1 | |
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2 | \chapter{Electromagnetic Dissociation Model} |
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3 | |
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4 | \section{The Model} |
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5 | The relative motion of a projectile nucleus travelling at relativistic |
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6 | speeds with respect to another nucleus can give rise to an increasingly |
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7 | hard spectrum of virtual photons. The excitation energy associated |
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8 | with this energy exchange can result in the liberation of nucleons or |
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9 | heavier nuclei ({\normalsize\it{}i.e.} deuterons, $\alpha$-particles, |
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10 | {\normalsize\it{}etc.}). The contribution of this source to the total |
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11 | inelastic cross section can be important, especially where the proton |
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12 | number of the nucleus is large. The electromagnetic dissociation (ED) |
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13 | model is implemented in the classes G4EMDissociation, |
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14 | G4EMDissociationCrossSection and G4EMDissociationSpectrum, with the |
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15 | theory taken from Wilson {\normalsize\it{}et al} \cite{ed.Wilson}, and |
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16 | Bertulani and Baur \cite{ed.BandB}. |
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17 | |
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18 | \noindent The number of virtual photons \(N(E_{\gamma},b)\) per unit |
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19 | area and energy interval experienced by the projectile due to the |
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20 | dipole field of the target is given by the expression \cite{ed.BandB}: |
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21 | |
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22 | \begin{equation} |
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23 | N\left( {E_\gamma ,b} \right) = \frac{{\alpha Z_T^2 }}{{\pi ^2 \beta ^2 b^2 E_\gamma }}\left\{ {x^2 k_1^2 (x) + \left( {\frac{{x^2 }}{{\gamma ^2 }}} \right)k_0^2 (x)} \right\} |
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24 | \label{ed.eqn1} |
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25 | \end{equation} |
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26 | |
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27 | \noindent where \(x\) is a dimensionless quantity defined as: |
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28 | |
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29 | \begin{equation} |
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30 | x = \frac{{bE_\gamma }}{{\gamma \beta \bar hc}} |
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31 | \label{ed.eqn2} |
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32 | \end{equation} |
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33 | |
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34 | \noindent and: |
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35 | |
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36 | \(\alpha\) \indent = fine structure constant |
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37 | |
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38 | \(\beta\) \indent = ratio of the velocity of the projectile in the |
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39 | laboratory frame to the velocity of light |
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40 | |
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41 | \(\gamma\) \indent = Lorentz factor for the projectile in the laboratory |
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42 | frame |
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43 | |
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44 | \(b\) \indent = impact parameter |
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45 | |
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46 | \(c\) \indent = speed of light |
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47 | |
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48 | \(\bar h\) \indent = quantum constant |
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49 | |
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50 | \(E_{\gamma}\) \indent = energy of virtual photon |
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51 | |
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52 | \(k_0\) and \(k_1\) \indent = zeroth and first order modified Bessel |
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53 | functions of the second kind |
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54 | |
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55 | \(Z_T\) \indent = atomic number of the target nucleus |
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56 | |
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57 | |
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58 | \noindent Integrating Eq. \ref{ed.eqn1} over the impact parameter from |
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59 | \(b_{min}\) to \(\infty \) produces the virtual photon spectrum for the |
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60 | dipole field of: |
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61 | |
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62 | \begin{equation} |
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63 | N_{E1} \left( {E_\gamma } \right) = \frac{{2\alpha Z_T^2 }}{{\pi \beta ^2 E_\gamma }}\left\{ {\xi k_0 (\xi )k_1 (\xi ) - \frac{{\xi ^2 \beta ^2 }}{2}\left( {k_1^2 (\xi ) - k_0^2 (\xi )} \right)} \right\} |
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64 | \label{ed.eqn3} |
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65 | \end{equation} |
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66 | |
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67 | \noindent where, according to the algorithm implemented by Wilson |
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68 | {\normalsize\it{}et al} in NUCFRG2 \cite{ed.Wilson}: |
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69 | |
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70 | \begin{equation} |
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71 | \begin{array}{c} |
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72 | \xi = \frac{{E_\gamma b_{\min } }}{{\gamma \beta \bar hc}} \\ |
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73 | \\ |
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74 | b_{\min } = (1 + x_d )b_c + \frac{{\pi \alpha _0 }}{{2\gamma }} \\ |
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75 | \\ |
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76 | \alpha _0 = \frac{{Z_P Z_T e^2 }}{{\mu \beta ^2 c^2 }} \\ |
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77 | \\ |
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78 | b_c = 1.34\left[ {A_P^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ |
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79 | {\vphantom {1 3}}\right.\kern-\nulldelimiterspace} |
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80 | \!\lower0.7ex\hbox{$3$}}} + A_T^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ |
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81 | {\vphantom {1 3}}\right.\kern-\nulldelimiterspace} |
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82 | \!\lower0.7ex\hbox{$3$}}} - 0.75\left( {A_P^{ - {\raise0.7ex\hbox{$1$} \!\mathord{\left/ |
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83 | {\vphantom {1 3}}\right.\kern-\nulldelimiterspace} |
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84 | \!\lower0.7ex\hbox{$3$}}} + A_T^{ - {\raise0.7ex\hbox{$1$} \!\mathord{\left/ |
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85 | {\vphantom {1 3}}\right.\kern-\nulldelimiterspace} |
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86 | \!\lower0.7ex\hbox{$3$}}} } \right)} \right] \\ |
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87 | \end{array} |
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88 | \label{ed.eqn4} |
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89 | \end{equation} |
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90 | |
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91 | \noindent and \(\mu\) is the reduced mass of the projectile/target system, |
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92 | \(x_d=0.25\), and \(A_P\) and \(A_T\) are the projectile and target nucleon |
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93 | numbers. For the last equation, the units of \(b_c\) are fm. Wilson |
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94 | {\normalsize\it{}et al} state that there is an equivalent virtual photon |
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95 | spectrum as a result of the quadrupole field: |
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96 | |
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97 | \begin{equation} |
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98 | N_{E2} \left( {E_\gamma } \right) = \frac{{2\alpha Z_T^2 }}{{\pi \beta ^4 E_\gamma }}\left\{ {2\left( {1 - \beta ^2 } \right)k_1^2 (\xi ) + \xi \left( {2 - \beta ^2 } \right)^2 k_0 (\xi )k_1 (\xi ) - \frac{{\xi ^2 \beta ^4 }}{2}\left( {k_1^2 (\xi ) - k_0^2 (\xi )} \right)} \right\} |
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99 | \label{ed.eqn5} |
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100 | \end{equation} |
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101 | |
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102 | \noindent The cross section for the interaction of the dipole and quadrupole |
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103 | fields is given by: |
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104 | |
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105 | \begin{equation} |
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106 | \sigma _{ED} = \int {N_{E1} \left( {E_\gamma } \right)\sigma _{E1} \left( {E_\gamma } \right)dE_\gamma } + \int {N_{E2} \left( {E_\gamma } \right)\sigma _{E2} \left( {E_\gamma } \right)dE_\gamma } |
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107 | \label{ed.eqn6} |
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108 | \end{equation} |
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109 | |
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110 | \noindent Wilson {\normalsize\it{}et al} assume that |
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111 | \(\sigma_{E1}(E_{\gamma})\) and \(\sigma_{E2}(E_{\gamma})\) are sharply peaked |
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112 | at the giant dipole and quadrupole resonance energies: |
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113 | |
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114 | \begin{equation} |
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115 | \begin{array}{c} |
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116 | E_{GDR} = \bar hc \left[ {\frac{{m^* c^2 R_0^2 }}{{8J}}\left( {1 + u - \frac{{1 + \varepsilon + 3u}}{{1 + \varepsilon + u}}\varepsilon } \right)} \right]^{- \frac {1} {2}} \\ |
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117 | \\ |
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118 | E_{GQR} = \frac{{63}}{{A_P^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ |
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119 | {\vphantom {1 3}}\right.\kern-\nulldelimiterspace} |
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120 | \!\lower0.7ex\hbox{$3$}}} }} \\ |
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121 | \label{ed.eqn7} |
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122 | \end{array} |
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123 | \end{equation} |
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124 | |
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125 | \noindent so that the terms for \(N_{E1}\) and \(N_{E2}\) can be taken |
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126 | out of the integrals in Eq. \ref{ed.eqn6} and evaluated at the resonances. |
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127 | |
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128 | \noindent In Eq. \ref{ed.eqn7}: |
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129 | |
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130 | \begin{equation} |
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131 | \begin{array}{c} |
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132 | u = \frac{{3J}}{{Q'}}A_P^{ - {\raise0.7ex\hbox{$1$} \!\mathord{\left/ |
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133 | {\vphantom {1 3}}\right.\kern-\nulldelimiterspace} |
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134 | \!\lower0.7ex\hbox{$3$}}} \\ |
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135 | \\ |
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136 | R_0 = r_0 A_P^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ |
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137 | {\vphantom {1 3}}\right.\kern-\nulldelimiterspace} |
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138 | \!\lower0.7ex\hbox{$3$}}} \\ |
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139 | \end{array} |
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140 | \label{ed.eqn8} |
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141 | \end{equation} |
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142 | |
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143 | \noindent \(\epsilon=0.0768\), \(Q'=17\)MeV, \(J=36.8\)MeV, \(r_0=1.18\)fm, |
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144 | and \(m^*\) is 7/10 of the nucleon mass (taken as 938.95 MeV/c$^2$). |
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145 | (The dipole and quadrupole energies are expressed in units of MeV.) |
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146 | |
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147 | \noindent The photonuclear cross sections for the dipole and quadrupole |
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148 | resonances are assumed to be given by: |
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149 | |
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150 | \begin{equation} |
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151 | \int {\sigma _{E1} \left( {E_\gamma } \right)dE_\gamma = 60\frac{{N_P Z_P }}{{A_P }}} |
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152 | \label{ed.eqn9} |
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153 | \end{equation} |
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154 | |
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155 | \noindent in units of MeV-mb (\(N_P\) being the number of neutrons in the |
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156 | projectile) and: |
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157 | |
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158 | \begin{equation} |
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159 | \int {\sigma _{E2} \left( {E_\gamma } \right)\frac{{dE_\gamma }}{{E_\gamma ^2 }} = 0.22fZ_P A_P^{{\raise0.7ex\hbox{$2$} \!\mathord{\left/ |
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160 | {\vphantom {2 3}}\right.\kern-\nulldelimiterspace} |
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161 | \!\lower0.7ex\hbox{$3$}}} } |
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162 | \label{ed.eqn10} |
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163 | \end{equation} |
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164 | \noindent in units of $\mu$b/MeV. In the latter expression, \(f\) is |
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165 | given by: |
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166 | |
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167 | \begin{equation} |
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168 | f = \left\{ {\begin{array}{*{20}c} |
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169 | {0.9} \hfill & {A_P > 100} \hfill \\ |
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170 | {0.6} \hfill & {40 < A_P \le 100} \hfill \\ |
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171 | {0.3} \hfill & {40 \le A_P } \hfill \\ |
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172 | \end{array}} \right. |
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173 | \end{equation} |
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174 | |
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175 | \noindent The total cross section for electromagnetic dissociation is |
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176 | therefore given by Eq. \ref{ed.eqn6} with the virtual photon spectra for |
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177 | the dipole and quadrupole fields calculated at the resonances: |
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178 | |
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179 | \begin{equation} |
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180 | \sigma _{ED} = N_{E1} \left( {E_{GDR} } \right)\int {\sigma _{E1} \left( {E_\gamma } \right)dE_\gamma } + N_{E2} \left( {E_{GQR} } \right)E_{GQR}^2 \int {\frac{{\sigma _{E2} \left( {E_\gamma } \right)}}{{E_\gamma ^2 }}dE_\gamma } |
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181 | \label{ed.eqn11} |
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182 | \end{equation} |
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183 | |
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184 | \noindent where the resonance energies are given by Eq. \ref{ed.eqn7} and |
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185 | the integrals for the photonuclear cross sections given by |
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186 | Eq. \ref{ed.eqn9} and Eq. \ref{ed.eqn10}. |
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187 | |
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188 | \noindent The selection of proton or neutron emission is made according to |
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189 | the following prescription from Wilson {\normalsize\it{}et al}. |
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190 | |
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191 | \begin{equation} |
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192 | \begin{array}{l} |
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193 | \sigma _{ED,p} = \sigma _{ED} \times \left\{ {\begin{array}{*{20}c} |
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194 | {0.5} \hfill & {Z_P < 6} \hfill \\ |
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195 | {0.6} \hfill & {6 \le Z_P \le 8} \hfill \\ |
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196 | {0.7} \hfill & {8 < Z_P < 14} \hfill \\ |
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197 | {\min \left[ {\frac{{Z_P }}{{A_P }},1.95\exp ( - 0.075Z_P )} \right]} \hfill & {Z_P \ge 14} \hfill \\ |
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198 | \end{array}} \right\} \\ |
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199 | \sigma _{ED,n} = \sigma _{ED} - \sigma _{ED,p} \\ |
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200 | \end{array} |
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201 | \label{ed.eqn12} |
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202 | \end{equation} |
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203 | |
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204 | \indent Note that this implementation of ED interactions only treats |
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205 | the ejection of single nucleons from the nucleus, and currently does |
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206 | not allow emission of other light nuclear fragments. |
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207 | |
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208 | \section{Status of this document} |
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209 | 19.06.04 created by Peter Truscott \\ |
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210 | |
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211 | |
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212 | \begin{latexonly} |
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213 | |
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214 | \begin{thebibliography}{99} |
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215 | |
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216 | \bibitem{ed.Wilson} |
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217 | J. W. Wilson, R. K. Tripathi, F. A. Cucinotta, J. K. Shinn, |
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218 | F. F. Badavi, S. Y. Chun, J. W. Norbury, C. J. Zeitlin, L. Heilbronn, |
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219 | and J. Miller, |
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220 | "NUCFRG2: An evaluation of the semiempirical nuclear fragmentation database," |
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221 | NASA Technical Paper 3533, 1995. |
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222 | |
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223 | \bibitem{ed.BandB} |
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224 | C. A. Bertulani, and G. Baur, Electromagnetic processes in |
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225 | relativistic heavy ion collisions, |
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226 | Nucl Phys, A458, 725-744, 1986. |
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227 | |
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228 | \end{thebibliography} |
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229 | |
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230 | \end{latexonly} |
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231 | |
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232 | |
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233 | \begin{htmlonly} |
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234 | |
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235 | \section{Bibliography} |
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236 | |
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237 | \begin{enumerate} |
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238 | \item |
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239 | J. W. Wilson, R. K. Tripathi, F. A. Cucinotta, J. K. Shinn, |
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240 | F. F. Badavi, S. Y. Chun, J. W. Norbury, C. J. Zeitlin, L. Heilbronn, |
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241 | and J. Miller, |
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242 | "NUCFRG2: An evaluation of the semiempirical nuclear fragmentation database," |
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243 | NASA Technical Paper 3533, 1995. |
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244 | |
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245 | \item |
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246 | C. A. Bertulani, and G. Baur, Electromagnetic processes in |
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247 | relativistic heavy ion collisions, |
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248 | Nucl Phys, A458, 725-744, 1986. |
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249 | |
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250 | \end{enumerate} |
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251 | |
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252 | \end{htmlonly} |
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253 | |
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