\chapter{Electromagnetic Dissociation Model} \section{The Model} The relative motion of a projectile nucleus travelling at relativistic speeds with respect to another nucleus can give rise to an increasingly hard spectrum of virtual photons. The excitation energy associated with this energy exchange can result in the liberation of nucleons or heavier nuclei ({\normalsize\it{}i.e.} deuterons, $\alpha$-particles, {\normalsize\it{}etc.}). The contribution of this source to the total inelastic cross section can be important, especially where the proton number of the nucleus is large. The electromagnetic dissociation (ED) model is implemented in the classes G4EMDissociation, G4EMDissociationCrossSection and G4EMDissociationSpectrum, with the theory taken from Wilson {\normalsize\it{}et al} \cite{ed.Wilson}, and Bertulani and Baur \cite{ed.BandB}. \noindent The number of virtual photons \(N(E_{\gamma},b)\) per unit area and energy interval experienced by the projectile due to the dipole field of the target is given by the expression \cite{ed.BandB}: \begin{equation} 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\} \label{ed.eqn1} \end{equation} \noindent where \(x\) is a dimensionless quantity defined as: \begin{equation} x = \frac{{bE_\gamma }}{{\gamma \beta \bar hc}} \label{ed.eqn2} \end{equation} \noindent and: \(\alpha\) \indent = fine structure constant \(\beta\) \indent = ratio of the velocity of the projectile in the laboratory frame to the velocity of light \(\gamma\) \indent = Lorentz factor for the projectile in the laboratory frame \(b\) \indent = impact parameter \(c\) \indent = speed of light \(\bar h\) \indent = quantum constant \(E_{\gamma}\) \indent = energy of virtual photon \(k_0\) and \(k_1\) \indent = zeroth and first order modified Bessel functions of the second kind \(Z_T\) \indent = atomic number of the target nucleus \noindent Integrating Eq. \ref{ed.eqn1} over the impact parameter from \(b_{min}\) to \(\infty \) produces the virtual photon spectrum for the dipole field of: \begin{equation} 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\} \label{ed.eqn3} \end{equation} \noindent where, according to the algorithm implemented by Wilson {\normalsize\it{}et al} in NUCFRG2 \cite{ed.Wilson}: \begin{equation} \begin{array}{c} \xi = \frac{{E_\gamma b_{\min } }}{{\gamma \beta \bar hc}} \\ \\ b_{\min } = (1 + x_d )b_c + \frac{{\pi \alpha _0 }}{{2\gamma }} \\ \\ \alpha _0 = \frac{{Z_P Z_T e^2 }}{{\mu \beta ^2 c^2 }} \\ \\ b_c = 1.34\left[ {A_P^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$3$}}} + A_T^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$3$}}} - 0.75\left( {A_P^{ - {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$3$}}} + A_T^{ - {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$3$}}} } \right)} \right] \\ \end{array} \label{ed.eqn4} \end{equation} \noindent and \(\mu\) is the reduced mass of the projectile/target system, \(x_d=0.25\), and \(A_P\) and \(A_T\) are the projectile and target nucleon numbers. For the last equation, the units of \(b_c\) are fm. Wilson {\normalsize\it{}et al} state that there is an equivalent virtual photon spectrum as a result of the quadrupole field: \begin{equation} 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\} \label{ed.eqn5} \end{equation} \noindent The cross section for the interaction of the dipole and quadrupole fields is given by: \begin{equation} \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 } \label{ed.eqn6} \end{equation} \noindent Wilson {\normalsize\it{}et al} assume that \(\sigma_{E1}(E_{\gamma})\) and \(\sigma_{E2}(E_{\gamma})\) are sharply peaked at the giant dipole and quadrupole resonance energies: \begin{equation} \begin{array}{c} 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}} \\ \\ E_{GQR} = \frac{{63}}{{A_P^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$3$}}} }} \\ \label{ed.eqn7} \end{array} \end{equation} \noindent so that the terms for \(N_{E1}\) and \(N_{E2}\) can be taken out of the integrals in Eq. \ref{ed.eqn6} and evaluated at the resonances. \noindent In Eq. \ref{ed.eqn7}: \begin{equation} \begin{array}{c} u = \frac{{3J}}{{Q'}}A_P^{ - {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$3$}}} \\ \\ R_0 = r_0 A_P^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$3$}}} \\ \end{array} \label{ed.eqn8} \end{equation} \noindent \(\epsilon=0.0768\), \(Q'=17\)MeV, \(J=36.8\)MeV, \(r_0=1.18\)fm, and \(m^*\) is 7/10 of the nucleon mass (taken as 938.95 MeV/c$^2$). (The dipole and quadrupole energies are expressed in units of MeV.) \noindent The photonuclear cross sections for the dipole and quadrupole resonances are assumed to be given by: \begin{equation} \int {\sigma _{E1} \left( {E_\gamma } \right)dE_\gamma = 60\frac{{N_P Z_P }}{{A_P }}} \label{ed.eqn9} \end{equation} \noindent in units of MeV-mb (\(N_P\) being the number of neutrons in the projectile) and: \begin{equation} \int {\sigma _{E2} \left( {E_\gamma } \right)\frac{{dE_\gamma }}{{E_\gamma ^2 }} = 0.22fZ_P A_P^{{\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 3}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$3$}}} } \label{ed.eqn10} \end{equation} \noindent in units of $\mu$b/MeV. In the latter expression, \(f\) is given by: \begin{equation} f = \left\{ {\begin{array}{*{20}c} {0.9} \hfill & {A_P > 100} \hfill \\ {0.6} \hfill & {40 < A_P \le 100} \hfill \\ {0.3} \hfill & {40 \le A_P } \hfill \\ \end{array}} \right. \end{equation} \noindent The total cross section for electromagnetic dissociation is therefore given by Eq. \ref{ed.eqn6} with the virtual photon spectra for the dipole and quadrupole fields calculated at the resonances: \begin{equation} \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 } \label{ed.eqn11} \end{equation} \noindent where the resonance energies are given by Eq. \ref{ed.eqn7} and the integrals for the photonuclear cross sections given by Eq. \ref{ed.eqn9} and Eq. \ref{ed.eqn10}. \noindent The selection of proton or neutron emission is made according to the following prescription from Wilson {\normalsize\it{}et al}. \begin{equation} \begin{array}{l} \sigma _{ED,p} = \sigma _{ED} \times \left\{ {\begin{array}{*{20}c} {0.5} \hfill & {Z_P < 6} \hfill \\ {0.6} \hfill & {6 \le Z_P \le 8} \hfill \\ {0.7} \hfill & {8 < Z_P < 14} \hfill \\ {\min \left[ {\frac{{Z_P }}{{A_P }},1.95\exp ( - 0.075Z_P )} \right]} \hfill & {Z_P \ge 14} \hfill \\ \end{array}} \right\} \\ \sigma _{ED,n} = \sigma _{ED} - \sigma _{ED,p} \\ \end{array} \label{ed.eqn12} \end{equation} \indent Note that this implementation of ED interactions only treats the ejection of single nucleons from the nucleus, and currently does not allow emission of other light nuclear fragments. \section{Status of this document} 19.06.04 created by Peter Truscott \\ \begin{latexonly} \begin{thebibliography}{99} \bibitem{ed.Wilson} J. W. Wilson, R. K. Tripathi, F. A. Cucinotta, J. K. Shinn, F. F. Badavi, S. Y. Chun, J. W. Norbury, C. J. Zeitlin, L. Heilbronn, and J. Miller, "NUCFRG2: An evaluation of the semiempirical nuclear fragmentation database," NASA Technical Paper 3533, 1995. \bibitem{ed.BandB} C. A. Bertulani, and G. Baur, “Electromagnetic processes in relativistic heavy ion collisions,” Nucl Phys, A458, 725-744, 1986. \end{thebibliography} \end{latexonly} \begin{htmlonly} \section{Bibliography} \begin{enumerate} \item J. W. Wilson, R. K. Tripathi, F. A. Cucinotta, J. K. Shinn, F. F. Badavi, S. Y. Chun, J. W. Norbury, C. J. Zeitlin, L. Heilbronn, and J. Miller, "NUCFRG2: An evaluation of the semiempirical nuclear fragmentation database," NASA Technical Paper 3533, 1995. \item C. A. Bertulani, and G. Baur, “Electromagnetic processes in relativistic heavy ion collisions,” Nucl Phys, A458, 725-744, 1986. \end{enumerate} \end{htmlonly}