*3.2. Shape Coexistence, Triaxiality, and the N = 50 Shell Closure in Germanium and Zinc Isotopes*

Detailed low-energy Coulomb-excitation studies were performed to investigate quadrupole properties of stable and exotic Ge and Zn isotopes, which are important in the context of the numerous Shell-Model calculations developed for this region. While extensive sets of electromagnetic matrix elements were extracted for the stable nuclei and interpreted within the quadrupole sum rules approach, in neutron-rich isotopes these measurements provided the first access to *B*(*E*2) values and, in some cases, also excitation energies.

In the stable Ge isotopes, the *Q*2 invariants extracted for the ground state and the 0<sup>+</sup> 2 state via low-energy Coulomb excitation represent one of the strongest signatures of shape coexistence [21,22]. As shown in Figure 3a, the ground-state *Q*2 values in 70–76Ge are similar, 0.2–0.3 *e*2b2, while those of the 0<sup>+</sup> <sup>2</sup> states evolve as a function of the neutron number. The 0<sup>+</sup> <sup>2</sup> state in 70Ge is more deformed than the ground state [23], in 72Ge both states seem to have comparable overall deformations and considerable triaxiality [24], while those for the 0<sup>+</sup> <sup>2</sup> states in 74,76Ge point to nearly spherical shapes [25,26]. Based on the similarity of the 0<sup>+</sup> <sup>2</sup> energy systematics in Ge and Zn nuclei (see Figure 3b), one could speculate that shape coexistence is present also in the latter isotopic chain.

The first hints of the intruder character of the 0<sup>+</sup> <sup>2</sup> states in the Zn isotopes came from *E*0 measurements in the stable even–even 64–68Zn isotopes [27], a feature further supported by the results of multi-step Coulomb-excitation experiments on 66,68Zn [14,28]. However, only for 68Zn has the key 2<sup>+</sup> <sup>3</sup> *E*20<sup>+</sup> <sup>2</sup> matrix element been determined, which, when combined with other matrix elements involving the 0<sup>+</sup> <sup>2</sup> state, leads to a *Q*2 invariant significantly different from that of the ground state [28]. On the other hand, multiple lowenergy Coulomb-excitation studies of stable Ge and Zn isotopes [14,25,28] demonstrated the importance of the triaxial degree of freedom in their structure, which was also evoked for the neighbouring 76,78Se nuclei [29,30]. Particularly relevant is the study of 76Ge [31], which yielded (*β*2, *γ*) parameters for the 0<sup>+</sup> <sup>1</sup> , 2<sup>+</sup> <sup>1</sup> and 2<sup>+</sup> <sup>2</sup> states and their dispersions, which are consistent with rigid triaxial deformation. This is particularly important considering that 76Ge is a candidate for searches of neutrinoless double-*β* decay, and the nuclear shape is predicted to play a significant role in this process [32,33].

**Figure 3.** (**a**) *Q*2 and cos <sup>3</sup>*δ* quantities for the 0<sup>+</sup> g.s. and 0<sup>+</sup> <sup>2</sup> states in Ge isotopes. Data are taken from [21,23–26]. (**b**) Systematics of excitation energies for the 0<sup>+</sup> <sup>2</sup> states in Ge, Zn and Ni isotopes from neutron number *N* = 32 to *N* = 46. Tentative spin assignments are shown by open symbols. Data are taken from the ENSDF database [34–37]. See text for details.

Shell-model calculations focusing on the Ge, Zn, and Se isotopes well reproduced the features related to their triaxial shapes [14,29,38,39], even though the degree of *γ* softness and the presence of static triaxial deformation are still debated [38]. The V-shaped pattern of the 0<sup>+</sup> <sup>2</sup> excitation energies in the Ge isotopes between the neutron numbers *N* = 36 and *N* = 44 (see Figure 3b) was related to shape coexistence by Shell-Model calculations [39] using the JUN45 effective interaction in a model space consisting of the 56Ni inert core and up to the 1*g*9/2 orbital for both neutrons and protons. The collectivity of the deformed ground states was linked to strong correlations (arising from pairing and the quadrupole –quadrupole force), which offset the *N* = 40 gap and lead to the enhanced occupation of the 1*g*9/2 neutron orbital that has a maximum predicted for *N* = 40. In contrast, the role of neutron excitations from the *p f* shell into the 1*g*9/2 orbital is smaller for the 0<sup>+</sup> <sup>2</sup> states, with, on average, two additional neutrons promoted through the *N* = 40 gap with respect to the normal-order configuration. In particular, the wave function of the 0<sup>+</sup> <sup>2</sup> state in 72Ge is dominated by the normal-order configuration, i.e., neutrons completely filling the *p f* shell, with a contribution of 37%, which suggests a nearly spherical shape.

As shown in Figure 3b, a decrease of the 0<sup>+</sup> <sup>2</sup> state energy between *N* = 36 and *N* = 40, similar to those observed in the Ge and Zn chains, is evident also in the Ni isotopes. According to Monte-Carlo Shell-Model (MCSM) calculations with the A3DA effective interaction in the *pfg*9/2*d*5/2 model space [40], the 0<sup>+</sup> <sup>2</sup> states in 64,66,68Ni are oblate deformed and result from neutron 2*p*–2*h* excitation across the *N* = 40 gap, similar to their counterparts in the Ge isotones. The V-shaped trend of the 0<sup>+</sup> <sup>2</sup> excitation energies with the vertex at *N* = 40 does not persist for 70Ni and beyond, as different configurations start to appear at low excitation energy. Specifically, proton 2*p*–2*h* excitations across the energy gap at *Z* = 28 are suggested [40,41] to dominate the structure of the 0<sup>+</sup> 4 state in 64,66Ni, the 0<sup>+</sup> <sup>3</sup> state in 68Ni and the 0<sup>+</sup> <sup>2</sup> state in 70Ni. MCSM calculations predict that these predominantly *π*(2*p*–2*h*) states have well-deformed prolate shapes, resulting from an interplay of type-I and type-II shell evolution. The experimental verification of this multiple shape-coexistence scenario through the quadrupole sum rules approach represents a challenge for future low-energy Coulomb-excitation studies. Unfortunately, the population of excited 0<sup>+</sup> states in both stable and radioactive Ni nuclei will be severely limited due to the high excitation energies involved, which is further complicated by the prohibitively low intensities of radioactive Ni beams that are currently available at energies suitable for low-energy Coulomb excitation.

On the neutron-rich side, low-energy Coulomb excitation has provided valuable structure information in the Ge and Zn isotopes. Experiments at ISOLDE identified the first excited 2<sup>+</sup> <sup>1</sup> state in 78,80Zn and yielded the *<sup>B</sup>*(*E*2; 2<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> g.s) values in 74–80Zn and the *B*(*E*2; 4<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>2</sup><sup>+</sup> <sup>1</sup> ) values in 74,76Zn [42,43]. The obtained *<sup>B</sup>*(*E*2) values hint at the importance of triaxiality also in neutron-rich Zn isotopes, whose ground states were suggested to be rather diffuse in the *γ* degree of freedom [42]. Furthermore, the energy of the first excited state in 80Zn confirms the persistence of the *N* = 50 shell closure two protons away from the doubly-magic 78Ni. The same conclusion was reached for the neutron-rich Ge isotopes from the *B*(*E*2; 2<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> g.s) values of the radioactive 78,80Ge measured using low-energy Coulomb excitation at ORNL [44].

The measured *B*(*E*2; 2<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> <sup>1</sup> ) values in 74–80Zn were found in good agreement with those deduced from the experimental 2<sup>+</sup> <sup>1</sup> excitation energies via the Grodzins rule [45], provided that a renormalization factor (0.92) was applied to the calculated values [42]. The experimental results for 74–80Zn and 78,80Ge were compared with Shell-Model calculations comprising the 2*p*3/2, 1 *f*5/2, 2*p*1/2, and 1*g*9/2 orbitals for both protons and neutrons outside of an inert 56Ni core. Effective charges significantly different from the standard *e<sup>ν</sup>* = 0.5*e*, *e<sup>π</sup>* = 1.5*e* values were adopted to compensate for the enhanced 56Ni core polarization reported in [46,47]. The persistence of the *N* = 50 shell closure in neutron-rich Zn and Ge isotopes, emerging from the experimental and calculated *B*(*E*2) values and excitation energies, anticipated the more recent results for 78Ni , in which the first excited 2<sup>+</sup> <sup>1</sup> state was ultimately identified [48].
