Constraints on the neutron skin and symmetry energy from the anti-analog giant dipole resonance in ${}^{208}\mathrm{Pb}$

We investigate the impact of the neutron skin thickness, $\mathrm{\Delta }{R}_{np}$, on the energy difference between the anti-analog giant dipole resonance (AGDR), $E_{\mathrm{AGDR}}$, and the isobaric analog state (IAS), $E_{\mathrm{IAS}}$, in a heavy nucleus such as ${}^{208}\mathrm{Pb}$. For guidance, we first develop a simple and analytic, yet physical, approach based on the droplet model that linearly connects the energy difference $E_{\mathrm{AGDR}}-E_{\mathrm{IAS}}$ with $\mathrm{\Delta }{R}_{np}$. To test this correlation on more fundamental grounds, we employ a family of systematically varied Skyrme energy density functionals where variations on the value of the symmetry energy at saturation density $J$ are explored. The calculations have been performed within the fully self-consistent Hartree-Fock (HF) plus charge-exchange random phase approximation (RPA) framework. We confirm the linear correlation within our microscopic approach and we can compare our results with available experimental data in ${}^{208}\mathrm{Pb}$ in order to extract a preferred value for $\mathrm{\Delta }{R}_{np}$ and, in turn, for the symmetry energy parameters. Averaging the results from two available experimental data, our analysis gives $\mathrm{\Delta }{R}_{np}$ = 0.$236\pm 0.018$ fm, $J$ = 33.$2\pm 1.0$ MeV, and a slope parameter of the symmetry energy at saturation $L$ = 97.$3\pm 11.2$ MeV. The errors include the experimental uncertainties and a lower-limit estimate of model uncertainties. These results are consistent with those extracted from different experimental data albeit $L$ and $\mathrm{\Delta }{R}_{np}$ are somewhat large when compared to previous estimations based on giant resonance studies. Possible hints whether model dependence can explain this difference are provided.

Figure 1

Various states related to the target and daughter nucleus, including the ground state, the IVGDR and $M1$ excitations of the target nucleus (isospin = $T_0)$, and the IAS (isospin = $T_0)$, anti-analog states (isospin = $T_0-1)$ in the daughter nucleus excited in a $\left(p,n\right)$ reaction.

Figure 2

The (a) IAS and (b) AGDR response functions calculated by using the SAMi-J Skyrme energy density functionals. The discrete RPA peaks have been smeared out by using Lorentzian functions with (a) 300 keV and (b) 3 MeV width.

Figure 3

Excitation energies of the IAS and AGDR in ${}^{208}\mathrm{Pb}$ as a function of effective mass and symmetry energy at saturation density. The filled boxes are the results of the SAMi-m family, while the filled circles are those of the SAMi-J family. The top circle in the line of circles of each window corresponds to the lowest $J$ value $(J=27$ MeV) and by going down one increases the $J$ value in steps of 1 MeV. The lowest circle then corresponds to the maximum value, $J=35$ MeV.

Figure 4

The energy difference $E_{\mathrm{AGDR}}-E_{\mathrm{IAS}}$ of AGDR and IAS as a function of neutron skin thickness, obtained by using the SAMi-J family of Skyrme functionals consistently. The calculated values are presented as solid circles. Two different experimental data [54, 57] are also shown as solid (magenta) and dashed (blue) lines, respectively. The arrows indicate the neutron skin constrained by these experimental data. We also display results obtained with the covariant DD-ME Lagrangians of Refs. [52, 54]. Interestingly, such models predict the same kind of correlation although with a different slope.

Figure 5

(a) and (b) show the correlations between the neutron skin thickness and either the symmetry energy $J$ at saturation density or the corresponding slope parameter $L$, respectively. The constraints provided by the experimental data have been already shown in Fig. 4.

Figure 6

The values of the slope parameter $L$ and symmetry energy $J$ at saturation density extracted in the current work are compared with the values extracted from other experimental data with several different methods.