# 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

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**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 |

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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