Microscopic analysis of shape evolution and triaxiality in germanium isotopes

Background: The motivation for this study is the experimental evidence for rigid triaxial deformation at low energy in ${}^{76}\mathrm{Ge}$ that was recently observed.

Purpose: Quadrupole shapes and low-energy spectra of the isotopes ${}^{72–82}\mathrm{Ge}$ are analyzed using a theoretical framework based on nuclear density functional theory.

Method: The relativistic functional DD-PC1, supplemented by a finite-range pairing force, is used to perform constrained triaxial mean-field calculations of energy surfaces as functions of quadrupole deformation parameters. The corresponding collective Hamiltonian, based on DD-PC1, is employed in the calculation of excitation spectra and transition rates.

Results: Model calculations reproduce the empirical trend of collective observables and predict the evolution of shapes from weakly triaxial in ${}^{74}\mathrm{Ge}$ to $\gamma$ soft in ${}^{78,80}\mathrm{Ge}$. For ${}^{76}\mathrm{Ge}$, in particular, the theoretical excitation spectrum is in good agreement with available data, the experimental ratio $E\left(2_2^{+}\right)/E\left(2_1^{+}\right)$ is reproduced, as well as the pattern and amplitude of the staggering in energy between odd- and even-spin states in the $\gamma$ band.

Conclusions: The mean-field potential of ${}^{76}\mathrm{Ge}$ appears to be $\gamma$ soft. Collective correlations drive the nucleus toward triaxiality but do not stabilize a rigid triaxial shape. Both the experimental and theoretical staggering of levels in the $\gamma$ band display a pattern consistent with triaxial shapes but the amplitudes are negligible and do not present evidence for rigid triaxiality.

Figure 1

Self-consistent RHB triaxial energy surfaces of even-even Ge nuclei in the $\beta \text{−}\gamma$ plane ($0\le \gamma \le {60}^{\circ }$). For each nucleus energies are normalized with respect to the binding energy of the absolute minimum.

Figure 2

Single-proton and single-neutron energy levels of ${}^{74}$Ge as functions of the deformation parameters along closed paths in the $\beta \text{−}\gamma$ plane. Solid (blue) and dashed (red) curves correspond to levels with positive and negative parity, respectively. The dot-dashed (green) curves denote the Fermi level. The panels on the left and right display prolate ($\gamma =0^{\circ }$) and oblate ($\gamma ={60}^{\circ }$) axially symmetric single-nucleon levels, respectively. In the middle panel of each figure the proton and neutron levels are plotted as functions of $\gamma$ for a fixed value of the axial deformation $\left|\beta \right|$ that corresponds to the mean-field minimum.

Figure 3

Same as described in the caption to Fig. 2 but for the nucleus ${}^{76}\mathrm{Ge}$.

Figure 4

Same as described in the caption to Fig. 2 but for the nucleus ${}^{78}$Ge.

Figure 5

Self-consistent RHB constrained energy curves of ${}^{74}$Ge, ${}^{76}\mathrm{Ge}$, and ${}^{78}$Ge as functions of the deformation parameter $\gamma$, at fixed values of the axial deformation: $\beta =0.25$, $\beta =0.20$, and $\beta =0.20$, respectively. For each nucleus energies are normalized with respect to the binding energy of the absolute minimum.

Figure 6

Evolution of the ratio between the excitation energies of the first $4^{+}$ and $2^{+}$ states (left panel), and the $B(E2;2_1^{+}\to 0_1^{+})$ values in Weisskopf units (right panel), with mass number for the Ge isotopes.

Figure 7

The excitation spectrum of ${}^{76}\mathrm{Ge}$ calculated with the collective Hamiltonian based on the DD-PC1 relativistic density functional (left) compared to data [12, 25] (right). The $B\left(E2\right)$ values are given in Weisskopf units.

Figure 8

Distribution of $K$ components (projection of the angular momentum on the body-fixed symmetry axis) in the collective wave functions of the nucleus ${}^{76}\mathrm{Ge}$.

Figure 9

Staggering $S\left(J\right)$ [Eq. (15)] in the $\gamma$ bands of ${}^{74–80}$Ge isotopes.