*3.3. One-Proton and Two-Proton Decays in* <sup>16</sup>*Ne and* <sup>18</sup>*Mg Unbound Nuclei*

Two-proton decay is one of the most important drip-line phenomena. It occurs in proton drip line nuclei, such as 48Ni, 54F, 54Zn, 76K, 16Ne, and 19Mg (see a review of this topic in Ref. [4] ). While 18Mg has not been observed, it can decay in principle by proton and/or two-proton emissions. The GSM is then a suitable method to study these types of particle emissions. We carried out GSM calculations of the proton-rich carbon isotones of 14O, which are all resonance [10], using 14O as an inner core. The obtained energy spectra of carbon isotones are depicted in Figure 10 with respect to the ground state of 14O. One can see that both the energies and widths of experimentally known eigenstates are well reproduced for the low-lying states in 15F and 16Ne [10]. We also provide predictions for the 17Na and 18Mg nuclear spectra, of which, there are no experimental data. Our calculations show that the 16Ne and 18Mg isotopes are unbound nuclei, where both oneproton separation energies *Sp* and two-proton separation energies *S*2*<sup>p</sup>* are negative, thereby indicating that two different particle-emission channels are open therein.

**Figure 10.** Excitation energies, *Ex*, and widths (in keV) of the ground and excited states of carbon isotones. The GSM calculations are compared to available experimental data [10,100–102]. Energies are given with respect to the 14O core. Widths are represented by green striped squares, and their explicit values are written above (with permissions from Ref. [48]).

To evaluate one-proton and two-proton decay widths, we changed the central potential depth *V*<sup>0</sup> of the WS core potential in order for the *Sp* to become positive or very negative (see details in Ref. [48]). Consequently, it is possible to find a central potential depth for which only the two-proton decay channel is open, so that the obtained width is that of the two-proton emission. The obtained results are shown in Figure 11. As 15F and 17Ne are one-proton resonances, their width increases steadily with the Hamiltonian central potential depth. In contrast, one can see that the widths of 16Ne and 18Mg increase abruptly when the one-proton channel opens. The width of two-proton decay is almost constant with respect to the central potential depth below the one-proton emission threshold, and is also about 500 keV to 1 MeV above (see Figure 11). It is reasonable to assume that the two-proton decay width is almost independent of energy. Therefore, the GSM results shown in Figure 11, where only the two-proton channel is open, can be extrapolated to the physical case (indicated by an arrow in Figure 11). This two-proton decay width is about 10-15 keV for both 16Ne and 18Mg nuclei. The one-proton width can be assumed as the difference between the total width and the two-proton emission width of 10–15 keV. Then, our calculations show that one-proton emission is negligible for 16Ne, whereas the one-proton decay width in 18Mg is estimated to be about 85-90 keV. The obtained data for 16Ne are also in agreement with experimental data [10,100–102].

**Figure 11.** Calculated energies and widths (in MeV) of 15F, 16Ne (**upper panel**), 17Na, and 18Mg (**lower panel**) as a function of the difference <sup>Δ</sup>*V*<sup>0</sup> <sup>=</sup> *<sup>V</sup>*<sup>0</sup> <sup>−</sup> *<sup>V</sup>*(fit) <sup>0</sup> (fit) of the WS central potential depths (see details in Ref. [48]). Energies are depicted by blue disks and red lozenges for even and odd nuclei, respectively. Widths are represented by segments centered on disks and lozenges. The widths of 16Ne and 18Mg have been multiplied by 20 for readability. Energies are given with respect to the 14O core. The physical GSM calculation, for which *<sup>V</sup>*<sup>0</sup> = *<sup>V</sup>*(fit) <sup>0</sup> , is indicated by an arrow (with permissions from Ref. [48]).

#### **4. Summary**

The Gamow shell model (GSM) is a powerful method for the description of the weakly bound and resonance properties of drip line nuclei. In the present review, we presented several recent applications of GSM dedicated to the study of drip line nuclei, including GSM calculations of neutron-rich oxygen and fluorine drip line nuclei, of the long chain of neutron-rich calcium isotopes, and of the unbound proton-rich 16Ne and 18Mg isotopes. For the neutron-rich oxygen and fluorine drip line nuclei, both the realistic GSM and GSM with phenomenological forces have been utilized. Our calculations have described the weakly-bound and unbound properties of drip line nuclei well. Furthermore, the unbound properties of the 28O are obtained within the two both types of GSM calculations, and the two-neutron halo property of 31F has been predicted in GSM calculations as well. The realistic GSM calculations provide good agreements of the neutron-rich calcium isotopes with experimental data, as GSM calculations predict that the one- and two-neutron drip line nuclei of calcium isotopes are 57Ca and 70Ca, respectively. For the unbound proton-rich 16Ne and 18Mg nuclei, GSM calculations provide calculations and predictions for their lowlying spectra. Added to that, the one- and two-proton emission widths could be estimated for 16Ne and 18Mg isotopes. Our calculations have shown that 16Ne decays only by twoproton emission, whereas 18Mg can decay through both one- and two-proton emission channels, whose widths are estimated to be about 85–90 and 10–15 keV, respectively.

GSM has thus been shown to be the tool of choice for the study of drip line nuclei. Many challenges remain to be overcome for the future applications of GSM:


**Author Contributions:** Writing—original draft preparation, J.L.; writing—review and editing, F.X. and N.M.; critically review, Y.M., B.H., Z.S. and W.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work has been funded by the National Key R&D Program of China under Grant No. 2018YFA0404401; the National Natural Science Foundation of China under Grants No. 11835001, NO. 11921006, NO. 12035001, and NO. 11975282; the State Key Laboratory of Nuclear Physics and Technology, Peking University under Grant NPT2020KFY13; the China Postdoctoral Science Foundation under Grant No. BX20200136 and 2020M682747; the Strategic Priority Research Program of Chinese Academy of Sciences under Grant No. XDB34000000; the Key Research Program of the Chinese Academy of Sciences under Grant No. XDPB15; and the CUSTIPEN (China-U.S. Theory Institute for Physics with Exotic Nuclei) funded by the U.S. Department of Energy, Office of Science

under Grant No. de-sc0009971. The High-Performance Computing Platform of Peking University is acknowledged.

**Data Availability Statement:** All of the relevant data are available from the corresponding author upon reasonable request.

**Conflicts of Interest:** The authors declare no conflict of interest.
