Quantum rotational band formulas from a two-parameter potential and the microscopic explanation from the fermion dynamical symmetry model.

Abstract

In this paper, we first demonstrate the applicability of a phenomenological two-parameter formula, as introduced by Holmberg and Lipas from the Bohr Hamiltonian in a way that is different from Wu and Zeng. Second, for the first time, we show microscopically that Holmberg's two-parameter formula can be applied to diatomic molecules and that it can fit the experimental data of rotational spectra of HCl, HBr, and HF very well when the parameters are determined by two arbitrary experimental levels. Third, we derive a two-parameter formula describing \ensuremath{\gamma}-soft rotational spectra which is similar to the Holmberg formula, called Holmberg-like formula in this paper. The experimental yrast lines of nine nuclei in the light rare-earth region are fitted by this formula. For the nuclear \ensuremath{\gamma} stiffness, \ensuremath{\gamma} softness, and for molecular rotational spectra, all the two-parameter formulas are obtained by making use of a single potential function. It is demonstrated that the reason why one can give a unified description for those three systems is the common rotational features like the widely used harmonic oscillator approximation. More importantly, from the more microscopic nuclear fermion dynamical symmetry model (FDSM), we may derive the variable moment of inertia (VMI) model, and further obtain the Holmberg formula and Holmberg-like formula under a certain approximation, as from the nuclear geometric description within the Bohr-Mottelson model (BM). It is shown that the bridge between the descriptions of the FDSM and of the BM is the effect of stretched alignment (stretching effect). According to another interpretation of the FDSM for the nuclear stretching effect, we also give a simple formula to explain the \ensuremath{\gamma}-soft rotational spectra and compare the formula with the above one. Finally, we give a phenomenological generalization to the combination of the Holmberg and Holmberg-like formulas, which may describe a transition from \ensuremath{\gamma}-stiff rotations to \ensuremath{\gamma}-soft rotations.