Nuclear Structure Investigations of Even–Even Hf Isotopes

Abstract

The mass region of rare-earth nuclei in the nuclear chart is riddled with well-deformed nuclei, exhibiting rotational properties and many interesting nuclear structure-related phenomena. The scarcity of experimental data as the neutron number increases and the exotic phenomena such as shape coexistence, which are strongly connected with the underlying symmetries of the Hamiltonian and are predicted to take place in this region, make this mass region a fertile ground for experimental and theoretical studies of nuclear structure. In this work, we investigate the structure of the even–even ^162–184Hf (hafnium) isotopes through a calculation of various observables such as $B(E2;0_1^{+}\to 2_1^{+})$ reduced transition matrix elements and quadrupole moments. Six different nuclear models are employed in the calculations of the observables for these nuclei, the shapes of which deviate from spherical symmetry, and as such, are characterized by Hamiltonians, which break the rotational invariance of the exact nuclear many-body Hamiltonian. The results of the present study are expected to establish some concrete guidelines for current and future experimental endeavors. Along these lines, the results for the ^162–180Hf isotopes are compared with existing experimental data where available, showing an overall good agreement.

1. Introduction

Over the past decade, the study of nuclear structure has regained a dominant role in nuclear physics, mainly due to the discovery of new physics, occurring far outside the traditionally studied valley of stability. The discoveries, which resulted from ground-breaking innovations in radioactive beam production, have posed multiple questions regarding the fundamental interactions among the nucleons, i.e., the protons and neutrons under extreme conditions. Filling the large gaps in the knowledge has a critical impact on understanding the evolution of the Universe driven by nuclear reaction networks in bodies in the Universe (such as the s- or r-processes) [1] and putting together the puzzle of the particle constituents that shape nuclear matter.

The massive shift in the nuclear physics scientific community toward using Radioactive Ion Beams (RIBs) to explore new phenomena and test existing symmetries, investing significant effort in establishing the outer limits of the nuclear chart and the properties of the exotic species inhabiting those mass regimes (see, for instance, Ref. [2]), has left large gaps in the knowledge of the mass region lying between the valley of stability and the nuclear driplines. In particular, the isotopes located in the $A\approx 140$–180 mass regime to the east of the valley of stability are of special interest, as they feature low-lying isomeric states [3,4], shape coexistence [5,6], and sizeable deformations, which can be attributed to the shape of nuclear potential, new prevailing symmetries, and, often, the appearance of octupole collectivity [6,7,8,9,10].

Deformed nuclei can be schematically categorized as prolate, oblate, and triaxial—a direct consequence of the nuclear dynamic symmetries of the deformed Hamiltonian—based on the three principal axes of rotation in the ellipsoid. These symmetries regarding the nuclear shapes are clearly reflected on the $\beta \gamma$ plane formed by the set of intrinsic shape variables ($\beta ,\gamma$) entering the Bohr Hamiltonian. This plane is subdivided into six equivalent parts based on the symmetries. All of the shapes are uniquely contained in the $\gamma =(0^{\circ },{60}^{\circ })$ sector, which can be taken as the representative one. Prolate nuclei lie on the $\gamma =0^{\circ }$ axis, oblate nuclei lie on the $\gamma ={60}^{\circ }$ axis, whereas the nuclei located in between the $\gamma =0^{\circ }$ and $\gamma ={60}^{\circ }$ are triaxial. These shapes are repeated in each of the remaining subsectors of the $\beta \gamma$ plane, alternating between the principal symmetry axes [11,12].

Many nuclei in the rare-earth region of the nuclear chart are known to be well deformed, often exhibiting quadrupole deformation parameter values ${\beta }_2>0.2$ for the ground or low-lying states [4]. It is common for the nuclei in this mass region to show a collective character with rotational properties, where the energy spectra can be approximated by the relation $E\propto I(I+1)$ (I: total spin) [11]. As nuclear rotations can alter the microscopic structure of nuclei, the preferred shapes of nuclei typically change as the spin increases. This is very common for the “soft” nuclei. In any case, a mixed character (both rotational and vibrational) can be present in many nuclei.

Over the past few decades, numerous theoretical approaches have attempted to study such phenomena including semi-empirical methods, mean-field and beyond mean-field models such as the particle-rotor model [13], the cranked shell model [14,15,16], the projected shell model [17], the cranking covariant density functional theory [18], and more. All such theoretical directions, one way or another, provide the means to study the nuclear structure evolution imposed by the underlying symmetries.

From an experimental standpoint, several observables can provide stringent tests of the existing theoretical models in an effort to gain insight into the nuclear structure. $B\left(E2\right)$ reduced electric transition probabilities are some of the most prominent among them, having been proven particularly useful in the task of understanding the collective behavior of deformed nuclei. Their direct relation to the nuclear state lifetimes allows for their determination through a variety of experiments, providing valuable information on the interplay between collective and single-particle degrees of freedom.

In the 1960s, at Oak Ridge National Laboratory, P. H. Stelson and L. Grodzins recognized the importance of the compilation and evaluation of $B\left(E2\right)$ transition probabilities for even–even nuclei and moved on to produce the first compilation of $B\left(E2\right)\uparrow$ values for the $2_1^{+}$ nuclear states [19]. The next two compilations and evaluations were carried out in 1987 [20] and 2001 [21] by Raman et al. in the framework of the Oak Ridge Nuclear Data Project prior to the successful launch of the Brookhaven $B\left(E2\right)\uparrow$ project [22] in 2005. The NNDC database currently contains a compilation of updated $B(E2;0_1^{+}\to 2_1^{+})$ experimental results and evaluated values. A detailed upgrade of the database with the experimental reduced matrix elements, $B\left(E2\right)\uparrow$s, and mean lifetimes, $\tau \left(2_1^{+}\right)$, was carried out by Pritychenko et al. [23,24]. Table I in [23] contains experimental values for nuclei with $Z=2–104$, whereas their adopted values, together with deformation ${\beta }_2$, are shown in Table III of the same reference.

In this work, the rotational properties of the well-deformed even–even ${}^{162–184}\mathrm{Hf}$ nuclei are investigated through calculations of their electric quadrupole moments and reduced electric quadrupole probabilities using various theoretical models. Neutron-rich Hf isotopes are located in the upper half of the $Z=50–82$ proton shell and the $N=82–126$ neutron shell, and in this region, multi-particle excitations are expected to be increasingly preferred. Regarding the theoretical models employed in this work, except for the phenomenological model and finite-range droplet model described in Section 3.1 and Section 3.2, respectively, they are based on a mean-field description (non-relativistic or relativistic) and include a pairing. Mean-field Hamiltonians break the symmetries that the many-body Hamiltonians have. The symmetries that are, in general, broken include the translational, rotational, reflection, and isospin symmetries [25,26]. As for the methods used to incorporate the pairing (Bardeen–Cooper–Schrieffer or Bogoliubov), they do not conserve the particle number. Several techniques have been proposed to restore the broken symmetries to some extent (see, for example, Refs. [25,27]).

The results for the Hf isotopes are extended to $A=182,184$ for which there currently exist no experimental data for the $2_1^{+}$ states other than the energies $E\left(2_1^{+}\right)$ (it should be noted that even the spin and parity of the $2_1^{+}$ state are questionable in the case of ¹⁸⁴Hf) [23]. The results for the various calculated physical quantities are compared with previous works [28,29,30], as well as with the available experimental data [4,21,23,31].

In recent years, many theoretical studies have been conducted that are centered around the Hf isotopic chain and employ several different models (see, for example, Refs. [32,33]). This work aims to provide a complete set of predictions for the examined observables, with an emphasis on experimentally measurable quantities (such as lifetimes and $B\left(E2\right)$s), using a variety of theoretical models. These results are expected to guide future experimental and theoretical studies in the unstable neutron-rich Hf isotopes [34,35], as was the case for a similar study of Yb isotopes [36,37].

2. Physical Quantities and Global Best Fit

For nuclei featuring rotational spectra, the $B\left(E2\right)$ can be related directly to the intrinsic electric quadrupole moment, $Q_0$, and, in turn, be associated with the ${\beta }_2$ deformation parameters [11]. Furthermore, the $B\left(E2\right)$ is connected to the root mean square (rms) of the deformation ${\beta }_2$ in non-rotational regions. In the following paragraphs, we give some brief definitions of the various observables studied in this work, among which are the intrinsic quadrupole moments $Q_0$, the electric quadrupole moments Q, and the ${\beta }_2$ deformation parameters.

2.1. The Intrinsic Quadrupole Moment $Q_0$

The intrinsic quadrupole moment $Q_0$ is defined in the intrinsic frame of reference of the nucleus. Deviations from the spherical shape are associated with the intrinsic quadrupole moment $Q_0$, which is, in turn, related to the reduced electric quadrupole transition probability $B\left(E2\right)$ through the relation [11]

$$Q_0={\left[\frac{16\pi }{5}\frac{B\left(E2\right)\uparrow }{e^2}\right]}^{1/2}$$

(1)

where the upward-pointing arrow represents the excitation.

2.2. The Electric Quadrupole Moment Q

The electric quadrupole moment Q describes the apparent shape of the nuclear charge distributed over the ellipsoid volume. As a convention, negative values of Q in the laboratory frame are associated with prolate nuclear shapes, whereas positive values of Q are associated with oblate nuclei. A zero value for the electric quadrupole moment Q indicates spherically symmetric nuclear shapes [12]. In the framework of the nuclear shell model, spherical nuclear charge distributions are expected for closed shells; thus, the measurement of the electric quadrupole moment can serve as a test of the shell model itself. Since the quadrupole moment depends on the size and charge of the nucleus, a better comparison is obtained after normalization for those factors, resulting in what is called the “reduced quadrupole moment”. A plot of measured values [38] reveals that magic numbers of neutrons and protons are associated with near-zero values of the quadrupole moment.

A single-nucleon transition model fails to accurately describe the quadrupole moments of strongly deformed nuclei. Such cases are more adequately described in the framework of a collective nuclear model, involving the collective motion of many nucleons within a given nucleus. In this model [11], the spectroscopic and intrinsic quadrupole moments are interconnected via the relation

$$Q=\frac{3K^2-I(2I-1)}{(I+1)(2I+3)}{Q}_0$$

(2)

where I is the total spin of the nucleus and K is the total projection of I onto the z-axis in the body fixed frame (symmetry axis of the nucleus). For $K=0$, $I=2$ one obtains

$$Q\left(2_1^{+}\right)=-\frac{2}{7}{Q}_0$$

(3)

If the intrinsic deformation is prolate ($Q_0>0$), the corresponding quadrupole moment in the laboratory frame turns negative, reflecting the result of time averaging a twirled cigar shape, which becomes an oblate apparent shape [12]. We note that the ratio $R_{QB}\equiv Q\left(2_1^{+}\right)/\sqrt{B\left(E{2}^{+}\right)\uparrow }=-0.906$ in this model [39].

2.3. The Deformation Parameter ${\beta }_2$

The $B\left(E2\right)\uparrow$ values are basic experimental quantities, which are model independent. Another useful quantity, which is model dependent, is the ${\beta }_2$ deformation parameter. Under the assumptions that the charge distribution is uniform from the center to a distance $R(\theta ,\varphi )$ and zero beyond and that the deformations are small, we find that ${\beta }_2$, $B\left(E2\right)$, and $Q_0$ are related by the formula

$${\beta }_2=\left(\frac{4\pi }{3Z{R}_0^2}\right){\left[\frac{B\left(E2\right)\uparrow }{e^2}\right]}^{1/2}=\frac{\sqrt{5\pi }}{3Z{R}_0^2}{Q}_0$$

(4)

where $R_0=1.2A^{1/3}$ [fm]. In the theoretical description of the excitation of collective states through direct reactions, a similar parameter is involved to account for the deformation of the average potential. The latter is analogous to ${\beta }_2$, which involves single-particle effects, indicative of the existence of quadrupole collective motion in a nucleus. Thus, the ratio ${\beta }_2/{\beta }_{SP}$ is often considered, where ${\beta }_{SP}$ refers to the single-particle model given in [23]:

$${\beta }_{SP}=\frac{1.59}{Z}$$

(5)

Before concluding this introductory subsection, we give a final relation between the observables studied in this work, namely the relation between the lifetimes $\tau$ and the $B\left(E2\right)$ reduced electric quadrupole transition probabilities. This relation, given below, is used to calculate the lifetimes $\tau \left(2_1^{+}\right)$ based on the $B\left(E2\right)$ values:

$$\tau =\frac{{\tau }_{\gamma }}{1+a_T}=40.81\times {10}^{13}{E}_{\gamma }^{-5}{\left[\frac{B\left(E2\right)\uparrow }{e^2{b}^2}\right]}^{-1}{(1+a_T)}^{-1}\left[\mathrm{ps}\right]$$

(6)

where $E_{\gamma }$ are the energies of the emitted photons in keV and $a_T$ are the internal conversion coefficients [21,23].

2.4. Global Best Fit

Based on the global systematics of available data, knowing the energy of the first $2_1^{+}$ state, $E\left(2_1^{+}\right)$ is sufficient for a prediction of the corresponding $B\left(E2\right)\uparrow$ [e${}^2$b${}^2$] and ${\tau }_{\gamma }$ [ps] values [21]. Within the framework of the hydrodynamic model with irrotational flow, Bohr and Mottelson [40,41] derived simple expressions for the ${\tau }_{\gamma }$ values, namely

$${\tau }_{\gamma }\approx 0.6\times {10}^{14}{E}^{-4}{Z}^{-2}{A}^{1/3}\left[\mathrm{ps}\right]$$

(7)

for the small harmonic vibrations of spherical nuclei, and

$${\tau }_{\gamma }\approx 1.4\times {10}^{14}{E}^{-4}{Z}^{-2}{A}^{1/3}\left[\mathrm{ps}\right]$$

(8)

for the rotational degrees of freedom of axially symmetric nuclei [21]. Grodzins [42] adopted this $E^{-4}{Z}^{-2}$ dependence in the above expressions to perform empirical fits on all even–even nuclei and further replaced $A^{1/3}$ with A.

The above expressions have been updated and revised to establish the functional relationship between $(E,A)$ and $(E,{\tau }_{\gamma })$ that best describes the experimental data in Ref. [20]. The adopted values for ${\tau }_{\gamma }$, excluding those for closed-shell nuclei, lead to the expression [21]:

$${\tau }_{\gamma }=(1.59\pm 0.28)\times {10}^{14}{E}^{-4}{Z}^{-2}{A}^{2/3}\left[\mathrm{ps}\right]$$

(9)

Using Equations (4) and (6), the corresponding $B\left(E2\right)\uparrow$ and ${\beta }_2$ predictions are given by [21]

$$B\left(E2\right)\uparrow =(2.57\pm 0.45)E^{-1}{Z}^2{A}^{-2/3}[\mathrm{e}{}^2\mathrm{b}{}^2]$$

(10)

and

$${\beta }_2=(466\pm 41)E^{-1/2}{A}^{-1}$$

(11)

We should mention that in Ref. [43], an analysis of the evaluated data for the $B\left(E2\right)\uparrow$s in Ref. [23] has been performed using the “elemental” fit of Habs et al. [44], who introduced a modification to the $B\left(E2\right)$ formula of Raman et al. for nuclei located in the $50\le Z\le 82$ region of the nuclear chart. Their fit performs better than the Global Best Fit of Raman et al. in some cases, however, it can lead to predictions of negative $B\left(E2\right)\uparrow$ values, with large uncertainties for neutron-rich nuclei (e.g., ²⁶O, ^214,216,218Po) (for a detailed discussion on the fits of Habs et al., see Refs. [43,44]). These non-physical predictions pose limits for the Habs formalism outside the $50\le Z\le 82$ mass region, whereas Raman’s formalism, albeit less accurate, is more robust and can safely be adopted across the nuclear chart.

The resulting values for the lifetimes $\tau \left(2_1^{+}\right)$, reduced transition probabilities $B\left(E2\right)\uparrow$, (intrinsic) quadrupole moments $\left(Q_0\right)$Q, and ${\beta }_2/{\beta }_{SP}$ ratios are calculated using Equations (1)–(6) and are compared with the experimental data of Pritychenko et al. [23], as well as with the theoretical predictions presented in Section 3.

3. Theoretical Models

In this section, we briefly discuss each of the models employed in this study and refer occasionally to their symmetries. For a complete and detailed description of the models, please refer to the original works (see the References section). These models have been used for the calculation of reduced electric quadrupole transition probabilities, $B\left(E2\right)$, electric quadrupole moments $Q_0$ and Q, and the ${\beta }_2$ deformation parameters in the framework of the nuclear collective model. The results presented later in this section are compared with the available experimental data. In the case of the phenomenological model (see Section 3.1 below), the energy levels for the ground-state band of the ^162–184Hf isotopes have been additionally determined. The structure of the excited levels is discussed in Section 3.1 and Section 4.

3.1. Phenomenological Model (PhM)

The phenomenological nuclear adiabatic model described by A. Bohr and B.R. Mottelson [11] has been central to explaining the properties of deformed nuclei. In their model, the low-excitation states in even–even deformed nuclei are connected with the collective rotations in axially symmetric nuclei. Despite its simplicity, this phenomenological explanation allows for a description of a large set of experimental data for even–even deformed nuclei while offering predictions of the many new properties of these deformed nuclei.

In Refs. [28,29,45], a phenomenological model (PhM), which takes into account the non-adiabaticity effects, is proposed by considering the Coriolis mixing of the low-lying states of positive parity in rotational bands and thus breaking the axial symmetry. The aforementioned model has been implemented in this work to determine the energies of the excited levels of the ground-state bands in the deformed ^162–184Hf nuclei.

The starting point for this model is a nuclear Hamiltonian of the form [28]

$$H=H_{rot}+H_{K,K^{\prime }}$$

(12)

where $H_{rot}$ is the rotational part of the Hamiltonian and

$$H_{K,K^{\prime }}={\omega }_K{\delta }_{K,K^{\prime }}-{\omega }_{rot}\left(I\right){\left(j_x\right)}_{K,K^{\prime }}\chi (I,K){\delta }_{K,K^{\prime }+1}$$

(13)

with ${\left(j_x\right)}_{K,K^{\prime }}=\langle K|j_x|K^{\prime }\rangle$ being the matrix element of the Coriolis coupling of the rotational band members, ${\omega }_{rot}\left(I\right)$ the rotational frequency of the core, ${\omega }_K$ the energies of the band heads, and

$$\chi (I,0)=1,\chi (I,1)={\left[1-\frac{2}{I(I+1)}\right]}^{1/2}$$

(14)

The complete energy of a state is found by solving the Schrödinger equation for the nuclear Hamiltonian of Equation (12). It is equal to

$$E_{\nu }\left(I\right)=E_{rot}\left(I\right)+{\epsilon }_{\nu }\left(I\right)$$

(15)

where $E_{rot}\left(I\right)$ is the energy of the rotational core and ${\epsilon }_{\nu }\left(I\right)$ is the eigenenergy corresponding to the second term of the Hamiltonian of Equation (12) [28] (a detailed description of the model’s Hamiltonian, eigenfunctions, and eigenenergies can be found in Refs. [28,29,30]). The rotational core energy, $E_{rot}\left(I\right)$, agrees with the ground-state energy of the rotational bands in even–even deformed nuclei at the lower values of spin I [29].

For the rotational core, an effective value of the rotational frequency can be determined from the available experimental data using the classical definition of the rotational frequency

$$\omega =\frac{dE}{dI}$$

(16)

giving [46]

$${\omega }_{eff}=\frac{E^{exp.}\left(I\right)-E^{exp.}(I-2)}{\sqrt{I(I+1)}-\sqrt{(I-1)(I-2)}}$$

(17)

which, for higher levels of spin I, reduces to [29,46]

$${\omega }_{eff}=\frac{E^{exp.}\left(I\right)-E^{exp.}(I-2)}{2}$$

(18)

Hence, the effective moment of inertia becomes

$${\mathcal{J}}_{eff}=\frac{d\sqrt{I(I+1)}}{d{\omega }_{eff}}$$

(19)

From Equations (17) and (19), we calculate the effective rotational frequency ${\omega }_{eff}$ and effective moment of inertia ${\mathcal{J}}_{eff}$, respectively. For low rotational frequencies, i.e., at low spin values $I\lesssim 8\hslash$, ${\mathcal{J}}_{eff}$ depends almost linearly on ${\omega }_{eff}^2$ so we can write

$${\mathcal{J}}_{eff}={\mathcal{J}}_0+{\mathcal{J}}_1{\omega }_{eff}^2$$

(20)

where ${\mathcal{J}}_0$ and ${\mathcal{J}}_1$ are the inertial parameters of the rotational core, determined by the least-squares fit in Equation (20).

Having determined the inertial parameters ${\mathcal{J}}_0$, ${\mathcal{J}}_1$, we can employ the Harris parameterization for the calculation of the energy of the rotational ground-state band [47]:

$$E_{rot}\left(I\right)=\frac{1}{2}{\mathcal{J}}_0{\omega }_{rot}^2\left(I\right)+\frac{3}{4}{\mathcal{J}}_1{\omega }_{rot}^4\left(I\right)$$

(21)

$$\tilde{I}\equiv \sqrt{I(I+1)}={\mathcal{J}}_0{\omega }_{rot}\left(I\right)+{\mathcal{J}}_1{\omega }_{rot}^3\left(I\right)$$

(22)

where ${\omega }_{rot}$ is the rotational angular frequency of the nuclei defined by the real root of the cubic Equation (22) [28,29]:

$$\begin{array}{cc}\hfill {\omega }_{rot}\left(I\right)& ={\left\{\frac{\tilde{I}}{2{\mathcal{J}}_1}+{\left[{\left(\frac{{\mathcal{J}}_0}{3{\mathcal{J}}_1}\right)}^3+{\left(\frac{\tilde{I}}{2{\mathcal{J}}_1}\right)}^2\right]}^{1/2}\right\}}^{1/3}\hfill \\ \hfill & +{\left\{\frac{\tilde{I}}{2{\mathcal{J}}_1}-{\left[{\left(\frac{{\mathcal{J}}_0}{3{\mathcal{J}}_1}\right)}^3+{\left(\frac{\tilde{I}}{2{\mathcal{J}}_1}\right)}^2\right]}^{1/2}\right\}}^{1/3}\hfill \end{array}$$

(23)

In the present work, we calculate ${\omega }_{eff}$ and ${\mathcal{J}}_{eff}$ from Equations (17) and (19) for the first low-lying levels (up to $I=8^{+}$) of the g.s. band of the even–even ^162–184Hf isotopes. A linear least-squares fit of Equation (20) is then performed to the aforementioned quantities in order to obtain the values of ${\mathcal{J}}_0$, ${\mathcal{J}}_1$, which are presented in

Table 1. ${\mathcal{J}}_0$, ${\mathcal{J}}_1$ values obtained within the framework of the phenomenological model for ^162–184Hf (see text for details). A comparison of the experimental $E\left(2_1^{+}\right)$ values of Ref. [23] and the corresponding values calculated within the framework of the phenomenological model (PhM) is also shown.

Isotope	${\mathcal{J}}_0$	${\mathcal{J}}_1$	${\mathbf{E}}^{\mathbf{th}.}\left(2_1^{+}\right)$	${\mathbf{E}}^{\mathbf{exp}.}\left(2_1^{+}\right)$
($\mathbf{Z}=72$)	[$\times {10}^{-2}$ keV${}^{-1}$ ${\mathit{\hslash }}^{-2}$]	[$\times {10}^{-8}$ keV${}^{-3}$ ${\mathit{\hslash }}^{-4}$]	[keV]	[keV]
162Hf	0.880	13.804	262.249	285.000
164Hf	1.327	14.623	199.610	210.700
166Hf	1.821	17.266	153.912	158.640
168Hf	2.373	17.522	122.068	124.100
170Hf	2.933	20.097	100.055	100.800
172Hf	3.128	12.837	94.750	95.220
174Hf	3.276	11.366	90.728	90.985
176Hf	3.381	8.958	88.126	88.349
178Hf	3.206	6.822	93.019	93.180
180Hf	3.211	3.584	93.137	93.324
182Hf	3.063	4.440	97.485	97.790
184Hf	2.800	5.465	106.349	107.100

. Based on these values and Equation (21), we proceed to calculate the energies for the $I=0_1^{+},\cdots {20}_1^{+}$ states and compare them with the available experimental data [4]. These are shown in

Table 2. Level energies (in MeV) and rotational frequencies ${\omega }_{rot}^{th}$ (in MeV ${\hslash }^{-1}$) for the even–even ^162–184Hf isotopes calculated in the framework of the phenomenological model (PhM) using Equations (20) and (23), with the ${\mathcal{J}}_0$, ${\mathcal{J}}_1$ values determined in this work (see Table 1). The $E^{exp.}$ values are taken from Refs. [4,23].

Isotope	162Hf			164Hf			166Hf
I	${\mathbf{E}}^{\mathbf{exp}.}$	${\mathbf{E}}^{\mathbf{th}}$	${\mathbf{\omega }}_{\mathit{rot}}^{\mathit{th}}$	${\mathbf{E}}^{\mathbf{exp}.}$	${\mathbf{E}}^{\mathbf{th}}$	${\mathbf{\omega }}_{\mathit{rot}}^{\mathit{th}}$	${\mathbf{E}}^{\mathbf{exp}.}$	${\mathbf{E}}^{\mathbf{th}}$	${\mathbf{\omega }}_{\mathit{rot}}^{\mathit{th}}$
$2_1^{+}$	0.285	0.262	0.183	0.211	0.200	0.148	0.158	0.154	0.119
$4_1^{+}$	0.730	0.709	0.253	0.587	0.577	0.220	0.470	0.465	0.185
$6_1^{+}$	1.293	1.269	0.302	1.085	1.072	0.270	0.897	0.889	0.234
$8_1^{+}$	1.940	1.915	0.341	1.669	1.655	0.310	1.406	1.398	0.273
${10}_1^{+}$	2.635	2.632	0.374	2.304	2.311	0.344	1.972	1.978	0.306
${12}_1^{+}$	3.185	3.409	0.402	2.995	3.028	0.372	2.566	2.619	0.334
${14}_1^{+}$	3.567	4.239	0.427	3.618	3.799	0.398	—	3.312	0.359
${16}_1^{+}$	4.068	5.116	0.450	—	4.619	0.421	—	4.053	0.381
${18}_1^{+}$	4.653	6.036	0.470	—	5.482	0.442	—	4.836	0.402
${20}_1^{+}$	5.310	6.996	0.489	—	6.386	0.461	—	5.658	0.421
Isotope	168Hf			170Hf			172Hf
I	${\mathbf{E}}^{\mathbf{exp}.}$	${\mathbf{E}}^{\mathbf{th}}$	${\mathbf{\omega }}_{\mathit{rot}}^{\mathit{th}}$	${\mathbf{E}}^{\mathbf{exp}.}$	${\mathbf{E}}^{\mathbf{th}}$	${\mathbf{\omega }}_{\mathit{rot}}^{\mathit{th}}$	${\mathbf{E}}^{\mathbf{exp}.}$	${\mathbf{E}}^{\mathbf{th}}$	${\mathbf{\omega }}_{\mathit{rot}}^{\mathit{th}}$
$2_1^{+}$	0.124	0.122	0.097	0.101	0.100	0.080	0.095	0.095	0.076
$4_1^{+}$	0.386	0.383	0.159	0.322	0.320	0.135	0.309	0.308	0.133
$6_1^{+}$	0.757	0.753	0.207	0.643	0.639	0.181	0.628	0.626	0.182
$8_1^{+}$	1.214	1.209	0.247	1.043	1.039	0.218	1.037	1.035	0.225
${10}_1^{+}$	1.736	1.737	0.280	1.504	1.509	0.250	1.521	1.523	0.262
${12}_1^{+}$	2.306	2.327	0.309	2.016	2.039	0.278	2.064	2.080	0.294
${14}_1^{+}$	2.858	2.971	0.334	2.567	2.621	0.303	2.654	2.700	0.324
${16}_1^{+}$	3.310	3.664	0.358	3.151	3.251	0.326	3.277	3.375	0.351
${18}_1^{+}$	3.833	4.401	0.379	3.768	3.923	0.346	3.919	4.101	0.375
${20}_1^{+}$	4.440	5.178	0.398	4.421	4.635	0.365	4.576	4.874	0.397
Isotope	174Hf			176Hf			178Hf
I	${\mathbf{E}}^{\mathbf{exp}.}$	${\mathbf{E}}^{\mathbf{th}}$	${\mathbf{\omega }}_{\mathit{rot}}^{\mathit{th}}$	${\mathbf{E}}^{\mathbf{exp}.}$	${\mathbf{E}}^{\mathbf{th}}$	${\mathbf{\omega }}_{\mathit{rot}}^{\mathit{th}}$	${\mathbf{E}}^{\mathbf{exp}.}$	${\mathbf{E}}^{\mathbf{th}}$	${\mathbf{\omega }}_{\mathit{rot}}^{\mathit{th}}$
$2_1^{+}$	0.091	0.091	0.073	0.088	0.088	0.071	0.093	0.093	0.075
$4_1^{+}$	0.297	0.296	0.129	0.290	0.289	0.127	0.307	0.306	0.134
$6_1^{+}$	0.608	0.606	0.178	0.597	0.595	0.177	0.632	0.631	0.188
$8_1^{+}$	1.010	1.008	0.221	0.998	0.996	0.222	1.059	1.057	0.237
${10}_1^{+}$	1.486	1.490	0.260	1.481	1.482	0.262	1.570	1.575	0.280
${12}_1^{+}$	2.021	2.044	0.294	2.035	2.044	0.299	2.150	2.177	0.320
${14}_1^{+}$	2.598	2.663	0.324	2.647	2.676	0.332	2.778	2.854	0.356
${16}_1^{+}$	3.209	3.340	0.352	3.308	3.370	0.362	3.435	3.600	0.389
${18}_1^{+}$	3.857	4.070	0.378	4.011	4.123	0.390	4.119	4.409	0.420
${20}_1^{+}$	4.551	4.850	0.401	—	4.929	0.416	4.837	5.277	0.448
Isotope	180Hf			182Hf			184Hf
I	${\mathbf{E}}^{\mathbf{exp}.}$	${\mathbf{E}}^{\mathbf{th}}$	${\mathbf{\omega }}_{\mathit{rot}}^{\mathit{th}}$	${\mathbf{E}}^{\mathbf{exp}.}$	${\mathbf{E}}^{\mathbf{th}}$	${\mathbf{\omega }}_{\mathit{rot}}^{\mathit{th}}$	${\mathbf{E}}^{\mathbf{exp}.}$	${\mathbf{E}}^{\mathbf{th}}$	${\mathbf{\omega }}_{\mathit{rot}}^{\mathit{th}}$
$2_1^{+}$	0.093	0.093	0.076	0.098	0.097	0.079	0.107	0.106	0.086
$4_1^{+}$	0.309	0.308	0.136	0.322	0.322	0.142	0.350	0.349	0.153
$6_1^{+}$	0.641	0.640	0.194	0.666	0.666	0.200	0.717	0.717	0.213
$8_1^{+}$	1.084	1.083	0.247	1.122	1.121	0.253	1.200	1.198	0.266
${10}_1^{+}$	1.631	1.629	0.297	1.680	1.678	0.302	—	1.780	0.314
${12}_1^{+}$	2.274	2.271	0.344	2.332	2.329	0.347	—	2.452	0.357
${14}_1^{+}$	3.005	3.003	0.387	3.065	3.065	0.388	—	3.207	0.396
${16}_1^{+}$	3.814	3.817	0.427	3.869	3.881	0.426	—	4.036	0.432
${18}_1^{+}$	4.682	4.709	0.464	4.734	4.769	0.461	—	4.933	0.465
${20}_1^{+}$	5.554	5.674	0.499	—	5.725	0.494	—	5.893	0.495

. Regarding ${\mathcal{J}}_0$, it increases as the number of nucleons grows, exhibiting a maximum at the middle of the shell ($A=176$). For isotopes ^166–172Hf, the values of ${\mathcal{J}}_0$, ${\mathcal{J}}_1$ calculated in this work are in good agreement with those in the earlier work of Ref. [47].

Figure 1 and Figure 2 show the dependence of the moment of inertia, $\mathcal{J}$, (determined through the known relation $2\mathcal{J}=(4I-2)/(E_I-E_{I-2})$ [48], with $E=E^{exp},E^{th}$ for the experimental and theoretical moments of inertia, respectively) on the squared angular frequency of rotation ${\omega }^2$ (calculated using Equation (18) for $E=E^{exp},E^{th}$, as in the case of $\mathcal{J}$). The treatment with the phenomenological model leads to a centrifugal stretching of the nucleus and an increase in $\mathcal{J}$ with the rotational frequency. This increase is almost linear for the lower values of spin (i.e., for $I\lesssim 10\hslash$). For high spin values, the emerging nonlinearity is associated with the mixture of the ground-state band with other rotational bands and with the decrease in nucleon pairing. Similar results are presented in Ref. [49], where the cranked shell model Hamiltonian with pairing correlations, treated by the particle-conserving method (PNC–CSM), is employed to investigate the upbending of the experimental moments of inertia appearing in the neutron-rich ^170–184Hf isotopes.

A more in-depth consideration of the phenomenological model would require us to consider the eigenfunctions of the Hamiltonian (12), which contain the mixing amplitudes of the various underlying states, for the calculation of the $B\left(E2\right)\uparrow$ values. However, in our simplified approach to this model, we instead calculated the $\tau \left(2_1^{+}\right)$ lifetimes, the $B\left(E2\right)\uparrow$ values, and the ${\beta }_2$ deformation parameters using the Global Best Fit of Raman et al. [21] from Equations (9)–(11), using the theoretical $E\left(2_1^{+}\right)$ energies resulting from the phenomenological model. The rest of the observables ($Q_0$, Q, and ${\beta }_2/{\beta }_{SP}$) were then calculated with Equations (1), (2), and (5). The results are presented in Table 3 and Table 4 and plotted in Figure 3.

3.2. Finite-Range Droplet Model (FRDM)

The finite-range droplet model (FRDM) is a global microscopic–macroscopic model of nuclear structure, which was introduced in 1988 [50], and soon incorporated improvements based on the AME1989 mass evaluation (FRDM92) [51]. The latest enhancement (FRDM12) [52] included adjustments due to the AME2003 mass evaluation (see Figure 1 in Ref. [52]). For our calculations, we use FRDM12 (hereafter referred to as FRDM in Figures and Tables).

In the macroscopic–microscopic method, the total nuclear potential energy, calculated as a function of shape, Z and N, can be expressed as the sum of a microscopic and a macroscopic term. The microscopic terms represent the shell-plus-pairing correction [52], namely

$$E_{pot}(Z,N,shape)=E_{mac}(Z,N,shape)+E_{s+p}(Z,N,shape)$$

(24)

The finite-range droplet model, an improved version of the droplet model [53], is employed for the calculation of the macroscopic contribution of Equation (24). For the microscopic part, proton/neutron shell and pairing corrections have to be taken into account. The matrix elements of the single-particle Hamiltonian are generated from the basis of the axial-symmetric harmonic-oscillator eigenfunctions. The shell correction is carried out using Strutinsky’s method [54], whereas for the pairing correction, the Lipkin–Nogami [55,56] version of the Bardeen–Cooper–Schrieffer (BCS) method is employed, which takes into account in lowest order the effects associated with particle number fluctuations and restores approximately the violation of the particle number that occurs in the BCS. A folded Yukawa single-particle potential is assumed in both cases and a zero-point energy is added to the calculated potential energy at the ground-state shape.

This version of the FRDM (FRDM12) allows for the calculation of several nuclear properties in addition to the nuclear ground-state masses. Among these are the ground-state deformation multipoles, calculated through the minimization of the nuclear potential energy function with respect to the parameters ${\epsilon }_2$, ${\epsilon }_3$, ${\epsilon }_4$, and ${\epsilon }_6$, as these are defined in the perturbed spheroid parameterization by Nilsson [57]. The $\beta$–shape parameters can then be derived using the relation

$${\beta }_{lm}=\sqrt{4\pi }\frac{\int r(\theta ,\varphi )Y_m^l(\theta ,\varphi )d\Omega }{\int r(\theta ,\varphi )Y_0^0(\theta ,\varphi )d\Omega }$$

(25)

with the radius vector r expressed in the $\epsilon$ parametrization [52].

In this work, we used the values for the deformation parameters ${\beta }_2$ and ${\beta }_4$ given in Ref. [52] to calculate the intrinsic quadrupole moments for the $2_1^{+}$ state in even–even ^162–184Hf isotopes using the relation [52]

$$Q_0=\frac{3Z{R}_0^2}{\sqrt{5\pi }}\left({\beta }_2+\frac{2}{7}\sqrt{\frac{5}{\pi }}{\beta }_2^2+\frac{20}{77}\sqrt{\frac{5}{\pi }}{\beta }_4^2+\frac{12}{7\sqrt{\pi }}{\beta }_2{\beta }_4\right)+\mathcal{O}\left({\beta }^3\right)$$

(26)

The above derivation requires $Q_0$ to assume the original value of the considered model when the protons are distributed uniformly inside a liquid drop with a radius $R_0$ and deformations ${\beta }_2$, ${\beta }_4$ (sharp-edged density model).

Having calculated the intrinsic electric quadrupole moments, the $B\left(E2\right)\uparrow$ values can be deduced using Equation (1). The rest of the calculated quantities are obtained using Equations (2), (5) and (6). All the quantities are shown in Table 3 and Table 4 and plotted in Figure 3a–f.

3.3. Hartree–Fock BCS with Skyrme MSk7 Model (HFBCS–MSk7)

In the HFBCS–MSk7 model, the nuclear ground-state properties can be deduced using the conventional HF+BCS model coupled with the Skyrme forces [58,59]. In this framework, the ground-state wavefunction is expressed as an expansion of the single-particle wavefunctions in a harmonic-oscillator basis. In addition, a 10-parameter Skyrme interaction (MSk7) is employed, along with a 2-parameter Wigner interaction and a 4-parameter $\delta$-function pairing force. The Skyrme and pairing parameters for the MSk7 force are determined in Ref. [59] by fitting to the same dataset of nuclear masses as its predecessor, the MSk6 force. These parameters are listed in Table A in Ref. [59].

For the case of deformed nuclei (see Ref. [59] for more details), the ETFSI-2 method [60] (where ETFSI stands for extended Thomas–Fermi plus Strutinsky integral) is implemented to derive the deformation parameters, which are subsequently used as a starting point in the deformed HF calculations [58]. A correction is made for the spurious center-of-mass motion and the spurious rotational energy of deformed nuclei is subtracted from the total computed amount of energy, as calculated in Ref. [58].

The Skyrme part of the MSk7 force on which the HFBCS-1 table of the different quantities in Ref. [59] is based has the usual form

$$\begin{array}{cc}\hfill u_{ij}& =t_0(1+x_0{P}_{\sigma })\delta \left({\overrightarrow{r}}_{ij}\right)\hfill \\ \hfill & +t_1(1+x_1{P}_{\sigma })\frac{1}{2{\hslash }^2}\{p_{ij}^2\delta \left({\overrightarrow{r}}_{ij}\right)+h.c.\}\hfill \\ \hfill & +t_2(1+x_2{P}_{\sigma })\frac{1}{{\hslash }^2}{\overrightarrow{p}}_{ij}·\delta \left({\overrightarrow{r}}_{ij}\right){\overrightarrow{p}}_{ij}\hfill \\ \hfill & +\frac{1}{6}{t}_3(1+x_3{P}_{\sigma }){\rho }^{\gamma }\delta \left({\overrightarrow{r}}_{ij}\right)\hfill \\ \hfill & +\frac{i}{{\hslash }^2}{W}_0({\overrightarrow{\sigma }}_i+{\overrightarrow{\sigma }}_j)·{\overrightarrow{p}}_{ij}\times \delta \left({\overrightarrow{r}}_{ij}\right){\overrightarrow{p}}_{ij}\hfill \end{array}$$

(27)

In the above equation, $P_{\sigma }$ is the two-body spin-exchange operator, whereas the $\delta$-function pairing force can be expressed as [59]

$${\upsilon }_{pair}\left(r_{ij}\right)=V_{\pi q}\delta \left(r_{ij}\right)$$

(28)

where the pairing-strength parameter $V_{\pi q}$ receives different values for protons and neutrons, whereas it is slightly stronger for odd nucleons ($V_{\pi q}^{-}$) than for even nucleons ($V_{\pi q}^{+}$) (see Ref. [58] for more details). As for the Wigner correction term, it has the form

$$E_W=V_W{e}^{-\lambda /N-Z/A}$$

(29)

The Skyrme pairing $V_{\pi q}^{+}$, $V_{\pi q}^{-}$ and Wigner parameters $V_W$ and $\lambda$ are determined via a fit on the 1995 mass compilation (Audi and Wapstra). The rather standard HF–MSk7 formalism is summarized in Ref. [58].

Using the ${\beta }_2$ and ${\beta }_4$ deformation parameter values obtained from the HFBCS-1 table in Ref. [59], we proceed to calculate the intrinsic quadrupole deformation $Q_0$ for the even–even ^162–184Hf isotopes using Equation (26). The rest of the physical quantities studied in this work are subsequently deduced from Equations (1), (2), (5) and (6). The results are shown in Table 3 and Table 4 and plotted in Figure 3a–f.

3.4. Hartree–Fock–Bogoliubov with Gogny D1S interaction (HFB–Gogny D1S)

The microscopic Hartree–Fock–Bogoliubov (HFB) method with the Gogny D1S effective nucleon–nucleon interaction [61,62] is a powerful approach with good predictive power in various aspects of nuclear structure [63]. Incorporating the mean-field approach, a many-body, effective Hamiltonian for the nucleus is expressed as

$$H=\sum _i^A{T}_i+\frac{1}{2}\sum _{i\ne j}^A{u}_{ij}$$

(30)

with $T_i$ being the kinetic energy of the nucleon i and $u_{ij}$ being the Gogny effective nucleon–nucleon interaction. The D1S parameterization of the Gogny force (Gogny D1S) [62]—a widely tested effective interaction—is used in the works of Refs. [63,64] to calculate various nuclear properties, among which are the ${\beta }_2$ ground-state deformation parameters.

Here, we give a brief description of the HFB–Gogny D1S model. Due to its finite range, the D1S interaction can be used in the framework of the full Hartree–Fock–Bogoliubov theory (HFB) to generate the nuclear pairing field in addition to the nuclear mean field [61], which gives it an advantage over the Skyrme interaction. The D1S effective nucleon–nucleon interaction is parameterized as

$$\begin{array}{cc}\hfill u_{12}& =\sum _{j=1}^{2}exp\left[-\frac{{({\overrightarrow{r}}_1-{\overrightarrow{r}}_2)}^2}{{\mu }_j^2}\right]\times (W_j+B_j{P}_{\sigma }-H_j{P}_{\tau }-M_j{P}_{\sigma }{P}_{\tau })\hfill \\ \hfill & +t_3(1+x_0{P}_{\sigma })\delta ({\overrightarrow{r}}_1-{\overrightarrow{r}}_2){\left[\rho \left(\frac{{\overrightarrow{r}}_1+{\overrightarrow{r}}_2}{2}\right)\right]}^a\hfill \\ \hfill & +i{W}_{LS}{\overleftarrow{\nabla }}_{12}\delta ({\overrightarrow{r}}_1-{\overrightarrow{r}}_2)\times {\overrightarrow{\nabla }}_{12}·({\overrightarrow{\sigma }}_1+{\overrightarrow{\sigma }}_2)\hfill \\ \hfill & +(1+2{\tau }_{1z})(1+2{\tau }_{2z})\frac{e^2}{|{\overrightarrow{r}}_1-{\overrightarrow{r}}_2|}\hfill \end{array}$$

(31)

In the above expression, the first term includes two finite ranges and the common mixing of spin, isospin, and space exchange operators ($P_{\sigma }$, $P_{\tau }$ and $-P_{\sigma }{P}_{\tau }$, respectively). A functional of the nuclear density $\rho$ is the second term; a zero-range, two-body spin-orbit interaction is the third term; and the Coulomb repulsion among protons is the fourth term.

For the self-consistent solution of the HFB equations [64,65], an iterative method is employed that is based on the minimization of the total energy of the nucleus

$$\delta \left(\langle \Phi \left|\left(H-{\lambda }_2{Q}_{20}-{\lambda }_{Z}Z-{\lambda }_{N}N\right)\right|\Phi \rangle \right)=0$$

(32)

In this expression:

• H is the nuclear Hamiltonian of Equation (30).
• $|\Phi \rangle$ is the HFB wavefunction.
• ${\lambda }_Z$, ${\lambda }_N$ are the Lagrange parameters fixing the proton and neutron numbers, respectively.
• ${\lambda }_2$ is the Lagrange parameter to fix the quadrupole moment $q_{20}$, defined as

  $$q_{20}=\langle \Phi |Q_{20}|\Phi \rangle$$

  (33)

  with the operator $Q_{20}$

  $$Q_{20}=\sqrt{\frac{16\pi }{5}}{r}^2{Y}_{20}$$

  (34)

A harmonic-oscillator basis with axial symmetry is used in this approach and the deformation parameters ${\beta }_2$ are given by

$${\beta }_2=\frac{1}{A{R}^2}\sqrt{\frac{5\pi }{9}}{q}_{20}$$

(35)

In the above equation, $R=1.2A^{1/3}$ fm is the nuclear radius and $q_{20}$ is the quadrupole moment of Equation (33), expressed in units of fm${}^2$ [63]. The results obtained with the above method are included in the AMEDEE database [64], where large-scale, axial mean-field calculations from the proton to the neutron dripline were carried out.

In the case of the HFB–Gogny D1S model, the deformation parameters ${\beta }_{min}$ were obtained from Ref. [64] and used as the ${\beta }_2$ deformation parameters to calculate the quantities shown in Table 3 and Table 4 with the help of Equations (1), (2), and (4)–(6). The results are plotted in Figure 3a–f.

3.5. Hatree–Fock–Bogoliubov UNEDF–1 (HFB–UNEDF–1)

In the framework of the Hartree–Fock–Bogoliubov theory, M. Kortelainen et al. [66] proposed a new Skyrme-like energy density functional, UNDEF-1, suitable for the description of nuclei with strong elongation. In the nuclear density functional theory (DFT), the total binding energy E of the nucleus is a functional of the one-body density $\rho$ and the pairing matrices $\tilde{\rho }$. In its quasilocal approximation, it can be expressed as a three-dimensional (3D) spatial integral [66]:

$$\begin{array}{cc}\hfill E[\rho ,\tilde{\rho }]=& \int d^3\overrightarrow{r}\mathcal{H}\left(\overrightarrow{r}\right)\hfill \\ \hfill =& \int d^3\overrightarrow{r}[{\mathcal{E}}^{kin}\left(\overrightarrow{r}\right)+{\chi }_0\left(\overrightarrow{r}\right)+{\chi }_1\left(\overrightarrow{r}\right)\hfill \\ \hfill & +\tilde{\chi }\left(\overrightarrow{r}\right)+{\mathcal{E}}_{Dir}^{Coul}\left(\overrightarrow{r}\right)+{\mathcal{E}}_{Exc}^{Coul}\left(\overrightarrow{r}\right)]\hfill \end{array}$$

(36)

where $\mathcal{H}\left(\overrightarrow{r}\right)$ is the energy density, which is quasilocal, time-even, scalar, isoscalar, and real. It is usually broken down into the kinetic energy ${\mathcal{E}}^{kin}\left(\overrightarrow{r}\right)$; isoscalar and isovector particle-hole energy densities ${\chi }_i\left(\overrightarrow{r}\right)$, $i=0,1$; pairing energy $\chi \left(\overrightarrow{r}\right)$; and Coulomb terms (${\mathcal{E}}_{Dir}^{Coul}\left(\overrightarrow{r}\right)$, ${\mathcal{E}}_{Exc}^{Coul}\left(\overrightarrow{r}\right)$). We will not present here the full theoretical framework related to UNEDF-1 but refer the reader to Ref. [66] and the references therein. The results were obtained using the solver HFBTHO in the axial-symmetric harmonic-oscillator basis and the Lipkin–Nogami version of the BCS theory. These results, which are available on the Mass Explorer [67] website, were obtained with large-scale DFT calculations focusing on the ground-state properties of even–even nuclei across the nuclear chart.

In this work, we use the ${\beta }_2$ deformation parameters obtained from the HFB formalism with the UNEDF-1 energy density functional [67] to calculate the quantities shown in Table 3 and Table 4 using Equations (1), (2), and (4)–(6). The results are plotted in Figure 3a–f.

3.6. Relativistic Hartree–Bogoliubov Covariant Energy Density Functional NL3* (RHB–NL3*)

Density functional theory has proven to be a universal and powerful tool in nuclear structure theory, showing great success in the description of various nuclear phenomena spanning the full range of the periodic table [68]. The form of DFT is determined by symmetry arguments and simplicity. The remaining parameter sets are deduced by fitting them to the experimental data. The relativistic (covariant) DFTs (CDFTs) provide some of the most interesting cases among the existing nuclear DFTs, respecting the Lorentz covariance while taking advantage of the basic properties of QCD at low energies such as symmetries and separation of scales. In this work, we consider the large-scale axial relativistic Hartree–Bogoliubov calculations of Abgemava et al. [69,70] and employ the covariant density functional NL3* [71], a nonlinear nucleon–meson coupling model.

Relativistic mean-field (RMF) theory [72], which is based on the Walecka model [73,74], is the starting point in the development of a covariant density functional (CDF). In this model, the exchange of phenomenological mesons drives the interaction of nucleons. In all the CDFs developed so far, two assumptions have been essential [75,76]: (i) the mean-field approximation in which only the nucleonic fields are quantized and the nucleons move independently in classical meson fields depending on the nuclear densities and currents in a self-consistent way and (ii) the no-sea approximation in which vacuum polarization and the contributions arising from the negative energy solutions are not explicitly taken into account.

The starting point of the NL3* CDFT is a standard Lagrangian density of the form

$$\begin{array}{cc}\hfill \mathcal{L}& =\overline{\psi }[\gamma (i\partial -g_{\omega }\omega -g_{\rho }\overrightarrow{\rho }\overrightarrow{\tau }-eA)-m-g_{\sigma }\sigma ]\psi \hfill \\ \hfill & +\frac{1}{2}{(\partial \sigma )}^2-\frac{1}{2}{m}_{\sigma }^2{\sigma }^2-\frac{1}{4}{\Omega }_{\mu \nu }{\Omega }^{\mu \nu }+\frac{1}{2}{m}_{\omega }^2{\omega }^2\hfill \\ \hfill & -\frac{1}{4}{\overrightarrow{R}}_{\mu \nu }{\overrightarrow{R}}^{\mu \nu }+\frac{1}{2}{m}_{\rho }^2{\overrightarrow{\rho }}^2-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }\hfill \end{array}$$

(37)

containing nucleons of mass m described by the Dirac spinors $\psi$ and various relativistic fields characterized by the spin, parity, and isospin quantum numbers. These are effective fields that are mediated by mesons characterizing the properties of the possible relativistic fields in the effective Dirac equation. The latter corresponds to the Kohn–Sham equation (KS) in the non-relativistic case. It is only for simplicity that the conventional names $\sigma (I^{\pi }=0^{+},T=0)$, $\omega (I^{\pi }=1^{-},T=0)$ and $\rho (I^{\pi }=1^{-},T=1)$ are used. The electromagnetic field A is also included in the above expression [75] (for a more detailed definition and description of the various quantities used in the above Lagrangian, see Ref. [72]).

Table 3. Values of the reduced transition matrix elements $B(E2;0_1^{+}\to 2_1^{+})$ and the lifetimes $\tau \left(2_1^{+}\right)$ (see text for more details about the relevant calculations). The results are compared with the experimental values [4,22,23]. Predictions for isotopes ^182,184Hf, for which no experimental data exist, are denoted in bold.

Isotope	Exp.	Global Fit	PhM	FRDM	HFBCS–MSk7	HFB–Gogny D1S	HFB–UNEDF–1	RHB–NL3*
$B(E2;0_1^{+}\to 2_1^{+})$ [e${}^2$ b${}^2$]
162Hf	1.34(10)	1.6(3)	1.710	2.109	2.230	2.208	2.287	2.143
164Hf	1.82(17)	2.1(4)	2.228	2.774	3.046	3.383	2.953	2.809
166Hf	3.46${}_{-0.15}^{+0.17}$	2.8(5)	2.866	3.452	3.407	4.766	3.744	3.692
168Hf	4.393(36)	3.5(6)	3.585	4.240	4.815	5.925	5.118	5.283
170Hf	5.11(18)	4.3(8)	4.339	5.148	5.719	6.514	6.138	6.132
172Hf	5.77(10)	4.5(8)	4.546	5.172	6.257	6.935	6.511	6.421
174Hf	5.38(20)	4.7(8)	4.711	5.592	7.086	7.002	6.480	6.132
176Hf	5.42(17)	4.8(8)	4.814	5.138	5.364	6.620	5.967	5.695
178Hf	4.736(63)	4.5(8)	4.526	5.103	4.633	6.087	5.145	5.375
180Hf	4.6470(30)	4.5(8)	4.487	4.638	3.333	5.567	4.656	5.093
182Hf	—	4.2(7)	4.255	4.656	3.949	5.061	4.332	4.813
184Hf	—	3.8(7)	3.872	4.191	4.376	4.435	3.934	4.401
$\tau \left(2_1^{+}\right)$ [ps]
162Hf	148(11)	126(22)	176	94	89	90	87	93
164Hf	435(41)	376(66)	467	286	261	235	269	283
166Hf	717(33)	895(157)	1010	721	731	522	665	674
168Hf	1239(10)	1548(271)	1654	1287	1134	921	1067	1033
170Hf	1740(61)	2074(363)	2137	1735	1562	1371	1455	1457
172Hf	1710${}_{-39}^{+31}$	2199(385)	2243	1924	1590	1435	1528	1550
174Hf	1986${}_{-71}^{+77}$	2291(401)	2317	1925	1519	1537	1661	1755
176Hf	2069${}_{-63}^{+67}$	2350(411)	2373	2196	2103	1704	1891	1981
178Hf	2168(29)	2288(401)	2304	2026	2231	1698	2009	1923
180Hf	2203.9(14)	2303(403)	2322	2224	3094	1853	2215	2025
182Hf	—	2236(392)	2265	2038	2403	1875	2190	1971
184Hf	—	2063(376)	2122	1893	1813	1789	2017	1803

Table 3. Values of the reduced transition matrix elements $B(E2;0_1^{+}\to 2_1^{+})$ and the lifetimes $\tau \left(2_1^{+}\right)$ (see text for more details about the relevant calculations). The results are compared with the experimental values [4,22,23]. Predictions for isotopes ^182,184Hf, for which no experimental data exist, are denoted in bold.

Isotope	Exp.	Global Fit	PhM	FRDM	HFBCS–MSk7	HFB–Gogny D1S	HFB–UNEDF–1	RHB–NL3*
$B(E2;0_1^{+}\to 2_1^{+})$ [e${}^2$ b${}^2$]
162Hf	1.34(10)	1.6(3)	1.710	2.109	2.230	2.208	2.287	2.143
164Hf	1.82(17)	2.1(4)	2.228	2.774	3.046	3.383	2.953	2.809
166Hf	3.46${}_{-0.15}^{+0.17}$	2.8(5)	2.866	3.452	3.407	4.766	3.744	3.692
168Hf	4.393(36)	3.5(6)	3.585	4.240	4.815	5.925	5.118	5.283
170Hf	5.11(18)	4.3(8)	4.339	5.148	5.719	6.514	6.138	6.132
172Hf	5.77(10)	4.5(8)	4.546	5.172	6.257	6.935	6.511	6.421
174Hf	5.38(20)	4.7(8)	4.711	5.592	7.086	7.002	6.480	6.132
176Hf	5.42(17)	4.8(8)	4.814	5.138	5.364	6.620	5.967	5.695
178Hf	4.736(63)	4.5(8)	4.526	5.103	4.633	6.087	5.145	5.375
180Hf	4.6470(30)	4.5(8)	4.487	4.638	3.333	5.567	4.656	5.093
182Hf	—	4.2(7)	4.255	4.656	3.949	5.061	4.332	4.813
184Hf	—	3.8(7)	3.872	4.191	4.376	4.435	3.934	4.401
$\tau \left(2_1^{+}\right)$ [ps]
162Hf	148(11)	126(22)	176	94	89	90	87	93
164Hf	435(41)	376(66)	467	286	261	235	269	283
166Hf	717(33)	895(157)	1010	721	731	522	665	674
168Hf	1239(10)	1548(271)	1654	1287	1134	921	1067	1033
170Hf	1740(61)	2074(363)	2137	1735	1562	1371	1455	1457
172Hf	1710${}_{-39}^{+31}$	2199(385)	2243	1924	1590	1435	1528	1550
174Hf	1986${}_{-71}^{+77}$	2291(401)	2317	1925	1519	1537	1661	1755
176Hf	2069${}_{-63}^{+67}$	2350(411)	2373	2196	2103	1704	1891	1981
178Hf	2168(29)	2288(401)	2304	2026	2231	1698	2009	1923
180Hf	2203.9(14)	2303(403)	2322	2224	3094	1853	2215	2025
182Hf	—	2236(392)	2265	2038	2403	1875	2190	1971
184Hf	—	2063(376)	2122	1893	1813	1789	2017	1803

The meson masses $m_{\sigma }$, $m_{\omega }$, and $m_{\rho }$ and the respective coupling constants $g_{\sigma }$, $g_{\omega }$, and $g_{\rho }$ are entered into the Lagrangian of Equation (37), which was first introduced by Walecka [73,74]. This model, however, was unsuccessful in accurately describing the surface properties of finite nuclei and, in particular, the incompressibility. Therefore, in Ref. [77] Boguta and Bodmer introduced an additional density dependence using a nonlinear meson coupling. In that scheme, the $\frac{1}{2}{m}_{\sigma }^2{\sigma }^2$ term in Equation (37) was replaced with

$$U\left(\sigma \right)=\frac{1}{2}{m}_{\sigma }^2{\sigma }^2+\frac{1}{3}{g}_2{\sigma }^3+\frac{1}{4}{g}_3{\sigma }^4$$

(38)

The NL3* parameter fit leads to a good description of nuclear masses, simultaneously providing good results for the collective properties of the rotational and vibrational character [69,70,75]. The parameters of NL3*, along with their values, are given in Refs. [70,75].

Following a similar path to the case of the HFB–Gogny D1S model, we proceed to calculate the various observables. The starting point for our calculations is the ${\beta }_2$ deformation parameters obtained from Ref. [67] in the framework of the relativistic Hartree–Bogoliubov (RHB) theory using the NL3* covariant energy density functional (labeled RHB–NL3* in the tables and graphs). The results are presented in Table 3 and Table 4 and plotted in Figure 3a–f.

Table 4. Values of the ${\beta }_2/{\beta }_{SP}$ ratios, intrinsic quadrupole moments $Q_0$, and electric quadrupole moments Q (see text for more details about the relevant calculations). The “experimental” values are calculated from Equations (1), (2), (4), and (5) using the experimental $B\left(E2\right)$s of Ref. [23] (see also Table 3). Predictions for isotopes ^182,184Hf, for which no experimental data exist, are denoted in bold.

Isotope	“Exp.”	Global Fit	PhM	FRDM	HFBCS–MSk7	HFB–Gogny D1S	HFB–UNEDF–1	RHB–NL3*
${\beta }_2/{\beta }_{SP}$
162Hf	7.13(27)	7.7(7)	8.044	8.287	8.604	9.147	9.311	9.011
164Hf	8.2(4)	8.9(8)	9.107	9.328	9.962	11.230	10.493	10.234
166Hf	11.27${}_{-0.24}^{+0.28}$	10.1(9)	10.247	10.325	10.415	13.223	11.720	11.638
168Hf	12.59(5)	11.3(10)	11.369	11.366	12.226	14.626	13.594	13.811
170Hf	13.48(24)	12.4(11)	12.409	12.408	13.132	15.215	14.770	14.762
172Hf	14.22(14)	12.6(11)	12.604	12.453	13.585	15.577	15.093	14.989
174Hf	13.63(27)	12.7(11)	12.732	13.042	14.491	15.532	14.943	14.536
176Hf	13.54(23)	12.8(11)	12.772	12.589	12.679	14.989	14.230	13.902
178Hf	12.58(8)	12.3(11)	12.292	12.589	11.774	14.264	13.115	13.404
180Hf	12.371(4)	12.1(11)	12.148	12.091	9.962	13.540	12.383	12.951
182Hf	—	11.7(10)	11.743	12.136	10.868	12.815	11.856	12.498
184Hf	—	11.1(10)	11.121	11.592	11.321	11.909	11.217	11.864
$Q_0$ [b]
162Hf	3.67(27)	4.0(7)	4.146	4.604	4.735	4.711	4.795	4.641
164Hf	4.28(40)	4.6(8)	4.732	5.280	5.534	5.831	5.449	5.314
166Hf	5.90${}_{-0.26}^{+0.29}$	5.3(9)	5.368	5.891	5.852	6.922	6.135	6.092
168Hf	6.650(54)	6.0(10)	6.003	6.529	6.958	7.718	7.173	7.288
170Hf	7.17(25)	6.6(12)	6.605	7.194	7.582	8.092	7.856	7.851
172Hf	7.62(13)	6.7(12)	6.761	7.211	7.931	8.350	8.090	8.034
174Hf	7.35(27)	6.9(12)	6.882	7.498	8.440	8.390	8.071	7.852
176Hf	7.38(23)	6.9(12)	6.957	7.187	7.344	8.158	7.745	7.567
178Hf	6.900(92)	6.7(12)	6.746	7.162	6.825	7.823	7.192	7.351
180Hf	6.8300(44)	6.7(12)	6.716	6.828	5.788	7.481	6.842	7.155
182Hf	—	6.5(11)	6.541	6.841	6.300	7.133	6.599	6.956
184Hf	—	6.2(11)	6.239	6.491	6.633	6.677	6.289	6.652
Q [b]
162Hf	−1.05(8)	−1.14(20)	−1.184	−1.316	−1.353	−1.346	−1.370	−1.326
164Hf	−1.22(11)	−1.32(23)	−1.352	−1.509	−1.581	−1.666	−1.557	−1.518
166Hf	−1.69${}_{-0.7}^{+0.8}$	−1.51(26)	−1.534	−1.683	−1.672	−1.978	−1.753	−1.741
168Hf	−1.899(16)	−1.7(3)	−1.715	−1.865	−1.988	−2.205	−2.049	−2.082
170Hf	−2.05(7)	−1.9(3)	−1.887	−2.055	−2.166	−2.312	−2.244	−2.243
172Hf	−2.18(4)	−1.9(3)	−1.932	−2.060	−2.266	−2.386	−2.311	−2.295
174Hf	−2.10(8)	−2.0(3)	−1.966	−2.142	−2.411	−-2.397	−2.306	−2.243
176Hf	−2.11(7)	−2.0(3)	−1.988	−2.053	−2.098	−2.331	−2.213	−2.162
178Hf	−1.971(26)	−1.9(3)	−1.927	−2.046	−1.950	−2.235	−2.055	−2.100
180Hf	−1.9528(13)	−1.9(3)	−1.919	−1.951	−1.654	−2.137	−1.955	−2.044
182Hf	—	−1.9(3)	−1.869	−1.955	−1.800	−2.038	−1.885	−1.988
184Hf	—	−1.8(3)	−1.783	−1.854	−1.895	−1.908	−1.797	−1.901

Table 4. Values of the ${\beta }_2/{\beta }_{SP}$ ratios, intrinsic quadrupole moments $Q_0$, and electric quadrupole moments Q (see text for more details about the relevant calculations). The “experimental” values are calculated from Equations (1), (2), (4), and (5) using the experimental $B\left(E2\right)$s of Ref. [23] (see also Table 3). Predictions for isotopes ^182,184Hf, for which no experimental data exist, are denoted in bold.

Isotope	“Exp.”	Global Fit	PhM	FRDM	HFBCS–MSk7	HFB–Gogny D1S	HFB–UNEDF–1	RHB–NL3*
${\beta }_2/{\beta }_{SP}$
162Hf	7.13(27)	7.7(7)	8.044	8.287	8.604	9.147	9.311	9.011
164Hf	8.2(4)	8.9(8)	9.107	9.328	9.962	11.230	10.493	10.234
166Hf	11.27${}_{-0.24}^{+0.28}$	10.1(9)	10.247	10.325	10.415	13.223	11.720	11.638
168Hf	12.59(5)	11.3(10)	11.369	11.366	12.226	14.626	13.594	13.811
170Hf	13.48(24)	12.4(11)	12.409	12.408	13.132	15.215	14.770	14.762
172Hf	14.22(14)	12.6(11)	12.604	12.453	13.585	15.577	15.093	14.989
174Hf	13.63(27)	12.7(11)	12.732	13.042	14.491	15.532	14.943	14.536
176Hf	13.54(23)	12.8(11)	12.772	12.589	12.679	14.989	14.230	13.902
178Hf	12.58(8)	12.3(11)	12.292	12.589	11.774	14.264	13.115	13.404
180Hf	12.371(4)	12.1(11)	12.148	12.091	9.962	13.540	12.383	12.951
182Hf	—	11.7(10)	11.743	12.136	10.868	12.815	11.856	12.498
184Hf	—	11.1(10)	11.121	11.592	11.321	11.909	11.217	11.864
$Q_0$ [b]
162Hf	3.67(27)	4.0(7)	4.146	4.604	4.735	4.711	4.795	4.641
164Hf	4.28(40)	4.6(8)	4.732	5.280	5.534	5.831	5.449	5.314
166Hf	5.90${}_{-0.26}^{+0.29}$	5.3(9)	5.368	5.891	5.852	6.922	6.135	6.092
168Hf	6.650(54)	6.0(10)	6.003	6.529	6.958	7.718	7.173	7.288
170Hf	7.17(25)	6.6(12)	6.605	7.194	7.582	8.092	7.856	7.851
172Hf	7.62(13)	6.7(12)	6.761	7.211	7.931	8.350	8.090	8.034
174Hf	7.35(27)	6.9(12)	6.882	7.498	8.440	8.390	8.071	7.852
176Hf	7.38(23)	6.9(12)	6.957	7.187	7.344	8.158	7.745	7.567
178Hf	6.900(92)	6.7(12)	6.746	7.162	6.825	7.823	7.192	7.351
180Hf	6.8300(44)	6.7(12)	6.716	6.828	5.788	7.481	6.842	7.155
182Hf	—	6.5(11)	6.541	6.841	6.300	7.133	6.599	6.956
184Hf	—	6.2(11)	6.239	6.491	6.633	6.677	6.289	6.652
Q [b]
162Hf	−1.05(8)	−1.14(20)	−1.184	−1.316	−1.353	−1.346	−1.370	−1.326
164Hf	−1.22(11)	−1.32(23)	−1.352	−1.509	−1.581	−1.666	−1.557	−1.518
166Hf	−1.69${}_{-0.7}^{+0.8}$	−1.51(26)	−1.534	−1.683	−1.672	−1.978	−1.753	−1.741
168Hf	−1.899(16)	−1.7(3)	−1.715	−1.865	−1.988	−2.205	−2.049	−2.082
170Hf	−2.05(7)	−1.9(3)	−1.887	−2.055	−2.166	−2.312	−2.244	−2.243
172Hf	−2.18(4)	−1.9(3)	−1.932	−2.060	−2.266	−2.386	−2.311	−2.295
174Hf	−2.10(8)	−2.0(3)	−1.966	−2.142	−2.411	−-2.397	−2.306	−2.243
176Hf	−2.11(7)	−2.0(3)	−1.988	−2.053	−2.098	−2.331	−2.213	−2.162
178Hf	−1.971(26)	−1.9(3)	−1.927	−2.046	−1.950	−2.235	−2.055	−2.100
180Hf	−1.9528(13)	−1.9(3)	−1.919	−1.951	−1.654	−2.137	−1.955	−2.044
182Hf	—	−1.9(3)	−1.869	−1.955	−1.800	−2.038	−1.885	−1.988
184Hf	—	−1.8(3)	−1.783	−1.854	−1.895	−1.908	−1.797	−1.901

4. Results and Discussion

The various observables calculated in this work for the even–even ^162–184Hf isotopes with the six different models described in Section 3 are shown in Table 3 and Table 4 and plotted in Figure 3a–f. These are compared with existing experimental data and with Raman’s Global Best Fit predictions [21]. We should clarify that in the case of the quantities labeled “Exp.” in Table 4, the “experimental” values refer to the values resulting from Equations (1), (2), (4), and (5) using the experimental $B\left(E2\right)$s of Ref. [23]. We should also mention that experimental data for Q exist for the cases of isotopes ^176,178,180Hf [78]. Those values are close to the ones presented in Table 4. All of the theoretical predictions of the models considered in this work seem to be able to reproduce the trend of the experimental data fairly well.

Regarding the energies of the low-lying excited states of the ground-state bands in the even–even ^162–184Hf, the phenomenological model employed in this work led to a very good description of the first low-lying energy levels, yielding an excellent agreement with the experimental values of Refs. [23,43] for the $E\left(2_1^{+}\right)$ levels (see Table 1 and Table 2), as well as with the theoretical results in the earlier works of Refs. [28,29,30,47]. The anti-correlation effect between the $E\left(2_1^{+}\right)$ energies and the deformation parameters ${\beta }_2$ was observed as expected. However, the energy difference $\Delta E\left(I\right)=E^{th}\left(I\right)-E^{exp}\left(I\right)$ presented an increase with the increasing angular momentum I. This is due to the occurrence of the non-adiabaticity of the energy rotational bands in large spin [28,29].

For the $B(E2;0_1^{+}\to 2_1^{+})$ reduced transition rates, the intrinsic quadrupole moments $Q_0$ and the electric quadrupole moments Q, the FRDM led to an improved description of the available experimental data compared to the other models considered in this study. The HFB formalism slightly overestimated the Q, $Q_0$, and $B\left(E2\right)\uparrow$ values when the Gogny D1S interaction was used (HFB–Gogny D1S model, green squares in graphs). This behavior seems to improve slightly in the case of the Skyrme-type energy density functional UNEDF-1 (orange triangles in graphs). Similar behavior was observed in the case of the relativistic (covariant) energy density functional NL3* (RHB–NL3*, purple triangles in graphs). The aforementioned models (HFB–Gogny D1S, HFB–UNEDF–1, RHB–NL3*) are based on the HFB formalism, differing in the implementation (nonrelativistic, relativistic) and the effective interactions used. It is interesting that, besides their differences, they exhibited a similar trend of slightly overestimating the $Q_0$, Q, and $B\left(E2\right)\uparrow$ values compared to the experimental data in Ref. [23]. Similar conclusions regarding various quantities were drawn, for instance, in the case of Hg isotopes [79] regarding the comparison of self-consistent Skyrme and Gogny forces and for heavier nuclei in the transactinide region around nobelium regarding the performance of Skyrme, Gogny, and covariant energy density functionals [80].

Regarding the ${\beta }_2/{\beta }_{SP}$ ratios, the “experimental” values in Table 4 were calculated from the adopted $B\left(E2\right)\uparrow$ values of Ref. [23] using the rotational model (Equation (4)). The “experimental” values of Ref. [23] are almost equal to those of Ref. [21], which is to be expected since the $B\left(E2\right)\uparrow$ values obtained by Pritychenko et al. and Raman et al. coincided up to the first decimal digit. Nuclear deformations are difficult to determine experimentally; therefore, deducing the deformation values from experimental data is associated with a considerable model dependence. However, the work of Raman and collaborators [20,21] in which Equation (4) was used seems to be the most common pathway for deducing deformations from experiments and conducting a systematic comparison of various model calculations [52]. All of the examined models were successful in reproducing the trend of the “experimental” ${\beta }_2/{\beta }_{SP}$ ratios (Figure 3b), which were well over unity, revealing the collective quadrupole motion in the neutron-rich even–even Hf isotopes and indicating a correlation between deformation and the filling of major shells. Depending on the model, the maximum deformation was observed in either ¹⁷²Hf or ¹⁷⁴Hf, whereas the maximum value was observed “experimentally” for $A=172$, four neutrons away from the mid-shell $A=176$. This differs from the case of the Yb isotopes [37] in which depending on the model, the maximum deformation was observed four or two neurons away from the mid-shell (¹⁷²Yb or ¹⁷⁰Yb) but the “experimental” one was observed two neutrons away from the mid-shell ¹⁷⁴Yb. We should note that the ${\beta }_2$ values used to determine the ${\beta }_2/{\beta }_{SP}$ ratios in this work were taken from the relevant references for each model. Furthermore, it should be stressed that depending on the availability of data, the ${\beta }_2$ values refer to either (i) the quadratic deformation of the mass distribution (for models FRDM, HFB–Gogny–D1S, and HFBCS–MSk7), or (ii) the quadratic deformation of the nuclear charge distribution. However, in this mass region, the two deformation parameters are expected to differ by less than 5%.

Figure 3. ${\beta }_2$ deformation parameters (a), ${\beta }_2/{\beta }_{SP}$ ratios (b), intrinsic $Q_0$ (c) and electric quadrupole moments Q (d), $B(E2;0_1^{+}\to 2_1^{+})$ reduced electric transition probabilities (e), and lifetimes $\tau \left(2_1^{+}\right)$ (f) calculated for the even–even ^162–184Hf isotopes using the models presented in Section 3 (for abbreviations, see text). The theoretical predictions are compared with the global fit [21] values (cyan-shaded areas in the graphs), as well as the experimental data where available [4,23].

Figure 3. ${\beta }_2$ deformation parameters (a), ${\beta }_2/{\beta }_{SP}$ ratios (b), intrinsic $Q_0$ (c) and electric quadrupole moments Q (d), $B(E2;0_1^{+}\to 2_1^{+})$ reduced electric transition probabilities (e), and lifetimes $\tau \left(2_1^{+}\right)$ (f) calculated for the even–even ^162–184Hf isotopes using the models presented in Section 3 (for abbreviations, see text). The theoretical predictions are compared with the global fit [21] values (cyan-shaded areas in the graphs), as well as the experimental data where available [4,23].

Based on the good agreement between the theoretical models studied in this work and the experimental data, we attempted to further extend the results to the neutron-rich nuclei ¹⁸²Hf and ¹⁸⁴Hf for which no experimental data other than the $E\left(2_1^{+}\right)$ exist. The predictions regarding the ${\beta }_2$ deformation parameters, ${\beta }_2/{\beta }_{SP}$ ratios, intrinsic quadrupole moments $Q_0$ and electric quadrupole moments Q, $B(E2;0_1^{+}\to 2_1^{+})$ reduced electric quadrupole transition probabilities, and lifetimes $\tau \left(2_1^{+}\right)$ are presented in Table 3 and Table 4 and plotted in Figure 3a–f. It is encouraging that the spread of the values among the different theoretical models for each quantity we examined decreases toward the more neutron-rich isotopes and is smaller than that of the global model, thus reducing the uncertainty of our predictions.

5. Conclusions

In the framework of the collective model, we conclude that the quadrupole moments are successfully calculated for a number of permanently deformed even–even nuclei belonging to the Hf isotopic chain in the rare-earth part of the nuclear chart. A $B(E2;0_1^{+}\to 2_1^{+})$ data compilation was assembled for the even–even ^162–184Hf isotopes using six different models. Based on the deformation parameters ${\beta }_2$, other physical quantities were additionally calculated, providing further insight into the phenomena related to the nuclear symmetries defining the shape of the nucleus. The ${\beta }_2/{\beta }_{SP}$ ratio is considerably greater than the unity, indicating that these nuclei demonstrate greater quadrupole deformations than would be expected from shell model predictions.

In this context, the sub-shell structure seems to be important. Some theoretical investigations [51,81,82] predict that the maximum quadrupole deformation occurs below $N=104$ within an isotopic chain, whereas some available experimental data suggest that the deformations increase as the proton number decreases below the mid-shell [83,84]. The strength of the $E2$ transitions between successive levels is of importance for clarifying some ambiguities in the structure of the observed states.

Based on the good agreement between the results of the theoretical models and the experimental data for the ^162–184Hf isotopes, we made predictions for the lifetimes of the $2_1^{+}$ state, the $B(E2;0_1^{+}\to 2_1^{+})$ reduced transition matrix elements, the intrinsic quadrupole moments $Q_0$, the electric quadrupole moments Q, and the ${\beta }_2/{\beta }_{SP}$ ratios for isotopes ¹⁸²Hf and ¹⁸⁴Hf (denoted in bold in Table 3 and Table 4) for which no information exists other than the energy of the $2_1^{+}$ state. This newly acquired information can serve as a comprehensive guide for current and future experiments focused on neutron-rich hafnium isotopes.