Microscopic analysis of prolate-oblate shape phase transition and shape coexistence in the Er-Pt region

The shape transition and possible occurrence of low-energy shape coexistence and rigid triaxial deformation are analyzed in the six even-even Er, Yb, Hf, W, Os, and Pt isotopic chains with the neutron number $102\ensuremath{\le}N\ensuremath{\le}124$, using a five-dimensional collective Hamiltonian (5DCH) based on covariant density-functional theory. The potential-energy surfaces display a transition from prolate to oblate or triaxial, and then to near spherical shapes as the neutron number increases. The corresponding 5DCH model calculations reproduce the empirical isotopic trend of the characteristic collective observables and confirm the overall shape transition in this region. It is emphasized that a rapid shape transition between prolate and oblate shapes is predicted in Er and Yb isotopic chains while it becomes smooth for higher-$Z$ isotopic chains and signature for rigid triaxial deformation is found in the transitional isotopes, e.g., $^{194}\mathrm{W}$ and $^{192\text{--}196}\mathrm{Os}$ by analyzing the energy staggering and probability density distribution in the $\ensuremath{\gamma}$ bands. Finally, the calculated low-lying spectra for $^{184}\mathrm{Er}$ and $^{186}\mathrm{Yb}$ demonstrate a remarkable multishape coexistence of medium-deformed oblate, medium- and large-deformed prolate shapes in both nuclei.