Analytical study of the $$ \gamma $$γ -unstable Bohr Hamiltonian with quasi-exactly solvable decatic potential
In this work, the $$\gamma $$-unstable Bohr Hamiltonian with quasi-exactly solvable decatic potential with a centrifugal barrier is considered to describe nuclei near the critical point of the vibrational to $$\gamma $$-unstable (shape) phase transition. Analytical expression of the wavefunctions and the corresponding eigen-energies are derived under the two quasi-exactly solvable constraints on parameters of the potential. The theoretical results are compared with those of the E(5) model and those of the same Hamiltonian with the quasi-exactly solvable sextic potential. Low-lying level energy ratios and B(E2) ratios of some nuclei near the critical point are fitted and compared with the experimental data and the fitting quality of previous models. The fitting results show that the E(5) model is indeed the best in description of nuclei near or at this critical point, while the decatic model seems a little better than the sextic one near the critical point. Furthermore, though further check on the fitting quality to the B(E2) values is needed, the fitting results for even-even $$^{118{-}128}$$Xe show that the decatic model seems the best in fitting the energy ratios, while the fitting qualities of both the decatic model and the E(5) model to the B(E2) ratios are quite the same, which are always better than the sextic model.