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Quartic oscillator potential in the γ-rigid regime of the collective geometrical model

A prolate $γ$-rigid version of the Bohr-Mottelson Hamiltonian with a quartic anharmonic oscillator potential in $β$ collective shape variable is used to describe the spectra for a variety of vibrational-like nuclei. Speculating the exact separation between the two Euler angles and the $β$ variable, one arrives to a differential Schrödinger equation with a quartic anharmonic oscillator potential and a centrifugal-like barrier. The corresponding eigenvalue is approximated by an analytical formula depending only on a single parameter up to an overall scaling factor. The applicability of the model is discussed in connection to the existence interval of the free parameter which is limited by the accuracy of the approximation and by comparison to the predictions of the related $X(3)$ and $X(3)$-$β^{2}$ models. The model is applied to qualitatively describe the spectra for nine nuclei which exhibit near vibrational features.