Extended analytical solutions of the Bohr Hamiltonian with the sextic oscillator

Low-lying collective quadrupole states in even-even nuclei are studied for the particular case of a gamma-unstable potential within the Bohr Hamiltonian. In particular, the quasi-exactly solvable beta-sextic potential is extended to cover the most relevant part of the low-lying spectra in nuclei. In previous papers (2004 Phys. Rev. C 69 014304, 2010 Phys Rev. C 81 044304), the same situation was solved for beta-wavefunctions with up to one node (M = 0, 1), which are relevant for the first few low-lying states. Here, the model space is enlarged by including beta-wavefunctions also with two nodes (M = 2), which generate many more states, in order to make it useful for actual fittings and more detailed checking of shape phase transitions between spherical and gamma-unstable beta-deformed shapes in nuclei. In addition to the energy eigenvalues and wavefunctions, closed analytical formulas are obtained for electric quadrupole and monopole transition probabilities too. The model is applied to the chains of even Ru and Pd isotopes to illustrate the transition between the spherical and deformed gamma-unstable phases. These applications indicate that the optional extension of the model with a phenomenologic rotational term L . L is consistent with the experimental data.