Shape phase transition, coexistence and mixing in the $^{98-106}$Ru isotopes

The deformation properties within the $^{98-106}$Ru even-even isotopic chain, are investigated by means of the Covariant Density Functional Theory with a Density-Dependent Point-Coupling X parametrization. The considered nuclei are found to exhibit very shallow prolate and triaxial ground state deformation. This information is used to ascertain their dynamical behavior within prolate $\gamma$-stable and $\gamma$-unstable instances of a phenomenological Bohr-Mottelson Hamiltonian with an octic potential in the axial deformation variable. The comparative study of the low-lying collective states, revealed the presence of a shape phase transition from low to high deformation, as well as evidence of shape coexistence and mixing between spherical vibrator, $\gamma$-unstable or prolate configurations in ground and excited states. It is also shown that the effect of shape coexistence and mixing on the $\gamma$-band states can account to some extent for the typical $\gamma$-unstable staggering even in prolate $\gamma$-stable conditions.