The proxy-SU(3) symmetry predicts, in a parameter-free way, based only on the Pauli principle and the short-range nature of the nucleon–nucleon interaction, non-vanishing values of the collective variable γ almost everywhere across the nuclear chart. Substantial triaxiality with γ between 15° and 45° is proved to be expected along horizontal and vertical stripes on the nuclear chart, covering the nucleon numbers 22–26, 34–48, 74–80, 116–124, 172–182. Empirical support for these stripes is found by collecting all even–even nuclei for which the first two excited 2⁺ states are known, along with the B(E2)s connecting them, as well as the second 2⁺ state to the ground state. The stripes are related to regions in which oblate SU(3) irreducible representations appear, bearing similarity to the appearance of triaxiality within the SU(3)* dynamical symmetry of the interacting boson model-2. Detailed comparisons of the proxy-SU(3) predictions to the data and to predictions by state-of-the-art Monte Carlo shell model calculations for deformed N = 94, 96, 98 isotones in the rare earth region show good overall agreement, with the exception of Z = 70 and N = 94, which correspond to fully symmetric proxy-SU(3) irreps, suggesting that the latter are an artifact of the method which can be amended by considering the influence of the neighboring irreps.

An interacting-boson-model-1 (IBM-1) Hamiltonian, derived from self-consistent mean-field calculations using a Skyrme energy density functional is employed for the study of energy spectra and $B\left(E2\right)$ transition strengths in the even-even ${}^{162–184}\mathrm{Hf}$ and ${}^{168–186}\mathrm{W}$. An intrinsic triaxial deformation, derived from fermionic proxy-SU(3) irreps, is incorporated into the IBM-1 potential energy curve, which is subsequently mapped to the fermionic one, in order to derive the parameters of the IBM-1 Hamiltonian. It is shown that the inclusion of the intrinsic triaxial deformation derived from the proxy-SU(3) irreps leads to a significantly improved agreement between the theoretical predictions and experimental data for the low-lying quadrupole bands in the examined isotopes, without the need of higher-order terms in the IBM-1 Hamiltonian. The calculated $B\left(E2\right)$ transition strengths are also improved, compared to the axially symmetric case. The recently suggested preponderance of triaxial deformation over extended regions of the nuclear chart is obtained as a byproduct. Future potential improvements and extensions to this mapping approach are also discussed.

Triaxial shapes in even–even nuclei have been considered since the early days of the nuclear collective model. Although many theoretical approaches have been used over the years for their description, no effort appears to have been made for grouping them together and identifying regions on the nuclear chart where the appearance of triaxiality might be favored. In addition, over the last few years, discussion has started on the appearance of small triaxiality in nuclei considered so far as purely axial rotors. In the present work, we collect the predictions made by various theoretical approaches and show that pronounced triaxiality appears to be favored within specific stripes on the nuclear chart, with low triaxiality being present in the regions between these stripes, in agreement with parameter-free predictions made by the proxy-SU(3) approximation to the shell model, based on the Pauli principle and the short-range nature of the nucleon–nucleon interaction. The robustness of triaxiality within these stripes is supported by global calculations made in the framework of the Finite-Range Droplet Model (FRDM), which is based on completely different assumptions and possesses parameters fitted in order to reproduce fundamental nuclear properties.

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.