Self-Consistent Tilted-Axis-Cranking Study of Triaxial Strongly Deformed Bands in ${}^{158}\mathrm{Er}$ at Ultrahigh Spin

Stimulated by recent experimental discoveries, triaxial strongly deformed (TSD) states in ${}^{158}\mathrm{Er}$ at ultrahigh spins have been studied by means of the Skyrme-Hartree-Fock model and the tilted-axis-cranking method. Restricting the rotational axis to one of the principal axes—as done in previous cranking calculations—two well-defined TSD minima in the total Routhian surface are found for a given configuration: one with positive and another with negative triaxial deformation $\gamma$. By allowing the rotational axis to change direction, the higher-energy minimum is shown to be a saddle point. This resolves the long-standing question of the physical interpretation of the two triaxial minima at a very similar quadrupole shape obtained in the principal-axis-cranking approach. Several TSD configurations have been predicted, including a highly deformed band, which is a candidate for the structure observed in experiment.

Figure 1

Schematic picture of a TSD shape. The angles $\theta$ (between the $x$ axis and the rotational axis) and $\alpha$ (between $\mathit{\omega }$ and $\mathit{J}$) are defined in the $x\mathrm{\text{−}}y$ plane. The short, medium, and long axes are denoted by $x$, $y$, and $z$, respectively; that is, the plotted shape corresponds to $\gamma >0$.

Figure 2

Lowest Routhians in ${}^{158}\mathrm{Er}$ calculated by means of SCTAC (Nilsson potential) at fixed ${\epsilon }_2$ and $\omega$ as a function of $\gamma$ for rotation about the $x$ axis (solid line) compared to those calculated at $\gamma =15°$ and 22° as functions of $\theta$ (dashed lines). In the latter case, the Routhians are drawn by uniformly scaling the range of $0°\le \theta \le 90°$ into the corresponding ranges of $\gamma$.

Figure 3

Total Routhian surfaces in the $(Q_{20},Q_{22})$ plane for the $A$ configuration of Table  at $\omega =0.7\mathrm{MeV}$ and $\theta =0$ (principal-axis cranking) obtained with SkM* (left) and $\mathrm{SLy}{4}_L$ (right) functionals. The corresponding values of $\gamma$ are marked.

Figure 4

Total Routhians of ${}^{158}\mathrm{Er}$ as a function of $\theta$ at $\omega =0.5$, 0.6, and 0.7 MeV for SkM* (left) and $\mathrm{SLy}{4}_L$ (right) functionals. Dashed, solid, dotted, and dot-dashed lines represent TSD configurations $A$, $B$, $C$, and $D$, respectively (see Table ). Note that the labeling of the configurations is only valid when $\theta =0°$ or 90°. Otherwise, the signature is no longer a good quantum number. For the $\mathrm{SLy}{4}_L$ functional, converged solutions for configurations $C$ and $D$ could not be obtained.

Figure 5

Relative energies of configurations $A$ (dashed lines), $B$ (solid lines), and $C$ (dotted lines) with respect to $D$ (dash-dotted line at 0 MeV).