Toroidal high-spin isomers in the nucleus ${}^{304}\mathrm{120}$

Background: Strongly deformed oblate superheavy nuclei form an intriguing region where the toroidal nuclear structures may bifurcate from the oblate spheroidal shape. The bifurcation may be facilitated when the nucleus is endowed with a large angular moment about the symmetry axis with $I=I_z$. The toroidal high-$K$ isomeric states at their local energy minima can be theoretically predicted using the cranked self-consistent Skyrme-Hartree-Fock method.

Purpose: We use the cranked Skyrme-Hartree-Fock method to predict the properties of the toroidal high-spin isomers in the superheavy nucleus ${}^{304}120_{184}$.

Method: Our method consists of three steps: First, we use the deformation-constrained Skyrme-Hartree-Fock-Bogoliubov approach to search for the nuclear density distributions with toroidal shapes. Next, using these toroidal distributions as starting configurations, we apply an additional cranking constraint of a large angular momentum $I=I_z$ about the symmetry $z$ axis and search for the energy minima of the system as a function of the deformation. In the last step, if a local energy minimum with $I=I_z$ is found, we perform at this point the cranked symmetry- and deformation-unconstrained Skyrme-Hartree-Fock calculations to locate a stable toroidal high-spin isomeric state in free convergence.

Results: We have theoretically located two toroidal high-spin isomeric states of ${}^{304}120_{184}$ with an angular momentum $I=I_z=81\hslash$ (proton 2p-2h, neutron 4p-4h excitation) and $I=I_z=208\hslash$ (proton 5p-5h, neutron 8p-8h) at the quadrupole moment deformations $Q_{20}=-297.7$ b and $Q_{20}=-300.8$ b with energies 79.2 and 101.6 MeV above the spherical ground state, respectively. The nuclear density distributions of the toroidal high-spin isomers ${}^{304}120_{184}(I_z=81\hslash$ and $208\hslash )$ have the maximum density close to the nuclear matter density, 0.$16{\mathrm{fm}}^{-3}$, and a torus major to minor radius aspect ratio $R/d=3.25$.

Conclusions: We demonstrate that aligned angular momenta of $I_z=81\hslash$ and $208\hslash$ arising from multiparticle-multihole excitations in the toroidal system of ${}^{304}120_{184}$ can lead to high-spin isomeric states, even though the toroidal shape of ${}^{304}120_{184}$ without spin is unstable. Toroidal energy minima without spin may be possible for superheavy nuclei with higher atomic numbers, $Z\gtrsim 122$, as reported previously [A. Staszczak and C. Y. Wong, Acta Phys. Pol. B 40, 753 (2008)].

Figure 1

Total HFB energy surface of ${}^{304}120_{184}$ as a function of the quadrupole $Q_{20}$ and octupole $Q_{30}$ moments. The HFB energy is normalized to the ground state energy. The dashed lines show the symmetric (sEF) and asymmetric elongated fission (aEF) paths along different valleys.

Figure 2

Total HFB energy curve of ${}^{304}120_{184}$ as a function of the quadrupole moment. The thick solid (blue color) and gray dashed (orange) lines show the symmetric (sEF) and asymmetric (aEF) elongated fission pathways along different valleys, respectively. The effects of triaxiality on the inner and outer barriers are shown in the inset, where the axially symmetric sEF $\left(\gamma =0^{\circ }\right)$ fission pathway is marked by solid thin (black) line. The nuclear matter density distributions with toroidal shapes appear at the region of large oblate deformation $Q_{20}\le -158$ b dark gray (red) solid circles.

Figure 3

Proton single-particle levels in the canonical basis for ${}^{304}120_{184}$ in the toroidal configuration as a function of the constraining quadrupole moment $Q_{20}$, obtained in the Skyrme-HFB calculations. The levels with positive parity are drawn with solid (black) lines, while those with negative parity are drawn with gray dashed (blue color) lines. The circled numbers denote the occupation numbers at regions of spare single-particle energy level density (“shells”).

Figure 4

The same as in Fig. 3, but for the neutron single-particle levels.

Figure 5

The proton single-particle energy levels of ${}^{304}120_{184}$ in the toroidal configuration at $Q_{20}=-300$ b, as a function of $2{\mathrm{\Omega }}_z$. The thin gray dashed (red color) lines give the tilted proton Fermi surfaces which lead to the proton spin value $I_z=26$ for $\hslash {\omega }_1\approx 0.1\mathrm{MeV}$ and $I_z=79$ at $\hslash {\omega }_2\approx 0.28\mathrm{MeV}$. In the case of $I_z=79$, the occupied states are shown as solid circular points, and the unoccupied states as open circles.

Figure 6

The neutron single-particle energy levels of ${}^{304}120_{184}$ in the toridal configuration at $Q_{20}=-300$ b, as a function of $2{\mathrm{\Omega }}_z$. The thin dashed lines give the tilted neutron Fermi surfaces which lead to the neutron spin value $I_z=55$ for $\hslash {\omega }_1\approx 0.1\mathrm{MeV}$, and $I_z=129$ for $\hslash {\omega }_2\approx 0.28\mathrm{MeV}$. In the case of $I_z=129$, the occupied states are shown as solid circular points, and the unoccupied states as open circles.

Figure 7

Proton single-particle Routhians of ${}^{304}120_{184}$ in the toroidal configuration with $Q_{20}=-300$ b, as a function of the cranking frequency $\hslash \omega$. The states are labeled by the Nilsson quantum numbers $[N,n_z,\mathrm{\Lambda }]\mathrm{\Omega }$. Solid (black) and dark gray dashed (red color) curves are used to distinguish even and odd principal quantum number states, respectively. The aligned angular momenta $I_z$ for $Z=120$ protons are shown at various $\hslash \omega$ locations.

Figure 8

The same as in Fig. 7, but for the neutron single-particle Routhians of ${}^{304}120_{184}$ in the toroidal configuration. The aligned angular momenta $I_z$ for $N=184$ neutrons are shown at various $\hslash \omega$ locations.

Figure 9

The deformation energies of ${}^{304}120_{184}$ in the toroidal configuration as a function of the quadrupole moment $Q_{20}$ for $I_z=0$, 71, 81, 126, 144, and 208. The locations of the toroidal high-spin-isomers (THSIs) for $I_z=81$ and 208 are indicated by star symbols. All deformation energies are measured relative to the energy of the spherical ground state.

Figure 10

Neuron, proton, and total density profiles of the THSIs ${}^{304}120_{184}(I=81$ and 208) as a function of $x$ for a cut in $y=0$ and $z=0$.

Figure 11

Contours of the total nuclear densities of ${}^{304}120_{184}(I_z=81)$ in cuts $x-y$ (a) and $x-z$ (b).