**4. The Region of 60Ca**

Many Hamiltonians have been developed for the calcium isotopes for the *f p* model space. Near 42Ca, it is known that Δ = 2 *sd*–*p f* proton excitations are necessary for the lowlying intruder states and their mixing with the *f p* configurations which greatly increase the B(E2) values compared to those obtained in the *f p* model space [95]. In the doubly-magic nucleus 48Ca, the *sd* <sup>−</sup> *p f* intruder states start with the 0<sup>+</sup> state at 4.28 MeV [104]. The <sup>48</sup>−55Ca nuclei exhibit low-lying spectra which are dominated by *f p* configurations [12]. There are weak magic numbers at *N* = 32 and 34 as shown in Figure 2. The reason for the low value of the pairing for the 1*p*1/2 at *N* = 33 was discussed in [104].

The KB3G [105] and GXPF1A [88,89] Hamiltonians have provided predictions for the spectra in this region which have been a source of comparison for many experiments over the last 20 years. Both of these are "universal" Hamiltonians for the *p f* model space. Recently, it has been shown that a data-driven Hamiltonian for the calcium isotopes improves the description of all of the known data [12]; this is called the UFP-CA (universal *f p* for calcium) Hamiltonian. All of the known energy data for *N* ≥ 28 can be described by an SVD-derived Hamiltonian that is close to the starting IMSRG Hamiltonian for 48Ca. UFP-CA is able to describe the energy data for *N* ≥ 28 with an rms error of 120 keV. In particular, the calculated *D*(*N*) values, shown by the red line in Figure 2, agree extremely well with the data (the black points).

The UFP-CA Hamiltonian does not explicitly involve the 2*s*–1*d*–0*g* orbitals, but the influence of these orbitals are present in their contributions to the renormalization into the *f p* model space. This renormalization is contained microscopically in the IMSRG starting point, as well as empirically in the SVD fit.

The success of UFA-CA is similar to the success of the USD-type Hamiltonians in the *sd* model space for all nuclei except those in the island of inversion. If the UFP-CA predictions for <sup>55</sup>−59Ca turn out to be in relatively good agreement with experiment, the implication is that 60Ca will be a doubly-magic nucleus similar to that of 68Ni [12]. If that is the case, 60Ca will be the last doubly-magic nucleus to be discovered. In [12], EDF models are used to estimate the 0 *f*5/2 0*g*9/2 shell gap at *N* = 40 to be approximately 3.0 MeV. The implication of this for *D*(*N*) is shown by the red dashed line in Figure 2. The 0*g*9/2 orbital will first appear as intruder states in the low-lying spectra of <sup>55</sup>−60Ca. These nuclei can be reached by proton knockout on the scandium and titatium isotopes. The proton knockout will be dominated by 0 *f*7/2 removal to the low-lying *f p* neutron configurations. An example of this is the population of the ground state of 54Ca from 55Sc [106]. Protons will also be removed from the 1*s*1/2 and 0*d*3/2 orbitals to populate states at higher energy such as the negative parity state in 54Ca. These will mix with the 2*s*–1*d*–0*g* configurations and neutron decay to the lighter calcium isotopes. For example, in 57,59Ca, a 9/2<sup>+</sup> (0*g*9/2) state just above the neutron separation energy *Sn* value would neutron decay to the (0+, 2+, 4+) multiplet predicted in 56,58Ca; see Figure 1 in [12]. Calculations that include proton excited from *sd* to *p f* and neutrons excited from *p f* to *sdg* will be needed to understand the neutron dacays of these states.

The position of the 0*g*9/2 orbital is crucial for the structure of nuclei around 60Ca [107]. Lenzi et al. [108] have extrapolated the neutron effective single-particle energies from *Z* = 28 down to *Z* = 20 based on their LNPS Hamiltonian. Their 0 *f*5/2-0*g*9/2 ESPE gap for 60Ca is close to zero (see Figure 1 in [108]) and the structure of 60Ca is dominated by Δ = 4 (*f p* to *sdg*) configurations (see Table 1 in [108]). With LNPS, 60Ca is very different from 68Ni which is dominated by the closed *f p*-shell configuration (Δ = 0). Below 68Ni, the nuclei <sup>66</sup>−70Fe, [11] <sup>64</sup>−66Cr [11] and 62Ti [10], have deformed spectra coming from *f p* − *sdf* island of inversion for *N* = 40. Calculations with the LNPS Hamiltonian [108] show that these are all dominated by Δ = 4. The *N* = 40 island of inversion is the topic of another contribution to this series of papers [109].

The existence of 60Ca, confirmed only recently, agrees with UFP-CA as well as with most of the other predictions [36]. It will be exciting to have more complete experimental data for nuclei around 60Ca from FRIB and other radioactive-beam facilities.
