Global performance of covariant energy density functionals: Ground state observables of even-even nuclei and the estimate of theoretical uncertainties

Covariant density functional theory is a modern theoretical tool for the description of nuclear structure phenomena. The current investigation aims at the global assessment of the accuracy of the description of the ground state properties of even-even nuclei. We also estimate theoretical uncertainties defined here as the spreads of predictions within four covariant energy density functionals in known regions of the nuclear chart and their propagation towards the neutron drip line. Large-scale axial relativistic Hartree-Bogoliubov calculations are performed for all $Z\le 104$ even-even nuclei between the two-proton and two-neutron drip lines with four modern covariant energy density functionals such as NL3*, DD-ME2, DD-ME$\delta$, and DD-PC1. The physical observables of interest include the binding energies, two-particle separation energies, charge quadrupole deformations, isovector deformations, charge radii, neutron skin thicknesses, and the positions of the two-proton and two-neutron drip lines. The predictions for the two-neutron drip line are also compared in a systematic way with the ones obtained in nonrelativistic models.

Figure 1

The nuclei (solid squares), shown in the $(N,Z)$ plane, which were used in the fit of indicated CDFT parametrizations. Their total number is shown below the parametrization label. Magic shell closures are shown by dashed lines.

Figure 2

Experimental and calculated neutron three-point indicators ${\Delta }^{\left(3\right)}\left(N\right)$ and calculated pairing gaps ${\Delta }_{\mathrm{uv}}$ and ${\Delta }_{\mathrm{lcs}}$ as functions of the neutron number $N$. Theoretical ${\Delta }^{\left(3\right)}\left(N\right)$ indicators shown by open red circles are derived from calculated binding energies of odd- and even-even nuclei; they are obtained in RHB calculations with the CEDF NL3* and the Gogny force D1S of Eq. (24) in the pairing channel. The calculated pairing gaps ${\Delta }_{\mathrm{uv}}$ and ${\Delta }_{\mathrm{lcs}}$ are shown by lines. They are calculated in even-even nuclei with the Gogny force D1S (labeled as “Gogny D1S”) and its separable approximation in Eq. (25) (labeled as “separable” in the figure).

Figure 3

The same as Fig. 2 but for proton three-point indicators ${\Delta }^{\left(3\right)}\left(Z\right)$ and proton pairing gaps ${\Delta }_{\mathrm{uv}}$ and ${\Delta }_{\mathrm{lcs}}$ as a function of proton number $Z$. Note that it was not possible to get a convergence for a few odd-mass nuclei in the $N=28$ and $N=82$ isotone chains in the RHB calculations with Gogny D1S force in pairing channel. This leads to the absence of theoretical ${\Delta }^{\left(3\right)}$ values in some proton number range.

Figure 4

Calculated neutron pairing gaps ${\Delta }_{\mathrm{uv}}$ and ${\Delta }_{\mathrm{lcs}}$ as a function of the neutron number $N$ for different isotonic chains. The results of RHB calculations with the separable pairing force (25) are presented for the indicated CEDF's.

Figure 5

The same as Fig. 4 but for calculated proton pairing gaps ${\Delta }_{\mathrm{uv}}$ and ${\Delta }_{\mathrm{lcs}}$ as a function of the proton number $Z$ for different isotopic chains.

Figure 6

The difference between theoretical and experimental masses of 835 even-even nuclei investigated in RHB calculations with indicated CEDF's. If $E_{\text{th}}-E_{\text{exp}}<0$, the nucleus is more bound in the calculations than in experiment.

Figure 7

The relative accuracy of the description of experimental masses in our model calculations. The same set of data as in Fig. 6 is used. Dashed lines show the $\pm 0.5\%$ error band.

Figure 8

The binding energy spreads $\Delta E(Z,N)$ as a function of proton and neutron number. $\Delta E(Z,N)=|E_{\max }(Z,N)-E_{\min }(Z,N)|$, where $E_{\max }(Z,N)$ and $E_{\min }(Z,N)$ are the largest and the smallest binding energies for each ($N,Z$) nucleus obtained with the four CEDF's used in this investigation.

Figure 9

The same as Fig. 8, but only for DD-ME2 and DD-ME$\delta$.

Figure 10

Two-neutron separation energies $S_{2n}(Z,N)$ given for different isotopic chains as a function of neutron number. To facilitate the comparison between theory and experiment, five different colors are used periodically as a function of neutron number. Black, red, green, orange, and blue colors are used for isotope chains with proton numbers ending with 2, 4, 6, 8, and 0, respectively.

Figure 11

Two-proton separation energies $S_{2p}(Z,N)$ given for different isotonic chains as a function of proton number. To facilitate the comparison between theory and experiment, five different colors are used periodically as a function of proton number. Black, red, green, orange, and blue colors are used for isotonic chains with neutron numbers ending with 2, 4, 6, 8, and 0, respectively.

Figure 12

Schematic illustration of the dependence of the accuracy of the prediction for the position of the two-particle drip line on the slope of the two-particle separation energy curve as a function of the relevant particle number. The error bars for the calculated results show typical rms deviations (1 MeV) between theory and experiment (Table 3). If these error bars would be taken into account (as it is effectively done when different CEDF's are compared), they would lead to the possible ranges of particle numbers corresponding to the two-particle drip line shown by arrows. For particle-bound nuclei the results for DD-PC1 are used. The separation energies for particle unbound nuclei ($S_{2n,2p}<0$) represent extrapolations. They are used here only for illustration purposes.

Figure 13

The calculated two-proton drip lines versus experimental data. For each isotope chain, the four experimentally known most proton-rich nuclei are shown by squares. Cyan shading of the squares is used for the nuclei located beyond the two-proton drip line ($S_{2p}<0)$. The experimental data are from Ref. [71]. The borderline between shaded and open squares delineates the known two-proton drip lines. Only in the cases of the $Z=4$, 6, 8, 80, 82, and 84 isotope chains the locations of two-proton drip lines are firmly established because the masses of the nuclei on both sides of the drip line are directly and accurately measured. The two-proton drip line is only tentatively delineated for other isotope chains because either the masses of beyond the drip line nuclei are only estimated in Ref. [71] or beyond the drip line nuclei are not known experimentally. The red lines with small symbols show the calculated two-proton drip lines which go along the last two-proton bound nuclei.

Figure 14

Two-neutron drip lines obtained in state-of-the-art DFT calculations. The regions of well-defined localization of the two-neutron drip line are encircled.

Figure 15

The same as in Fig. 14 but with the four most neutron-rich two-neutron drip lines shown in color and the rest in black.

Figure 16

The same as in Fig. 14 but with the four least neutron-rich two-neutron drip lines shown in color and the rest in black.

Figure 17

Charge quadrupole deformations ${\beta }_2$ obtained in the RHB calculations with indicated CEDF's.

Figure 18

Proton quadrupole deformation spreads $\Delta {\beta }_2(Z,N)$ as a function of proton and neutron number. $\Delta {\beta }_2(Z,N)=|{\beta }_2^{\max }(Z,N)-{\beta }_2^{\min }(Z,N)|$, where ${\beta }_2^{\max }(Z,N)$ and ${\beta }_2^{\min }(Z,N)$ are the largest and smallest proton quadrupole deformations obtained with four employed CEDF's for the $(Z,N)$ nucleus.

Figure 19

Proton hexadecapole deformations ${\beta }_4$ obtained in the RHB calculations with the indicated CEDF's.

Figure 20

Proton hexadecapole deformation spreads $\Delta {\beta }_4(Z,N)$ as a function of the proton and neutron numbers. $\Delta {\beta }_4(Z,N)=|{\beta }_4^{\max }(Z,N)-{\beta }_4^{\min }(Z,N)|$, where ${\beta }_4^{\max }(Z,N)$ and ${\beta }_4^{\min }(Z,N)$ are the largest and smallest proton hexadecapol deformations obtained with four employed CDFT parametrizations for the $(Z,N)$ nucleus.

Figure 21

Isovector ${\beta }_2^{IV}={\beta }_2\left(\nu \right)-{\beta }_2\left(\pi \right)$ deformations obtained in the RHB calculations with the indicated CDFT parametrizations.

Figure 22

Isovector quadrupole deformation spreads $\Delta {\beta }_2^{IV}(Z,N)$ as a function of proton and neutron number. $\Delta {\beta }_2^{IV}(Z,N)=|{\beta }_{2,\max }^{IV}(Z,N)-{\beta }_{2,\min }^{IV}(Z,N)|$, where ${\beta }_{2,\max }^{IV}(Z,N)$ and ${\beta }_{2,\min }^{IV}(Z,N)$ are the largest and smallest isovector quadrupole deformations, respectively, obtained with four CDFT parametrizations for the $(Z,N)$ nucleus.

Figure 23

Experimental and theoretical charge radii as a function of neutron number. The calculations are performed with DD-PC1. Black, red, green, orange, and blue colors are used for isotope chains with proton numbers ending with 2, 4, 6, 8, and 0, respectively. The experimental data are taken from Ref. [128]. Panels (b), (c), and (d) show the comparison in an enlarged scale.

Figure 24

Charge radii spread $\Delta r_{\mathrm{ch}}(Z,N)$ as a function of proton and neutron number. $\Delta r_{\mathrm{ch}}(Z,N)=|r_{\mathrm{ch}}^{\max }(Z,N)-r_{\mathrm{ch}}^{\min }(Z,N)|$, where $r_{\mathrm{ch}}^{\max }(Z,N)$ and $r_{\mathrm{ch}}^{\min }(Z,N)$ are the largest and the smallest charge radii obtained with the four CDFT parameterizations for the $(Z,N)$ nucleus.

Figure 25

Neutron skin thicknesses obtained in RHB calculations with several CEDF's.

Figure 26

Neutron skin thickness spreads $\Delta r_{\mathrm{skin}}(Z,N)$ as a function of proton and neutron number. $\Delta r_{\mathrm{skin}}(Z,N)=|r_{\mathrm{skin}}^{\max }(Z,N)-r_{\mathrm{skin}}^{\min }(Z,N)|$, where $r_{\mathrm{skin}}^{\max }(Z,N)$ and $r_{\mathrm{skin}}^{\min }(Z,N)$ are the largest and smallest proton hexadecapol deformations obtained with four CDFT parametrizations for the $(Z,N)$ nucleus.

Figure 27

The same as Fig. 26 but for neutron skin thickness spreads obtained with exclusion of the NL3*.