Covariant density functional theory: Reexamining the structure of superheavy nuclei

A systematic investigation of even-even superheavy elements in the region of proton numbers $100\le Z\le 130$ and in the region of neutron numbers from the proton-drip line up to neutron number $N=196$ is presented. For this study we use the five most up-to-date covariant energy density functionals of different types, with a nonlinear meson coupling, with density-dependent meson couplings, and with density-dependent zero-range interactions. Pairing correlations are treated within relativistic Hartree-Bogoliubov theory based on an effective separable particle-particle interaction of finite range and deformation effects are taken into account. This allows us to assess the spread of theoretical predictions within the present covariant models for the binding energies, deformation parameters, shell structures, and $\alpha$-decay half-lives. Contrary to the previous studies in covariant density functional theory, it was found that the impact of $N=172$ spherical shell gap on the structure of superheavy elements is very limited. Similar to nonrelativistic functionals, some covariant functionals predict the important role played by the spherical $N=184$ gap. For these functionals (NL3*, DD-ME2, and PC-PK1) there is a band of spherical nuclei along and near the $Z=120$ and $N=184$ lines. However, for other functionals (DD-PC1 and DD-$\mathrm{ME}\delta$) oblate shapes dominate at and in the vicinity of these lines. Available experimental data are, in general, described with comparable accuracy and do not make it possible to discriminate between these predictions.

Figure 1

Neutron (bottom panels) and proton (top panels) single-particle states at spherical shape in the SHEs ${}^{292}120$ and ${}^{304}120$. They are determined with the indicated CEDFs in RMF calculations without pairing. Solid and dashed connecting lines are used for positive- and negative-parity states. Spherical gaps are indicated. The different CEDFs have been arranged in such a way that we first start from the functionals with a nonlinear $\sigma$ coupling (NL1 [16], NL3 [17], NL3* [20]), i.e., with a density dependence in the isoscalar channel; then we continue with the functional FSUGold [49], which has, in addition, a nonlinear coupling between $\omega$ and $\rho$ mesons and therefore also a density dependence in the isovector channel; then we plot the results for the functionals with explicit density dependent meson-nucleon couplings in all channels (DD-ME2 [18] and DD-$\mathrm{ME}\delta$ [22]) and finally we end with the point coupling functionals (DD-PC1 [19] and PC-PK1 [21]).

Figure 2

(a) Neutron and proton shell gaps $\mathrm{\Delta }{E}_{\mathrm{gap}}$ of the indicated nuclei. The average (among ten used CEDFs) size of the shell gap is shown by a solid circle. Thin and thick vertical lines are used to show the spread of the sizes of the calculated shell gaps; the tops and bottoms of these lines correspond to the largest and smallest shell gaps among the considered set of CEDFs. Thin lines show this spread for all employed CEDFs, while thick lines are used for the subset of four CEDFs (NL3*, DD-ME2, DD-$\mathrm{ME}\delta$, and DD-PC1). Particle numbers corresponding to the shell gaps are indicated. (b) The same as in panel (a), but with the sizes of the shell gaps and the spreads in their predictions scaled with mass factor $A^{1/3}$.

Figure 3

Deformation energy curves for the chain of $Z=120$ isotopes obtained in axial RHB calculations with the indicated CEDFs. The energy of the spherical or oblate ground states are set to zero.

Figure 4

The same as in Fig. 3 but for $N=184$ isotones.

Figure 5

Single-particle energies, i.e., the diagonal elements of the single-particle Hamiltonian $h$ in the canonical basis [61], for the lowest in total energy solution in the nucleus ${}^{304}120$ calculated as a function of the quadrupole equilibrium deformation ${\beta }_2$ for the two indicated functionals. Solid and dashed lines are used for positive- and negative-parity states, respectively. Relevant spherical and deformed gaps are indicated. Note that the transition from spherical to deformed shapes removes the $2j+1$ degeneracy of the spherical orbitals. The selected range in deformation is representative for ground-state deformations of the SHEs in the vicinity of the $Z=120$ and $N=184$ lines and beyond them.

Figure 6

Charge quadrupole deformations ${\beta }_2$ obtained in the RHB calculations with the indicated CEDFs. Note that ground-state equilibrium deformations of the $N=182$ nuclei with $Z=106$ and 116 and of the $N=184$ nuclei with $Z=108,114,116$, and 118 obtained with DD-PC1 [Fig. 6] differ from the ones shown in the bottom panel of Fig. 4 in Ref. [72]. This is a consequence of the use of a smaller deformation range $\left(-0.4\le {\beta }_2\le 1.0\right)$ and a larger step in deformation $\left(\mathrm{\Delta }{\beta }_2=0.05\right)$ in the RHB calculations of Ref. [72] as compared with the present paper.

Figure 7

The same as Fig. 6 but with experimentally known nuclei shown by open circles. Only the results with PC-PK1 and DD-PC1 are shown. The information on experimentally known nuclei is taken from Refs. [26, 73]. Note that the region of experimentally known superheavy nuclei is broader at high $Z$ when odd and odd-odd mass nuclei are included (see Fig. 2 in Ref. [26]).

Figure 8

Proton quadrupole deformation spreads $\mathrm{\Delta }{\beta }_2$ as a function of proton and neutron number. $\mathrm{\Delta }{\beta }_2(Z,N)=|{\beta }_2^{\text{max}}(Z,N)-{\beta }_2^{\text{min}}(Z,N)|$, where ${\beta }_2^{\text{max}}(Z,N)$ and ${\beta }_2^{\text{min}}(Z,N)$ are the largest and smallest proton quadrupole deformations obtained with the set of CEDFs used for the nucleus $(Z,N)$. Panel (a) shows the results for all functionals, while DD-$\mathrm{ME}\delta$ is excluded in the results shown on panel (b).

Figure 9

The comparison of the quantities ${\delta }_{2n}(Z,N)$ (top panels) and ${\delta }_{2p}(Z,N)$ (bottom panels) obtained in spherical (left panels) and deformed (right panels) RHB calculations with DD-PC1. Note that for clarity of the presentation, the color maps for the quantities ${\delta }_{2p}(Z,N)$ for protons and ${\delta }_{2n}(Z,N)$ for neutrons are different.

Figure 10

Two-neutron separation energies $S_{2n}(Z,N)$ given for different isotopic chains as a function of neutron number. The experimental data are shown by the symbols and the calculated results by the lines; the same color is used for both quantities belonging to the same isotopic chain. Solid symbols are used for the $S_{2n}(Z,N)$ values determined from measured masses, and open symbols for those including at least one estimated mass. The measured and estimated masses are from Ref. [75]. The transition to a prolate minimum with ${\beta }_2\sim 0.35$ in nuclei with $N\sim 192$ (see Fig. 6) creates the jumps in $S_{2n}(Z,N)$; these jumps are not shown and are detectable in the graphs by the breakage of the line.

Figure 11

The same as Fig. 10 but for two-proton separation energies $S_{2p}(Z,N)$ given for different isotonic chains as a function of the proton number. The shapes of the nuclei on the different sides of the $N=188\text{–}196$ chains (see Fig. 6) are indicated; the circles envelop the isotone chains which are affected by sharp prolate-oblate transitions. This transition creates the jumps in $S_{2p}(Z,N)$; these jumps are not shown and are detectable in the graphs by the breakage of the line.

Figure 12

The quantity ${\delta }_{2n}(Z,N)$ for the Cf, Fm, and No isotope chains. The experimental data (circles) are compared with the results (open symbols) obtained in deformed RHB calculations with the indicated CEDFs. Solid circles are used for the ${\delta }_{2n}(Z,N)$ values which are determined from measured masses and open circles for those the definition of which involves at least one estimated mass. The measured and estimated masses are from Ref. [75].

Figure 13

The same as Fig. 12 but for the quantity ${\delta }_{2p}(Z,N)$ in the $N=150,152$, and 154 isotone chains.

Figure 14

The comparison of experimental and calculated $Q_{\alpha }$ values for even-even superheavy nuclei. The experimental and calculated values are shown by symbols and lines, respectively. For a given isotope chain, the same color is used for both types of values. Experimental $Q_{\alpha }$ values are from Ref. [75]. Solid symbols are used for experimentally measured $Q_{\alpha }$ values [75], which are determined either from measured masses (for low-$Z$ values) or from $\alpha$-decays (for high-$Z$ values). Open symbols are used for the $Q_{\alpha }$ values, the determination of which involves at least one estimated mass.

Figure 15

Experimental and calculated half-lives for $\alpha$ decays of even-even superheavy nuclei. The experimental and calculated values are shown by symbols and lines, respectively. For a given isotope chain, the same color is used for both types of values. The experimental data are from Ref. [81].