Low-lying dipole response: Isospin character and collectivity in ${}^{68}$Ni, ${}^{132}$Sn, and ${}^{208}$Pb

The isospin character, the collective or single-particle nature, and the sensitivity to the slope of the nuclear symmetry energy of the low-energy isovector dipole response (known as pygmy dipole resonance) are nowadays under debate. In the present work we study, within the fully self-consistent nonrelativistic mean field (MF) approach based on Skyrme Hartree-Fock plus random phase approximation (RPA), the measured even-even nuclei ${}^{68}$Ni, ${}^{132}$Sn, and ${}^{208}$Pb. To analyze the model dependence in the predictions of the pygmy dipole strength, we employ three different Skyrme parameter sets. We find that both the isoscalar and the isovector dipole responses of all three nuclei show a low-energy peak that increases in magnitude, and is shifted to larger excitation energies, with increasing values of the slope of the symmetry energy at saturation. We highlight the fact that the collectivity associated with the RPA state(s) contributing to this peak is different in the isoscalar and isovector case, or in other words it depends on the external probe. While the response of these RPA states to an isovector operator does not show a clear collective nature, the response to an isoscalar operator is recognizably collective, for all analyzed nuclei and all studied interactions.

Figure 1

Strength function in the case of the SLy5 interaction [40] for a test nucleus (${}^{208}$Pb) as a function of the excitation energy for three cases: (i) the spurious state has not been subtracted (solid line), (ii) the spurious state has been subtracted by correcting the isoscalar dipole operator Eq. (9) (dashed line), and (iii) the spurious state has been subtracted as explained in the Appendix (dot-dashed line).

Figure 2

Strength function corresponding to the isovector (a) and isoscalar (b) dipole response of ${}^{208}$Pb as a function of the excitation energy. The inset in (a) displays in a larger scale the pygmy region. In both figures the predictions of SGII, SLy5, and SkI3 are depicted. Black arrows indicate the experimental centroid energies for the PDS ($E=7.37$ MeV within a window of 6–8 MeV) [20], for the ISGDR ($E=20.3\pm 2$ MeV [58]) and the energy peak for the IVGDR ($E=13.43$ MeV and a total width of $2.42$ MeV [59]).

Figure 3

Same as Fig. 2 for ${}^{132}$Sn. The experimental value for the peak energy of the PDS ($E=9.8\pm 0.7$ MeV) is indicated by a black arrow [18].

Figure 4

Same as Fig. 2 for ${}^{68}$Ni. The experimental value for the peak energy of the PDS ($E=11$ MeV and an energy width estimated to be less than 1 MeV) is indicated by a black arrow [17].

Figure 5

Reduced transition probabilities for the isovector dipole response [(a), (c), and (e)] and isoscalar dipole response [(b), (d), and (f)], in the case of ${}^{208}$Pb [(a) and (b)], ${}^{132}$Sn [(c) and (d)] and ${}^{68}$Ni [(e) and (f)] in s.p. units, as a function of the excitation energy and as predicted by the selected MF interactions. Note that we only show the energy region relevant for our study of the RPA-pygmy state.

Figure 6

Neutron and proton [(a), (c), and (e)] and isoscalar and isovector [(b), (d), and (f)] transition densities of the RPA-pygmy state, as a function of the radial distance, for ${}^{208}$Pb [(a) and (b)], ${}^{132}$Sn [(c) and (d)], and ${}^{68}$Ni [(e) and (f)]. The predictions of SGII, SLy5, and SkI3 are shown. Proton ($r_p$) and neutron ($r_n$) rms radii are indicated for each interaction by the edges of the grey region.

Figure 7

Strength function corresponding to the isovector dipole response of ${}^{208}$Pb (a), ${}^{132}$Sn (b) and ${}^{68}$Ni (c) as a function of the excitation energy (solid lines), and partial contribution due to those states which are at least 70$\%$ isoscalar (dashed line): for each of the panels, the RPA states that are at least 70$\%$ isoscalar between 0 and $R$ are considered in the panel at the left hand side, those which have this feature between 0 and $R/2$ are considered in the central panel, and those which are like that between $R/$2 and $R$ are considered in the panel at the right hand side. See text for further explanations.

Figure 8

Proton and neutron single particle levels of ${}^{208}$Pb as predicted by the different MF models (a). The Fermi level is indicated by a dashed black line. All ph contributions to the isovector reduced amplitude corresponding to the ${}^{208}$Pb RPA-pygmy state as a function of the ph excitation energy (c). All studied models are shown. Largest neutron ph contributions are also listed in decreasing order from top to bottom. Same as (b) but for the isoscalar reduced amplitude (c).

Figure 9

Same as Fig. 8 for the case of ${}^{132}$Sn.

Figure 10

Same in Fig. 8 for the case of ${}^{68}$Ni.