Collective Hamiltonian for wobbling modes

The simple, longitudinal, and transverse wobblers are systematically studied within the framework of a collective Hamiltonian, where the collective potential and mass parameter included are obtained based on the tilted axis cranking approach. Solving the collective Hamiltonian by diagonalization, the energies and the wave functions of the wobbling states are obtained. The obtained results are compared with those by the harmonic approximation formula and particle rotor model. The wobbling energies calculated by the collective Hamiltonian are closer to the exact solutions by the particle rotor model than the harmonic approximation formula. It is confirmed that the wobbling frequency increases with the rotational frequency in simple and longitudinal wobbling motions while decreases in transverse wobbling motion. These variation trends are related to the stiffness of the collective potential in the collective Hamiltonian.

Figure 1

Orientation of the rotational frequency $\mathit{\omega }$ with respect to the principal axes.

Figure 2

Lower panels: Contour plots of the total Routhian surface calculation $E^{\prime }(\theta ,\phi )$ for a triaxial rigid body rotor with $\gamma =-{30}^{\circ }$ at the frequencies $\hslash \omega =0.1$, $0.2$, $0.3$, and $0.4\mathrm{MeV}$. All energies at each rotational frequency are normalized with respect to the absolute minimum. Upper panels: The collective potential $V\left(\phi \right)$ as a function of $\phi$ extracted from the corresponding total Routhian surface calculation.

Figure 3

The collective levels and wave functions obtained from the collective Hamiltonian. Upper panel: The ten lowest energy levels labeled as $n=0$–9 (left) and the corresponding wave functions for even-$n$ (middle) and odd-$n$ (right) states at the frequency $\hslash \omega =0.1\mathrm{MeV}$. Lower panel: The ten lowest energy levels labeled as $n=0$–9 (left) and the corresponding wave functions for even-$n$ (middle) and odd-$n$ (right) states at the frequency $\hslash \omega =0.4\mathrm{MeV}$.

Figure 4

The simple wobbling frequency $\hslash {\Omega }_{\mathrm{wob}}$ obtained by the collective Hamiltonian in comparison with those of the HA formula (14).

Figure 5

The collective potentials obtained by HA (12) in comparison with those by TAC (11) at frequencies $\hslash \omega =0.1$ and $0.4\mathrm{MeV}$.

Figure 6

Energy spectra of four simple wobbling bands $n=1,2,3,4$ relative to the $n=0$ yrast sequence obtained by collective Hamiltonian in comparison with TRM and HA. In the TRM, the wobbling energies for even-$n$ wobbling bands are calculated as $E_{\mathrm{wob}}^n=E_n\left(I\right)-E_0\left(I\right)$, while for odd-$n$ wobbling bands $E_{\mathrm{wob}}^n\left(I\right)=E_n\left(I\right)-[E_0(I+1)+E_0(I-1)]/2$.

Figure 7

Same as Fig. 2 but for longitudinal wobbling motion, where a proton $h_{11/2}$ particle coupled to a triaxial rigid body rotor with $\gamma =-{30}^{\circ }$.

Figure 8

The calculated mass parameter as a function of rotational frequency $\hslash \omega$ for longitudinal wobbling motion.

Figure 9

Same as Fig. 3 but for longitudinal wobbling motion.

Figure 10

The longitudinal wobbling frequency $\hslash {\Omega }_{\mathrm{wob}}$ obtained by collective Hamiltonian in comparison with those of the HFA approximation (20).

Figure 11

The collective potentials obtained by HFA (18) in comparison with those by TAC (17) at frequencies $\hslash \omega =0.1$ and $0.4\mathrm{MeV}$.

Figure 12

Energy spectra of two longitudinal wobbling bands $n=1,2$ relative to the $n=0$ yrast sequence obtained by collective Hamiltonian in comparison with PRM and HFA. In the PRM, the wobbling energies for even-$n$, the wobbling energies are calculated as $E_{\mathrm{wob}}^n=E_n\left(I\right)-E_0\left(I\right)$, while for odd-$n$ $E_{\mathrm{wob}}^n\left(I\right)=E_n\left(I\right)-[E_0(I+1)+E_0(I-1)]/2$.

Figure 13

Same as Fig. 2 but for transverse wobbling, where a proton $h_{11/2}$ particle coupled to a triaxial irrotational flow rotor with $\gamma =-{30}^{\circ }$.

Figure 14

Same as Fig. 8 but for transverse wobbling motion.

Figure 15

Same as Fig. 3 but for transverse wobbling motion.

Figure 16

Same as Fig. 10 but for transverse wobbling motion.

Figure 17

Same as Fig. 12 but for transverse wobbling motion.