
\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}

