Rotational bands in the quadrupole-octupole collective model

A collective band of positive as well as negative parity could be composed of vibrational and rotational motions. The octupole vibrational configurations can be based either on axial or nonaxial octupole excitations. A consistent approach to the quadrupole-octupole collective vibrations coupled with the rotational motion enables us to distinguish between various scenarios of disappearance of the $E2$ transitions in negative-parity bands observed in several nuclei. The theoretical estimates presented here are compared with the very recent experimental energies and transition probabilities in and between the ground-state and low-energy negative-parity bands in ${}^{156}\mathrm{Dy}$. A realistic collective Hamiltonian contains the potential-energy term obtained through the macroscopic-microscopic Strutinsky-like method with a particle-number-projected BCS approach and a deformation-dependent mass tensor. The potential energy and the inertia parameters are defined in the vibrational-rotational, nine-dimensional collective space of the multipole-deformation parameters and Euler angles. The symmetrization procedure applied to the eigenstates of the collective Hamiltonian ensures their uniqueness with respect to the laboratory coordinate system. This quadrupole-octupole collective approach may also allow us to find and/or verify some fingerprints of possible high-rank symmetries (e.g., tetrahedral, octahedral, ...) in nuclear collective bands.

Figure 1

The potential energy of ${}^{156}\mathrm{Dy}$ in the quadrupole plane $\left({\alpha }_{20},{\alpha }_{22}\right)$.

Figure 2

Potential-energy maps generated for quadrupole $({\alpha }_{20},{\alpha }_{22}=0)$ versus octupole $({\alpha }_{30},{\alpha }_{31},{\alpha }_{32},{\alpha }_{33})$ deformations.

Figure 3

The predicted ground-state and odd-spin negative-parity bands of ${}^{156}\mathrm{Dy}$. Arrows mark the $E1$ and $E2$ transitions. The branching ratios $B\left(E2\right)/B\left(E1\right)$ are written in parentheses.

Figure 4

Similar to Fig. 3 but for the predicted ground-state and even-spin negative-parity bands of ${}^{156}\mathrm{Dy}$.

Figure 5

The negative-parity states built on octupole axial and nonaxial one-phonon excitations in ${}^{156}\mathrm{Dy}$. Colors mark $(J+1)/2$ states with different $\kappa$ numbers for given spin $J$.

Figure 6

The distribution of rotational basis states of given $K$ for the ground-state band of the ${}^{156}\mathrm{Dy}$.

Figure 7

Distribution of rotational basis states of given $K$ for two model bands based on ${\alpha }_{30}$ and ${\alpha }_{32}$ one-phonon excitations of ${}^{156}\mathrm{Dy}$ for odd-spin negative-parity-level scheme describing the experimental band 2.

Figure 8

Similar to Fig. 7 but for model band based on ${\alpha }_{32}$ one-phonon excitation as a candidate for even-spin negative-parity-level scheme of experimental band 4.