1/3}_{p}+A^{1/3}_{t}+\delta_{E})^{2}[1-\frac{B}{E_{CM}}], |
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162 | \label{eqn15.4} |
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163 | \end{equation} |
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164 | where $r_0$ = 1.1 fm. In the Tripathi formula $B$ and $R$ are given by |
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165 | \begin{eqnarray*} |
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166 | B=\frac{1.44Z_{t}Z_{p}}{R} |
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167 | \end{eqnarray*} |
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168 | \begin{eqnarray*} |
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169 | R=r_{p}+r_{t}+\frac{1.2(A^{1/3}_{p}+A^{1/3}_{t})}{E^{1/3}_{CM}} |
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170 | \end{eqnarray*} |
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171 | where $r_i$ is the equivalent sphere radius and is related to the $r_{rms,i}$ |
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172 | radius by |
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173 | |
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174 | \[ |
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175 | r_{i}=1.29r_{rms,i} ~ (i=p,t) . |
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176 | \] |
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177 | |
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178 | $\delta_{E}$ represents the energy-dependent term of the reaction cross section |
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179 | which is due mainly to transparency and Pauli blocking effects. It is given by |
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180 | \begin{eqnarray*} |
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181 | \delta_{E}=1.85S+(0.16S/E^{1/3}_{CM})-C_{KE}+[0.91(A_{t}-2Z_{t})Z_{p}/(A_{p}A_{t})], |
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182 | \end{eqnarray*} |
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183 | where $S$ is the mass asymmetry term given by |
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184 | \begin{eqnarray*} |
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185 | S=\frac{A^{1/3}_{p}A^{1/3}_{t}}{A^{1/3}_{p}+A^{1/3}_{t}}. |
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186 | \end{eqnarray*} |
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187 | This is related to the volume overlap of the colliding system. The last term |
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188 | accounts for the isotope dependence of the reaction cross section and |
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189 | corresponds to the $D$ term in the Kox formula and the second term of $R$ in |
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190 | the Shen formula. |
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191 | |
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192 | The term $C_{KE}$ corresponds to $c$ in Kox and $C'(KE)$ in Shen and is given |
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193 | by |
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194 | \begin{eqnarray*} |
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195 | C_{E}=D_{Pauli}[1-\exp(-KE/40)]-0.292\exp(-KE/792)\times\cos(0.229KE^{0.453}) .\,\\ |
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196 | \end{eqnarray*} |
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197 | Here D$_{Pauli}$ is related to the density dependence of the colliding system, |
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198 | scaled with respect to the density of the $^{12}$C+$^{12}$C colliding |
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199 | system: \\ |
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200 | \begin{eqnarray*} |
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201 | D_{Pauli} = 1.75 \frac{\rho_{A_p}+\rho_{A_t}}{\rho_{A_{\boldmath C}}+\rho_{A_{\boldmath C}}} . |
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202 | \end{eqnarray*} |
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203 | The nuclear density is calculated in the hard sphere model. $D_{Pauli}\,$ |
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204 | simulates the modifications of the reaction cross sections caused by Pauli |
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205 | blocking and is being introduced to the Tripathi formula for the first time. |
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206 | The modification of the reaction cross section due to Pauli blocking plays an |
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207 | important role at energies above 100 MeV/nucleon. Different forms of |
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208 | $D_{Pauli}\,$ are used in the Tripathi formula for alpha-nucleus and |
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209 | lithium-nucleus collisions. |
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210 | \noindent |
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211 | For alpha-nucleus collisions, |
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212 | \begin{eqnarray*} |
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213 | D_{Pauli}=2.77 - (8.0\times 10^{-3} A_t) + (1.8\times 10^{-5}A^{2}_t) \\ |
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214 | - 0.8/\{1+\exp[(250-KE)/75]\}\, |
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215 | \end{eqnarray*} |
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216 | \noindent |
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217 | For lithium-nucleus collisions, |
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218 | \begin{eqnarray*} |
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219 | D_{Pauli}=D_{Pauli}/3. |
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220 | \end{eqnarray*} |
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221 | Note that the Tripathi formula is not fully implemented in Geant4 and can only |
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222 | be used for projectile energies less than 1 GeV/nucleon. |
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223 | |
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224 | |
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225 | \section{Representative Cross Sections} |
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226 | |
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227 | |
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228 | \noindent |
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229 | Representative cross section results from the Sihver, Kox, Shen and Tripathi |
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230 | formulae in Geant4 are displayed in Table I and compared to the experimental |
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231 | measurements of Ref.~\cite{nnc.Kox87}. |
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232 | |
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233 | \section{Tripathi Formula for "light" Systems} |
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234 | \label{TripathiLight} |
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235 | For nuclear-nuclear interactions in which the projectile and/or target are |
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236 | light, Tripathi {\normalsize\it{et al}} \cite{RefTripathiLight} propose an |
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237 | alternative algorithm for determining the interaction cross section |
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238 | (implemented in the new class G4TripathiLightCrossSection). For such systems, |
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239 | Eq.\ref{eqn15.4} becomes: |
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240 | |
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241 | \begin{equation} |
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242 | \sigma _R = \pi r_0^2 [ A_p^{1/3} + A_t^{1/3} + \delta _E ]^2 |
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243 | (1 - R_C \frac{B}{E_{CM}})X_m |
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244 | \label{eqn15.6} |
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245 | \end{equation} |
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246 | |
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247 | \noindent $R_C$ is a Coulomb multiplier, which is added since for light |
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248 | systems Eq. \ref{eqn15.4} overestimates the interaction distance, |
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249 | causing $B$ (in Eq. \ref{eqn15.4}) to be underestimated. Values for $R_C$ are |
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250 | given in Table \ref{tab15.1}. |
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251 | |
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252 | \begin{equation} |
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253 | X_m = 1 - X_1 \exp \left( { - \frac{E}{{X_1 S_L }}} \right) |
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254 | \label{eqn15.7} |
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255 | \end{equation} |
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256 | |
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257 | \noindent where: |
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258 | |
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259 | \begin{equation} |
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260 | X_1 = 2.83 - \left( {3.1 \times 10^{ - 2} } \right)A_T + \left( {1.7 \times 10^{ - 4} } \right)A_T^2 |
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261 | \label{eqn15.8} |
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262 | \end{equation} |
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263 | |
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264 | \noindent except for neutron interactions with $^4$He, for which $X_1$ is |
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265 | better approximated to 5.2, and the function $S_L$ is given by: |
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266 | |
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267 | \begin{equation} |
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268 | S_L = 1.2 + 1.6\left[ {1 - \exp \left( { - \frac{E}{{15}}} \right)} \right] |
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269 | \label{eqn15.9} |
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270 | \end{equation} |
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271 | |
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272 | \noindent For light nuclear-nuclear collisions, a slightly more general |
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273 | expression for $C_E$ is used: |
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274 | |
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275 | \begin{equation} |
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276 | C_E = D\left[ {1 - \exp \left( { - \frac{E}{{T_1 }}} \right)} \right] - 0.292\exp \left( { - \frac{E}{{792}}} \right) \cdot \cos \left( {0.229E^{0.453} } \right) |
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277 | \label{eqn15.10} |
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278 | \end{equation} |
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279 | |
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280 | \noindent $D$ and $T_1$ are dependent on the interaction, and are defined |
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281 | in table \ref{tab15.2}. |
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282 | |
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283 | \begin{table} |
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284 | \begin{center} |
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285 | \caption{Representative total reaction cross sections} |
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286 | \label{nn-x-section-tb} |
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287 | \begin{tabular}{|c|c|c|c|c|c|c|c|} |
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288 | \hline |
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289 | Proj.&Target&Elab&Exp. Results&Sihver&Kox&Shen&Tripathi\\ |
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290 | &&[MeV/n]&[mb]&&&&\\ |
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291 | \hline |
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292 | &&&&&&&\\ |
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293 | $^{12}$C&$^{12}$C&30&1316$\pm$40&---&1295.04&1316.07&1269.24\\ |
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294 | &&83&965$\pm$30&---&957.183&969.107&989.96\\ |
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295 | &&200&864$\pm$45&868.571&885.502&893.854&864.56\\ |
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296 | &&300&858$\pm$60&868.571&871.088&878.293&857.414\\ |
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297 | &&870$^1$&939$\pm$50&868.571&852.649&857.683&939.41\\ |
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298 | &&2100$^1$&888$\pm$49&868.571&846.337&850.186&936.205\\ |
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299 | &$^{27}$Al&30&1748$\pm$85&---&1801.4&1777.75&1701.03\\ |
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300 | &&83&1397$\pm$40&---&1407.64&1386.82&1405.61\\ |
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301 | &&200&1270$\pm$70&1224.95&1323.46&1301.54&1264.26\\ |
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302 | &&300&1220$\pm$85&1224.95&1306.54&1283.95&1257.62\\ |
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303 | &$^{89}$Y&30&2724$\pm$300&---&2898.61&2725.23&2567.68\\ |
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304 | &&83&2124$\pm$140&---&2478.61&2344.26&2346.54\\ |
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305 | &&200&1885$\pm$120&2156.47&2391.26&2263.77&2206.01\\ |
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306 | &&300&1885$\pm$150&2156.47&2374.17&2247.55&2207.01\\ |
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307 | &&&&&&&\\ |
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308 | $^{16}$O&$^{27}$Al&30&1724$\pm$80&---&1965.85&1935.2&1872.23\\ |
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309 | &$^{89}$Y&30&2707$\pm$330&---&3148.27&2957.06&2802.48\\ |
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310 | &&&&&&&\\ |
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311 | $^{20}$Ne&$^{27}$Al&30&2113$\pm$100&---&2097.86&2059.4&2016.32\\ |
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312 | &&100&1446$\pm$120&1473.87&1684.01&1658.31&1667.17\\ |
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313 | &&300&1328$\pm$120&1473.87&1611.88&1586.17&1559.16\\ |
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314 | &$^{108}$Ag&300&2407$\pm$200$^2$&2730.69&3095.18&2939.86&2893.12\\ |
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315 | \hline |
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316 | \end{tabular} |
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317 | \end{center} |
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318 | 1. Data measured by Jaros et al. \cite{nnc.Jaros78} \\ |
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319 | 2. Natural silver was used in this measurement. |
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320 | \end{table} |
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321 | |
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322 | \begin{table} |
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323 | \begin{center} |
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324 | \caption{Coulomb multiplier for light systems \cite{RefTripathiLight}.} |
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325 | \label{tab15.1} % Give a unique label |
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326 | \begin{tabular}{cc} |
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327 | \hline\noalign{\smallskip} |
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328 | System & \(R_C\) \\ |
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329 | \noalign{\smallskip}\hline\noalign{\smallskip} |
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330 | p + d & 13.5 \\ |
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331 | p + $^3$He & 21 \\ |
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332 | p + $^4$He & 27 \\ |
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333 | p + Li & 2.2 \\ |
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334 | d + d & 13.5 \\ |
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335 | d + $^4$He & 13.5 \\ |
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336 | d + C & 6.0 \\ |
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337 | $^4$He + Ta & 0.6 \\ |
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338 | $^4$He + Au & 0.6 \\ |
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339 | \noalign{\smallskip}\hline |
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340 | \end{tabular} |
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341 | \end{center} |
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342 | \vspace*{2cm} % with the correct table height |
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343 | \end{table} |
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344 | |
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345 | \begin{table} |
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346 | \caption{Parameters D and T1 for light systems \cite{RefTripathiLight}.} |
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347 | \label{tab15.2} % Give a unique label |
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348 | % For LaTeX tables use |
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349 | \begin{tabular}{cccc} |
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350 | \hline\noalign{\smallskip} |
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351 | System & T1 [MeV] & D & G [MeV] \\ |
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352 | & & & ($^4$He + X only) \\ |
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353 | \noalign{\smallskip}\hline\noalign{\smallskip} |
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354 | p + X & 23 & |
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355 | \(1.85 + \frac{{0.16}}{{1 + \exp \left( {\frac{{500 - E}}{{200}}} \right)}}\) & |
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356 | (Not applicable) \\ |
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357 | |
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358 | n + X & 18 & |
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359 | \(1.85 + \frac{{0.16}}{{1 + \exp \left( {\frac{{500 - E}}{{200}}} \right)}}\) & |
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360 | (Not applicable) \\ |
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361 | |
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362 | d + X & 23 & |
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363 | \(1.65 + \frac{{0.1}}{{1 + \exp \left( {\frac{{500 - E}}{{200}}} \right)}}\) & |
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364 | (Not applicable) \\ |
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365 | |
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366 | $^3$He + X & 40 & 1.55 & (Not applicable) \\ |
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367 | |
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368 | $^4$He + $^4$He & 40 & |
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369 | \( |
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370 | \begin{array}{l} |
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371 | D = 2.77 - 8.0 \times 10^{ - 3} A_T \\ |
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372 | + 1.8 \times 10^{ - 5} A_T^2 \\ |
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373 | - \frac{{0.8}}{{1 + \exp \left( {\frac{{250 - E}}{G}} \right)}} \\ |
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374 | \end{array} |
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375 | \) & |
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376 | 300 \\ |
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377 | |
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378 | $^4$He + Be & 25 & (as for $^4$He + $^4$He) & 300 \\ |
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379 | $^4$He + N & 40 & (as for $^4$He + $^4$He) & 500 \\ |
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380 | $^4$He + Al & 25 & (as for $^4$He + $^4$He) & 300 \\ |
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381 | $^4$He + Fe & 40 & (as for $^4$He + $^4$He) & 300 \\ |
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382 | $^4$He + X (general) & 40 & (as for $^4$He + $^4$He) & 75 \\ |
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383 | |
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384 | \noalign{\smallskip}\hline |
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385 | \end{tabular} |
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386 | % Or use |
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387 | \vspace*{5cm} % with the correct table height |
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388 | \end{table} |
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389 | |
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390 | \section{Status of this document} |
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391 | \noindent |
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392 | 25.11.03 created by Tatsumi Koi\\ |
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393 | 28.11.03 grammar check and re-wording by D.H. Wright\\ |
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394 | 18.06.04 light system section added by Peter Truscott \\ |
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395 | |
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396 | \begin{latexonly} |
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397 | |
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398 | \begin{thebibliography}{99} |
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399 | |
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400 | \bibitem{nnc.Sihver93} |
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401 | L. Sihver et al., Phys. Rev. C47, 1225 (1993). |
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402 | |
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403 | \bibitem{nnc.Kox87} |
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404 | Kox et al. Phys. Rev. C35, 1678 (1987). |
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405 | |
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406 | \bibitem{nnc.Shen89} |
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407 | Shen et al. Nucl. Phys. A491, 130 (1989). |
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408 | |
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409 | \bibitem{nnc.Tripathi97} |
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410 | Tripathi et al, NASA Technical Paper 3621 (1997). |
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411 | |
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412 | \bibitem{nnc.Jaros78} |
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413 | Jaros et al, Phys. Rev. C 18 2273 (1978). |
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414 | |
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415 | \bibitem{RefTripathiLight} |
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416 | % Format for Journal Reference |
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417 | R K Tripathi, F A Cucinotta, and J W Wilson, "Universal parameterization of absorption cross-sections - Light systems," NASA Technical Paper TP-1999-209726, 1999. |
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418 | |
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419 | \end{thebibliography} |
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420 | |
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421 | \end{latexonly} |
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422 | |
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423 | \begin{htmlonly} |
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424 | |
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425 | \section{Bibliography} |
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426 | |
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427 | \begin{enumerate} |
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428 | \item |
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429 | L. Sihver et al., Phys. Rev. C47, 1225 (1993). |
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430 | |
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431 | \item |
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432 | Kox et al. Phys. Rev. C35, 1678 (1987). |
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433 | |
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434 | \item |
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435 | Shen et al. Nucl. Phys. A491, 130 (1989). |
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436 | |
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437 | \item |
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438 | Tripathi et al, NASA Technical Paper 3621 (1997). |
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439 | |
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440 | \item |
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441 | Jaros et al, Phys. Rev. C 18 2273 (1978). |
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442 | |
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443 | \item |
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444 | % Format for Journal Reference |
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445 | R K Tripathi, F A Cucinotta, and J W Wilson, "Universal parameterization of absorption cross-sections - Light systems," NASA Technical Paper TP-1999-209726, 1999. |
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446 | |
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447 | \end{enumerate} |
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448 | |
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449 | \end{htmlonly} |
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450 | |
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