Full leading-order nuclear polarization in highly charged ions

The nuclear-polarization corrections to the energy levels of highly charged ions are systematically investigated to leading order in the fine-structure constant. To this end, the notion of effective photon propagators with nuclear-polarization insertions is employed, where the nuclear excitation spectrum is calculated by means of the Hartree-Fock-based random-phase approximation. The effective Skyrme force is used to describe the interaction between nucleons, and the model dependence is analyzed. To leading order, the formalism predicts two contributions given by the effective vacuum-polarization and self-energy diagrams. The existing ambiguity around the vacuum-polarization term is resolved by demonstrating that it is effectively absorbed in the standard finite-nuclear-size correction. The self-energy part is evaluated with the full electromagnetic electron-nucleus interaction taken into account, where the importance of the effects of the nuclear three-currents is emphasized.

Figure 1

Feynman rule for evaluating the NP correction to the photon propagator.

Figure 2

(a) Effective self-energy and (b) vacuum-polarization diagrams as a result of applying the NP insertion (shaded circle) to the photon propagator (wavy line). The former can be interpreted as a two-photon exchange between the bound atomic electron (double line) and the nucleus (single solid line), which includes the ladder, the cross, and the seagull diagrams; while the latter can be understood as the interaction of the induced vacuum polarization with the bound electron via additional virtual nuclear excitations.

Figure 3

Contributions from different intermediate electronic states to the sum of the ladder and the cross parts of the SE-NP correction to the $1s_{1/2}$ state of ${}^{208}\mathrm{Pb}^{81+}$ for single fixed (a) $1^{-}$ and (b) $2^{+}$ nuclear excitations with the energies of 12.8 MeV and 12.2 MeV, respectively. Dashed lines represent the Coulomb approximation, while the solid lines correspond to the full electron-nucleus interaction including the transverse part. Different colors are used for the two allowed values of the relativistic angular number $\kappa$ of the electronic states in both cases. The nuclear part of the calculations is performed using the SLy5 Skyrme parametrization.