Bohr-Mottelson Hamiltonian with octic potential applied to the $^{106-116}$Cd isotopes

The Bohr-Mottelson Hamiltonian, with an octic potential in the $\beta$-deformation variable, is numerically solved for a $\gamma$-unstable symmetry of the nuclear system. The analytical structure of the model allows the description of multiple phenomena of great interest for the nuclear structure such as ground-state shape phase transitions and their critical points, dynamical shape phase transitions, shape coexistence with and without mixing, anomalous in-band $E2$ transitions, large $E2$ intra-band transitions and large monopole transition between the first excited $0^+$ state and the ground state, respectively. As a first application of the present model is selected the $^{106-116}$Cd isotope chain known in literature to manifest shape phase transition, respectively shape coexistence and mixing.