4.1.2. *N* = Z Chain from 100Sn to 80Zr

The region of the heaviest and bound *N* = *Z* nuclei is below 100Sn. This fact makes it particularly attractive for studies of proton-neutron correlations and their dependance on the spin of the nucleus. Already decades ago theoretical calculations predicted the existence of *I* <sup>π</sup> = 16+ and *I* <sup>π</sup> = 25/2+ high-spin isomers in 96,97Cd, respectively. Only much later, technical advances allowed the confirmation of these isomers experimentally [22,106].

Early shell-model studies employing empirical interactions in the πν(*p*1/2, *g*9/2) model space were reviewed in Ref. [1]. Realistic interactions with LSSM calculations were presented for the full πν(*f* 5/2,*p*3/2,*p*1/2,*g*9/2) model space [78,79,107–109] as well as for the upper πν(*gds*) shell using the SDG interaction [110]. The strength of the πν interaction in the πν*g*9/2 orbits manifests itself best in the strongly-binding *T* = 0, *I* <sup>π</sup> = 9<sup>+</sup> TBME, which is comparable in strength with the "normal", *T* = 1 pairing mode [85,86]. With the identification of excited states in 92Pd [23], the role of the πν*g*9/2 pairs with maximum spin of *I* <sup>π</sup> = 9+

in the *N* = Z nuclei 96Cd, 94Ag and 92Pd has been investigated in a series of multi-step shell model and IBM studies with respect to the "fully aligned" 9+ -TBME [24,111–114].

The experimental yrast states for the even–even *N* = Z nuclei in the *g*9/2 shell are shown with colored symbols in Figure 4. For the lower mass *N* = Z even–even nuclei the level structure implies an onset of deformation. The spectrum of 88Ru [42] exhibits a rotational band similar to the known states of 84Mo [115]. This could intuitively understood, similarly to the *N* = 50 chain, as even stronger influence by the lower shell *fp*-shell therefore driving deformation. However, recent developments in the calculation power could disprove this hypothesis, as shown below.

**Figure 4.** Excitation energies of the even–even *N=Z* nuclei in the *g*9/2 shell with *A* (atomic mass number) = 84–96 for spins *I* <sup>π</sup> = 2+–16+. The spins of excited states in all isotopes are assigned tentatively [101]. Experimental energies are given with colored symbols distinguished by their spin values. Shell-model calculations are shown with solid lines with different colors for each spin for the GF [85] calculation, dashed line for the JUN45 calculation [78,107] and present work for the higher-spin states and colored and short dashed-dotted colored lines for SDG [110] calculation for 96Cd.

Various shell-model approaches in different model spaces have been reported for this region. The calculation presented with solid colored lines is GF [85], which includes all yrast spins states up to the 16+ for all the *N* = Z nuclei in g9/2 shell nuclei and in particular an isomeric trap, known to exist experimentally in 96Cd [110]. Calculation using JUN45 interactions [78] are shown with dashed lines whenever possible for the computer power available within this work or in the literature [78,107].

The LSSM SDG [22] approach with up to 5*p*5*h* excitations (t = 5) across the Z, *N* = 50 closed shell are available only for 96Cd up to high spins and is shown with short, colored dashed-dotted lines. A consistent description for all the calculations is obtained in the upper g9/2 shell, as shown for 96Cd. However, towards the mid shell, the *N* = Z results are hampered by severe truncation. The advantage of including a larger model space, particularly the lower-lying orbitals, is evident. Nevertheless, with increasing spin the lower-shell nuclei follow very different trend than the predicted one.

According to the GF calculation presented in Figure 4, the spin trap at *I* <sup>π</sup> = 16+ in 84Mo will stay yrast even if the spherical 0<sup>+</sup> state in the small model space of GF calculation is shifted by ~2 MeV up in energy with respect to the known deformed ground state and the rotational band member energies extrapolated with a constant moment of inertia to higher spins [115].

However, a common conclusion of various shell model approaches with Z ≥ 40, *N* = Z nuclei is the quest to include excitations across the Z, *N* = 50 shell closure [79,116–119] in order to attempt the description of the deformation. 84Mo marks the transitional region where the shape-driving role of the *r*3*g* space is replaced/enhanced by the *gds* space [79]. The spectra of 96Cd is a benchmark for both model spaces and yield similar results. In addition, the 84Mo spectrum was calculated with the help of the nucleon-pair approximation method [120], which also could be expanded to higher spins and other *N* = *Z* nuclei in the *g*9/2 shell in the future.

Recently, a new approach was proposed in the mean field [95], where within a simple SO(8) pairing model, it was shown that the symmetry-projected condensates of mixed isovector and isoscalar pairs very accurately describe properties of the exact solutions, including the coexistence of the isovector and isoscalar pairing. Lack of symmetry restoration thus explains the limited success in describing such a coexistence in the standard mean-field approaches to date. It was concluded that the future work investigating properties of the proton-neutron nuclear pairing can be carried out within the variation-after-projection approach to mean-field pairing methods.
