Rotational motion of triaxially deformed nuclei studied by the microscopic angular-momentum-projection method. I. Nuclear wobbling motion

Rotation of triaxially deformed nucleus has been an interesting subject in the study of nuclear structure. In the present series of work, we investigate wobbling motion and chiral rotation by employing the microscopic framework of angular-momentum projection from cranked triaxially deformed mean-field states. In this first part, the wobbling motion is studied in detail. The consequences of the three-dimensional cranking are investigated. It is demonstrated that the multiple wobbling rotational bands naturally appear as a result of fully microscopic calculation. They have the characteristic properties that are expected from the macroscopic triaxial-rotor model or the phenomenological particle-triaxial-rotor model, although quantitative agreement with the existing data is not achieved. It is also found that the excitation spectrum reflects dynamics of the angular-momentum vector in the intrinsic frame of the mean field (transverse vs longitudinal wobbling). The results obtained by using the Woods-Saxon potential and the schematic separable interaction are mainly discussed, while some results with the Gogny D1S interaction are also presented.

Figure 1

Cranking moments of inertia of the three intrinsic axes, $x,y$, and $z$, which are the short, medium, and long axes and denoted by dotted, solid, and dashed lines, respectively, as functions of the triaxiality parameter $\gamma$ for the even-even core nucleus ${}^{162}\mathrm{Yb}$ of ${}^{163}\mathrm{Lu}$. The deformation parameters ${\beta }_2=0.42,{\beta }_4=0.02$ and the pairing gaps ${\mathrm{\Delta }}_{\mathrm{n}}={\mathrm{\Delta }}_{\mathrm{p}}=0.5$ MeV are employed for the upper panels, (a) and (b), and the larger pairing gaps ${\mathrm{\Delta }}_{\mathrm{n}}={\mathrm{\Delta }}_{\mathrm{p}}=1.0$ MeV for panel (c). The $\gamma$ parameter of the Woods-Saxon potential is utilized in panel (a) and that of the density distribution, Eq. (14), in panels (b) and (c). Panel (d) shows the irrotational-flow moments of inertia in arbitrary units.

Figure 2

The calculated spectrum (left panels) and the relative excitation energies (right panels) for ${}^{162}\mathrm{Yb}$ obtained by the angular-momentum-projection method. The rigid-rotor reference energy $0.007I(I+1)$ MeV is subtracted in the left panels. The upper panels are the result of projection from the noncranked mean field, while the lower panels are from the infinitesimally cranked mean field, with ${\omega }_x={\omega }_y={\omega }_z=0.01\mathrm{MeV}/\hslash$.

Figure 3

The calculated behavior of the angular-momentum vector in the body-fixed frame for the yrast and the first excited bands in ${}^{162}\mathrm{Yb}$. Here the mean field is infinitesimally cranked with ${\omega }_x={\omega }_y={\omega }_z=0.01\mathrm{MeV}/\hslash$ corresponding to the lower panels of Fig. 2. The left panel shows the expectation values of the components of squared angular-momentum operator defined by Eq. (19), while the right panel shows the square root of them.

Figure 4

The calculated excitation energy (left panels) and the expectation values of the angular-momentum vector (right panels) in ${}^{162}\mathrm{Yb}$ calculated by the projection from the cranked mean field with the frequencies ${\omega }_x=0.30,{\omega }_y={\omega }_z=0.01\mathrm{MeV}/\hslash$ (upper panels) and those with the frequencies ${\omega }_x=0.40,{\omega }_y={\omega }_z=0.01\mathrm{MeV}/\hslash$ (lower panels).

Figure 5

The spectrum of the angular-momentum projection obtained from the cranked mean field with the frequencies ${\omega }_x=0.40,{\omega }_y={\omega }_z=0.01\mathrm{MeV}/\hslash$ in ${}^{162}\mathrm{Yb}$, corresponding to the lower panels of Fig. 4.

Figure 6

Wobbling spectrum for ${}^{163}\mathrm{Lu}$ calculated by the angular-momentum-projection method from the noncranked mean field (left panel) and from the infinitesimally cranked mean field with the frequencies ${\omega }_x={\omega }_y={\omega }_z=0.01\mathrm{MeV}/\hslash$ (right panel). The rigid-rotor reference energy $0.007I(I+1)$ MeV is subtracted for both the calculated and experimental energies. The energy of the experimental TSD1 [37] is also included in each panel.

Figure 7

The calculated excitation energy (left panels) and the expectation values of the angular-momentum vector (right panels) in ${}^{163}\mathrm{Lu}$ calculated by the projection from the noncranked mean field (upper panels) and those from the infinitesimally cranked mean field with ${\omega }_x={\omega }_y={\omega }_z=0.01\mathrm{MeV}/\hslash$ (lower panels). The experimental excitation energies of TSD1, TSD2, and TSD3 [37] are also included in the left panels.

Figure 8

The calculated excitation energy (left panels) and the expectation values of the angular-momentum vector (right panels) in ${}^{163}\mathrm{Lu}$ calculated by the projection from the cranked mean field with the frequencies ${\omega }_x=0.20,{\omega }_y={\omega }_z=0.01\mathrm{MeV}/\hslash$ (upper panels), and those with the frequencies ${\omega }_x=0.40,{\omega }_y={\omega }_z=0.01\mathrm{MeV}/\hslash$ (lower panels).

Figure 9

The calculated excitation energy (left panels) and the expectation values of the angular-momentum vector (right panels) in ${}^{163}\mathrm{Lu}$ calculated by the projection from the cranked mean-field with the frequencies ${\omega }_x=0.01,{\omega }_y=0.20,{\omega }_z=0.01\mathrm{MeV}/\hslash$ (upper panels), and those with the frequencies ${\omega }_x={\omega }_y=0.01,{\omega }_z=0.20\mathrm{MeV}/\hslash$ (lower panels).

Figure 10

Wobbling spectrum for ${}^{163}\mathrm{Lu}$ calculated by the projection from the cranked mean field with ${\omega }_x=0.20,{\omega }_y={\omega }_z=0.01\mathrm{MeV}/\hslash$, corresponding to the upper panels of Fig. 8 (left panel), and with ${\omega }_x={\omega }_y=0.01,{\omega }_z=0.20\mathrm{MeV}/\hslash$, corresponding to the lower panels of Fig. 9 (right panel). The energy of the experimental TSD1 is also included in both panels.

Figure 11

The calculated excitation energy (left panel) and the expectation values (right panel) of the angular-momentum vector for the cranked mean-field with ${\omega }_x=0.20,{\omega }_y=0.0,{\omega }_z=0.01\mathrm{MeV}/\hslash$ in ${}^{163}\mathrm{Lu}$. The experimental excitation energies of TSD1, TSD2, and TSD3 are also included in the left panel.

Figure 12

The $I\to I-2$ in-band $E2$ transition probabilities (left panel) and the $I\to I\pm 1$ out-of-band to in-band $B\left(E2\right)$ ratios (right panel) are compared with the experimental data in ${}^{163}\mathrm{Lu}$. The cranked mean field with the frequencies ${\omega }_x=0.20,{\omega }_y=0.0,{\omega }_z=0.01\mathrm{MeV}/\hslash$ are used, corresponding to Fig. 11. The experimental data are taken from Refs. [37, 38, 42].

Figure 13

The $I\to I\pm 1$ out-of-band $B\left(M1\right)$ to in-band $B\left(E2\right)$ ratios are compared with the experimental data in ${}^{163}\mathrm{Lu}$. The cranked mean field with the frequencies ${\omega }_x=0.20,{\omega }_y=0.0,{\omega }_z=0.01\mathrm{MeV}/\hslash$ are used, corresponding to Fig. 11. The experimental data are taken from [42].

Figure 14

The $I\to I-2$ in-band $E2$ transition probabilities (left panel) and the $I\to I\pm 1$ out-of-band to in-band $B\left(E2\right)$ ratios (right panel) in ${}^{163}\mathrm{Lu}$. The cranked mean field with the frequencies, ${\omega }_x={\omega }_y={\omega }_z=0.01\mathrm{MeV}/\hslash$, are used, corresponding to the lower panels of Fig. 7. Note that the scale for the $B\left(E2\right)$ ratios is enlarged from that in Fig. 12.

Figure 15

Same as Fig. 6 but the result of projection from the mean field with larger triaxial deformation $\gamma \left(\mathrm{WS}\right)={30}^{\circ }$.

Figure 16

Same as Fig. 7 but the result of projection from the mean field with larger triaxial deformation $\gamma \left(\mathrm{WS}\right)={30}^{\circ }$.

Figure 17

Same as Fig. 11 but the result of projection from the mean field with larger triaxial deformation $\gamma \left(\mathrm{WS}\right)={30}^{\circ }$.

Figure 18

Same as Fig. 12 but the result of projection from the mean field with larger triaxial deformation $\gamma \left(\mathrm{WS}\right)={30}^{\circ }$.

Figure 19

Same as Fig. 15 but the odd-proton particle occupies low-$j$ orbit in this calculation.

Figure 20

Same as Fig. 16 but the odd-proton particle occupies low-$j$ orbit in this calculation.

Figure 21

Wobbling spectrum for ${}^{163}\mathrm{Lu}$ calculated by the angular-momentum-projection method with the Gogny D1S effective interaction. The cranked mean field with ${\omega }_x=0.10\left({\omega }_x=0.20\right)$, ${\omega }_y={\omega }_z=0.0\mathrm{MeV}/\hslash$ is employed in the left (right) panel. The energy of the experimental TSD1 [37] is also included in each panel.

Figure 22

The calculated excitation energy (left panels) and the expectation values of the angular-momentum vector (right panels) with using the Gogny-D1S interaction in ${}^{163}\mathrm{Lu}$. The mean field is cranked with the frequencies, ${\omega }_x=0.10,{\omega }_y={\omega }_z=0.0\mathrm{MeV}/\hslash$ (upper panels) and ${\omega }_x=0.20,{\omega }_y={\omega }_z=0.0\mathrm{MeV}/\hslash$ (lower panels), The experimental excitation energies of TSD1, TSD2, and TSD3 [37] are also included in the left panels.

Figure 23

The $I\to I-2$ in-band $E2$ transition probabilities (left panel) and the $I\to I\pm 1$ out-of-band to in-band $B\left(E2\right)$ ratios (right panel), which are calculated with the Gogny-D1S interaction, are compared with the experimental data in ${}^{163}\mathrm{Lu}$. The cranked mean field with the frequencies, ${\omega }_x=0.20,{\omega }_y={\omega }_z=0.0\mathrm{MeV}/\hslash$, are used corresponding to the lower panels of Fig. 22.

Figure 24

The $I\to I\pm 1$ out-of-band $B\left(M1\right)$ to in-band $B\left(E2\right)$ ratios calculated with the Gogny-D1S interaction, are compared with the experimental data in ${}^{163}\mathrm{Lu}$. The cranked mean field with the frequencies ${\omega }_x=0.20,{\omega }_y={\omega }_z=0.0\mathrm{MeV}/\hslash$, are used, corresponding to the lower panels of Fig. 22.

Figure 25

Wobbling spectrum for ${}^{163}\mathrm{Lu}$ calculated by the angular-momentum-projection method with multicranked configuration mixing employing the Gogny-D1S effective interaction. The four cranked HFB states with ${\omega }_x=0.10,0.25,0.40,0.55\mathrm{MeV}/\hslash$ are configuration mixed. The energy of the experimental TSD1 [37] is also included.

Figure 26

Left panel: Absolute energy of the yrast band obtained by the multicranked configuration mixing in comparison with those calculated with a single HFB state with ${\omega }_x=0.10,0.25,0.40,0.55\mathrm{MeV}/\hslash$. Right panel: Moments of inertia defined in Eq. (18) for the lowest two TSD bands calculated by the angular-momentum-projection method with multicranked configuration mixing corresponding to Fig. 25. The experimental data are also included for comparison.

Figure 27

The excitation energy (left) and the expectation values of the angular-momentum vector (right) in ${}^{163}\mathrm{Lu}$ calculated by the multicranked configuration mixing corresponding to Fig. 25. The experimental excitation energies of TSD1, TSD2, and TSD3 [37] are also included in the left panel.

Figure 28

The $I\to I-2$ in-band $E2$ transition probabilities (left panel) and the $I\to I\pm 1$ out-of-band to in-band $B\left(E2\right)$ ratio (right panel), which are calculated by the multicranked configuration mixing with the Gogny-D1S interaction in ${}^{163}\mathrm{Lu}$, corresponding to Fig. 25.