*3.2. Realistic Gamow Shell Model Calculations of Neutron-Rich Calcium Isotopes*

The long chain of calcium isotopes provides an ideal laboratory for both theoretical and experimental investigations of unstable isotopes. With two typical doubly magic isotopes, 40Ca and 48Ca, and two new magic isotopes discovered in the neutron-rich region, 52Ca [90] and 54Ca [91], the calcium chain is speculated to end the 70Ca isotope. Its rich nuclear structure data [10] attract continued theoretical interest, especially using methods that include continuum coupling. The realistic GSM based on the realistic CD-Bonn [55] interaction has also been performed to investigate the properties of neutron-rich calcium isotopes up to the drip line.

**Figure 8.** Calculated one-neutron separation energies *Sn* (**a**) and two-neutron separation energies *S*2*<sup>n</sup>* (**b**), compared with data [78,92], and calculations obtained with SV-min density-functional theory (DFT) [93] and multireference IM-SRG (*S*2*<sup>n</sup>* only) [94]. The *Sn* calculations end at 60Ca because odd isotopes heavier than 60Ca become unbound in our GSM calculations (with permissions from Ref. [47]).

The calculated one-neutron separation energies *Sn* and two-neutron separation energies *S*2*<sup>n</sup>* are shown in Figure 8 and compared with experimental data [78,92], DFT [93], and IM-SRG [94] calculations. The calculated one-neutron separation energies *Sn* show that 57Ca is the heaviest odd-mass bound calcium isotope, which is consistent with MBPT calculations [95]. 59Ca is weakly unbound with a small one-neutron separation energy *Sn* = −326 keV in our GSM calculations. For the two-neutron separation energy *S*2*n*, the GSM calculations are performed with two different cores, 48Ca and 54Ca. For 56,58,60Ca, the two calculations give similar results. The calculated two-neutron separation energy *S*2*<sup>n</sup>* is in good agreement with experimental data [78,92] and other theoretical calculations, e.g., with DFT [93] and IM-SRG [94] calculations. The large decrease in *S*2*<sup>n</sup>* at neutron number *N* = 32 and 34 indicate that subshell closures occur therein, which has also been suggested from experiments [90,91] and theoretical calculations [94–96]. Moreover, the calculated two-neutron separation energy *S*2*<sup>n</sup>* using GSM predicts that the two-neutron drip line of the calcium isotopes should be located at 70Ca. This is consistent with the recent mean-field calculations of Ref. [97].

**Figure 9.** Neutron effective single-particle energies (ESPE) with respect to the 48Ca core, as a function of neutron number. The Vlow−*<sup>k</sup>* <sup>Λ</sup> <sup>=</sup> 2.6 fm−<sup>1</sup> CD-Bonn interaction is utilized (with permissions from Ref. [47]).

In order to see the shell evolution of the calcium isotopes around the neutron number *N* = 32, 34, 40, and 50, we calculated effective single-particle energies (ESPE) based on the GSM effective Hamiltonian. Figure 9 shows the evolution of the valence neutron ESPEs when increasing the neutron number. The calculations show that large shell gaps between 1*p*3/2 and 1*p*1/2 and between 1*p*1/2 and 0 *f*5/2 exist, indicating that shell closures occur at *N* = 32 and 34, respectively. These results are consistent with experimental observations [90,91] and theoretical calculations [94–96,98]. The shell gap above the 0 *f*5/2 orbit is reduced at around *N* = 40, implying a weakening of the *N* = 40 shell closure in the calcium chain. The 0*g*9/2 shell becomes bound at *N* ≥ 40, which can enhance the stability of the heavy calcium isotopes. The observed 60Ca isotope in experiments [99] may be an indication of this enhanced stability. Moreover, the calculated ESPEs show a significant shell gap at *N* = 50, implying a shell closure at 70Ca.
