*3.2. A Doubly-Closed Nucleus* <sup>68</sup>*Ni*

The Type-II shell evolution was first discussed in [58] for 68Ni as an example. Figure 10 shows its theoretical and experimental energy levels. The theoretical results were obtained for the A3DA-m interaction by the MCSM [59–62], which is a powerful methodology for the shell model calculation but is not discussed in this article due to the length limitation.

**Figure 10.** Level scheme of 68Ni. Taken from Figure 2 of [58].

Because *Z* = 28 is an SO magic number and *N* = 40 is an HO magic number (see Figure 1), the ground state of 68Ni is primarily a doubly closed shell. Indeed, in the theoretical ground state, the occupation of the neutron *g*9/2 orbit is negligibly small. In contrast, the 0<sup>+</sup> <sup>3</sup> state located at the excitation energy, Ex∼3 MeV, is the band head of a rotational band of an ellipsoidal shape, and its neutron *g*9/2 occupation number is as large as ∼4. The mechanism shown in Figure 5e is then switched on, reducing the proton *f*5/2– *f*7/2 splitting. A reduced splitting facilitates more configuration mixing between these two orbits, which can produce notable effects on the quadrupole deformation as stated below.

#### *3.3. Coexistence between Spherical and Deformed Shapes*

We here quickly overview the quadrupole deformation or the shape deformation from a sphere to an ellipsoid [13]. The quadrupole deformation is driven by the quadrupole interaction, a part of the multipole interaction in Equation (11). The quadrupole interaction is a somewhat vague idea because of a certain mathematical complication, but its main effects can be simulated by the (scalar) coupling of the quadrupole moment operators. If the quadrupole moments are larger, i.e., a stronger quadrupole deformation occurs, the nucleus gains more binding energy from the quadrupole interaction. This is a very general phenomenon, and because of this the ground states of many nuclei are deformed, although 68Ni is not among them.

The energy of 68Ni (intrinsic state) is graphically illustrated in Figure 11 left for various ellipsoidal shapes, *spherical*, *prolate*, *oblate* and in between (called *triaxial*). The energy is calculated by the constraint Hartree–Fock (CHF) calculation with the same shell–model Hamiltonian as in Figure 10. The imposed constraints are given by the quadrupole moments in the intrinsic (body-fixed) frame, represented usually by *Q*<sup>0</sup> and *Q*<sup>2</sup> [13]. This plot is usually called the Potential Energy Surface (PES). The minimum energy occurs at the spherical shape (red sphere), with *Q*<sup>0</sup> = *Q*<sup>2</sup> = 0. The constraints are changed to a more prolate deformed ellipsoid (blue object) along the upper-right axis ("prolate deformation" in the figure), where *Q*<sup>0</sup> increases but *Q*<sup>2</sup> = 0. (Between two axes in Figure 11, *Q*<sup>2</sup> = 0. We come back to this point below.) The energy relative to the minimum energy climbs up by 6 MeV first. This is because protons and neutrons must be excited across the magic gaps from the doubly closed shell in order to create states of deformed shapes (see Figure 1). The energy then starts to come down, as the quadrupole moments increase, thanks to the quadrupole interaction. It is lowered by 3 MeV from the local peak to the local minimum. Beyond the local-minimum area, the effect of the quadrupole interaction is saturated, and it cannot compete with the energy needed for exciting more protons and neutrons across the gaps required by the constraints. This energy variation appears as the basin in the three-dimensional PES. This is the usual explanation of the local deformed minimum. The appearance of two (or more) different shapes with a rather small energy difference is one of the phenomena frequently seen and is called the shape coexistence [63]. The quadrupole interaction is undoubtedly among the essential factors of the shape coexistence. However, this may not be a full story.

**Figure 11. Left**: Potential energy surface (PES) of 68Ni. Taken from Figure 5 of [24]. **Right**: PES of 68Ni with axially symmetric shapes. The solid line shows the PES of the full Hamiltonian, whereas the dashed line is the PES with practically no tensor-force contribution. Taken from Figure 6 of [24].

Figure 11 right exhibits the same energy along the axis lines of Figure 11 left, where *Q*<sup>0</sup> is varied from −400 fm2 to 400 fm2 while *<sup>Q</sup>*<sup>2</sup> = 0 is kept. The positive (intrinsic) quadrupole moments (*Q*<sup>0</sup> > 0) imply prolate shapes (blue object in Figure 11 left), whereas the negative ones imply (*Q*<sup>0</sup> < 0) oblate shapes (green object). The red solid line shows the CHF results of the full Hamiltonian, whereas for the dashed line, the tensor monopole interactions between the neutron *g*9/2 orbit and the proton *f*5/2,7/2 orbits are practically removed. This removal means no effects depicted in Figure 5d,e. The dashed line displays a less-pronounced prolate local minimum at weaker deformation with much higher excitation energy. The significant difference between the solid and dashed lines suggests that the monopole effects are crucial to lower this local minimum and stabilize it. We now discuss the mechanism for this difference. With the tensor monopole interaction, once sufficient neutrons are in *g*9/2, the proton *f*5/2– *f*7/2 splitting is reduced, and this reduced splitting facilitates the mixing between these two orbits driven by the quadrupole interaction. The resulting deformation is stronger compared to no tensor-force case. In parallel to this, the tensor monopole interaction involving the neutron *g*9/2 orbit produces extra binding energy, if more protons are in *f*5/2 and less are in *f*7/2. This extra binding energy lowers the deformed states, otherwise they are high in energy because of the energy cost for promoting neutrons from the *p f* shell to *g*9/2. Thus, a strong interplay emerges between the monopole interaction and the quadrupole interaction, and type-II shell evolution materializes this interplay in the present case. It enhances the deformation and lowers the energy of deformed states. Without this interplay, as indicated by blue dashed line in Figure 11 right, the rotational band corresponding to the local minimum is pushed up by 4 MeV and may be dissolved into the sea of many other states. It is obvious that this interplay mechanism works self-consistently.
