Insight into nuclear midshell structures by studying $K$ isomers in rare-earth neutron-rich nuclei

Inspired by the newly discovered isomeric states in the rare-earth neutron-rich nuclei, high-$K$ isomeric states in neutron-rich samarium and gadolinium isotopes are investigated within the framework of the cranked shell model (CSM) with pairing correlation treated by a particle-number-conserving (PNC) method. The experimental multiparticle state energies and moments of inertia are reproduced quite well by the PNC-CSM calculations. A remarkable effect from the high-order deformation ${\varepsilon }_6$ is demonstrated. Based on the occupation probabilities, the configurations are assigned to the observed high-$K$ isomeric states. The lower $5^{-}$ isomeric state in ${}^{158}\mathrm{Sm}$ is preferred as the two-proton state with configuration $\pi {\frac{5}{2}}^{+}\left[413\right]\otimes \pi {\frac{5}{2}}^{-}\left[532\right]$. More low-lying two-particle states are predicted. The systematics of the electric quadrupole transition probabilities $B\left(E2\right)$ among the neodymium, samarium, gadolinium, and dysprosium isotopes and $N=96,98,100,102$ isotones is investigated to reveal the midshell collectivities.

Figure 1

Cranked Nilsson levels near the Fermi surface of ${}^{160}\mathrm{Sm}$ for protons (left) and neutrons (right) with signature $\alpha =+1/2$ (solid) and $\alpha =-1/2$ (dashed).

Figure 2

The experimental (left) and calculated bandhead energy with ${\varepsilon }_6\ne 0$ (middle) and ${\varepsilon }_6=0$ (right) for samarium isotopes. The experimental data are taken from Refs. [4, 8, 11, 12, 44, 45]. $K^{\pi }=5^{-}\left(1\right)$ and $K^{\pi }=5^{-}\left(2\right)$ in ${}^{158}\mathrm{Sm}$ denote the states taken from Simpson et al. in Ref. [12] and from Wang et al. in Ref. [44], respectively. The positive-parity (negative-parity) levels are denoted by red (black) lines.

Figure 3

The same as Fig. 2, but for gadolinium isotopes. The experimental data are taken from Refs. [4, 11].

Figure 4

Occupation probabilities $n_{\mu }$ of cranked Nilsson orbital $\mu$ (including both $\alpha =\pm 1/2$) near the Fermi surface of the samarium and gadolinium isotopes for the two-particle bands. The thick solid (dashed) lines denote positive (negative) parity orbitals. Fully occupied $n_{\mu }\approx 2$ and empty $n_{\mu }\approx 0$ orbitals denoted by thin lines are not labeled.

Figure 5

Calculated kinematic moments of inertia for samarium and gadolinium isotopes, compared with available experimental data [8, 9, 11, 12, 44, 45]. Experimental data are denoted by symbols, and the multiparticle state configurations of these bands are labeled by $K^{\pi }$. Data for the $K^{\pi }=5^{-}\left(1\right)$ and $K^{\pi }=5^{-}\left(2\right)$ bands in ${}^{158}\mathrm{Sm}$ are taken from Simpson et al. in Ref. [12] and from Wang et al. in Ref [44], respectively. Theoretical calculations are denoted by lines and their configurations are labeled as $J^{\left(1\right)}(\mu \otimes \nu )$.

Figure 6

Calculated $B(E2,\omega =0)$ systematics, connected by the dashed lines, for the GSB and two-particle states of samarium and gadolinium isotopes.

Figure 7

The same as Fig. 6, but for the GSB of $N=96,98,100,102$ isotones.

Figure 8

Left panel: Calculated $\omega$-dependent $B\left(E2\right)$ systematics, connected by the dashed lines, for the GSB of neodymium, samarium, gadolinium, and dysprosium isotopes. The thickness of the dashed lines denotes the $\omega$ dependence of $B\left(E2\right)$. Right panel: $B\left(E2\right)$ values versus rotational frequency $\hslash \omega$ for the GSB of neodymium, samarium, gadolinium, and dysprosium isotopes.