Possible triaxial deformation in <i>N</i> = <i>Z</i> nucleus germanium-64
Authors: Shen Shui-Fa , Wang Hua-Lei , Meng Hai-Yan , Yan Yu-Peng , Shen Jie-Jie , Wang Fei-Peng , Jiang Hai-Bin , Bao Li-Na ,   School of Intelligent Manufacturing, Zhejiang Guangsha Vocational and Technical University of Construction, Zhejiang, Jinhua 322100, China,   Institute of Nuclear Energy Safety Technology, Hefei Institutes of Physical Science, Chinese Academy of Sciences, Anhui, Hefei 230031, China,   School of Electronic, Electrical Engineering and Physics, Fujian University of Technology, Fujian, Fuzhou 350118, China,   Key Laboratory of High Precision Nuclear Spectroscopy, Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China,   School of Physics, Zhengzhou University, Zhengzhou 450001, China,   School of Physics, Suranaree University of Technology, Nakhon Ratchasima 30000, Thailand,   Thailand Center of Excellence in Physics (ThEP), Commission on Higher Education, Bangkok 10400, Thailand,   Division of Health Sciences, Hangzhou Normal University, Zhejiang, Hangzhou 310012, China,   Department of Basic Sciences, Army Academy of Artillery and Air Defense, Anhui, Hefei 230031, China
Published: September 2021

Evidence for nonaxial <i>γ</i> deformations has been widely found in collective rotational states. The <i>γ</i> deformation has led to very interesting characteristics of nuclear motions, such as wobbling, chiral band, and signature inversion in rotational states. There is an interesting question; why the nonaxial <i>γ</i> deformation is not favored in the ground states of even-even (e-e) nuclei. The quest for stable triaxial shapes in the ground states of e-e nuclei, with a maximum triaxial deformation of <inline-formula><tex-math id="M2">\begin{document}$ \left| \gamma \right| $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210187_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210187_M2.png"/></alternatives></inline-formula> ≈ 30°, is still a major theme in nuclear structure. In the present work, we use the cranked Woods-Saxon (WS) shell model to investigate possible triaxial shapes in ground and collective rotational states. Total-Routhian-surface calculations by means of the pairing-deformation-frequency self-consistent cranked shell model are carried out for even-even germanium and selenium isotopes, in order to search for possible triaxial deformations of nuclear states. Calculations are performed in the lattice of quadrupole (<i>β</i><sub>2</sub>, <i>γ</i>) deformations with the hexadecapole <i>β</i><sub>4</sub> variation. In fact, at each grid point of the quadrupole deformation (<i>β</i><sub>2</sub>, <i>γ</i>) lattice, the calculated energy is minimized with respect to the hexadecapole deformation <i>β</i><sub>4</sub>. The shape phase transition from triaxial shape in <sup>64</sup>Ge, oblate shape in <sup>66</sup>Ge, again through triaxiality, to prolate deformations is found in germanium isotopes. In general, the Ge and Se isotopes have <i>γ</i>-soft shapes, resulting in significant dynamical triaxial effect. There is no evidence in the calculations pointing toward rigid triaxiality in ground states. The triaxiality of <inline-formula><tex-math id="M3">\begin{document}$ \gamma = - 30^\circ $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210187_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="19-20210187_M3.png"/></alternatives></inline-formula> for the ground and collective rotational states, that is the limit of triaxial shape, is found in <sup>64, 74</sup>Ge. One should also note that the depth of the triaxial minimum increases with rotational frequency increasing in these two nuclei. The present work focuses on the possible triaxial deformation of <i>N</i> = <i>Z</i> nucleus <sup>64</sup>Ge. Single-particle level diagrams can give a further understanding of the origin of the triaxiality. Based on the information about single-particle levels obtained with the phenomenological Woods-Saxon (WS) potential, the mechanism of triaxial deformation in <i>N</i> = <i>Z</i> nucleus <sup>64</sup>Ge is discussed, and caused surely by a deformed <i>γ</i>≈30° shell gap at <i>Z</i>(<i>N</i>) = 32. At <i>N</i> = 34, however, an oblate shell gap appears, which results in an oblate shape in <sup>66</sup>Ge (<i>N</i> = 34). With neutron number increasing, the effect from the <i>N</i> = 34 oblate gap decreases, and hence the deformations of heavier Ge isotopes change toward the triaxiality (or prolate).