The atomic nucleus is a self-organized quantum manybody system comprising specific numbers of protons Z and neutrons N. The nuclear force, which binds protons and neutrons together, favors equal numbers for each particle. In contrast, the long-range Coulomb repulsion discourages the accumulation of protons in the nucleus. The competition between these two forces determines the valley of stability. However, the strong internucleon interaction binds nuclei even when many more neutrons than protons are added. Extreme N/Z asymmetry has been observed in light-neutronrich nuclei. The particle-stable nuclei are classified as bound nuclei, which indicates that the protons and neutrons of the isotopes can be bound by nuclear forces unless radioactive decay (β decay) transforms them into other nuclei. The existence limit of the bound isotopes for a given proton number is known as the neutron or proton dripline at the neutronrich or proton-rich edge, respectively. Beyond the driplines, nuclei may momentarily exist as resonances before emitting neutrons or protons, with very short lifetimes.

As extreme examples of particle-unstable exotic nuclei, unbound ^27,28O were recently produced by Kondo et al. using a proton-induced nucleon knockout with a highenergy ²⁹F beam at RIKEN [1]. The heaviest oxygen isotopes, ^27,28O, occur beyond the neutron dripline, ²⁴O, and appear in resonances. They quickly decay into ²⁴O through ²⁶O via sequential neutron emissions. As illustrated in Fig. 1, the ²⁷O and the ²⁸O undergo a 1n-2n and a 2n-2n emission process, respectively. Measurements of the emitted neutrons yielded ²⁸O at a ground-state energy of approximately ${0.46}_{-0.04}^{+0.05}$ (stat) ± 0.02 (syst) MeV above that of ²⁴O with an upper limit on the resonance width of 0.7 MeV [1]. Likewise, ²⁷O was obtained at an energy of approximately 1.09 ± 0.04(stat) ± 0.02(syst) MeV above that of ²⁴O with an upper limit on the width of 0.18 MeV, see Fig. 2.

Fig. 1Full-screen View

(Color online) Schematic of the decay processes of the particle-unstable isotopes ²⁵⁻²⁸O

Fig. 2Full-screen View

(Color online) Experimental [1] and calculated ²⁵⁻²⁸O groundstate (g.s.) energies with respect to ²⁴O. Our calculations are labeled by GSM EFT(LO) [8] and GSM NN+3N [9]. Other calculations marked by EEdf3, CC, USDB, CSM, GSM, SDPF-M, VS-IMSRG, SCGF NN+3N(lnl), SCFG NNLO_sat, and Λ-CCSD(T) are explained in the original paper [1]

The heaviest ²⁸O, which contains eight protons and 20 neutrons, is of special interest as it touches the ‘expected’ N = 20 magic shell. However, the experiment [1] supports an unclosed N = 20 shell in the oxygen chain, which is consistent with previous predictions [2–7]. Further experiments on a possible 2⁺ excited state may be required to conclude the unclosed shell at N = 20. In the experimental study [1], the data of the ^27,28O ground-state energies were compared with different calculations or predictions, showing that none of the calculations reproduced the data well. In addition to the theoretical calculations mentioned in the Nature publication [1], Gamow shell model (GSM) calculations performed by a theoretical group at Peking University (PKU) [8, 9] show excellent predictions [8, 9] compared with the data reported in Ref. [1], as shown in Fig. 2. Unfortunately, the PKU predictions [8, 9] are missing in Ref. [1].

In the published calculations [8, 9], we used the advanced GSM, which takes into account coupling to the continuum. Continuum coupling is important for weakly bound and unbound nuclei as open quantum systems, whereas the continuum effect is absent in the models mentioned in Ref. [1] except for the continuum shell model [10] and GSM with phenomenological interactions [11]. In our calculations [8], a leading-order pionless effective field theory force denoted as EFT(LO) was used with only one parameter that was fitted by the binding energies of ²⁴⁻²⁶O. Various regulators for the interaction were used, showing that regulator cutoffs larger than a certain value resulted in stable calculations, as shown in Fig. 2, where the pink band indicates the range of calculated energies with cutoffs of Λ = 356, 390, and 436 MeV [8]. In another study [9], we performed ab initio GSM calculations based on chiral two- and three-nucleon interactions, which showed that both continuum coupling and three-nucleon forces were important for weakly bound and unbound nuclei. These two studies [8, 9] yielded similar results, as shown in Fig. 2, grouped by PKU labels. The predicted energies [8, 9] of ^27,28O were consistent with the experimental data [1] and clearly better than the other calculations reported in Ref. [1]. On the other hand, both of our studies [8, 9] showed that the inclusion of pf shell is necessary in the calculations for sd-shell weakly bound and unbound nuclei. Unbound ^27,28O have significant pf components that may be associated with the disappearance or weakening of the N = 20 shell closure, as claimed in Ref. [1]. Based on our calculated energies and one-body densities, we predicted [8] that ²⁸O should undergo four-neutron decay, more specifically a 2n-2n emission process through ²⁶O, which is consistent with experiments [1]. The observation of ²⁸O has opened up a new window to probe into the elaborate internucleon interplay under extreme conditions.