Quantum Choreography of the Nucleus: Rotations, Vibrations, and Emergent Structure

Abstract

Nuclei are complex many-body quantum systems where interactions of the neutrons and protons via the strong, the weak, and the electromagnetic forces lead to the emergence of simple patterns of energy states that have been described by various theoretical approaches. One of the goals of all the theoretical models is the development of a universal theory that can be applied across the entire chart of nuclides. Significant progress has been made by experiments as well as the increasing sophistication of models, but a universal theory has yet to be established. A recent reviewof nuclei in the Z = 50–82 region of the chart of nuclides has analyzed all the available compiled data from several decades of studies towards a clarification of the low-lying structure of nuclei. Other reviews have reported and explained the emergence of multiple different shapes in nuclei at somewhat higher excitation energies than the ground state. Somehave challenged the interpretation of the first excited K^π $=0^{+}$ band as a vibration of ground state. This work attempts to provide a guide to determining the nature of the first excited K^π $=0^{+}$ band in nuclei by the combined use of nuclear lifetimes, energy level evolutions, dynamic moments of inertia, and intrinsic quadrupole moments extracted from transition probabilities. The result is that for a subset of the nuclei in this region, the K^π $=0^{+}$ band is consistent with the traditional $\beta$-vibration description of an oscillation built on the ground state.

1. Introduction

Rotations and vibrations of any physical system reveal the underlying structure, the governing forces, the symmetries, and the emergent behaviors of that system. Macroscopic scales include astrophysical objects, such as stars, planets, disks, and galaxies, which evolve under gravity, rotations, and magnetic fields over vast ranges of length and time. Atomic nuclei at the other end of the dimension scale evolve under the strong, the weak, and the electromagnetic forces via complex many-body interactions of protons and neutrons to yield simple emergent patterns of excitation levels (level schemes) and the electromagnetic transition probabilities linking the excited states. While the relevant forces and scales differ substantially between astrophysical bodies, solids, molecules, and nuclei, the general concepts of dynamical observables that provide significant information about the internal structures remain and provide a useful point of comparison. Rotations and vibrations encode information about shapes, rotational bands, coexisting minima, collective behavior, and moments of inertia.

Historically, the connection between particle motion and collective degrees of freedom was established by Bohr, Mottelson, and Rainwater, recognized by the 1975 Nobel Prize in Physics. In this framework, the nucleus was described in terms of intrinsic shapes that support rotational and vibrational excitations. The lowest-order shape affecting the degree of freedom is quadrupole ($\lambda =2$). Near spherical shapes, quadrupole oscillations are expected to produce excitation spectra that can be interpreted in terms of one- and multi-phonon quadrupole vibrations of a near spherical shape. For well-deformed nuclei, rotational sequences dominate. The expectations and observations show well deformed shapes away from sphericity with a level scheme that can be described as a rotor with energy spacings given by J(J + 1) factor of the moments of inertia. A long-standing question in nuclear structure concerns the extent to which a well-deformed nucleus can sustain vibrational excitations built on the ground-state shape in addition to the rotational motion. Figure 1 shows the chart of nuclides as a function of the quadrupole deformation $\beta$ where the largest region of deformation is the region of nuclei from Z = 50–82.

Low lying quadrupole oscillations of a nucleus along the symmetry axis are commonly classified as $\beta$ vibrations with no projection of angular momentum on the symmetry axis to form bands of excited levels with K^π $=0^{+}$, while $\gamma$ vibrations, associated with oscillations that break axial symmetry, yield a projection of 2 on the symmetry axis, and rotational bands built on K^π $=2^{+}$ excitations. The focus of this paper is specifically on the nature of the first excited 0⁺ states in deformed nuclei. These $0^{+}$ excitations are difficult to observe in experiments, and therefore, difficult to characterize; more recently, improved experimental capabilities have revealed a rich density of $0^{+}$ states in many deformed nuclei. Figure 2 shows the number of 0⁺ states observed below 3 MeV in excitation energy. A few of them were observed previously, but a large percentage of them were measured at the Technical University of Munich’s nuclear physics laboratory using the high precision system of a focal plane detector paired with the Q3D spectrometer [1].

The Z = 50–82 nuclei are also in the region of the chart of nuclides where an abundance of data, including level schemes, lifetime measurements, and transfer reactions, are available. The interpretation of $\gamma$-vibrational bands as small-amplitude oscillations about the ground-state minimum is broadly supported by experimental systematics. In contrast, the nature of excited K^π $=0^{+}$ bands has remained unsettled. The observations of the 0⁺ excitations and their reduced transition probabilities, B($E2$) values, connecting candidate K^π $=0^{+}$ bands to the ground-state band are often weaker than those observed for established K^π $=2^{+}$ bands, complicating a simple $\beta$-vibrational interpretation. The reduced transition probabilities are shown for several nuclei in Figure 3.

This paper uses the existing information reported in a recent review [2], and summarizes with examples, how to clearly identify the nature of the first excited state in regions of nuclear structure evolution. There were arguments made in favor of a coexisting minimum interpretation rather than a vibrational or $\beta$-type excitation. The data, along with numerous theoretical interpretations, have motivated renewed scrutiny of whether excited K^π $=0^{+}$ bands correspond to genuine $\beta$ vibrations or instead reflect a shape coexistence minimum, a pairing isomerism, or other non-collective or mixed configuration state [3,4,5,6]. We present a clear way of characterizing the first excited K^π $=0^{+}$ either as oscillations built on the ground state or a coexisting minimum of a different shape. A $\beta$-vibrational band built on the ground-state minimum is expected, to leading order, to exhibit a deformation comparable to that of the ground-state band, implying similar intrinsic quadrupole moments and dynamic moments of inertia. By contrast, a band associated with a competing shape minimum may display systematically different rotational properties and transition strengths. Modern geometric, microscopic, and algebraic approaches continue to revisit these expectations, and the ability of deformed nuclei to support vibrational motion remains a central problem in understanding collective behavior in finite quantum many-body systems.

2. Nuclear Models

While a fully universal and predictive nuclear model remains elusive, a range of complementary theoretical frameworks continues to provide valuable insights into nuclear structure across the chart of nuclides. Progress in nuclear structure physics relies somewhat on a pluralistic strategy, where diverse models—each grounded in different approximations and symmetries—collectively constrain, inform, and refine our understanding of complex nuclear phenomena. The approaches discussed include global calculations based on Hartree–Fock–Bogoliubov theory extended by the generator coordinate method and mapped onto a five-dimensional collective quadrupole Hamiltonian (CHFB+5DCH), the Interacting Boson Approximation (IBA) and its extensions incorporating partial dynamical symmetries, the pseudo-SU(3) shell model, proxy-SU(3), and models addressing shape coexistence [3,4,7,8,9,10,11]. Many additional theoretical frameworks exist but are not discussed here in detail. Our aim is not to provide an exhaustive survey, but rather to capture the essential ideas most relevant to understanding the nature of the first excited $0^{+}$ states. Approaches such as the Random Phase Approximation (RPA), the Quasiparticle Random Phase Approximation (QRPA) [12,13,14,15], and the Quasiparticle Phonon Model (QPM) [16,17,18,19] focus on multi-phonon vibrational excitations and their microscopic origins and are, therefore, not expanded upon in this paper.

In this paper, we point to the aspects of the various models used in the review to identify a subset of the nuclei in the Z = 50–82 region with viable vibrational character for the first K^π $=0^{+}$ band. These aspects are discussed in the following sections.

2.1. CHFB+5DCH: Structure of Even-Even Nuclei with the D1S Gogny Interaction

Delaroche et al. [7] performed a global, systematic study of nuclei with Z = 10–110 and N $\le 200$, covering nearly the entire nuclear chart from drip line to drip line. Their calculations were carried out within constrained Hartree–Fock–Bogoliubov (CHFB) theory using the D1S Gogny interaction, extended by the generator coordinate method and mapped onto a five-dimensional collective quadrupole Hamiltonian (5DCH). This framework is particularly valuable because it attempts a unified description of nuclear structure across the chart of nuclides. Calculated observables include ground-state charge radii, two-particle separation energies, correlation energies, intrinsic quadrupole deformation parameters, excitation spectra, and quadrupole and monopole transition matrix elements. Of particular relevance to this paper are the lowest excited $0^{+}$ states and the first two excited $2^{+}$ states. Many predicted properties depend strongly on intrinsic deformation. The results are especially reliable for well-deformed nuclei with $R_{4/2}\approx 3.33$ (the axial rotor limit), in good agreement with experimental data. For the first excited $0^{+}$ states, the theory reproduces the qualitative excitation-energy trends but sometimes overestimates the energies.

The calculated non-yrast $2^{+}$ states exhibit a clear separation between $\gamma$- and $\beta$-type excitations. The character of the first excited $0^{+}$ state ($0_2^{+}$) is interpreted either as a $\beta$ vibration or as a manifestation of shape coexistence, based primarily on quadrupole transition strengths. A $\beta$ vibration is identified when the excited configuration has nearly the same average deformation as the ground state but with a larger fluctuation in $\beta$. Alternatively, the $0_2^{+}$ state may arise from the coexistence of distinct intrinsic shapes at comparable energies. Shape coexistence is particularly prominent in light doubly magic nuclei [20,21] and in the actinide region, where superdeformed configurations occur at relatively low excitation energy [22,23,24]. Delaroche et al. identify four regions where the conditions for $\beta$-vibrational character are satisfied, including strongly deformed rare-earth and actinide nuclei. In such systems, where $R_{4/2}$ approaches the axial rotor limit, the $0_2^{+}\to 2_1^{+}$ transition matrix element is significantly larger than those involving higher-lying $2^{+}$ states, consistent with an interpretation of these states in terms of relatively pure vibrational phonons or oscillations on the ground state. The full experimental findings in the Z = 50–82 region as evolutions of elemental structure in comparison with the CHFB+5DCH predictions for the first excited $0^{+}$ states are shown in Aprahamian et al. [2].

2.2. Interacting Boson Approximation (IBA)

Low-lying collective excitations can also be described by algebraic models developed in the late 1970s and early 1980s, notably the Interacting Boson Approximation (IBA) [25,26,27]. In this framework, valence nucleon pairs outside closed shells are mapped onto s and d bosons forming a U(6) algebra. Only three dynamical symmetry limits of U(6) contain the rotational group O(3), corresponding to spherical (U(5)), axially deformed (SU(3)), and $\gamma$-soft (O(6)) structures. These limits are visualized as the vertices of a triangle, shown in Figure 4. The IBA successfully reproduces systematic trends in the evolution of low-lying collective motion with a relatively small number of parameters instead of using a set of distinct models for each type or symmetry.

The number of bosons is determined by counting valence nucleon pairs relative to the nearest closed shell, starting with holes beyond mid-shell. For instance, in ¹⁶⁸Er (Z = 68, N = 100), one counts 14 proton holes relative to Z = 82 (7 bosons) and 18 neutrons beyond N = 82 (9 bosons), giving a total boson number of 16. Structural evolution across the symmetry triangle is governed primarily by the quadrupole–quadrupole interaction, which lowers the energy of collective excitations. In the SU(3) limit, the first excited $0^{+}$ and $2^{+}$ bands belong to the same irreducible representation as the ground-state band and would not decay to it. However, realistic nuclei break SU(3) symmetry, leading to finite interband transitions. A study of B($E2$) dominance patterns [28] predicted that $\beta \to gs$ transitions are weaker than $\beta \to \gamma$ and $\gamma \to gs$ transitions by approximately a factor of six, largely independent of deformation. This result provides a benchmark for interpreting experimental transition rates.

Figure 4. The three symmetries of the IBA [29,30]. The U(5) symmetry represents the spherical limit, SU(3) the extreme well-deformed limit, and O(6) the $\gamma$-unstable limit. The color scale indicates the quadrupole collectivity’s contribution to the binding energies (in MeV). Constant ratios of the first $4^{+}$ to $2^{+}$ or ${\mathrm{R}}_{4/2}$ within the IBA limits are marked by black solid lines. Figure reproduced from Ref. [2].

Figure 4. The three symmetries of the IBA [29,30]. The U(5) symmetry represents the spherical limit, SU(3) the extreme well-deformed limit, and O(6) the $\gamma$-unstable limit. The color scale indicates the quadrupole collectivity’s contribution to the binding energies (in MeV). Constant ratios of the first $4^{+}$ to $2^{+}$ or ${\mathrm{R}}_{4/2}$ within the IBA limits are marked by black solid lines. Figure reproduced from Ref. [2].

2.3. IBA with Partial Dynamical Symmetries (PDS)

Partial dynamical symmetry (PDS) models extend the IBA by allowing selected subsets of states to retain exact symmetry properties even when the Hamiltonian breaks the symmetry globally [31]. In the SU(3)-PDS framework, the ground state and $\gamma$ bands preserve SU(3) symmetry, while the K^π $=0_2^{+}$ band is a mixture. Three types of PDS are identified:

• Type I—some states preserve the full dynamical symmetry
• Type II—all states preserve part of the symmetry
• Type III—some states preserve part of the symmetry

• PDS provides analytic insight into complex spectra while retaining flexibility for realistic modeling. It offers a useful intermediate description between exact symmetry and complete symmetry breaking, often enabling improved reproduction of observables in deformed nuclei.

2.4. Pseudo-SU(3) Shell Model

The pseudo-SU(3) model [10,32,33,34,35,36,37,38,39] combines microscopic shell-model structure with $SU\left(3\right)$ symmetry concepts, providing a bridge between algebraic and geometric descriptions. It relates $SU\left(3\right)$ invariants directly to the $\beta$ and $\gamma$ deformation variables of the collective model. Applied systematically to rare-earth nuclei (Nd, Sm, Gd, Dy, Er, Yb, Hf), the model reproduces trends in low-lying K^π $=0^{+}$ and K^π $=2^{+}$ bands while drastically reducing the configuration space relative to full shell-model calculations. The relative positions and collectivity of bands depend on the $SU\left(3\right)$ content of the wave functions, and mixing of different $SU\left(3\right)$ irreducible representations is essential for reproducing observed spectra.

2.5. Proxy-SU(3) and Coexistence

The proxy-SU(3) model [8,11,40,41,42,43] is inspired by pseudo-SU(3) and simplifies calculations in medium and heavy nuclei by approximating Nilsson orbitals with pairs differing by $0\left[110\right]$ quantum numbers, which have large spatial overlap and identical angular-momentum projection. This approximation enables simplified predictions of collective properties and deformation trends based primarily on valence nucleon counting. Within this framework, nuclei with $R_{4/2}<3.05$ and strong B($E2$) connections between K^π $=0^{+}$ and K^π $=2^{+}$ bands are identified as candidates for shape coexistence rather than pure $\beta$ vibrations.

2.6. First-Order Quantum Phase Transitions and Coexistence

The concept of quantum phase transitions in nuclei was introduced by Iachello [44,45] and further developed by Casten and collaborators [46,47,48,49]. These transitions describe abrupt changes in nuclear shape driven by competition between pairing and neutron–proton interactions. The strength of the valence neutron–proton interactions is given by $P=N_p{N}_n/(N_p+N_n)$ and the transition from spherical to deformed structure typically occurs near $P\gtrsim 5$ [50]. This framework eliminates special, separate theoretical models and allows the description of the onset of deformation as well as the potential for a coexisting minimum in one Hamiltonian, as shown in Figure 5. The coexistence regime corresponds to soft potential-energy surfaces with competing spherical and deformed minima.

Critical-point symmetries include E(5) and X(5), corresponding to phase transitions between U(5)–O(6) and U(5)–SU(3), respectively. Figure 6 shows the IBM triangle with these symmetries added. For well-deformed regions, X(5) is particularly relevant. Nuclei near N = 90 (e.g., Nd and Dy isotopes) exhibit excitation–energy ratios and B($E2$) patterns consistent with X(5) predictions. This can be seen in Figure 7, which shows the comparison of experimental energy ratios and B($E2$) values with X(5) predictions along with the vibrational or rotational limits.

2.7. Monte Carlo Shell Model and Shape Coexistence

Microscopic calculations, including the Monte Carlo Shell Model [51,52], emphasize the role of particle–hole excitations across shell or subshell gaps in generating competing minima and shape coexistence [9,53]. Shape coexistence is well established near closed shells (e.g., Pb region) and arises from the competition between shell gaps and residual interactions that lower intruder configurations. Experimental signatures include characteristic level schemes, enhanced $E0$ transitions, distinct rotational spacings, and selective population mechanisms [3,4,9] in transfer reactions. Although coexistence is firmly established near closed shells, its existence in well-deformed nuclei remains under debate. Several theoretical approaches—including the Triaxial Projected Shell Model, Monte Carlo Shell Model, proxy-SU(3), and global macroscopic–microscopic models [43,51,54,55]—suggest that multiple competing deformed configurations may occur. However, definitive experimental proof in strongly deformed nuclei remains elusive.

The variation in excitation energies, transition strengths, $E0$ matrix elements, and transfer-reaction cross sections for the first excited $0^{+}$ states continues to motivate detailed investigation of their underlying structure. Distinguishing between $\beta$ vibration and shape coexistence remains one of the central challenges in understanding collective excitations in atomic nuclei.

3. Experimental Evidence

A recent review [2] presented the experimental data available from decades of studies for nuclei between Z = 50–82. The review compiled energy levels, the number of 0⁺ states, level lifetimes, transition probabilities, and transfer reaction cross-sections. These data can be summarized by saying that the studies over the last two decades have led to an explosion of observed 0⁺ states below 3.1 MeV in excitation energies. These measurements were enabled by the high-precision focal plane detector + the Q3D spectrometer at the Technical University of Munich [1,56,57]. The high energy resolution of the Q3D allowed the identification of $13 0^{+}$ states in one nucleus (¹⁵⁸Gd) below 3.1 MeV [58]. Over the years, that list has grown much larger as shown in Figure 2 and reveals that, in some cases, tens of $0^{+}$ states exist in the low lying structure of deformed nuclei [57,59,60,61,62,63,64,65,66,67,68,69,70,71]. These findings changed the question about the nature of the $0^{+}$ states from absence in observation to characterization.

The energies of the first excited K^π $=0^{+}$ states evolve across isotopic chains and show some strong variations, unlike the simpler and more uniform patterns of the K^π $=2^{+}$ bands. The B($E2$) transition probabilities resulting from the lifetime measurements are the strongest experimental result. The B($E2$) values are sometimes weaker for the depopulation of the K^π $=0^{+}$ states in comparison to the ones depopulating the K^π $=2^{+}$ bands. Two-nucleon transfer reaction studies, (p,t) and (t,p), populating the 0⁺ states in the deformed nuclei of this region were expected to allow decisive discrimination, but no definitive conclusions were possible. Some of the 0⁺ states were populated strongly while others were not, opening the venues for systematic measurements that can, in fact, yield some conclusive results. Intrinsic quadrupole moments were extracted in cases where there was enough information to distinguish the low-lying bands in the spectra of select nuclei. Theoretically, true $\beta$-vibrations should have similar rotational properties as the ground state band. In cases where multiple 0⁺ bands exist, it is possible to extract the moments of inertia from the energy levels and determine if they indicate vibrations of the ground state band or a band with somewhat different moments of inertia.

Figure 8 shows the known 0⁺ states in the Gd isotopes. The energy of the first excited 0⁺ state increases from 600 keV in ¹⁵²Gd to approximately 1.4 MeV in ¹⁶⁰Gd as the isotopes evolve from a near spherical nucleus with a ratio of the first two excited states ($R_{4/2}=2.19$) to a well deformed one ($R_{4/2}=3.3$). We point out the characteristics that allow the discrimination of the first excited 0⁺ state between a vibration built on the ground state or a coexistence minimum. We demonstrate the characterization for ¹⁵⁶Gd as an example.

Dynamic Moments of Inertia: Studies of deformation and super-deformation in nuclei led to the characterization of bands by their dynamic moments of inertia [72,73,74]. The approach was used to identify bands that were identical in a number of deformed and super-deformed nuclei some decades ago. This is a concept that we have used in characterizing the low-lying excited bands in the deformed nuclei of interest. Defined as the first derivative of spin, the dynamic moment of inertia is an indicator of a band’s collective rotational properties and is identified by the slope of the relationship between the energy differences and the spins. Vibrational excitations built on the ground state should essentially have the same dynamic moments of inertia with some wider dispersions. Therefore, bands with the same vibrational structure have similar slopes within a few percent.

Figure 9 shows the dynamic moments of inertia for the ground state bands and several $0^{+}$ bands in ^154−160Gd. The slopes of the bands are listed in

Table 1. Slopes of the ground state and $0^{+}$ bands in ^154–160Gd.

	gs	$0_2^{+}$	$0_3^{+}$	$0_4^{+}$
154Gd	0.021	0.026	0.043	0.009
156Gd	0.022	0.028	0.019	0.030
158Gd	0.022	0.024	0.024	0.033
160Gd	0.022	0.030

. From ¹⁵⁴Gd to ¹⁵⁶Gd, the dynamic moments of inertia demonstrate the change in structure. The ¹⁵⁴Gd case shows that the slope of the $0_2^{+}$ band is similar to the value of the ground state band (24% higher), but the $0_{3,4}^{+}$ bands indicate a very different character. From ¹⁵⁶Gd, the dynamic moments of inertia begin to align with those of the ground state, a trend that continues to ¹⁶⁰Gd. Aprahamian et al. determined that the $0_4^{+}$ was an excitation built on the $0_2^{+}$ band in ¹⁵⁶Gd [75]. It is interesting to note that in this nucleus, the dynamic moments of inertia of the $0_{2,4}^{+}$ bands are larger than the ground state, which is expected for a vibrational band with somewhat larger dispersion built on the ground state.

Intrinsic quadrupole moments: The intrinsic quadrupole moments are a way to determine if the matrix elements for the transitions from the indicated bands are the same within error bars. These moments are extracted from the transition probabilities for the $0^{+}$ bands that have a sufficient number of B($E2$) values. A full list of the known transition probabilities in the $0^{+}$ bands is shown in

Table A1. Transition probabilities for the first ${\mathrm{K}}^{\pi }=2^{+}$ band and ${\mathrm{K}}^{\pi }=0^{+}$ bands in ¹⁵⁶Gd. Level energies, lifetimes, $\gamma$-ray energies, relative intensities, conversion coefficients, and mixing ratios are from ENSDF [84] unless otherwise noted. For mixed transitions ($M1/E2$) without a known mixing ratio ($\delta$), the B($E2$) is reported as an upper limit. All transition probabilities are reported in W.u. and calculated using TROPIC [85]. This table was originally published in Ref. [2].

${\mathrm{K}}_{\mathit{i}}^{\mathit{\pi }}$, ${\mathrm{J}}_{\mathit{i}}^{\mathit{\pi }}$	${\mathrm{E}}_{\mathit{L}}$ (keV)	$\mathit{\tau }$ (fs)	${\mathrm{E}}_{\mathit{\gamma }}$ (keV)	${\mathrm{K}}_{\mathit{f}}^{\mathit{\pi }}$, ${\mathrm{J}}_{\mathit{f}}^{\mathit{\pi }}$	${\mathrm{I}}_{\mathit{\gamma }}$ (rel.)	${\mathit{\alpha }}_{\mathit{T}}$	$\mathit{\pi }\mathit{\ell }$ or $\mathit{\delta }$	B($\mathit{\pi }\mathit{\ell }$) (W.u.)
$2_{\gamma }^{+}$, $2^{+}$	1154.15	0.82(3) ps	104.55(4)	$0_2^{+}$, $0^{+}$	0.0007(4)	–	$E2$	5.$6_{-3.3}^{+3.6}$
			865.968(21)	$0_{gs}^{+}$, $4^{+}$	3.78(12)	0.00374	$E2$	0.77(6)
			1065.1781(2)	$0_{gs}^{+}$, $2^{+}$	100.0(5)	0.00242	−16(5)	7.2(4)
			1154.1467(2)	$0_{gs}^{+}$, $0^{+}$	96.2(6)	0.00205	$E2$	4.7(2)
$2_{\gamma }^{+}$, $3^{+}$	1248.006	0.89 ps	118.56(4)	$0_2^{+}$, $2^{+}$	0.004	–	$M1/E2$	24.6(2)
			959.820(9)	$0_{gs}^{+}$, $4^{+}$	27.1(3)	0.00301	${-12}_{-5}^{+3}$	4.8(1)
			1159.031(8)	$0_{gs}^{+}$, $2^{+}$	100.0(5)	0.00204	−11.$8_{-0.7}^{+0.6}$	6.9(1)
$2_{\gamma }^{+}$, $4^{+}$	1355.422	0.9 ps	57.62(2)	$0_2^{+}$, $4^{+}$	0.003(2)	–	$E2$	${610}_{-410}^{+430}$
			107.41(1)	$2_{\gamma }^{+}$, $3^{+}$	0.022(4)	–	$E2$	${200}_{-39}^{+40}$
			201.269(4)	$2_{\gamma }^{+}$, $2^{+}$	0.27(2)	–	$E2$	${106}_{-9}^{+10}$
			225.88(4)	$0_2^{+}$, $2^{+}$	0.017(4)	0.1485	$E2$	3.$7_{-0.9}^{+1.0}$
			770.2(3)	$0_{gs}^{+}$, $6^{+}$	0.9(3)	–	$E2$	0.43(15)
			1067.2325(2)	$0_{gs}^{+}$, $4^{+}$	100(1)	0.00249	−4.$0_{-1.6}^{+0.9}$	8.$8_{-1.0}^{+0.4}$
			1266.446(12)	$0_{gs}^{+}$, $2^{+}$	39(1)	0.00172	$E2$	1.5(1)
$2_{\gamma }^{+}$, $5^{+}$	1506.863	1.6 ps	151.43(1)	$2_{\gamma }^{+}$, $4^{+}$	0.08(2)	–	$E2$	${75}_{-21}^{+24}$
			258.860(4)	$2_{\gamma }^{+}$, $3^{+}$	1.34(7)	0.0957	$E2$	${86}_{-8}^{+9}$
			922.183(10)	$0_{gs}^{+}$, $6^{+}$	35.6(10)	0.00327	$E2$	4.0(3)
			1218.708(13)	$0_{gs}^{+}$, $4^{+}$	100(6)	0.00185	$E2$	2.8(3)
$0_2^{+}$, $0^{+}$	1049.48	4.2 ps	960.50771(25)	$0_{gs}^{+}$, $2^{+}$	100	0.003	$E2$	4.8(1)
$0_2^{+}$, $2^{+}$	1129.44	2.3(2) ps	79.878(9)	$0_2^{+}$, $0^{+}$	0.0040(17)	5.83	$E2$	${52}_{-25}^{+30}$
			841.241(7)	$0_{gs}^{+}$, $4^{+}$	40.8(15)	0.00399	$E2$	4.$1_{-0.5}^{+0.6}$
			1040.470(8)	$0_{gs}^{+}$, $2^{+}$	100(3)	0.0143	-5.$9_{-2.8}^{+1.4}$	3.$3_{-0.6}^{+0.5}$
			1129.419(9)	$0_{gs}^{+}$, $0^{+}$	27.4(15)	0.00214	$E2$	0.${63}_{-0.09}^{+0.11}$
$0_2^{+}$, $4^{+}$	1297.82	3 ps	143.672(11)	$2_{\gamma }^{+}$, $2^{+}$	0.09(3)	–	$E2$	${39}_{-14}^{+16}$
			168.382(3)	$0_2^{+}$, $2^{+}$	1.11(9)	0.397	$E2$	${220}_{-26}^{+29}$
			713.102(8)	$0_{gs}^{+}$, $6^{+}$	11.3(10)	0.0058	$E2$	1.6(2)
			1009.619(11)	$0_{gs}^{+}$, $4^{+}$	90(5)	0.017	$E2$	2.3(2)
			1208.87(10)	$0_{gs}^{+}$, $2^{+}$	100(3)	0.00187	$E2$	1.0(1)
$0_3^{+}$, $0^{+}$	1168.19	13 ps	1079.226(8)	$0_{gs}^{+}$, $2^{+}$	100(4)	0.00235	$E2$	0.86(7)
$0_3^{+}$, $2^{+}$	1258.075	2.2(2) ps	103.89(2)	$2_{\gamma }^{+}$, $2^{+}$	0.006(2)	–	$E2$	${18}_{-7}^{+9}$
			208.54	$0_2^{+}$, $0^{+}$	<0.003	–	$E2$	<0.32
			969.865(8)	$0_{gs}^{+}$, $4^{+}$	100(2)	0.00294	$E2$	4.$3_{-0.5}^{+0.7}$
			1169.087(10)	$0_{gs}^{+}$, $2^{+}$	71(1)	0.0031(8)	0.38(6)	0.${15}_{-0.05}^{+0.07}$
			1258.087(14)	$0_{gs}^{+}$, $0^{+}$	26(1)	0.00174	$E2$	0.${31}_{-0.04}^{+0.05}$
$0_4^{+}$, $0^{+}$	1715.211	6.2 ps	348.726(7)	$1_2^{-}$, $1^{-}$	10.3(5)	0.01097	$E1$	8.$9_{-1.0}^{+1.2}$ $\times {10}^{-5}$
			472.699(5)	$1_1^{-}$, $1^{-}$	100(5)	0.00535	$E1$	3.$5_{-3.9}^{+4.6}$ $\times {10}^{-4}$
			585.830(15)	$0_2^{+}$, $2^{+}$	4.0(11)	0.0093	$E2$	1.$0_{-0.3}^{+0.4}$
			1625.19(21)	$0_{gs}^{+}$, $2^{+}$	33(4)	–	$E2$	5.$2_{-0.9}^{+1.1}$ $\times {10}^{-2}$
$0_4^{+}$, $2^{+}$	1771.09	0.64 ps	232.255(12)	$1_2^{-}$, $3^{-}$	0.11(1)	–	$E1$	4.1(6) $\times {10}^{-5}$
			404.634(16)	$1_2^{-}$, $1^{-}$	1.1(1)	0.00768	$E1$	7.$8_{-1.1}^{+1.2}$ $\times {10}^{-5}$
			494.941(6)	$1_1^{-}$, $3^{-}$	7.3(3)	0.00482	$E1$	2.$8_{-0.2}^{+0.3}$ $\times {10}^{-4}$
			513.020(13)	$0_3^{+}$, $2^{+}$	2.4(3)	0.0237	$M1$	7.$9_{-1.3}^{+1.5}$ $\times {10}^{-3}$
			528.626(22)	$1_1^{-}$, $1^{-}$	0.83(7)	0.00416	$E1$	2.$6_{-0.3}^{+0.4}$ $\times {10}^{-5}$
			1682.174(15)	$0_{gs}^{+}$, $2^{+}$	100(5)	0.00152	$M1$	9.$3_{-0.9}^{+1.0}$ $\times {10}^{-3}$
$0_4^{+}$, $4^{+}$	1893.40	0.40 ps	431.122(13)	$0_3^{+}$, $4^{+}$	1.0(1)	0.037	$M1$	8.$3_{-1.1}^{+1.2}$ $\times {10}^{-3}$
			485.273(11)	$1_1^{-}$, $5^{-}$	4.1(7)	0.00504	$E1$	2.$4_{-0.5}^{+0.6}$ $\times {10}^{-4}$
			537.953(15)	$2_{\gamma }^{+}$, $4^{+}$	10.2(5)	0.016(5)	$E2$	78(7)
			595.58(4)	$0_2^{+}$, $4^{+}$	1.16(14)	0.01626	$M1$	3.7(6) $\times {10}^3$
			617.24(3)	$1_1^{-}$, $3^{-}$	2.4(5)	–	$E1$	7.$2_{-1.7}^{+1.9}$ $\times {10}^{-5}$
			1605.208(19)	$0_{gs}^{+}$, $4^{+}$	100(3)	0.00141(23)	$E2$	3.2(2)

(reproduced from Ref. [2]). The intrinsic quadrupole moments ($Q_0^2$) are determined for both the ${\mathrm{K}}^{\pi }=2_{\gamma }^{+}$ and the ${\mathrm{K}}^{\pi }=0_2^{+}$ bands using the B($E2$) values [2] with the equations:

$$\mathrm{B}(E2:J_i\to J_f)={\langle J_{i}020|J_{f}0\rangle }^2{\displaystyle \frac{5}{4\pi }}{e}^2{Q}_0^2$$

(1)

$$\mathrm{B}(E2:J_i\to J_f)={\langle J_{i}220|J_{f}2\rangle }^2{\displaystyle \frac{5}{4\pi }}{e}^2{Q}_0^2$$

(2)

where $J_i$ and $J_f$ are the initial and final spins.

As an example, Figure 10 shows the intrinsic quadrupole moments for the ${\mathrm{K}}^{\pi }=0_{2,3}^{+}$ and ${\mathrm{K}}^{\pi }=2_{\gamma }^{+}$ bands in ¹⁵⁶Gd, extracted from the lifetime measurements. As expected, the intrinsic quadrupole moments of the ${\mathrm{K}}^{\pi }=0^{+}$ bands are lower than the ${\mathrm{K}}^{\pi }=2^{+}$ band. The ${\mathrm{K}}^{\pi }=0_2^{+}$ band has $Q_0$ of approximately 0.25 b while the ${\mathrm{K}}^{\pi }=2^{+}$ band has a $Q_0$ approximately twice that value. This is simply an angular momentum argument, which is the reason that transition probabilities from the $0^{+}$ bands to the ground state band are significantly lower than those of the ${\mathrm{K}}^{\pi }=2^{+}$ bands.

Mikhailov Plots: Mikhailov plots of $0^{+}$ bands show the mixing of the states of a given band with those of the ground state band. The plot for the K $=2_{\gamma }^{+}$ and the K $=0_{2,3}^{+}$ bands in the ¹⁵⁶Gd nucleus is shown in Figure 11. If there is no mixing in the Mikhailov framework, the result would be a straight horizontal line as observed in the first excited $0_2^{+}$ band. However, the same cannot be said for the $0_3^{+}$ band.

4. Interacting Boson Approximation (IBM) Framework

Several deformed nuclei were explored in case studies within the IBM framework. The ${\mathrm{K}}^{\pi }=0_2^{+}$ and $2_1^{+}$ bands are described in terms of $\beta$ and $\gamma$ vibrations in the IBM. We show that the intrinsic IBM automatically leads to vibrational energies and quadrupole transitions [76]. The main features of the IBM can be studied by considering a schematic Hamiltonian which is the sum of the d-boson energy and the quadrupole–quadrupole interaction

$$\begin{array}{ccc}\hfill H& =& (1-\xi ){\widehat{n}}_d-{\displaystyle \frac{\xi }{4(N-1)}}Q\left(\chi \right)·Q\left(\chi \right),\hfill \\ \hfill {\widehat{n}}_d& =& \sqrt{5}{\left(d^{†}\tilde{d}\right)}^{\left(0\right)},\hfill \\ \hfill Q\left(\chi \right)& =& {(s^{†}\tilde{d}+d^{†}\tilde{s})}^{\left(2\right)}+\chi {\left(d^{†}\tilde{d}\right)}^{\left(2\right)}.\hfill \end{array}$$

(3)

The IBM symmetries for all three cases (U(5), SU(3), and SO(6)) arise from special choices of the coefficients $\xi$ and $\chi$:

(i)

$\xi =0$: the U(5) (or vibrational) limit where the Hamiltonian reduces to that of a quadrupole oscillator

(ii)

$\xi =1$ and $\chi =0$: the SO(6) (or $\gamma$ unstable) limit

(iii)

$\xi =1$ and $\chi =\mp \sqrt{7}/2$: the SU(3) (or rotational) limit

• Axially deformed nuclei with prolate/oblate symmetry have the variable ∓ sign. The solutions can be obtained in closed, analytic form for these cases. In general, numerical diagonalization of the Hamiltonian matrix yields the eigenvalues. The results show that the weaker B($E2$) values from the $0^{+}$$\beta$-bands arise naturally.

A closed-form solution can be obtained by employing a large N (or classical) limit. This limit can be studied by introducing a coherent (or intrinsic) state in the form of a boson condensate, consisting of a superposition of monopole and quadrupole bosons [26,77,78,79].

$$\begin{array}{ccc}\hfill |gs\rangle & =& {\displaystyle \frac{1}{\sqrt{N!}}}{\left(b_c^{†}\right)}^N|0\rangle ,\hfill \\ \hfill b_c^{†}& =& {\displaystyle \frac{1}{\sqrt{1+{\beta }^2}}}\left(s^{†}+\beta cos\gamma d_0^{†}+\beta sin\gamma {\displaystyle \frac{1}{\sqrt{2}}}(d_2^{†}+d_{-2}^{†})\right),\hfill \end{array}$$

(4)

where $\beta$ and $\gamma$ are the deformation parameters in the IBM. The expectation value of the (normal-ordered) Hamiltonian in the intrinsic state gives the associated energy surface

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{N}}E(\beta ,\gamma )& =& {\displaystyle \frac{1}{N}}\langle gs|:H:|gs\rangle \hfill \\ & =& (1-\xi ){\displaystyle \frac{{\beta }^2}{1+{\beta }^2}}-\xi {\displaystyle \frac{\frac{2}{7}{\chi }^2{\beta }^4-4\sqrt{\frac{2}{7}}\chi {\beta }^{3}cos3\gamma +4{\beta }^2}{4{(1+{\beta }^2)}^2}}.\hfill \end{array}$$

(5)

The minimum in the energy surface is represented by ${\beta }_0$ and ${\gamma }_0$, which characterize the equilibrium shape.

Table 2. Special choices of $\xi$, $\chi$, ${\beta }_0$, and ${\gamma }_0$ that yield the dynamical symmetries.

Limit	$\mathit{\xi }$	$\mathit{\chi }$	${\mathit{\beta }}_0$	${\mathit{\gamma }}_0$	Shape
$U\left(5\right)$	0	−	0	−	Spherical
$SO\left(6\right)$	1	0	1	−	$\gamma$ unstable
$SU\left(3\right)$	1	$-{\displaystyle \frac{1}{2}}\sqrt{7}$	$\sqrt{2}$	$0^{\circ }$	Prolate deformed
$SU\left(3\right)$	1	$+{\displaystyle \frac{1}{2}}\sqrt{7}$	$\sqrt{2}$	${180}^{\circ }$	Oblate deformed

summarizes the special choices of $\xi$, $\chi$, ${\beta }_0$, and ${\gamma }_0$ that yield the dynamical symmetries. The z-axis is chosen along the symmetry axis.

The control parameters $\xi$ and $\chi$ determine the structure of the quadrupole operator. Varying $\xi$ over the range $0\le \xi \le 1$ allows the quantum phase transitions to be studied [77]. The Hamiltonian exhibits a second-order phase transition between the $U\left(5\right)$ and $SO\left(6\right)$ limits for $\chi =0$ with the critical point at ${\xi }_c={\displaystyle \frac{1}{2}}$. The equilibrium shape is spherical for $0\le \xi <{\xi }_c$ (${\beta }_0=0$), and deformed for ${\xi }_c<\xi \le 1$ (${\beta }_0>0)$. There is no dependence on the asymmetry parameter ${\gamma }_0$. For $\chi =\mp \sqrt{7}/2$, the Hamiltonian exhibits a first-order phase transition between the $U\left(5\right)$ and $SU\left(3\right)$ limits with the critical point at ${\xi }_c={\displaystyle \frac{8}{17}}$. The equilibrium shape is spherical for $0\le \xi <{\xi }_c$ (${\beta }_0=0$), and it is axially deformed for ${\xi }_c<\xi \le 1$ (${\beta }_0>0)$. It is prolate deformed for $\chi =-\sqrt{7}/2$ and oblate deformed for $\chi =+\sqrt{7}/2$.

For $0<\left|\chi \right|\le \sqrt{7}/2$, this corresponds to a first-order phase transition between the spherical and deformed phase. For the $\xi <{\xi }_c$, the equilibrium shape is spherical (${\beta }_0=0$), and for $\xi >{\xi }_c$, the equilibrium shape is prolate deformed (${\beta }_0>0$) for $\chi <0$ (${\gamma }_0=0^{\circ }$) while oblate deformation results for $\chi >0$ (${\gamma }_0={180}^{\circ }$). The excited $\beta$ and $\gamma$ bands are intrinsic excitations of the ground state band in which one of the condensate bosons, $b_c^{†}=(s^{†}+\beta d_0^{†})/\sqrt{1+{\beta }^2}$, is replaced by a deformed boson for $\beta$ and $\gamma$ vibrations, respectively

$$\begin{array}{ccc}\hfill |N,\beta \rangle ={\displaystyle \frac{1}{\sqrt{N}}}{b}_{\beta }^{†}{b}_c|N,gs\rangle ,& & b_{\beta }^{†}={\displaystyle \frac{1}{\sqrt{1+{\beta }^2}}}(-\beta s^{†}+d_0^{†}),\hfill \\ \hfill |N,\gamma \rangle ={\displaystyle \frac{1}{\sqrt{N}}}{b}_{\gamma }^{†}{b}_c|N,gs\rangle ,& & b_{\gamma }^{†}={\displaystyle \frac{1}{\sqrt{2}}}(d_2^{†}+d_{-2}^{†}).\hfill \end{array}$$

(6)

The properties of the $\beta$ and $\gamma$ bands are studied in terms of the ratio of excitation energies and the ratio of $E2$ decays to the ground state band within the intrinsic IBM. The $\beta$ and $\gamma$ bands can be obtained by evaluating the Hamiltonian [80]. In finite N calculations, the ratio of intrinsic energy excitations converges in the limit of large N to the following expression

$$E_{\gamma \beta }={\displaystyle \frac{\langle N,\gamma |:H:|N,\gamma \rangle -\langle N,gs|:H:|N,gs\rangle }{\langle N,\beta |:H:|N,\beta \rangle -\langle N,gs|:H:|N,gs\rangle }}$$

(7)

It should be noted that this is not the case with the expression used previously in [2], which was based on [81]. The intrinsic energies of the $\beta$ and $\gamma$ bands are the same ($E_{\gamma \beta }=1$) in the $SU\left(3\right)$ limit ($\chi =\mp \sqrt{7}/2$ and $\xi =1$). Outside of this limit in the $\xi \chi$-plane, the intrinsic excitation energy of the $\gamma$ band is less than that of the $\beta$ band ($E_{\gamma \beta }<1$).

The consistent Q quadrupole matrix elements for transitions have $E2$ operators with the same structure as the quadrupole operator in the Hamiltonian [82,83]. The transitions between the ground state band and the $\beta$ and $\gamma$ bands are used to obtain the following ratio [28]

$$R_{\beta \gamma }={\displaystyle \frac{R_{\beta }^2}{R_{\gamma }^2}}={\displaystyle \frac{{\left|〈g|T_0\left(E2\right)|\beta 〉\right|}^2}{{\left|〈g|T_2\left(E2\right)+T_{-2}\left(E2\right)|\gamma 〉\right|}^2}}={\displaystyle \frac{1}{2(1+{\beta }_0^2)}}{\left({\displaystyle \frac{1+\sqrt{\frac{2}{7}}\left|\chi \right|{\beta }_0-{\beta }_0^2}{1-\sqrt{\frac{2}{7}}\left|\chi \right|{\beta }_0}}\right)}^2.$$

(8)

For nearly the entire parameter space, the ratio of intrinsic transition rates, $R_{\beta \gamma }$, decreases with increasing $\xi$, ultimately vanishing in the limit of a pure quadrupole–quadrupole interaction ($\xi =1$). However, for values of $\chi$ approaching the $SU\left(3\right)$ limit, $\left|\chi \right|=\sqrt{7}/2$, the ratio initially rises before declining to zero. In contrast, at the exact $SU\left(3\right)$ point, $\left|\chi \right|=\sqrt{7}/2$, $R_{\beta \gamma }$ instead increases with $\xi$ and reaches the limiting value $R_{\beta \gamma }=3/2$. In general, $\gamma \to gs$ transitions are stronger than $\beta \to gs$ transitions. The two transition strengths become comparable only when $\chi$ approaches the $SU\left(3\right)$ limit of $\mp \sqrt{7}/2$, with the $\beta \to gs$ transitions potentially exceeding the $\gamma \to gs$ strength.

As an illustrative example, the nucleus ${}^{156}\mathrm{Gd}$, which has an $R_{4/2}$ ratio of 3.24, is considered within the intrinsic IBM framework for a deformed system. The intrinsic excitation energies are obtained by subtracting the rotational contribution from the experimental values, namely $E_{\beta }=E_{0_{\beta }^{+}}$ and $E_{\gamma }=E_{2_{\gamma }^{+}}-{\displaystyle \frac{6}{2{\mathcal{I}}_{\gamma }}}$, where ${\mathcal{I}}_{\gamma }$ denotes the moment of inertia of the $\gamma$ band. The intrinsic transition rates are then extracted from the ratios of the corresponding B($E2$) values

$$\begin{array}{c}\hfill R_{\beta \gamma }={\displaystyle \frac{B(E2;0_{gs}^{+}\to 2_{\beta }^{+})}{B(E2;0_{gs}^{+}\to 2_{\gamma }^{+})}}.\end{array}$$

(9)

The nucleus ${}^{156}\mathrm{Gd}$ is often regarded as a canonical example of $SU\left(3\right)$ symmetry, characterized by nearly degenerate $\beta$ and $\gamma$ bands [26]. Nevertheless, only a very narrow region in the $(\xi ,\chi )$ parameter space is capable of reproducing the experimental observables. In particular, the prediction of very small values of $R_{\beta \gamma }$ is robust only in the limit of a pure quadrupole–quadrupole interaction ($\xi =1$) with $\chi \ne \mp \sqrt{7}/2$. In this limit, the intrinsic matrix element for $\beta \to gs$ transitions vanishes in the large N limit. Figure 12 shows that a slight deviation from the $SU\left(3\right)$ symmetry, with $\xi \sim 0.98$ and $\chi \sim -1.31$, is necessary to reproduce the experimental value $R_{\beta \gamma }=0.34\left(17\right)$ [2].

In summary, the analysis of intrinsic energies and $E2$ transition probabilities within the consistent-Q formalism of the IBM captures the full richness of its dynamical symmetries and the quantum phase transition between spherical and deformed nuclear shapes. Across most of the $(\xi ,\chi )$ parameter space, $\gamma \to gs$ transitions are stronger than $\beta \to gs$ transitions, with the notable exception occurring for $\chi$ values approaching the $SU\left(3\right)$ limit, $\left|\chi \right|=\sqrt{7}/2$.

While this paper only discusses the case for ¹⁵⁶Gd, the intrinsic IBM has been employed as an example to describe several other well-deformed ${\mathrm{nuclei—}}^{162}\mathrm{Dy}$, ${}^{168}\mathrm{Er}$, ${}^{168}\mathrm{Yb}$, ${}^{178}\mathrm{Hf}$, and ${}^{184}\mathrm{W}$—all of which exhibit $R_{4/2}$ ratios close to the rotational limit of $10/3$ [2]. In nearly all of the cases, only a limited region of the $(\xi ,\chi )$ plane yields agreement with the experimental intrinsic energies and $E2$ transition rates. Improved lifetime measurements, particularly those that move beyond upper or lower bounds, would further constrain the allowed parameter space. Overall, this analysis demonstrates that the properties of the ${\mathrm{K}}^{\pi }=0_2^{+}$ and $2_1^{+}$ bands across a broad range of rare-earth nuclei can be consistently and accurately interpreted in terms of the $\beta$ and $\gamma$ band structures of the IBM.

5. Discussion

The compilation of experimental evidence acquired over decades has been presented in a review of nuclei in the Z = 50–82 region of the chart of nuclides in order to characterize the nature of the first excited $0^{+}$ band in deformed nuclei. The traditional interpretation of these states was as $\beta$-vibrational excitations. Recent reviews on coexistence have pointed to the overabundance and frequent occurrences of coexisting shapes in spherical nuclei, and predicted that they might also occur in deformed nuclei, but none have been observed yet. The IBA calculations and the CHFB+5DCH calculations predicted that the first excited ${\mathrm{K}}^{\pi }=0^{+}$ band is indeed $\beta$ vibrations. Coexistence as a phenomenon is common in the chart of nuclides, but not for well-deformed nuclei. Dynamic moments of inertia for excited bands in a number of well-deformed nuclei exhibit that some are indeed vibrational excitations built on the ground state band, while others are of a very different nature. We have analyzed the large number of observed $0^{+}$ bands, their excitation-energy systematics, the generally weaker and fragmented B($E2$) transition strengths, to point to cases where the nucleus is well deformed and the $0^{+}$ band is similar in moment of inertia to the ground state band, is not highly mixed with other excited bands, and has significant transition probabilities to the ground state levels. This is the first detailed characterization of a vibration or oscillation versus coexisting minima and other mixed excitations, including quasi-particles. The weaker ${\mathrm{K}}^{\pi }=0^{+}$ to ground state B($E2$) values were used as an argument against the vibrational nature of these $0^{+}$ states. IBA calculations in the consistent Q framework indicate that this arises naturally and is expected of $\beta$ vibrations until the extreme and pure $SU\left(3\right)$ limit is reached.

6. Results and Conclusions

Nuclei in the Z = 50–82 region of the chart of nuclides show first excited $0^{+}$ bands that vary strongly across isotopic chains with the evolution of nuclear structure and the onset of deformation. The data do not show a simple universal pattern for these excited bands. The B($E2$) values for transitions from these bands to the ground state are generally weaker than those of the $\gamma$ bands. Calculations in the Interacting Boson Approximation framework show that weaker B($E2$) values are not exclusive of the $\beta$ nature of the $0^{+}$ bands. A subset of the $0^{+}$ bands has been identified as $\beta$ vibrations based on B($E2$) values, moments of inertia, and intrinsic quadrupole moments. These nuclei include ^152,154Sm, ^154,156,158Gd, ¹⁶²Dy, ¹⁶⁸Er, ¹⁶⁸Yb, ¹⁷⁸Hf, and ^182,184W.

The CHFB+5DCH calculations show that there are $0^{+}$ bands predicted in the chart of nuclides which are not collective vibrations built on the ground state, but in specific cases where the nucleus is well deformed, the first excited $0^{+}$ bands are indeed vibrations with similar moments of inertia built on the ground state. The IBA calculations show an analysis of intrinsic energies and $E2$ transition probabilities in the consistent-Q formalism. A Hamiltonian has been proposed that contains all of the richness of the dynamical symmetries and quantum phase transitions between spherical and deformed nuclei. We show that for almost the entire parameter range of the control parameter $\xi$ and the structure of the quadrupole operator $\chi$, the $\gamma \to gs$ transitions are stronger than the $\beta \to gs$ transitions, with the only exception for $\chi$ close to the $SU\left(3\right)$ value $\left|\chi \right|=\sqrt{7}/2$. IBM calculations for several well-deformed nuclei: ¹⁵⁶Gd, ¹⁶²Dy, ¹⁶⁸Er, ¹⁶⁸Yb, ¹⁷⁸Hf and ¹⁸⁴W with $R_{4/2}$ values close to the rotational value of $10/3$ were made and the results are similar to the ¹⁵⁶Gd case. We describe the ¹⁵⁶Gd case in detail in this paper. It was found that there is a relatively small region in the parameter space of the $\xi \chi$-plane in which the experimental intrinsic energies and $E2$ transition rates can be reproduced. The ${\mathrm{K}}^{\pi }=0_2^{+}$ and $2_1^{+}$ bands in a wide range of rare earth nuclei can be described very well in terms of the $\beta$ and $\gamma$ bands of the IBM.