Possible triaxial deformation in <i>N</i> = <i>Z</i> nucleus germanium-64

Abstract

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).