Extension of the nuclear landscape to hyperheavy nuclei

The properties of hyperheavy nuclei and the extension of nuclear landscape to hyperheavy nuclei are extensively studied within covariant density functional theory. Axial reflection symmetric and reflection asymmetric relativistic Hartree-Bogoliubov (RHB) calculations are carried out. The role of triaxiality is studied within triaxial RHB and triaxial relativistic mean field + BCS frameworks. With increasing proton number beyond $Z\approx 130$ the transition from ellipsoidal-like nuclear shapes to toroidal ones takes place. The description of latter shapes requires the basis which is typically significantly larger than the one employed for the description of ellipsoidal-like shapes. Many hyperheavy nuclei with toroidal shapes are expected to be unstable toward multifragmentation. However, three islands of stability of spherical hyperheavy nuclei have been predicted for the first time in Afanasjev et al. [Phys. Lett. B 782, 533 (2018)]. Proton and neutron densities, charge radii, neutron skins, and underlying shell structure of the nuclei located in the centers of these islands have been investigated in detail. Large neutron shell gaps at $N=228,308,$ and 406 define approximate centers of these islands in neutron number. On the contrary, large proton gap appear only at $Z=154$ in the $(Z\approx 156,N\approx 310)$ island. As a result, this is the largest island of stability of spherical hyperheavy nuclei found in the calculations. The calculations indicate the stability of the nuclei in these islands with respect to octupole and triaxial distortions. The shape evolution of toroidal shapes along the fission path and the stability of such shapes with respect to fission have been studied. Fission barriers in neutron-rich superheavy nuclei are studied in triaxial RHB framework; the impact of triaxiality on the heights of fission barriers is substantial in some parts of this region. Based on the results obtained in the present work, the extension of nuclear landscape to hyperheavy nuclei is provided.

Figure 1

The dependence of total binding energy on the truncation of the basis and on the deformation of basis ${\beta }_0$ in the ${}^{208}\mathrm{Pb}$ and ${}^{466}156$ nuclei. Total binding energies are shown as a function of the ${\beta }_2$ values. Panels (a) and (b) show the dependence of total binding energies on the number of fermionic shells $N_F$ for the deformation of basis ${\beta }_0=0.5{\beta }_2$. Panels (c) and (d) show the dependence of total binding energies of hyperheavy ${}^{466}156$ nucleus on the deformation of basis ${\beta }_0$ for $N_F=20$ and $N_F=30$, respectively.

Figure 2

Neutron density distributions of the ${}^{466}156$ nucleus at the indicated ${\beta }_2$ values. They are plotted in the $yz$ plane at the position of the Gauss-Hermite integration points in the $x$ direction closest to zero. The density colormap starts at ${\rho }_n=0.005$ ${\mathrm{fm}}^3$ and shows the densities in ${\mathrm{fm}}^3$. Based on the results of axial RHB calculations for the lowest in energy solution obtained with $N_F=30$ [see Fig. 1]. Note that proton density (not shown here) is roughly half of the neutron one.

Figure 3

Proton ${\beta }_2$ values of the lowest in energy solutions of the $Z=140\text{–}180$ nuclei obtained in axial RHB calculations with $N_F$ = 20. The calculations cover the region between two-proton and two-neutron drip lines.

Figure 4

The evolution of proton and neutron densities with the transition from the ${}^{208}\mathrm{Pb}$ nucleus to the region of hyperheavy nuclei. The figure is based on the results of spherical RHB calculations; the employed CEDFs are indicated. The ${}^{368}138$, ${}^{466}156$, and ${}^{466}156$ nuclei are located in the centers of the islands of stability of spherical hyperheavy nuclei predicted in Ref. [15]. Note that it was verified that proton and neutron densities of the nuclei in these regions are very similar to the densities of above mentioned central nuclei. For comparison, the densities of spherical ${}^{208}\mathrm{Pb}$ and ${}^{292}120$ nuclei are presented. Note that latter nucleus is an example of substantial central depression in density distribution (see Ref. [39] for details).

Figure 5

Proton and neutron single-particle states at spherical shape in the ${}^{466}156$ nucleus determined with the indicated CEDFs in the calculations without pairing. Solid and dashed connecting lines are used for positive and negative parity states. Spherical gaps are indicated.

Figure 6

Two-neutron and two-proton separation energies for spherical nuclei located in the $(Z\approx 156,N\approx 310)$ region of stability of hyperheavy nuclei. They are obtained in the RHB calculations with the DD-PC1 functional. The lines are labeled by respective proton [panel (a)] and neutron [panel (b)] numbers.

Figure 7

The same as described in the caption of Fig. 5 but for the ${}^{366}138$ nucleus.

Figure 8

The same as described in the caption of Fig. 5 but for the ${}^{580}174$ nucleus.

Figure 9

Potential energy surfaces in the $({\beta }_2,{\beta }_3)$ plane of the central nuclei of the regions of potential stability of spherical hyperheavy nuclei. Spherical minimum is indicated by a white semicircle. Equipotential lines are shown in steps of 1.0 MeV. Note that the results are shown in different $({\beta }_2,{\beta }_3$) deformation ranges.

Figure 10

Potential energy surfaces of the nuclei located in the centers of the regions of stability of spherical hyperheavy nuclei obtained in the TRHB calculations with $N_F=18$. Note that the topology of potential energy surfaces is almost the same in the calculations with $N_F=18$ and $N_F=20$. Thus, to save computational time these figures are plotted with $N_F=18$. The energy difference between two neighboring equipotential lines is equal to 1.0 MeV. Spherical minimum is indicated by a circle. The colormaps show the excitation energies (in MeV) with respect to the energy of the deformation point with largest (in absolute value) binding energy. Note that the results for the ${}^{548}174$ nucleus are shown in a smaller deformation range (because of convergence problems at large ${\beta }_2$ values) and different colormap is used for this nucleus.

Figure 11

Neutron and proton pairing energies $E_{\text{pairing}}$ (a) and pairing gaps ${\mathrm{\Delta }}_{\text{uv}}$ (b) as a function of ${\beta }_2$ for the lowest in energy solution in the ${}^{466}156$ nucleus, obtained with $N_F=30$ and the deformation of basis ${\beta }_0=0.5{\beta }_2$, shown in Fig. 1.

Figure 12

Neutron (a) and proton (b) pairing energies $E_{\text{pairing}}$ for spherical minima of the nuclei forming the $(Z\approx 156,N\approx 310)$ island of stability of spherical hyperheavy nuclei.

Figure 13

The evolution of neutron (a) and proton (b) pairing energies $E_{\text{pairing}}$ as a function of the ${\beta }_2$ and $\gamma$ deformations in the ${}^{466}156$ nucleus.

Figure 14

Deformation energy curves of selected even-even superheavy nuclei obtained in axial RHB calculations performed with $N_F=26$. Arrows A and B indicate prolate superdeformed ${\beta }_2\approx 0.6$ and oblate ${\beta }_2\approx -0.5$ local minima, respectively.

Figure 15

Potential energy surfaces in the $({\beta }_2,{\beta }_3)$ plane for the nuclei shown in Fig. 14. Equipotential lines are shown in steps of 1.0 MeV.

Figure 16

Deformation energy curves of even-even $Z=138$ hyperheavy nuclei obtained in axial RHB calculations with DD-PC1 functional and the $N_F=26$ basis. The nuclei located between two-proton and two-neutron drip lines are shown in step of $\mathrm{\Delta }N=4$. The curves are plotted only for negative ${\beta }_2$ values since prolate solutions are unstable. The deformation energy curves are normalized in such a way that the minimum of total energy for negative ${\beta }_2$ values corresponds to zero energy.

Figure 17

Proton and neutron chemical potentials as a function of the ${\beta }_2$ values for the solutions displayed in Fig. 16. Blue dashed lines show the continuum threshold.

Figure 18

Three-dimensional potential energy surfaces with their two-dimensional projections (contour plots) for the solutions with minimum at ${\beta }_2\approx 2.3,{\beta }_4\approx +1.5,\gamma ={60}^{\circ }$ in indicated nuclei. Based on the results of the TRMF+BCS calculations of Ref. [15]. These solutions are excited ones in axial calculations, but they are the lowest in energy stable solutions in triaxial calculations. The red line shows static fission path from the minimum indicated by open white circle; the saddle point is shown by black solid circle. The energy difference between two neighboring equipotential lines in contour plot is 0.5 MeV.

Figure 19

The evolution of toroidal shapes along the fission path in the ${}^{354}134$ nucleus shown on left panel of Fig. 18. Neutron density distributions are shown at indicated (${\beta }_2,\gamma$)-deformations along this fission path. To give a full three-dimensional representation of the density distributions, they are plotted in the $xy,yz$, and $xz$ planes at the positions of the Gauss-Hermite integration points in the $z,x$, and $y$ directions closest to zero, respectively. The density colormap starts at ${\rho }_n=0.005{\mathrm{fm}}^{-3}$ and shows the densities in ${\mathrm{fm}}^{-3}$.

Figure 20

Three-dimensional potential energy surfaces with their two-dimensional projections (contour plots) for the nuclei with the ground states having ellipsoidal shape. They have been obtained in the TRHB calculations with $N_F=18$. The red line shows static fission path from respective minimum, while black dashed line the $\gamma =0^{\circ }$ axis. The energy difference between two neighboring equipotential lines in contour plot is 0.5 MeV.

Figure 21

Inner fission barrier heights $E_{\text{triax}}^B$ obtained in the TRHB calculations (a) and the decrease of the fission barrier height due to triaxiality $\mathrm{\Delta }{E}^{\text{gain}}$ (b) as a function of neutron number $N$.

Figure 22

(a) Proton quadrupole deformations ${\beta }_2$ of the lowest in energy minima for axial symmetry (LEMAS) obtained in axial RHB calculations with the DD-PC1 functional. Based on the results presented in Fig. 17(c) of Ref. [4] and Fig. 3 of Ref. [15]. Only the nuclei whose LEMAS have ellipsoidal-like shapes are included here; those who have toroidal shapes in LEMAS (see Fig. 3 in Ref. [15]) are neglected. The color map in the ${\beta }_2=-0.4\text{–}0.5$ range is equivalent to the one of Fig. 17(c) of Ref. [4] for consistency with previous results. (b) The same as panel (a) but with the nuclei, in which neither inner nor outer (if exist) fission barrier(s) have the height(s) higher than 2 MeV, excluded. Here the results of the calculations for fission barmvriers presented in Table 1 of supplemental material to Ref. [15] and in Table 2 of the present manuscript are used for approximate deliniation of the boundaries of the region of nuclear chart in which fission barriers satisfy above mentioned condition. (c) The same as panel (b) but with two-proton and two-neutron drip lines (shown by solid lines), defined from separation energies, for toroidal nuclei added. They are based on the results of axial RHB calculations with $N_F=26$.

Figure 23

Calculated Coulomb energy as a function of the ${\beta }_2$ values. The results are displayed for four indicated nuclei; the total deformation energy curves of these nuclei are shown in Fig. 1 of Ref. [15]. Orange vertical dashed line indicates spherical shapes. Horizontal dashed lines of different color start at the positions of respective Coulomb energy curves at ${\beta }_2=-4.0$ and end at vertical orange dashed line. The numbers above these horizontal lines indicate the difference $\mathrm{\Delta }{E}_{\text{Coul}}=E_{\text{Coul}}({\beta }_2=0.0)-E_{\text{Coul}}({\beta }_2=-4.0)$ (in MeV, rounded to closest value), which is the reduction of the Coulomb energy due to the transition from spherical shape to toroidal shape with typical ${\beta }_2=-4.0$ values seen at the minima of deformation energy curves of the ${}^{466}156$ and ${}^{426}176$ nuclei (see Figs. 1(c) and 2 in Ref. [15]).

Figure 24

The same as described in the caption of Fig. 22 but with extended proton and neutron ranges and added regions of relatively stable spherical hyperheavy nuclei shown in gray. Note that in the same nucleus two-neutron drip lines for spherical and toroidal shapes are different. This is a reason why some regions of stability of spherical nuclei extend beyond two-neutron drip line for toroidal shapes.