Octupole deformation in the ground states of even-even $Z\sim 96, N\sim 196$ actinides and superheavy nuclei

A systematic search for axial octupole deformation in the actinides and superheavy nuclei with proton numbers $Z=88–126$ and neutron numbers from the two-proton drip line up to $N=210$ was performed in covariant density functional theory (DFT) using four state-of-the-art covariant energy density functionals representing different model classes. The nuclei in the $Z\sim 96, N\sim 196$ region of octupole deformation were investigated in detail and the systematic uncertainties in the description of their observables were quantified. A similar region of octupole deformation exists also in Skyrme DFT and microscopic+macroscopic approaches but it is centered at somewhat different particle numbers. Theoretical uncertainties in the predictions of the regions of octupole deformation increase on going to superheavy nuclei with $Z\sim 120, N\sim 190$. There are no octupole deformed nuclei for $Z=112–126$ in covariant DFT calculations. This agrees with Skyrme DFT calculations, but disagrees with Gogny DFT and microscopic+macroscopic calculations which predict an extended $Z\sim 120, N\sim 190$ region of octupole deformation.

Figure 1

The dependence of calculated quadrupole and octupole deformations, total binding energy and the $|\mathrm{\Delta }{E}^{\mathrm{oct}}|$ quantity on the number of fermionic shells employed in the RHB calculations for ${}^{290}\mathrm{Cm}$ with the DD-PC1 functional. The results obtained in octupole and quadrupole RHB codes in respective local minima with $({\beta }_2\ne 0,{\beta }_3\ne 0)$ and $({\beta }_2\ne 0,{\beta }_3=0)$ are shown by solid black and dashed red curves, respectively.

Figure 2

Octupole deformed nuclei in the selected part of the nuclear chart. Only nuclei with nonvanishing $\mathrm{\Delta }{E}^{\mathrm{oct}}$ are shown by squares; the colors of the squares represent the values of $|\mathrm{\Delta }{E}^{\mathrm{oct}}|$ (see colormap). The two-proton and two-neutron drip lines are displayed by solid black lines; for the CEDFs NL3*, DD-ME2, and DD-PC1 they are taken from Ref. [34]. Two-proton drip line for PC-PK1 is taken from Ref. [27]. The two-neutron drip line of the NL3* functional is used in panel (d) since it is not defined for CEDF PC-PK1 at present.

Figure 3

The calculated equilibrium quadrupole ${\beta }_2$ (top panel of each figure) and octupole ${\beta }_3$ (middle panel of each figure) deformations as well as the $\mathrm{\Delta }{E}^{\mathrm{oct}}$ quantities (bottom panel of each figure). The employed functionals are indicated.

Figure 4

Potential energy surfaces of the Cm ($Z=96$) isotopes in the $({\beta }_2,{\beta }_3)$ plane calculated with the CEDF DD-PC1. The white circle indicates the global minimum. Equipotential lines are shown in steps of 0.5 MeV. The neutron number $N$ is shown in each panel in order to make the comparison between different isotones easier.

Figure 5

The same as Fig. 4, but for the $N=198$ isotones.

Figure 6

The calculated spreads in quadrupole and octupole deformations as well as in the $|\mathrm{\Delta }{E}^{\mathrm{oct}}|$ quantities. The nucleus is shown by a square if it has nonzero octupole deformation in the calculations with at least one CEDF.

Figure 7

Octupole deformed nuclei obtained in the CDFT calculations with CEDFs DD-ME2 (a) and PC-PK1 (b), in Skyrme DFT calculations with the SLy6 functional [50] (c), Gogny DFT calculations with EDF D1S [3, 26] (d), and in microscopic+macroscopic calculations of Ref. [2, 25] (e). Only nuclei with nonzero calculated octupole deformation are shown by squares. The two-proton and two-neutron drip lines are displayed by solid black lines in panels (a), (b), and (e). In panels (c) and (d), the regions in which the searches for octupole deformation were performed are outlined by dashed lines. The results of the SDFT calculations for the $Z\sim 100, N\sim 190$ region of octupole deformation are extracted from Fig. 11 of Ref. [50]. The $Z\sim 92, N\sim 134$ region of octupole deformation in panel (c) is shown schematically (based on Fig. 4 of Ref. [50]). Note that the results presented in panel (d) in two regions of octupole deformation were obtained in two independent calculations of Refs. [3, 26]. The nuclei which are octupole deformed in the mic+mac calculations of Ref. [25] are shown by solid blue squares and open diamonds in panel (e). Open squares indicate additional (as compared with Ref. [25]) octupole deformed nuclei obtained in Ref. [2], while open diamonds indicate the nuclei which cease to be octupole deformed (as compared with Ref. [25]) in the mic+mac calculations of Ref. [2].

Figure 8

Potential energy surfaces of ${}^{286}\mathrm{Cm}$ in the $({\beta }_2,{\beta }_3)$ plane calculated with the CEDF DD-PC1 for different values of scaling factor $f$ of the pairing strength. The white circle indicates the global minimum. Equipotential lines are shown in steps of 0.5 MeV.

Figure 9

The same as Fig. 8, but for ${}^{290}\mathrm{Cm}$.