Two-dimensional collective Hamiltonian for chiral and wobbling modes

A two-dimensional collective Hamiltonian (2DCH) on both azimuth and polar motions in triaxial nuclei is proposed to investigate the chiral and wobbling modes. In the 2DCH, the collective potential and the mass parameters are determined from three-dimensional tilted axis cranking (TAC) calculations. The broken chiral and signature symmetries in the TAC solutions are restored by the 2DCH. The validity of the 2DCH is illustrated with a triaxial rotor ($\gamma =-{30}^{\circ }$) coupling to one $h_{11/2}$ proton particle and one $h_{11/2}$ neutron hole. By diagonalizing the 2DCH, the angular momenta and energy spectra are obtained. These results agree with the exact solutions of the particle rotor model (PRM) at high rotational frequencies. However, at low frequencies, the energies given by the 2DCH are larger than those by the PRM due to the underestimation of the mass parameters. In addition, with increasing angular momentum, the transitions from the chiral vibration to chiral rotation and further to longitudinal wobbling motion have been presented in the 2DCH.

Figure 1

Total Routhian surface calculations for one $h_{11/2}$ proton particle and one $h_{11/2}$ neutron hole coupled to a triaxial rotor with $\gamma =-{30}^{\circ }$ at the frequencies $\hslash \omega =0.10$, 0.20, 0.30, 0.50, 0.70, 0.90 MeV. All energies are normalized with respect to the absolute minima (stars). Note that different scales have been used in different panels. In each panel, twenty contour lines are shown.

Figure 2

The mass parameters $B_{\theta \theta }, B_{\theta \phi }$, and $B_{\phi \phi }$ as functions of $\theta$ and $\phi$ at the frequencies $\hslash \omega =0.10$, 0.50, and 0.90 MeV calculated based on TAC. Note that different scales have been used in different panels. In each panel, twenty contour lines are shown.

Figure 3

The $B_{\theta \theta }$ and $B_{\phi \phi }$ and the curvatures $K_{\theta }$ and $K_{\phi }$ at the minimum of the potential energy surface as functions of rotational frequency.

Figure 4

Collective energy levels obtained from the two-dimensional collective Hamiltonian at the frequencies $\hslash \omega =0.10$, 0.20, 0.30, 0.50, 0.70, and 0.90 MeV. Note that different scales have been used in different panels.

Figure 5

The density profiles of of the lowest states in groups $(++), (+-), (-+)$, and $(--)$ at the frequencies $\hslash \omega =0.10$, 0.50, and 0.90 MeV calculated by the 2DCH.

Figure 6

Collective energy levels obtained from the two-dimensional collective Hamiltonian in comparison with those from the one-dimensional collective Hamiltonian for $\hslash \omega =0.10$, 0.50, and 0.90 MeV. Note that different scales have been used in different panels.

Figure 7

Left: The wave functions along the $\phi$ direction obtained by the 2DCH in comparison with those by the 1DCH. Right: The probability distributions $P\left(\theta \right)$ (31) along the $\theta$ direction obtained by the 2DCH.

Figure 8

The energy splittings between the chiral partner states (open and solid circles) and the signature partner states (open and solid squares) as functions of rotational frequency obtained by 2DCH. Inset: For visibility, the results at the rotational frequency region $0.50\le \hslash \omega \le 0.90\mathrm{MeV}$ are shown at a small scale.

Figure 9

The angular momenta of the lowest states in groups $(++), (+-), (-+)$, and $(--)$ as functions of rotational frequency obtained from the 2DCH in comparison with the yrast and yrare bands 1 and 2 in the PRM and the yrast band 1 in the TAC. The angular momenta are relative to a rigid-rotor reference $40\omega$.

Figure 10

The energy spectra of the lowest states in the groups $(++), (+-), (-+)$, and $(--)$ as functions of spin obtained by the 2DCH in comparison with the exact solutions of the PRM, and also the solution of the TAC. The energies are relative to a rigid-rotor reference $0.007I(I+1)$.

Figure 11

Same as Fig. 10, but with the mass parameter calculated by assuming constant vibrational frequencies $\hslash {\mathrm{\Omega }}_{\theta }=\hslash {\mathrm{\Omega }}_{\phi }=1.10\mathrm{MeV}$ (see text).

Figure 12

The mass parameters $B_{\theta \theta }$ and $B_{\phi \phi }$ calculated by Eq. (36) with the vibrational frequencies $\hslash {\mathrm{\Omega }}_{\theta }=\hslash {\mathrm{\Omega }}_{\phi }=1.10\mathrm{MeV}$ and the energy differences ${\varepsilon }_2-{\varepsilon }_1$ between the lowest two levels for the proton and neutron obtained by TAC at the minimum of the potential energy surface.