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 (i.e. deuterons, \(\alpha\)-particles, 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 et al [WTC+95], and Bertulani and Baur [BaGB86].

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 [BaGB86]:

(276)\[N(E_\gamma, b) = \frac{\alpha Z_T^2}{\pi^2 \beta^2 b^2 E_\gamma} \left\{ x^2 k_1^2 (x) + \left( \frac{x^2}{\gamma^2} \right) k_0^2 (x) \right\}\]

where \(x\) is a dimensionless quantity defined as:

\[x = \frac{bE_\gamma}{\gamma \beta \hbar c}\]

and:

\[\begin{split}\alpha &= \mbox{fine structure constant}\\ \beta &= \mbox{ratio of the velocity of the projectile in the laboratory frame to the velocity of light}\\ \gamma &= \mbox{Lorentz factor for the projectile in the laboratory frame}\\ b &= \mbox{impact parameter}\\ c &= \mbox{speed of light}\\ \hbar &= \mbox{quantum constant}\\ E_\gamma &= \mbox{energy of virtual photon}\\ k_0 \mbox{ and } k_1 &= \mbox{zeroth and first order modified Bessel functions of the second kind}\\ Z_T &= \mbox{atomic number of the target nucleus}\end{split}\]

Integrating Eq. (276) over the impact parameter from \(b_{min}\) to \(\infty \) produces the virtual photon spectrum for the dipole field of:

\[N_{E1} (E_\gamma) = \frac{2\alpha Z_T^2}{\pi \beta^2 E_\gamma} \left\{ \xi k_0 (\xi) k_1(\xi) - \frac{\xi^2 \beta^2}{2} \left( k_1^2(\xi) - k_0^2(\xi) \right) \right\}\]

where, according to the algorithm implemented by Wilson et al in NUCFRG2 [WTC+95]:

\[\begin{split}\xi &= \frac{E_\gamma b_{min}}{\gamma \beta \hbar c} \\ b_{min} &= (1 + x_d) b_c + \frac{\pi \alpha_0}{2\gamma} \\ \alpha_0 &= \frac{Z_P Z_T e^2}{\mu \beta^2 c^2} \\ b_c &= 1.34 \left[ A_P^{1/3} + A_T^{1/3} - 0.75 \left( A_P^{-1/3} + A_T^{-1/3} \right) \right]\end{split}\]

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 et al state that there is an equivalent virtual photon spectrum as a result of the quadrupole field:

\[N_{E2} (E_\gamma) = \frac{2\alpha Z_T^2}{\pi \beta^4 E_\gamma} \left\{ 2(1-\beta^2) k_1^2(\xi) + \xi (2-\beta^2)^2 k_0(\xi)k_1(\xi ) - \frac{\xi^2 \beta^4}{2} \left(k_1^2(\xi )-k_0^2(\xi) \right) \right\}\]

The cross section for the interaction of the dipole and quadrupole fields is given by:

(277)\[\sigma_{ED} = \int N_{E1}(E_\gamma) \sigma_{E1} (E_\gamma) dE_\gamma + \int N_{E2} (E_\gamma) \sigma_{E2} (E_\gamma) dE_\gamma\]

Wilson et al assume that \(\sigma_{E1}(E_{\gamma})\) and \(\sigma_{E2}(E_{\gamma})\) are sharply peaked at the giant dipole and quadrupole resonance energies:

(278)\[\begin{split}E_{GDR} &= \hbar c \left[ \frac{m^* c^2 R_0^2}{8J} \left( 1+u-\frac{1 + \varepsilon + 3u}{1+ \varepsilon + u} \varepsilon \right) \right]^{-1/2} \\ E_{GQR} &= \frac{63}{A_P^{1/3}}\end{split}\]

so that the terms for \(N_{E1}\) and \(N_{E2}\) can be taken out of the integrals in Eq. (277) and evaluated at the resonances.

In Eq. (278):

\[\begin{split}u &= \frac{3J}{Q'}A_P^{-1/3} \\ R_0 &= r_0 A_P^{1/3}\end{split}\]

\(\epsilon=0.0768\), \(Q'=17\) MeV, \(J=36.8\)eV, \(r_0=1.18\) fm, and \(m^*\) is 7/10 of the nucleon mass (taken as 938.95 MeV/c²). (The dipole and quadrupole energies are expressed in units of MeV.)

The photonuclear cross sections for the dipole and quadrupole resonances are assumed to be given by:

(279)\[\int \sigma_{E1} (E_\gamma) dE_\gamma = 60\frac{N_P Z_P}{A_P}\]

in units of MeV-mb (\(N_P\) being the number of neutrons in the projectile) and:

(280)\[\int \sigma_{E2} (E_\gamma) \frac{dE_\gamma}{E_\gamma^2} = 0.22 fZ_P A_P^{2/3}\]

in units of \(\mu\)b/MeV. In the latter expression, \(f\) is given by:

\[\begin{split}f = \left\{ \begin{array}{ll} 0.9 & A_P > 100 \\ 0.6 & 40 < A_P \le 100 \\ 0.3 & 40 \le A_P \\ \end{array} \right.\end{split}\]

The total cross section for electromagnetic dissociation is therefore given by Eq. (277) with the virtual photon spectra for the dipole and quadrupole fields calculated at the resonances:

\[\sigma_{ED} = N_{E1} (E_{GDR}) \int \sigma_{E1} (E_\gamma) dE_\gamma + N_{E2} (E_{GQR}) E_{GQR}^2 \int \frac{\sigma_{E2} (E_\gamma)}{E_\gamma^2} dE_\gamma\]

where the resonance energies are given by Eq. (278) and the integrals for the photonuclear cross sections given by Eq. (279) and Eq. (280).

The selection of proton or neutron emission is made according to the following prescription from Wilson et al.

\[\begin{split} \sigma _{ED,p} = \sigma _{ED} \times \left\{ \begin{array}{ll} 0.5 & Z_P < 6 \\ 0.6 & 6 \le Z_P \le 8 \\ 0.7 & 8 < Z_P < 14 \\ \min \left[ \frac{Z_P}{A_P}, 1.95 \exp (-0.075Z_P) \right] & Z_P \ge 14 \\ \end{array} \right.\end{split}\]

and

\[\sigma_{ED,n} = \sigma_{ED} - \sigma_{ED,p}\]

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.

Bibliography

BaGB86(1,2)

C. A. Bertulani and and G. Baur. Electromagnetic processes in relativistic heavy ion collisions. Nucl. Phys. A, 458:725–744, 1986.

WTC+95(1,2)

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 and J Miller. Nucfrg2: an evaluation of the semiempirical nuclear fragmentation database. Technical Report 3533, NASA Technical Paper, 1995.