[1208] | 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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