The nature of 0+ excitations in deformed nuclei

Abstract

This is a review on the nature of low-lying 0${}^{+}$ states in the excitation spectra of deformed nuclei. Early in the history of the field, Bohr–Mottelson–Rainwater won the 1975 Nobel prize in physics for connecting nucleon motion to the emergent collective behavior observed in nuclei. They essentially described the nucleus as a geometric shape with rotational and vibrational degrees of freedom. The lowest shape affecting vibrations in nuclei would be quadrupole ($\lambda$=2). In spherical nuclei, the oscillations of the ground state shape were expected to yield an energy spectrum that could be described in terms of single and multiple quadrupole phonons. In deformed nuclei, rotational motion is prominent and could be described in terms of a rigid rotor. The question in nuclear structure physics that has remained unanswered for decades is the viability of a deformed nucleus to sustain oscillations or vibrations built on the ground state. The quadrupole oscillations in deformed nuclei could result in two types of vibrations: $\beta$-vibrations resulting from oscillations along the symmetry axis with K${}^{\pi }=0^{+}$ and $\gamma$-vibrations breaking axial symmetry with a projection of K${}^{\pi }=2^{+}$ on the symmetry axis. The K${}^{\pi }=2^{+}$ or $\gamma$-vibrational bands are well characterized and accepted as oscillations around the g.s. The question which has remained open is the nature of the K${}^{\pi }=0^{+}$ bands. Historically, 0${}^{+}$ states were difficult to observe and to measure, more recently however, there has been a large abundance of states identified. The discussions have shifted towards the characterization of these 0${}^{+}$ states. The systematics of the observed B(E2) values depopulating the K${}^{\pi }=0^{+}$ bands were shown to be weaker than the K${}^{\pi }=2^{+}$ band decays. Questions arose about the nature of the K${}^{\pi }=0^{+}$ bands. Were they indeed vibrations built on the ground state? Or are they coexisting minima of other shapes? The debates and discussions have led to a reexamination of the nature of vibrational excitations. A $\beta$-vibrational band built on the ground state shape of a deformed nucleus is expected to show the same degree of deformation, hence the same dynamic moment of inertia, and perhaps even the same intrinsic quadrupole moment. Geometric, microscopic, and algebraic theoretical nuclear models have revisited the predictions and expectations of a $\beta$-vibration in contrast to a coexisting minimum of a different shape. The topic continues to be of great interest in nuclear structure studies as evidenced by the hundreds of theoretical and experimental publications on the topic. The ability of deformed nuclei to sustain oscillations or vibrations is fundamental to understanding the properties of the nuclear quantum system.

This review brings together the extensive data sets from the numerous 0${}^{+}$ states that have been observed in the past six decades, their lifetime measurements, transition probabilities, transfer reaction populations, dynamic moments of inertia, and the extracted intrinsic quadrupole moments to clearly identify $\beta$ vibrations. Two-neutron transfer reactions were expected to elucidate the nature of 0${}^{+}$ states. However as shown in this extensive data collection effort, they do not provide the definitive answers to the open question regarding the nature of these states. The studies reported here are confined to the Z=50-82 region of the chart of nuclides with the largest demonstrated regions of deformation. The discussion has specifically only focused on highly deformed nuclei in order to avoid any confusion with coexisting minima which are not expected in high deformation regions. The theory section explores and briefly presents a tour of the numerous relevant theoretical models and the resulting constraints or assertions with respect to the nature of vibrations built on a deformed ground state. The interpretation and discussion chapters present the analysis of the vast body of knowledge that has been developed. The result is the identification of a large number of 0${}^{+}$ bands as $\beta$-vibrations in the spectra of well-deformed nuclei. The list includes ${}^{152,154}$Sm, ${}^{154,156,158}$Gd, ¹⁶²Dy, ¹⁶⁸Er, ¹⁶⁸Yb, ¹⁷⁸Hf, and the ${}^{182,184}$W nuclei.