Giant quadrupole resonances in ${}^{208}$Pb, the nuclear symmetry energy, and the neutron skin thickness

Recent improvements in the experimental determination of properties of the isovector giant quadrupole resonance (IVGQR), as demonstrated in the $A=208$ mass region, may be instrumental for characterizing the isovector channel of the effective nuclear interaction. We analyze properties of the IVGQR in ${}^{208}$Pb, using both macroscopic and microscopic approaches. The microscopic method is based on families of nonrelativistic and covariant energy density functionals (EDF), characterized by a systematic variation of isoscalar and isovector properties of the corresponding nuclear matter equations of state. The macroscopic approach yields an explicit dependence of the nuclear symmetry energy at some subsaturation density, for instance $S(\rho =0.1$ fm${}^{-3})$, or the neutron skin thickness $\Delta r_{np}$ of a heavy nucleus, on the excitation energies of isoscalar and isovector GQRs. Using available data it is found that $S(\rho =0.1$ fm${}^{-3})=23.3\pm 0.6$ MeV. Results obtained with the microscopic framework confirm the correlation of the $\Delta r_{np}$ to the isoscalar and isovector GQR energies, as predicted by the macroscopic model. By exploiting this correlation together with the experimental values for the isoscalar and isovector GQR energies, we estimate $\Delta r_{np}=0.14\pm 0.03$ fm for ${}^{208}$Pb, and the slope parameter of the symmetry energy: $L=37\pm 18$ MeV.

Figure 1

Isoscalar (a) and isovector (b) quadrupole strength functions. The strengths are calculated within the RPA for SAMi, KDE, SkI3, NL3, and DD-ME2. The experimental energies for the ISGQR $(10.9\pm 0.1$ MeV$)$, and the IVGQR $(22.7\pm 0.2$ MeV$)$ (weighted averages) listed in Table are indicated by arrows.

Figure 2

Neutron and proton (a) and isoscalar and isovector (b) transition densities for the main peak of the isovector response, as a function of the radial distance. The predictions, calculated within the RPA, for the SAMi and DD-ME2 functionals are shown. The proton ($r_p$) and neutron ($r_n$) rms radii are indicated by the edges of the shaded region.

Figure 3

(a) Diagrams contributing to the strength function of the GR. (b) Probability $P$ to find the IVGQR state at an energy $E$. Different curves are obtained when the phonons listed in the legend are used as intermediate states. The label RPA [black-dashed line in (b)] refers to the curve calculated in the RPA with a Lorentzian width of 1 MeV.

Figure 4

Excitation energy of the ISGQR in ${}^{208}$Pb as a function of $\sqrt{m/m^{*}}$, calculated with the SAMi-m and SAMi-J family of functionals. On the horizontal upper axis we display the corresponding values for $m^{*}/m$. The data from Table are also included (square and shaded band).

Figure 5

Square of the excitation energy of the IVGQR in ${}^{208}$Pb as a function of the effective mass (a), and the neutron skin thickness (b),(c), predicted by the SAMi-m and SAMi-J [(a) and (b)], and DD-ME (c) families of energy density functionals.

Figure 6

Values of $\Delta r_{np}$ in ${}^{208}$Pb as functions of $[{\left(E_x^{\mathrm{IV}}\right)}^2-2{\left(E_x^{\mathrm{IS}}\right)}^2]/J$, calculated with the SAMi-J and DD-ME functionals. The dashed line and shaded band indicate the experimental value and corresponding uncertainty (see text).

Figure 7

Neutron skin thickness $\Delta r_{np}$ of ${}^{208}$Pb as a function of the slope parameter of the symmetry energy at saturation density, for the two families of functionals: SAMi-J and DDME.