# Nuclear Structure Investigations of Even–Even Hf Isotopes

^{*}

## Abstract

**:**

^{162–184}Hf (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

^{162–180}Hf isotopes are compared with existing experimental data where available, showing an overall good agreement.

## 1. Introduction

^{184}Hf) [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].

## 2. Physical Quantities and Global Best Fit

#### 2.1. The Intrinsic Quadrupole Moment ${Q}_{0}$

#### 2.2. The Electric Quadrupole Moment Q

#### 2.3. The Deformation Parameter ${\beta}_{2}$

#### 2.4. Global Best Fit

^{26}O,

^{214,216,218}Po) (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.

## 3. Theoretical Models

^{162–184}Hf 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)

^{162–184}Hf nuclei.

^{162–184}Hf 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. 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. 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

^{166–172}Hf, 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].

^{170–184}Hf isotopes.

#### 3.2. Finite-Range Droplet Model (FRDM)

^{162–184}Hf isotopes using the relation [52]

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

^{162–184}Hf 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)

- 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$${q}_{20}=\langle \Phi |{Q}_{20}|\Phi \rangle $$$${Q}_{20}=\sqrt{\frac{16\pi}{5}}{r}^{2}{Y}_{20}$$

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

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

**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

^{182,184}Hf, 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${}^{2}$] | ||||||||

^{162}Hf | 1.34(10) | 1.6(3) | 1.710 | 2.109 | 2.230 | 2.208 | 2.287 | 2.143 |

^{164}Hf | 1.82(17) | 2.1(4) | 2.228 | 2.774 | 3.046 | 3.383 | 2.953 | 2.809 |

^{166}Hf | 3.46${}_{-0.15}^{+0.17}$ | 2.8(5) | 2.866 | 3.452 | 3.407 | 4.766 | 3.744 | 3.692 |

^{168}Hf | 4.393(36) | 3.5(6) | 3.585 | 4.240 | 4.815 | 5.925 | 5.118 | 5.283 |

^{170}Hf | 5.11(18) | 4.3(8) | 4.339 | 5.148 | 5.719 | 6.514 | 6.138 | 6.132 |

^{172}Hf | 5.77(10) | 4.5(8) | 4.546 | 5.172 | 6.257 | 6.935 | 6.511 | 6.421 |

^{174}Hf | 5.38(20) | 4.7(8) | 4.711 | 5.592 | 7.086 | 7.002 | 6.480 | 6.132 |

^{176}Hf | 5.42(17) | 4.8(8) | 4.814 | 5.138 | 5.364 | 6.620 | 5.967 | 5.695 |

^{178}Hf | 4.736(63) | 4.5(8) | 4.526 | 5.103 | 4.633 | 6.087 | 5.145 | 5.375 |

^{180}Hf | 4.6470(30) | 4.5(8) | 4.487 | 4.638 | 3.333 | 5.567 | 4.656 | 5.093 |

^{182}Hf | — | 4.2(7) | 4.255 | 4.656 | 3.949 | 5.061 | 4.332 | 4.813 |

^{184}Hf | — | 3.8(7) | 3.872 | 4.191 | 4.376 | 4.435 | 3.934 | 4.401 |

$\tau \left({2}_{1}^{+}\right)$ [ps] | ||||||||

^{162}Hf | 148(11) | 126(22) | 176 | 94 | 89 | 90 | 87 | 93 |

^{164}Hf | 435(41) | 376(66) | 467 | 286 | 261 | 235 | 269 | 283 |

^{166}Hf | 717(33) | 895(157) | 1010 | 721 | 731 | 522 | 665 | 674 |

^{168}Hf | 1239(10) | 1548(271) | 1654 | 1287 | 1134 | 921 | 1067 | 1033 |

^{170}Hf | 1740(61) | 2074(363) | 2137 | 1735 | 1562 | 1371 | 1455 | 1457 |

^{172}Hf | 1710${}_{-39}^{+31}$ | 2199(385) | 2243 | 1924 | 1590 | 1435 | 1528 | 1550 |

^{174}Hf | 1986${}_{-71}^{+77}$ | 2291(401) | 2317 | 1925 | 1519 | 1537 | 1661 | 1755 |

^{176}Hf | 2069${}_{-63}^{+67}$ | 2350(411) | 2373 | 2196 | 2103 | 1704 | 1891 | 1981 |

^{178}Hf | 2168(29) | 2288(401) | 2304 | 2026 | 2231 | 1698 | 2009 | 1923 |

^{180}Hf | 2203.9(14) | 2303(403) | 2322 | 2224 | 3094 | 1853 | 2215 | 2025 |

^{182}Hf | — | 2236(392) | 2265 | 2038 | 2403 | 1875 | 2190 | 1971 |

^{184}Hf | — | 2063(376) | 2122 | 1893 | 1813 | 1789 | 2017 | 1803 |

**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

^{182,184}Hf, 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* |
---|---|---|---|---|---|---|---|---|

${\beta}_{2}/{\beta}_{SP}$ | ||||||||

^{162}Hf | 7.13(27) | 7.7(7) | 8.044 | 8.287 | 8.604 | 9.147 | 9.311 | 9.011 |

^{164}Hf | 8.2(4) | 8.9(8) | 9.107 | 9.328 | 9.962 | 11.230 | 10.493 | 10.234 |

^{166}Hf | 11.27${}_{-0.24}^{+0.28}$ | 10.1(9) | 10.247 | 10.325 | 10.415 | 13.223 | 11.720 | 11.638 |

^{168}Hf | 12.59(5) | 11.3(10) | 11.369 | 11.366 | 12.226 | 14.626 | 13.594 | 13.811 |

^{170}Hf | 13.48(24) | 12.4(11) | 12.409 | 12.408 | 13.132 | 15.215 | 14.770 | 14.762 |

^{172}Hf | 14.22(14) | 12.6(11) | 12.604 | 12.453 | 13.585 | 15.577 | 15.093 | 14.989 |

^{174}Hf | 13.63(27) | 12.7(11) | 12.732 | 13.042 | 14.491 | 15.532 | 14.943 | 14.536 |

^{176}Hf | 13.54(23) | 12.8(11) | 12.772 | 12.589 | 12.679 | 14.989 | 14.230 | 13.902 |

^{178}Hf | 12.58(8) | 12.3(11) | 12.292 | 12.589 | 11.774 | 14.264 | 13.115 | 13.404 |

^{180}Hf | 12.371(4) | 12.1(11) | 12.148 | 12.091 | 9.962 | 13.540 | 12.383 | 12.951 |

^{182}Hf | — | 11.7(10) | 11.743 | 12.136 | 10.868 | 12.815 | 11.856 | 12.498 |

^{184}Hf | — | 11.1(10) | 11.121 | 11.592 | 11.321 | 11.909 | 11.217 | 11.864 |

${Q}_{0}$ [b] | ||||||||

^{162}Hf | 3.67(27) | 4.0(7) | 4.146 | 4.604 | 4.735 | 4.711 | 4.795 | 4.641 |

^{164}Hf | 4.28(40) | 4.6(8) | 4.732 | 5.280 | 5.534 | 5.831 | 5.449 | 5.314 |

^{166}Hf | 5.90${}_{-0.26}^{+0.29}$ | 5.3(9) | 5.368 | 5.891 | 5.852 | 6.922 | 6.135 | 6.092 |

^{168}Hf | 6.650(54) | 6.0(10) | 6.003 | 6.529 | 6.958 | 7.718 | 7.173 | 7.288 |

^{170}Hf | 7.17(25) | 6.6(12) | 6.605 | 7.194 | 7.582 | 8.092 | 7.856 | 7.851 |

^{172}Hf | 7.62(13) | 6.7(12) | 6.761 | 7.211 | 7.931 | 8.350 | 8.090 | 8.034 |

^{174}Hf | 7.35(27) | 6.9(12) | 6.882 | 7.498 | 8.440 | 8.390 | 8.071 | 7.852 |

^{176}Hf | 7.38(23) | 6.9(12) | 6.957 | 7.187 | 7.344 | 8.158 | 7.745 | 7.567 |

^{178}Hf | 6.900(92) | 6.7(12) | 6.746 | 7.162 | 6.825 | 7.823 | 7.192 | 7.351 |

^{180}Hf | 6.8300(44) | 6.7(12) | 6.716 | 6.828 | 5.788 | 7.481 | 6.842 | 7.155 |

^{182}Hf | — | 6.5(11) | 6.541 | 6.841 | 6.300 | 7.133 | 6.599 | 6.956 |

^{184}Hf | — | 6.2(11) | 6.239 | 6.491 | 6.633 | 6.677 | 6.289 | 6.652 |

Q [b] | ||||||||

^{162}Hf | −1.05(8) | −1.14(20) | −1.184 | −1.316 | −1.353 | −1.346 | −1.370 | −1.326 |

^{164}Hf | −1.22(11) | −1.32(23) | −1.352 | −1.509 | −1.581 | −1.666 | −1.557 | −1.518 |

^{166}Hf | −1.69${}_{-0.7}^{+0.8}$ | −1.51(26) | −1.534 | −1.683 | −1.672 | −1.978 | −1.753 | −1.741 |

^{168}Hf | −1.899(16) | −1.7(3) | −1.715 | −1.865 | −1.988 | −2.205 | −2.049 | −2.082 |

^{170}Hf | −2.05(7) | −1.9(3) | −1.887 | −2.055 | −2.166 | −2.312 | −2.244 | −2.243 |

^{172}Hf | −2.18(4) | −1.9(3) | −1.932 | −2.060 | −2.266 | −2.386 | −2.311 | −2.295 |

^{174}Hf | −2.10(8) | −2.0(3) | −1.966 | −2.142 | −2.411 | −-2.397 | −2.306 | −2.243 |

^{176}Hf | −2.11(7) | −2.0(3) | −1.988 | −2.053 | −2.098 | −2.331 | −2.213 | −2.162 |

^{178}Hf | −1.971(26) | −1.9(3) | −1.927 | −2.046 | −1.950 | −2.235 | −2.055 | −2.100 |

^{180}Hf | −1.9528(13) | −1.9(3) | −1.919 | −1.951 | −1.654 | −2.137 | −1.955 | −2.044 |

^{182}Hf | — | −1.9(3) | −1.869 | −1.955 | −1.800 | −2.038 | −1.885 | −1.988 |

^{184}Hf | — | −1.8(3) | −1.783 | −1.854 | −1.895 | −1.908 | −1.797 | −1.901 |

## 4. Results and Discussion

^{162–184}Hf 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

^{176,178,180}Hf [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.

^{162–184}Hf, 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].

^{172}Hf or

^{174}Hf, 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 (

^{172}Yb or

^{170}Yb) but the “experimental” one was observed two neutrons away from the mid-shell

^{174}Yb. 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

^{162–184}Hf 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].

^{182}Hf and

^{184}Hf 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

^{162–184}Hf 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.

^{162–184}Hf 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

^{182}Hf and

^{184}Hf (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.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Drout, M.R.; Piro, A.L.; Shappee, B.J.; Kilpatrick, C.D.; Simon, J.D.; Contreras, C.; Coulter, D.A.; Foley, R.J.; Siebert, M.R.; Morrell, N.; et al. Light curves of the neutron star merger GW170817/SSS17a: Implications for r–process nucleosynthesis. Science
**2017**, 358, 1570. [Google Scholar] [CrossRef] [PubMed][Green Version] - Tarasov, O.B.; Ahn, D.S.; Bazin, D.; Fukuda, N.; Gade, A.; Hausmann, M.; Inabe, N.; Ishikawa, S.; Iwasa, N.; Kawata, K.; et al. Discovery of
^{60}Ca and Implications For the Stability of^{70}Ca. Phys. Rev. Lett.**2018**, 121, 022501. [Google Scholar] [CrossRef] [PubMed][Green Version] - Walker, P.; Dracoulis, G. Exotic Isomers in Deformed Atomic Nuclei. Hyperfine Interact.
**2001**, 135, 83–107. [Google Scholar] [CrossRef] - National Nuclear Data Center. Available online: https://www.nndc.bnl.gov/nudat2 (accessed on 1 December 2022).
- Kota, V. Low lying spectra and electromagnetic transitions for
^{164}Dy,^{166}Er, and^{168}Yb nuclei. Phys. Rev. C**1979**, 19, 521. [Google Scholar] [CrossRef] - Heyde, K.; Wood, J.L. Shape coexistence in atomic nuclei. Rev. Mod. Phys.
**2011**, 83, 1467–1521. [Google Scholar] [CrossRef] - Robledo, L.; Rodríguez-Guzmán, R.; Sarriguren, P. Role of triaxiality in the ground–state shape of neutron–rich Yb, Hf, W, Os and Pt isotopes. J. Phys. G Nucl. Part. Phys.
**2009**, 36, 115104. [Google Scholar] [CrossRef][Green Version] - Nomura, K.; Otsuka, T.; Rodríguez-Guzmán, R.; Robledo, L.; Sarriguren, P. Collective structural evolution in neutron–rich Yb, Hf, W, Os, and Pt isotopes. Phys. Rev. C
**2011**, 84, 054316. [Google Scholar] [CrossRef][Green Version] - Bonatsos, D.; Assimakis, I.; Minkov, N.; Martinou, A.; Sarantopoulou, S.; Cakirli, R.; Casten, R.; Blaum, K. Analytic predictions for nuclear shapes, prolate dominance, and the prolate-oblate shape transition in the proxy–SU(3) model. Phys. Rev. C
**2017**, 95, 064326. [Google Scholar] [CrossRef][Green Version] - Bonatsos, D. Prolate over oblate dominance in deformed nuclei as a consequence of the SU(3) symmetry and the Pauli principle. Eur. Phys. J. A
**2017**, 53, 148. [Google Scholar] [CrossRef][Green Version] - Bohr, A.N.; Mottelson, B.R. Nuclear Structure; Vol. II: Nuclear Deformations; World Scientific Publishing: Singapore, 1998. [Google Scholar]
- Greiner, W.; Maruhn, J.A. Nuclear Models; Springer: Berlin/Heidelberg, Germany, 1996. [Google Scholar]
- Ogle, W.; Wahlborn, S.; Piepenbring, R.; Fredriksson, S. Single–Particle Levels of Nonspherical Nuclei in the Region 150 < A < 190. Rev. Mod. Phys.
**1971**, 43, 424–478. [Google Scholar] - Satula, W.; Wyss, R.; Magierski, P. The Lipkin–Nogami formalism for the cranked mean field. Nucl. Phys. A
**1994**, 578, 45–61. [Google Scholar] [CrossRef] - Xu, F.; Satula, W.; Wyss, R. Quadrupole pairing interaction and signature inversion. Nucl. Phys. A
**2000**, 669, 119–134. [Google Scholar] [CrossRef][Green Version] - Zhang, Z.H.; He, X.T.; Zeng, J.Y.; Zhao, E.G.; Zhou, S.G. Systematic investigation of the rotational bands in nuclei with Z ≈ 100 using a particle–number conserving method based on a cranked shell model. Phys. Rev. C
**2012**, 85, 014324. [Google Scholar] [CrossRef][Green Version] - Hara, K.; Sun, Y. Projected shell model and high–spin spectroscopy. Int. J. Mod. Phys. E
**1995**, 04, 637–785. [Google Scholar] [CrossRef] - Shi, Z.; Zhang, H.; Chen, Q.; Zhang, S.; Meng, J. Shell–model–like approach based on cranking covariant density functional theory: Band crossing and shape evolution in
^{60}Fe. Phys. Rev. C**2018**, 97, 034317. [Google Scholar] [CrossRef][Green Version] - Stelson, P.; Grodzins, L. Nuclear transition probability, B(E2) for ${0}_{g.s.}^{+}$–${2}_{\mathit{first}}^{+}$ transitions and deformation parameter, β
_{2}. Nucl. Data Sheets Sect. A**1965**, 1, 21–102. [Google Scholar] [CrossRef] - Raman, S.; Malarkey, C.H.; Milner, W.T.; Nestor, C.W., Jr.; Stelson, P.H. Transition probability, B(E2)↑, from the ground to the first–excited 2
^{+}state of even–even nuclides. At. Data Nucl. Data Tables**1987**, 36, 1–128. [Google Scholar] [CrossRef] - Raman, S.; Nestor, C.W., Jr.; Tikkanen, P. Transition Probability from the Ground to the First–Excited 2
^{+}State of Even–Even Nuclides. At. Data Nucl. Data Tables**2001**, 78, 1–128. [Google Scholar] [CrossRef][Green Version] - Reduced Transition Probabilities or B(E2;0
^{+}→2^{+}) Values. Available online: https://www.nndc.bnl.gov/be2 (accessed on 1 December 2022). - Pritychenko, B.; Birch, M.; Singh, B.; Horoi, M. Tables of E2 transition probabilities from the first 2
^{+}states in even–even nuclei. At. Data Nucl. Data Tables**2016**, 107, 1–139. [Google Scholar] [CrossRef][Green Version] - Pritychenko, B.; Singh, B.; Verpelli, B. Systematic trends of ${0}_{2}^{+}$, ${1}_{1}^{-}$, ${3}_{1}^{-}$ and ${2}_{1}^{+}$ excited states in even–even nuclei. Nucl. Phys. A
**2022**, 1027, 122511. [Google Scholar] [CrossRef] - Ring, P.; Schuck, P. The Nuclear Many–Body Problem; Springer: New York, NY, USA, 1980. [Google Scholar]
- Frank, A.; Jolie, J.; van Isacker, P. Symmetries in Atomic Nuclei: From Isospin to Supersymmetry, 2nd ed.; Springer: Cham, Switzerland, 2019; ISBN 978-3-030-21930-7. [Google Scholar]
- Sheith, J.; Ali, R. Symmetry projection in atomic nuclei. Eur. Phys. J. Spec. Top.
**2020**, 229, 2555–2602. [Google Scholar] [CrossRef] - Usmanov, P.; Okhunov, A.; Salikhbaev, U.; Vdonin, A. Analysis of Electromagnetic Transitions in Nuclei
^{176,178}Hf. Phys. Part. Nucl. Lett.**2010**, 7, 185–191. [Google Scholar] [CrossRef] - Okhunov, A.; Turaeva, G.; Kassim, H.; Khandaker, M.; Rosli, N.B. Analysis of the energy spectra of ground states of deformed nuclei in the rare–earth region. Chin. Phys. C
**2015**, 39, 044101. [Google Scholar] [CrossRef][Green Version] - Okhunov, A.; Sharrad, F.; Al-Sammarraie, A.A.; Khandaker, M. Correspondence between phenomenological and IBM-1 models of even isotopes of Yb. Chin. Phys. C
**2015**, 39, 084101. [Google Scholar] [CrossRef][Green Version] - Wiederhold, J.; Werner, V.; Kern, R.; Pietralla, N.; Bucurescu, D.; Carroll, R.; Cooper, N.; Daniel, T.; Filipescu, D.; Florea, N.; et al. Evolution of E2 strength in the rare-earth isotopes
^{174,176,178,180}Hf. Phys. Rev. C**2019**, 99, 024316. [Google Scholar] [CrossRef][Green Version] - Qasim, H.N.; Al-Khudair, F.H. Nuclear shape phase transition in even-even
^{158–168}Hf isotopes. Nucl. Phys. A**2020**, 1002, 121962. [Google Scholar] [CrossRef] - Das, M.; Biswal, N.; Panda, R.; Bhuyan, M. Structural evolution and shape transition in even-even Hf-isotopes within the relativistic mean-field approach. Nucl. Phys. A
**2022**, 1019, 122380. [Google Scholar] [CrossRef] - Vasileiou, P.; Mertzimekis, T.; Chalil, A.; Zyriliou, A.; Pelonis, S.; Efstathiou, M.; Lagaki, V.; Siltzovalis, G.; Koseoglou, P.; Bonatsos, D.; et al. Experimental Investigation of the Nuclear Structure in the Neutron–Rich
^{180}Hf. Bulg. J. Phys.**2021**, 48, 618–624. [Google Scholar] [CrossRef] - Mertzimekis, T.; Vasileiou, P.; Zyriliou, A.; Efstathiou, M.; Chalil, A.; Pelonis, S.; Lagaki, V.; Siltzovalis, G.; Koseoglou, P.; Bonatsos, D.; et al. Experimental Investigations of Nuclear Structure around A = 180. Bulg. J. Phys.
**2021**, 48, 625–633. [Google Scholar] [CrossRef] - Zyriliou, A.; Mertzimekis, T.; Chalil, A.; Vasileiou, P.; Pelonis, S.; Efstathiou, M.; Mavrommatis, E.; Bonatsos, D.; Martinou, A.; Peroulis, S.; et al. Reviewing Nuclear Structure Properties of Even–Even Yb Isotopes. Bulg. J. Phys.
**2021**, 48, 608–617. [Google Scholar] [CrossRef] - Zyriliou, A.; Mertzimekis, T.J.; Chalil, A.; Vasileiou, P.; Mavrommatis, E.; Bonatsos, D.; Martinou, A.; Peroulis, S.; Minkov, N. A study of some aspects of the nuclear structure in the even–even Yb isotopes. Eur. Phys. J. Plus
**2022**, 137, 352. [Google Scholar] [CrossRef] - Krane, K.S. Introductory Nuclear Physics; Wiley: New York, NY, USA, 1988. [Google Scholar]
- Sharon, Y.; Benczer-Koller, N.; Kumbartzki, G.; Zamick, L.; Casten, R. Systematics of the ratio of electric quadrupole moments Q(${2}_{1}^{+}$) to the square root of the reduced transition probabilities B(E2;${0}_{1}^{+}$→${2}_{1}^{+}$) in even–even nuclei Nucl. Phys. A
**2018**, 980, 131–142. [Google Scholar] [CrossRef] - Bohr, A.; Mottelson, B. Collective and Individual–Particle Aspects of Nuclear Structure. Mat. Fys. Medd. Dan. Vid. Selsk.
**1953**, 27, 1–174. [Google Scholar] - Rowe, D.J. Nuclear Collective Motion; Metheuen: London, UK, 1970; p. 21. [Google Scholar]
- Grodzins, L. The uniform behaviour of electric quadrupole transition probabilities from first 2
^{+}states in even–even nuclei. Phys. Lett.**1962**, 2, 88–91. [Google Scholar] [CrossRef] - Pritychenko, B.; Birch, M.; Singh, B. Revisiting Grodzins systematics of B(E2) values. Nucl. Phys. A
**2017**, 962, 73–102. [Google Scholar] [CrossRef][Green Version] - Habs, D.; Kester, O.; Ames, F.; Sieber, T.; Bongers, H.; Emhofer, S.; Loewe, M.; Reiter, P.; Lutter, R.; Thirolf, P.; et al. CERN Proposal INTC-P-156. 2002. Available online: https://cds.cern.ch/record/000545918 (accessed on 1 December 2022).
- Usmanov, P.; Mikhailov, I. Non–adiabatic effects of collective motion in even–even deformed nuclei. Phys. Part. Nucl. Lett.
**1997**, 28, 348. [Google Scholar] [CrossRef] - Bertulani, C.A. Nuclear Physics in a Nutshell; Princeton University Press: Princeton, NJ, USA, 2007. [Google Scholar]
- Harris, S.M. Higher Order Corrections to the Cranking Model. Phys. Rev.
**1965**, 138, B509–B513. [Google Scholar] [CrossRef] - Casten, R.F. Nuclear Structure from a Simple Perspective; Oxford University Press: Oxford, UK, 2000. [Google Scholar]
- He, X.T.; Cao, Y.; Gan, X.L. Effects of high-j orbitals, pairing, and deformed neutron shells on upbendings of ground-state bands in the neutron-rich even-even isotopes
^{170–184}Hf. Phys. Rev. C**2020**, 102, 014322. [Google Scholar] [CrossRef] - Möller, P.; Nix, J. Nuclear Masses from a Unified Macroscopic–Microscopic Model. At. Data Nucl. Data Tables
**1988**, 39, 213–223. [Google Scholar] [CrossRef] - Möller, P.; Nix, J.; Myers, W.; Swiatecki, W. Nuclear ground–state masses and deformations. At. Data Nucl. Data Tables
**1995**, 59, 185–381. [Google Scholar] [CrossRef][Green Version] - Möller, P.; Sierk, A.; Ichikawa, T.; Sagawa, H. Nuclear ground–state masses and deformations: FRDM(2012). At. Data Nucl. Data Tables
**2016**, 109–110, 1–204. [Google Scholar] [CrossRef][Green Version] - Myers, W.; Swiatecki, W. Average nuclear properties. Ann. Phys.
**1969**, 55, 395–505. [Google Scholar] [CrossRef] - Strutinsky, V. Shell effects in nuclear masses and deformation energies. Nucl. Phys. A
**1967**, 95, 420–442. [Google Scholar] [CrossRef] - Lipkin, H. Collective motion in many–particle systems: Part 1. The violation of conservation laws. Ann. Phys.
**1960**, 9, 272–291. [Google Scholar] [CrossRef] - Nogami, Y. Improved Superconductivity Approximation for the Pairing Interaction in Nuclei. Phys. Rev.
**1964**, 134, B313–B321. [Google Scholar] [CrossRef] - Möller, P.; Nix, J. Calculation of fission barriers with the droplet model and folded Yukawa single–particle potential. Nucl. Phys. A
**1974**, 229, 269–291. [Google Scholar] [CrossRef] - Tondeur, F.; Goriely, S.; Pearson, J.; Onsi, M. Towards a Hartree–Fock Mass Formula. Phys. Rev. C
**2000**, 62, 024308. [Google Scholar] [CrossRef] - Goriely, S.; Tondeur, F.; Pearson, J. A Hartree–Fock Nuclear Mass Table. At. Data Nucl. Data Tables
**2001**, 77, 311–381. [Google Scholar] [CrossRef] - Goriely, S. Capture Gamma-Ray Spectroscopy and Related Topics. In Proceedings of the 10th International Symposium, Santa Fe, NM, USA, 30 August–3 September 1999; Volume 529, pp. 287–294. [Google Scholar]
- Dechargé, J.; Gogny, D. Hartree–Fock–Bogolyubov calculations with the D1 effective interaction on spherical nuclei. Phys. Rev. C
**1980**, 21, 1568–1593. [Google Scholar] [CrossRef] - Berger, J.; Girod, M.; Gogny, D. Time–dependent quantum collective dynamics applied to nuclear fission. Comput. Phys. Commun.
**1991**, 63, 365–374. [Google Scholar] [CrossRef] - Hilaire, S.; Girod, D. Large-scale mean-field calculations from proton to neutron drip lines using the D1S Gogny force. Eur. Phys. J. A
**2007**, 33, 237–241. [Google Scholar] [CrossRef] - Hartree–Fock–Bogoliubov Results Based on the Gogny Force. 2006. Available online: http://www-phynu.cea.fr/science_en_ligne/carte_potentiels_microscopiques/carte_potentiel_nucleaire_eng.htm (accessed on 9 March 2021).
- Delaroche, J.P.; Girod, M.; Libert, J.; Goutte, H.; Hilaire, S.; Péru, S.; Pillet, N.; Bertsch, G. Structure of even-even nuclei using a mapped collective Hamiltonian and the D1S Gogny interaction. Phys. Rev. C
**2010**, 81, 014303. [Google Scholar] [CrossRef] - Kortelainen, M.; McDonnell, J.; Nazarewicz, W.; Reinhard, P.G.; Sarich, J.; Schunck, N.; Stoitsov, M.; Wild, S. Nuclear energy density optimization: Large deformations. Phys. Rev. C
**2012**, 85, 024304. [Google Scholar] [CrossRef][Green Version] - Mass Explorer UNEDF Project. Available online: http://massexplorer.frib.msu.edu/content/DFTMassTables.html (accessed on 20 February 2021).
- Furnstahl, R. EFT for DFT. In Lecture Notes in Physics; Springer: Berlin/Heidelberg, Germany, 2012; pp. 133–191. [Google Scholar] [CrossRef][Green Version]
- Agbemava, S.; Afanasjev, A.; Ray, D.; Ring, P. Global performance of covariant energy density functionals: Ground state observables of even–even nuclei and the estimate of theoretical uncertainties. Phys. Rev. C
**2014**, 89, 054320. [Google Scholar] [CrossRef][Green Version] - Agbemava, S.; Afanasjev, A.; Tanina, A. Propagation of statistical uncertainties in covariant density functional theory: Ground state observables and single–particle properties. Phys. Rev. C
**2019**, 99, 014318. [Google Scholar] [CrossRef][Green Version] - Lalazissis, G.; Köning, J.; Ring, P. New parametrization for the Lagrangian density of relativistic mean field theory. Phys. Rev. C
**1997**, 55, 540–543. [Google Scholar] [CrossRef][Green Version] - Gambhir, X.; Ring, P.; Thimet, A. Relativistic Mean Field Theory for Finite Nuclei. Ann. Phys.
**1990**, 198, 132–179. [Google Scholar] [CrossRef] - Walecka, J. A theory of highly condensed matter. Ann. Phys.
**1974**, 83, 491–529. [Google Scholar] [CrossRef] - Serot, B.D.; Walecka, J.D. The Relativistic Nuclear Many Body Problem. Adv. Nucl. Phys.
**1986**, 16, 1–327. [Google Scholar] - Lalazissis, G.; Karatzikos, S.; Fossion, R.; Arteaga, D.P.; Afanasjev, A.; Ring, P. The effective force NL3 revisited. Phys. Lett. B
**2009**, 671, 36–41. [Google Scholar] [CrossRef][Green Version] - Afanasjev, A.V.; Agbemava, S.E.; Ray, D.; Ring, P. Neutron drip line: Single–particle degrees of freedom and pairing properties as sources of theoretical uncertainties. Phys. Rev. C
**2015**, 91, 014324. [Google Scholar] [CrossRef][Green Version] - Boguta, J.; Bodmer, A. Relativistic calculation of nuclear matter and the nuclear surface. Nucl. Phys. A
**1977**, 292, 413–428. [Google Scholar] [CrossRef] - Stone, N. Table of nuclear electric quadrupole moments. At. Data Nucl. Data Tables
**2016**, 111–112, 1–28. [Google Scholar] [CrossRef] - Warda, M.; Staszczak, A.; Próchniak, L. Comparison of Self–Consistent Skyrme and Gogny Calculations for Light Hg Isotopes. Int. J. Mod. Phys. E
**2010**, 19, 787. [Google Scholar] [CrossRef][Green Version] - Dobaczewski, J.; Afanasjev, A.; Bender, M.; Robledo, L.; Shi, Y. Properties of nuclei in the nobelium region studied within the covariant, Skyrme, and Gogny energy density functionals. Nucl. Phys. A
**2015**, 944, 388–414. [Google Scholar] [CrossRef][Green Version] - Vargas, C.E.; Velázquez, V.; Lerma, S. Microscopic study of neutron–rich dysprosium isotopes. Eur. Phys. J. A
**2013**, 49, 4. [Google Scholar] [CrossRef][Green Version] - Rath, A.K.; Stevenson, P.D.; Regan, P.H.; Xu, F.R.; Walker, P.M. Self-consistent description of dysprosium isotopes in the doubly midshell region. Nucl. Phys. Group
**2003**, 68, 10. [Google Scholar] [CrossRef][Green Version] - Patel, Z.; Söderström, P.A.; Podolyák, Z.; Regan, P.H.; Walker, P.M.; Watanabe, H.; Ideguchi, E.; Simpson, G.S.; Liu, H.L.; Nishimura, S.; et al. Isomer Decay Spectroscopy of
^{164}Sm and^{166}Gd: Midshell Collectivity Around N = 100. Phys. Rev. Lett.**2014**, 113, 262502. [Google Scholar] [CrossRef][Green Version] - Ideguchi, E.; Simpson, G.S.; Yokoyama, R.; Tanaka, M.; Nishimura, S.; Doornenbal, P.; Lorusso, G.; Söderström, P.A.; Sumikama, T.; Wu, J.; et al. μs isomers of
^{158,160}Nd. Phys. Rev. C**2016**, 94, 064322. [Google Scholar] [CrossRef]

**Figure 1.**Experimental vs. phenomenological model moments of inertia, $2\mathcal{J}=(4I-2)/({E}_{I}-{E}_{I-2})$, plotted against the squared angular frequencies of rotation ${\omega}^{2}={({E}_{I}-{E}_{I-2})}^{2}/4$ [48] for the even–even

^{162–172}Hf isotopes: (

**a**)

^{162}Hf, (

**b**)

^{164}Hf, (

**c**)

^{165}Hf, (

**d**)

^{168}Hf, (

**e**)

^{170}Hf, (

**f**)

^{172}Hf. Solid lines are drawn to guide the eye. Experimental uncertainties are smaller than the data symbols.

**Figure 2.**Same as in Figure 1 for the even–even

^{174–184}Hf isotopes: (

**a**)

^{174}Hf, (

**b**)

^{176}Hf, (

**c**)

^{178}Hf, (

**d**)

^{180}Hf, (

**e**)

^{182}Hf, (

**f**)

^{184}Hf.

**Table 1.**${\mathcal{J}}_{0}$, ${\mathcal{J}}_{1}$ values obtained within the framework of the phenomenological model for

^{162–184}Hf (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] |

^{162}Hf | 0.880 | 13.804 | 262.249 | 285.000 |

^{164}Hf | 1.327 | 14.623 | 199.610 | 210.700 |

^{166}Hf | 1.821 | 17.266 | 153.912 | 158.640 |

^{168}Hf | 2.373 | 17.522 | 122.068 | 124.100 |

^{170}Hf | 2.933 | 20.097 | 100.055 | 100.800 |

^{172}Hf | 3.128 | 12.837 | 94.750 | 95.220 |

^{174}Hf | 3.276 | 11.366 | 90.728 | 90.985 |

^{176}Hf | 3.381 | 8.958 | 88.126 | 88.349 |

^{178}Hf | 3.206 | 6.822 | 93.019 | 93.180 |

^{180}Hf | 3.211 | 3.584 | 93.137 | 93.324 |

^{182}Hf | 3.063 | 4.440 | 97.485 | 97.790 |

^{184}Hf | 2.800 | 5.465 | 106.349 | 107.100 |

**Table 2.**Level energies (in MeV) and rotational frequencies ${\omega}_{rot}^{th}$ (in MeV ${\hslash}^{-1}$) for the even–even

^{162–184}Hf 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 | ^{162}Hf | ^{164}Hf | ^{166}Hf | ||||||
---|---|---|---|---|---|---|---|---|---|

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 | ^{168}Hf | ^{170}Hf | ^{172}Hf | ||||||

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 | ^{174}Hf | ^{176}Hf | ^{178}Hf | ||||||

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 | ^{180}Hf | ^{182}Hf | ^{184}Hf | ||||||

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 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Vasileiou, P.; Mertzimekis, T.J.; Mavrommatis, E.; Zyriliou, A.
Nuclear Structure Investigations of Even–Even Hf Isotopes. *Symmetry* **2023**, *15*, 196.
https://doi.org/10.3390/sym15010196

**AMA Style**

Vasileiou P, Mertzimekis TJ, Mavrommatis E, Zyriliou A.
Nuclear Structure Investigations of Even–Even Hf Isotopes. *Symmetry*. 2023; 15(1):196.
https://doi.org/10.3390/sym15010196

**Chicago/Turabian Style**

Vasileiou, Polytimos, Theo J. Mertzimekis, Eirene Mavrommatis, and Aikaterini Zyriliou.
2023. "Nuclear Structure Investigations of Even–Even Hf Isotopes" *Symmetry* 15, no. 1: 196.
https://doi.org/10.3390/sym15010196