Relativistic nuclear energy density functionals: Adjusting parameters to binding energies

We study a particular class of relativistic nuclear energy density functionals in which only nucleon degrees of freedom are explicitly used in the construction of effective interaction terms. Short-distance (high-momentum) correlations, as well as intermediate- and long-range dynamics, are encoded in the medium (nucleon-density) dependence of the strength functionals of an effective interaction Lagrangian. Guided by the density dependence of microscopic nucleon self-energies in nuclear matter, a phenomenological ansatz for the density-dependent coupling functionals is accurately determined in self-consistent mean-field calculations of binding energies of a large set of axially deformed nuclei. The relationship between the nuclear matter volume, surface, and symmetry energies and the corresponding predictions for nuclear masses is analyzed in detail. The resulting best-fit parametrization of the nuclear energy density functional is further tested in calculations of properties of spherical and deformed medium-heavy and heavy nuclei, including binding energies, charge radii, deformation parameters, neutron skin thickness, and excitation energies of giant multipole resonances.

Figure 1

The equations of state of symmetric nuclear matter (binding energy as a function of nucleon density) for the eight point-coupling effective interactions of Table , in comparison with the EoS of the meson-exchange effective interaction DD-ME2 [30] and the microscopic EoS of Ref. [36]. The two points from the microscopic EoS on which the point-coupling effective interactions (sets A–H) were adjusted are denoted by larger filled circle symbols.

Figure 2

Surface energy of semi-infinite nuclear matter as a function of the surface thickness, for the eight point-coupling effective interactions of Table . The corresponding values of the strength ${\delta }_S$ of the derivative coupling term in the point-coupling Lagrangian [Eq. (2)] are displayed on the upper horizontal axis. The filled square symbol denotes the surface energy and surface thickness predicted by the meson-exchange effective interaction DD-ME2 [30].

Figure 3

Surface energies of semi-infinite nuclear matter that minimize the deviation of the calculated binding energies from data, for the set of nuclei listed in Table , plotted as functions of the volume energy at saturation, for six values of the symmetry energy $\langle S_2\rangle$.

Figure 4

${\chi }^2$ deviations [Eq. (17)] of the theoretical binding energies from data for the set of deformed nuclei listed in Table and for each combination of the surface, volume, and symmetry energy shown in Fig. 3.

Figure 5

The minimum ${\chi }^2$ deviation of the theoretical binding energies from data, as a function of the volume energy coefficient. Each point represents the minimum of the corresponding curve plotted in Fig. 4.

Figure 6

Absolute deviations of the calculated binding energies from the experimental values of the 64 axially deformed nuclei listed in Table , as functions of (a) the asymmetry coefficient and (b) the mass number. Lines connect nuclei that belong to the isotopic chains shown in the legend. The theoretical binding energies are calculated by using the point-coupling effective interaction characterized by the volume energy $a_v=-16.04$ MeV.

Figure 7

Same as in Fig. 6, but for the point-coupling effective interaction with volume energy $a_v=-16.06$ MeV.

Figure 8

Same as in Fig. 6, but for the point-coupling effective interaction with volume energy $a_v=-16.08$ MeV.

Figure 9

Same as in Fig. 6, but for the point-coupling effective interaction with volume energy $a_v=-16.10$ MeV.

Figure 10

Same as in Fig. 6, but for the point-coupling effective interaction with volume energy $a_v=-16.12$ MeV.

Figure 11

Same as in Fig. 6, but for the point-coupling effective interaction with volume energy $a_v=-16.14$ MeV.

Figure 12

Absolute deviations of the calculated binding energies from the experimental values of the 64 axially deformed nuclei listed in Table , as functions of (a) the asymmetry coefficient and (b) the mass number. Lines connect nuclei that belong to the isotopic chains shown in the legend. The theoretical binding energies are calculated by using the meson-exchange effective interaction DD-ME2 [30].

Figure 13

(a) Vector and (b) scalar nucleon self-energies in symmetric nuclear matter as functions of the nucleon density. The self-energies that correspond to the phenomenological density functional DD-PC1 are compared with the Hartree-Fock self-energies [33] calculated from the Idaho N${}^3\mathrm{LO}$ $\mathit{NN}$ potential [34].

Figure 14

Charge radii of Nd, Sm, Gd, Dy, Er, and Yb isotopic chains. The results of the RMF $+$ BCS calculation with the DD-PC1 and DD-ME2 interactions are compared with data [46].

Figure 15

DD-PC1 and DD-ME2 predictions for the ground-state quadrupole deformations ${\beta }_2$ of the Nd, Sm, Gd, Dy, Er, and Yb isotopes, in comparison with empirical values [48].

Figure 16

Absolute deviations of the DD-PC1 binding energies from experimental values of deformed nuclei in the mass regions $A\approx 120\text{−}130,A\approx 150\text{−}180$, and $A\approx 230\text{−}250$, as functions of (a) the asymmetry coefficient and (b) the mass number.

Figure 17

Absolute deviations of the calculated binding energies from experimental values for (a) the Pb and (b) the Sn isotopic chains, as functions of the mass number. The theoretical binding energies are calculated by using the RMF $+$ BCS model with the point-coupling effective interaction DD-PC1 and the finite-range meson-exchange interaction DD-ME2.

Figure 18

Comparison between experimental (left) and DD-PC1 (right) single-nucleon spectra of (a) protons and (b) neutrons for ${}^{208}\mathrm{Pb}$ and ${}^{132}\mathrm{Sn}$. The experimental spectra are from Ref. [49].

Figure 19

RMF $+$ BCS model predictions for the charge radii of (a) Pb and (b) Sn isotopes, calculated with the DD-PC1 and DD-ME2 effective interactions, and compared with data [46].

Figure 20

RMF $+$ BCS model predictions for the differences between the neutron and proton rms radii of (a) Pb and (b) Sn isotopes, calculated with the DD-PC1 and DD-ME2 effective interactions, in comparison with available data [56, 57, 58].

Figure 21

(a) Isoscalar monopole and (b) isovector dipole strength distributions in ${}^{208}\mathrm{Pb}$ calculated with the relativistic RPA by using the effective interaction DD-PC1. The experimental excitation energies are denoted by arrows: $13.96\pm 0.2$ [59] for the giant monopole resonance and $13.3\pm 0.1$ [60] for the giant dipole resonance, respectively.

Figure 22

The RQRPA isovector dipole strength functions in ${}^{116,118,120,124}\mathrm{Sn}$, calculated with the DDPC1 effective interaction. The experimental IVGDR excitation energies [61] are denoted by arrows.

Figure 23

The RQRPA isoscalar monopole strength distributions in even-even ${}^{112\text{−}124}\mathrm{Sn}$ nuclei, calculated with the DDPC1 effective interaction. Arrows denote the positions of experimental ISGMR excitation energies [62].