Triaxiality and shape coexistence in germanium isotopes

The ground-state deformations of the Ge isotopes are investigated in the framework of Gogny-Hartree-Fock-Bogoliubov (HFB) and Skyrme Hartree-Fock plus pairing in the BCS approximation. Five different Skyrme parametrizations are used to explore the influence of different effective masses and spin-orbit models. There is generally good agreement for binding energies and deformations (total quadrupole moment and triaxiality) with experimental data where available (i.e., in the valley of stability). All calculations agree in predicting a strong tendency for triaxial shapes in the Ge isotopes with only a few exceptions owing to neutron (sub)shell closures. The frequent occurrence of energetically very close shape isomers indicates that the underlying deformation energy landscape is very soft. The general triaxial softness of the Ge isotopes is demonstrated in the fully triaxial potential energy surface. The differences between the forces play an increasing role with increasing neutron number. This concerns particularly the influence of the spin-orbit model, which has a visible effect on the trend of binding energies toward the drip line. Different effective mass plays an important role in predicting the quadrupole and triaxial deformations. The pairing strength only weakly affects binding energies and total quadrupole deformations, but it considerably influences triaxiality.

Figure 1

Binding energies along the chain of Ge isotopes for the six forces, Gogny, SLy6, SkI3, SkT6, SkM${}^{*}$, and SkP, as indicated. The lower panel shows the energy differences from FRLDM values [40] and the upper panel the differences from the experimental values [41]. Filled symbols denote ground states and open symbols isomers.

Figure 2

Potential-energy surfaces (PES) versus axially symmetric quadrupole deformation computed with SLy6 for three selected Ge isotopes as indicated. Positive quadrupole moments correspond to prolate shapes and negative moments to oblate ones.

Figure 3

Triaxial PES with SLy6 for the nucleus ${}^{76}\mathrm{Ge}$. The contour lines are labeled with the energy in MeV. The two straight lines indicate the directions of triaxial deformation, $\gamma ={30}^{°}$ and ${60}^{°}$.

Figure 4

The shape parameters β for total quadrupole deformation (middle panels) as well as γ for the triaxiality (upper panels) and the pairing energy (lower panels) for the six forces: Gogny (upper left panels), SLy6 (upper middle), SkI3 (upper right), SkM${}^{*}$ (lower left), SkP (lower middle), and SkT6 (lower right). The ground-state values are shown with full symbols. Isomers are indicated by open symbols. Available experimental values for deformation and triaxiality are indicated by error bars [4, 5].