Triaxiality and the nature of low-energy excitations in ${}^{76}\mathrm{Ge}$

The deformation properties of the low-lying states in ${}^{76}\mathrm{Ge}$ have been investigated following a safe-energy Coulomb excitation measurement with the GRETINA tracking array and CHICO2 heavy-ion counter at the ATLAS accelerator facility at Argonne National Laboratory. A comprehensive set of transition and static $E2$ matrix elements were extracted from the measured differential Coulomb cross sections and compared with results of configuration-interaction shell-model calculations and computations carried out within the framework of the generalized triaxial rotor model. The remarkable agreement between the calculated and experimental data supports a near-maximum triaxial deformation for the ground state of ${}^{76}\mathrm{Ge}$. In addition, the degree of softness of the asymmetry in ${}^{76}\mathrm{Ge}$ and ${}^{76}\mathrm{Se}$ was investigated using rotational invariants generated from configuration-interaction shell-model wave functions computed with the jj44b and JUN45 effective interactions. The resulting invariants are shown to be consistent with a stiff triaxial deformation in ${}^{76}\mathrm{Ge}$ and a predominantly soft triaxial potential for ${}^{76}\mathrm{Se}$, in agreement with the conclusions of recent works by this collaboration.

Figure 1

Differences in the time-of-flight between the projectile and target recoils as a function of scattering angle measured with the CHICO2 detector. A clear separation between the ${}^{76}\mathrm{Ge}$ and ${}^{208}\mathrm{Pb}$ ions is observed.

Figure 2

Doppler-corrected $\gamma$-ray spectrum measured with GRETINA following Coulomb excitation of the ${}^{76}\mathrm{Ge}$ beam impinging on the ${}^{208}\mathrm{Pb}$ target. Peaks labeled in blue and marked with an asterisk correspond to transitions whose placement in the level scheme is unknown. The 1410-keV $\gamma$ ray is the sum peak arising from the simultaneous detection of the $2_1^{+}\to 0_1^{+}$, 563- and 847-keV, $4_1^{+}\to 2_1^{+}$ transitions. These three peaks were not included in the ${\chi }^2$ minimization (see text for more details).

Figure 3

Partial level scheme of ${}^{76}\mathrm{Ge}$ with all the transitions observed in the present Coulomb excitation measurement; red-colored transitions are those observed in Ref. [33], the most recent Coulomb excitation measurement prior to the present work. Transitions in black refer to $\gamma$ rays observed for the first time in Coulomb excitation. The subscripts on the spin quantum numbers refer to the sequence in which levels of the same spin and parity are observed in this work.

Figure 4

Spectra after Coulomb excitation of ${}^{76}\mathrm{Ge}$ on ${}^{208}\mathrm{Pb}$ for five subsets of data corresponding to different ranges of particle-scattering angles. The observation at backward angles of many more $\gamma$ rays associated with transitions between higher-spin states can be noticed.

Figure 5

Experimental transition matrix elements for intraband transitions, $\langle I_i\left|\right|M\left(E2\right)\left|\right|I_f\rangle$, within the (a) ground-state and (b) $\gamma$ bands in comparison with theoretical calculations with the generalized triaxial rotor (GTRM) and symmetric rotor models (SYMM). Note that all these states are of positive parity.

Figure 6

Experimental transition matrix elements for transitions linking the ground-state and $\gamma$ bands in comparison with theoretical calculations with the generalized triaxial rotor (GTRM) and symmetric rotor (SYMM) models. The failure of the asymmetric model to reproduce these matrix elements is evident, implying that a departure from axial symmetry is necessary to account for the data on these low-lying transitions.

Figure 7

Comparisons of the static quadrupole moments $\langle I_i\left|\right|M\left(E2\right)\left|\right|I_i\rangle$ in the ground and quasi-$\gamma$ bands with the generalized triaxial rotor (GTRM) and symmetric rotor models (SYMM). Note that all these states are of positive parity.

Figure 8

The relative moments of inertia for all three axes as a function of axial asymmetry, $\gamma$. The experimental values (circles) have been normalized to the irrotational values (lines) through the 1 axis. Reproduced from Ref. [51]. Note: the ${}^{76}\mathrm{Ge}$ values are represented by black squares.

Figure 9

Comparison of the experimental levels energies with theoretical ones computed with the jj44b and JUN45 interactions (see text for details).

Figure 10

The absolute quadrupole and asymmetry deformations mapped in the $(\langle Q^2\rangle ,\delta )$ space for the ${}^{76}\mathrm{Ge}$ ground state. (a) The experimental values are shown alongside quadrupole invariants computed with the configuration-interaction shell model using (b) the jj44b and (c) JUN45 effective interactions. Statistical fluctuations describing the measure of the relative stiffness of the deformations are presented as standard deviations of a normal distribution.

Figure 11

Configuration-interaction shell-model deformation parameters in the ($\langle Q^2\rangle$, $\delta$) deformation plane computed with the jj44b effective interaction for the ground states of ${}^{76}\mathrm{Ge}$ and ${}^{76}\mathrm{Se}$. The black dot represent the root-mean-square $\langle Q^2\rangle$ and $\delta$ value in each nucleus.

Figure 12

Configuration-interaction shell-model deformation parameters in the ($\langle Q^2\rangle$, $\delta$) deformation plane computed with the JUN45 effective interaction for the ground states of ${}^{76}\mathrm{Ge}$ and ${}^{76}\mathrm{Se}$. The black dot represent the root-mean-square $\langle Q^2\rangle$ and $\delta$ value in each nucleus.