Microscopic analysis of shape transition in neutron-deficient Yb isotopes

The development of nuclear collectivity in even-even ${}^{152\text{–}170}\mathrm{Yb}$ is studied with three types of mean-field calculations: the nonrelativistic Hartree-Fock plus BCS calculation using the Skyrme SLy4 force plus a density-dependent $\delta$ pairing force and the relativistic mean-field calculation using a point-coupling energy functional supplemented with either a density-independent $\delta$ pairing force or a separable pairing force. The low-lying states are obtained by solving a five-dimensional collective Hamiltonian with parameters determined from the three mean-field solutions. The energy surfaces, excitation energies, electric multiple transition strengths, and differential isotope shifts are presented in comparison with available data. Our results show that different treatments of pairing correlations have a significant influence on the speed of developing collectivity as the increase of neutron number. All the calculations demonstrate the important role of dynamic shape-mixing effects in resolving the puzzle in the dramatic increase of charge radius from ${}^{152}\mathrm{Yb}$ to ${}^{154}\mathrm{Yb}$ and the role of triaxiality in ${}^{160,162,164}\mathrm{Yb}$.

Figure 1

Mean-field energy $E_{\mathrm{tot}}$ of ${}^{152\text{–}170}\mathrm{Yb}$ isotopes in the $\beta \text{−}\gamma$ plane, from the constrained RMF + BCS calculations with the PC-PK1 parametrization for the particle-hole (ph) channel and density-independent $\delta$ force (DIDF) for the particle-particle (pp) channel. All energies are normalized to the absolute minimum. Each contour line is separated by 0.5 MeV.

Figure 2

Same as Fig. 1, but by the SLy4 force for the ph channel and density-dependent $\delta$ force (DDDF) for the pp channel.

Figure 3

Systematics of excitation energies and electric quadrupole transition strengths for the ground-state rotational band in neutron-deficient Yb isotopes, in comparison with the data [43].

Figure 4

Average pairing gaps ${\langle \mathrm{\Delta }\rangle }^{uv}$ (in MeV) for neutrons and protons in ${}^{160}\mathrm{Yb}$ from the RMF calculations with either the DIDF or the TMR force [39] for the pp channel.

Figure 5

The ratio $R_{42}=E_x\left({4^{+}}_1\right)/E_x\left(2{{}^{+}}_1\right)$ as a function of neutron number in Yb isotopes from the 5DCH calculations with different forces, in comparison with the data [43].

Figure 6

Systematics of excitation energies for the states in the $\gamma$-band for the neutron-deficient Yb isotopes, in comparison with the data [43].

Figure 7

Electric monopole transition strength ${\rho }_{E0}^2({0^{+}}_2\to {0^{+}}_1)$ and excitation energy of ${0^{+}}_2$ state as functions of neutron number in Yb isotopes from the 5DCH calculations with the PC-PK1 parametrization of covariant energy density functional, in comparison with the data [43].

Figure 8

Variation of the isotope shifts with respect to ${}^{152}\mathrm{Yb}$ (upper panel) and the differential isotope shifts (lower panel) $\delta {\langle r^2\rangle }_{N,N-2}={\langle r^2\rangle }_N-{\langle r^2\rangle }_{N-2}$ as a function of neutron number for the ground states of Yb isotopes, in comparison with the data [45].

Figure 9

Low-lying energy spectra of ${}^{156,158}\mathrm{Yb}$ by the 5DCH calculation with the PC-PK1 (DIDF), in comparison with available data. The absolute value of the reduced transition matrix element $\langle J\left|\right|E2\left|\right|J^{\prime }\rangle$ (in $eb$) is indicated on the arrows.