Quasiparticle and $\gamma$-band structures in ${}^{156}\mathrm{Dy}$

Excited band structures recently observed in ${}^{156}\mathrm{Dy}$ are investigated using the microscopic triaxial projected shell model (TPSM) approach and the quasiparticle random phase approximation (QRPA) based on the rotating mean field. It is demonstrated that new observed excited bands, tracking the ground-state band, are the $\gamma$ bands based on the excited two-quasineutron configurations as conjectured in the experimental work.

Figure 1

TPSM projected energies, before band mixing, of positive parity states for ${}^{156}\mathrm{Dy}$. Bands are labeled as ($K$, qp numbers) so that the labels $(0,0), (2,0), (4,0), (1,2n), (3,2n), (1,2p), (3,2p), (0,2n2p), (2,2n2p)$, and $(4,2n2p)$ correspond to ground, $\gamma , 2\gamma$, two-neutron aligned state, $\gamma$ band on this two-neutron aligned state, two-proton aligned state, $\gamma$-band on this two-proton aligned state, two-neutron plus two-proton aligned state, and $\gamma$ and $\gamma \gamma$ bands built on this four-quasiparticle state. The excited $K=0$ resulting from projection of two quasineutrons and two quasiprotons are denoted by $(0,2n)$ and $(0,2p)$, respectively. Some bands, having large signature splitting, are separated into even-spin (labeled as $\alpha =0$) and odd-spin states (labeled as $\alpha =1$). The first crossing at $I=14$ is due to the $(1,2n)$ aligned configuration that forms the $s$-band configuration. The second and third crossings at about $I=24$ and 28 are due to the aligning $4n$ configurations with $K=2$ and 4, respectively. The inset depicts the four-quasiparticle crossings with the ground-state band.

Figure 2

Comparison of the measured positive parity energy levels of the ${}^{156}\mathrm{Dy}$ nucleus [6] and the results of TPSM calculations.

Figure 3

Experimental energies (right panel) of the positive parity bands in ${}^{156}\mathrm{Dy}$ as reported in Fig. 4 of Ref. [6] compared with the TPSM calculations (left panel). Labeling: Eg - ground-state band, Es - $s$ band, Egae - even-spin $\gamma$-band, Egao - odd-spin $\gamma$ band, ESV - SV band, E12 - band 12, E17 - band 17, E20 - band 20. Even-spin states are connected by solid lines, odd-spin states by dashed lines. In the upper panel $I_{\mathrm{ref}}=\omega {\mathcal{J}}_0+{\omega }^3{\mathcal{J}}_1$ is subtracted. In the lower panel $E_{\mathrm{ref}}^{\prime }=-{\omega }^2{\mathcal{J}}_0/2-{\omega }^4{\mathcal{J}}_1/4$ is subtracted with ${\mathcal{J}}_0=23{\hslash }^2{\mathrm{MeV}}^{-1}$ and ${\mathcal{J}}_1=90{\hslash }^4{\mathrm{MeV}}^{-3}$.

Figure 4

Probability of various projected $K$ configurations in the wave functions of the band structures after diagonalization are plotted for the ${}^{156}\mathrm{Dy}$ nucleus.

Figure 5

Calculated $B\left(E2\right)$ transition probabilities for the ground-state band and the $s$ band using the TPSM approach. The experimental values have been taken from Refs. [38, 39, 40].

Figure 6

Ratios of the $B\left(E2\right)$ values for out-of-band to in-band transitions $R_{E2}=B(E2;\text{out})/B(E2;\text{in})$ for the $\gamma$ decays from the $\gamma$ band to the ground-state band and for the $\gamma$ decays from bands 17 and 20 to the $s$ band

Figure 7

Comparison of observed and TPSM calculated staggering parameter, Eq. (5), for the $\gamma$ band in ${}^{156}\mathrm{Dy}$.

Figure 8

Quasiparticle Routhians for ${}^{156}\mathrm{Dy}$.

Figure 9

Energies of the lowest positive parity bands in ${}^{156}\mathrm{Dy}$ calculated by means of the cranking model. The bands are labeled by the occupied quasiparticles in Fig. 8. “Eg” is the $\gamma$ band calculated by means of the tidal wave approach. Even-spin states are connected by straight lines, odd-spin states by dashed lines.

Figure 10

Upper panel: Angular-momentum expectation values calculated by means of the cranking model relative to the reference. Lower panel: Routhians relative to the reference. Reference as in Fig. 3.

Figure 11

Routhians of the lowest positive parity bands in ${}^{156}\mathrm{Dy}$ calculated by means of the QRPA extension of the rotating mean-field derived from the pairing + quadrupole quadrupole model Hamiltonian. Even-spin states are connected by straight lines, odd-spin states by dashed lines. The green line depicts the QRPA results for the $\beta$ band, i.e., the band based on the $0_2^{+}$ state. The figure was prepared using the results of Refs. [18, 19, 20]. Reference energy is as in Fig. 3.

Figure 12

Energies corresponding to angular momenta $J_3=0$ and $J_3=2\hslash$ calculated by the shell correction version of the cranking model.