Number of virtual photons in Coulomb excitation

We calculate the number of virtual photons for Coulomb dipole excitation (CDE; $l=1$), Coulomb quadrupole excitation (CQE; $l=2$), and Coulomb magnetic excitation (CME; $l=1$) in the $^{8}\mathrm{B}+ ^{208}\mathrm{Pb}$ system at a laboratory energy of ${E}_{\text{lab}}=83$ MeV/nucleon. Virtual photon counts are obtained using both general and relativistic methods for CDE, CQE, and CME. For the CDE, the number of virtual photons obtained using the general method is highly consistent with that derived using the relativistic method. In contrast, the CQE analysis reveals a marginal discrepancy between the two methods. Nonetheless, when retardation effects are included in the general method, the differences became negligible. The heuristically derived nonrelativistic expression matches the general method at low energies, and consistency among all three approaches (CDE, CQE, and CME) is achieved only under the simultaneous conditions $\ensuremath{\gamma} \ensuremath{\rightarrow}$ 1, $\ensuremath{\varepsilon} \ensuremath{\rightarrow}$ 1, and $\ensuremath{\zeta} \ensuremath{\rightarrow}$ 0. Subsequently, employing the virtual photon numbers and Coulomb dissociation cross sections, we compute the Coulomb strength distributions for CDE, CQE, and CME. The peak of the Coulomb strength distribution for CDE analysis demonstrates that Coulomb excitation-induced breakup reactions are more significant in neutron-rich nuclei than in proton-rich nuclei. Using the Coulomb electric dipole strength distributions, we calculate the imaginary CDE potential.