High-spin torus isomers and their precession motions

Background: In our previous study, we found that an exotic isomer with a torus shape may exist in the high-spin, highly excited states of ${}^{40}\mathrm{Ca}$. The $z$ component of the total angular momentum, $J_z=60\hslash$, of this torus isomer is constructed by totally aligning 12 single-particle angular momenta in the direction of the symmetry axis of the density distribution. The torus isomer executes precession motion with the rigid-body moments of inertia about an axis perpendicular to the symmetry axis. The investigation, however, has been focused only on ${}^{40}\mathrm{Ca}$.

Purpose: We systematically investigate the existence of exotic torus isomers and their precession motions for a series of $N=Z$ even-even nuclei from ${}^{28}\mathrm{Si}$ to ${}^{56}\mathrm{Ni}$. We analyze the microscopic shell structure of the torus isomer and discuss why the torus shape is generated beyond the limit of large oblate deformation.

Method: We use the cranked three-dimensional Hartree-Fock method with various Skyrme interactions in a systematic search for high-spin torus isomers. We use the three-dimensional time-dependent Hartree-Fock method for describing the precession motion of the torus isomer.

Results: We obtain high-spin torus isomers in ${}^{36}\mathrm{Ar},{}^{40}\mathrm{Ca},{}^{44}\mathrm{Ti},{}^{48}\mathrm{Cr}$, and ${}^{52}\mathrm{Fe}$. The emergence of the torus isomers is associated with the alignments of single-particle angular momenta, which is the same mechanism as found in ${}^{40}\mathrm{Ca}$. It is found that all the obtained torus isomers execute the precession motion at least two rotational periods. The moment of inertia about a perpendicular axis, which characterizes the precession motion, is found to be close to the classical rigid-body value.

Conclusions: The high-spin torus isomer of ${}^{40}\mathrm{Ca}$ is not an exceptional case. Similar torus isomers exist widely in nuclei from ${}^{36}\mathrm{Ar}$ to ${}^{52}\mathrm{Fe}$ and they execute the precession motion. The torus shape is generated beyond the limit of large oblate deformation by eliminating the $0s$ components from all the deformed single-particle wave functions to maximize their mutual overlaps.

Figure 1

Schematic picture for the precession motion of torus isomers taken from Ref. [2]. The bold solid line denotes the symmetry axis of the density distribution. The dashed line denotes the precession axis. The symbols $\theta$ and $\varphi$ denote the tilting and the rotational angles, respectively.

Figure 2

Density distributions on the $z=0$ plane of the obtained stable torus isomers. The contours correspond to multiple steps of 0.015 ${\mathrm{fm}}^{-3}$. The color is normalized by the largest density in each plot.

Figure 3

Single-particle energies versus the $z$ component of the total angular momentum, $\Omega$, for ${}^{36}\mathrm{Ar}$. Solid and open circles denote the single-particle energies of the positive- and negative-parity states, respectively. To illustrate the degeneracy of positive- and negative-parity states, some negative-parity states are shown by double open circles.

Figure 4

Single-particle energies versus $\Omega$ for ${}^{40}\mathrm{Ca}$. All symbols are the same as in Fig. 3.

Figure 5

Single-particle energies versus $\Omega$ for ${}^{44}\mathrm{Ti}$. All symbols are the same as in Fig. 3.

Figure 6

Single-particle energies versus $\Omega$ for ${}^{48}\mathrm{Cr}$. All symbols are the same as in Fig. 3.

Figure 7

Single-particle energies versus $\Omega$ for ${}^{52}\mathrm{Fe}$. All symbols are the same as in Fig. 3.

Figure 8

Regions of $\omega$ for which each of the torus isomers stably exists. The solid, dashed, and dotted lines denote the results calculated with the SLy6, SkI3, and ${\mathrm{SkM}}^{*}$ interactions, respectively.

Figure 9

Time evolution of the precession motion for the torus isomers of ${}^{36}\mathrm{Ar}$ calculated by solving the TDHF equation of motion for the SLy6 interaction. In the plot, panels (a), (b), and (c) denote the total angular momentum $I$, the tilting angle $\theta$, and the rotational angle $\varphi$, respectively.

Figure 10

Time evolution of the precession motion for the torus isomers of ${}^{40}\mathrm{Ca}$. All symbols are the same as Fig. 9.

Figure 11

Time evolution of the precession motion for the torus isomers of ${}^{44}\mathrm{Ti}$. All symbols are the same as Fig. 9.

Figure 12

Time evolution of the precession motion for the torus isomers of ${}^{48}\mathrm{Cr}$. All symbols are the same as Fig. 9.

Figure 13

Time evolution of the precession motion for the torus isomers of ${}^{52}\mathrm{Fe}$. All symbols are the same as Fig. 9.

Figure 14

Radial density distributions of individual single-particle states on the $z=0$ plane for ${}^{40}\mathrm{Ca}$ obtained by the cranked HF calculations (upper panel) and the RDHO model (lower pannel). The densities for the $\varphi$ direction in cylindrical coordinates are integrated.

Figure 15

Nilsson diagram versus $\eta =R_0/d$ of the RDHO model for ${}^{40}\mathrm{Ca}$. The solid and dashed lines denote the single-particle states with positive and negative parities, respectively. The single-particle energies are plotted in units of $\hslash {\omega }_0\left(\eta \right)$. To illustrate the degeneracy of the levels, the single-particle energies with higher $\Omega$ are slightly shifted.

Figure 16

Single-particle energies of the RDHO model versus $\Omega$ at various values of $\eta =R_0/d$. The solid and open circles denote the single-particle states with positive and negative parities, respectively.

Figure 17

Nilsson diagram versus the aspect ratio of the short (the $z$ direction) to long (the radial direction) axes for an ellipsoidal nuclear surface (oblate deformations). The solid and dashed lines denote the single-particle states with positive and negative parities, respectively. The aspect ratio $1:1$ corresponds to the spherical shape. The aspect ratio $1:5$ is close to that of a torus isomer obtained by the cranked HF calculation.

Figure 18

Single-particle energies of the deformed harmonic-oscillator versus $\Omega$ at various oblate deformations. In (e), the dashed line denotes the Fermi surface for ${}^{40}\mathrm{Ca}$ $\left(N=20\right)$ at $\omega =0$. The gray area denotes the occupied states in ${}^{40}\mathrm{Ca}$ at $\omega =1.6$ MeV/$\hslash$ for rotation about the symmetry axis.

Figure 19

(a) Density distributions of neutrons in ${}^{40}\mathrm{Ca}$ calculated for the deformed harmonic-oscillator model at an oblate deformation of the aspect ratio $1:5$ with $\omega =0$. The contours correspond to multiple steps of 0.05 ${\mathrm{fm}}^{-2}$. The densities for the $\varphi$ direction in the cylindrical coordinate are integrated. (b) The same as (a) but with $\omega =1.6$ MeV/$\hslash$. The colors are normalized by the largest density of (a).

Figure 20

Density distributions of the single-particle states of special interest for ${}^{40}\mathrm{Ca}$ on the $z=0$ plane in an oblate deformation of the aspect ratio $1:5$ calculated with the deformed harmonic-oscillator model. The densities in the $\varphi$ direction in the cylindrical coordinate are integrated. The solid lines show the density distributions of the aligned single-particle states with $\Lambda =2$, 4, and 5 that are occupied at $\omega =1.6$ MeV/$\hslash$. The dashed line shows the density distribution of the lowest $\Lambda =0$ state.