Abstract
Subjecting a physical system to extreme conditions is one of the means often used to obtain a better understanding and deeper insight into its organization and structure. In the case of the atomic nucleus, one such approach is to investigate isotopes that have very different neutrontoproton (N/Z) ratios than in stable nuclei. Light, neutronrich isotopes exhibit the most asymmetric N/Z ratios and those lying beyond the limits of binding, which undergo spontaneous neutron emission and exist only as very shortlived resonances (about 10^{−21} s), provide the most stringent tests of modern nuclearstructure theories. Here we report on the first observation of ^{28}O and ^{27}O through their decay into ^{24}O and four and three neutrons, respectively. The ^{28}O nucleus is of particular interest as, with the Z = 8 and N = 20 magic numbers^{1,2}, it is expected in the standard shellmodel picture of nuclear structure to be one of a relatively small number of socalled ‘doubly magic’ nuclei. Both ^{27}O and ^{28}O were found to exist as narrow, lowlying resonances and their decay energies are compared here to the results of sophisticated theoretical modelling, including a largescale shellmodel calculation and a newly developed statistical approach. In both cases, the underlying nuclear interactions were derived from effective field theories of quantum chromodynamics. Finally, it is shown that the crosssection for the production of ^{28}O from a ^{29}F beam is consistent with it not exhibiting a closed N = 20 shell structure.
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Main
One of the most active areas of presentday nuclear physics is the investigation of rare isotopes with large N/Z imbalances. The structure of such nuclei provides for strong tests of our theories, including—most recently—sophisticated ab initiotype approaches whereby the underlying interactions between the constituent nucleons are constructed from firstprinciples approaches (see, for example, ref. ^{3}).
Owing to the strong nuclear force, nuclei remain bound to the addition of many more neutrons than protons and the most extreme N/Z asymmetries are found for light, neutronrich nuclei (Fig. 1a). Here, beyond the limits of nuclear binding—the socalled neutron drip line—nuclei can exist as veryshortlived (about 10^{−21} s) resonances, which decay by spontaneous neutron emission, with their energies and lifetimes dependent on the underlying structure of the system. Experimentally, such nuclei can only be reached for the lightest systems (Fig. 1a), in which the location of the neutron drip line has been established up to neon (Z = 10)^{4} and the heaviest neutron unbound nucleus observed for fluorine (Z = 9) ^{28}F (ref. ^{5}). Arguably the most extreme system, if confirmed to exist as a resonance, would be the tetraneutron, for which a narrow nearthreshold continuum structure has been found in a recent missingmass measurement^{6}. Here we report on the direct observation of ^{28}O (N/Z = 2.5), which is unbound to fourneutron decay, and of neighbouring ^{27}O (threeneutron unbound).
The nucleus ^{28}O has long been of interest^{7,8} as, in the standard shellmodel picture of nuclear structure, it is expected to be ‘doubly magic’. Indeed, it is very well established that for stable and nearstable nuclei, the proton and neutron numbers 2, 8, 20, 28, 50, 82 and 126 correspond to spherical closed shells^{1,2}. Such nuclei represent a cornerstone in our understanding of the structure of the manybody nuclear system. In particular, as substantial energy is required to excite them owing to the large shell gaps, they can be considered, when modelling nuclei in their mass region, as an ‘inert’ core with no internal degrees of freedom. Such an approach has historically enabled more tractable calculations to be made than attempting to model an Abody (A = Z + N) nucleus from the full ensemble of nucleons. Indeed, this approach has been a fundamental premise of the shellmodel methods that have enabled an extremely wide variety of structural properties of nuclei to be described with good accuracy over several decades (see, for example, ref. ^{9}).
Of the very limited number of nuclei that are expected to be doubly magic based on the classical shell closures, ^{28}O is, given its extreme N/Z asymmetry, the only one that is—in principle—experimentally accessible that has yet to be observed. In recent years, the doubly magic character of the two other such neutronrich nuclei, ^{78}Ni (Z = 28, N = 50; N/Z = 1.8)^{10} and ^{132}Sn (Z = 50, N = 82; N/Z = 1.6)^{11}, has been confirmed. The remaining candidate, twoneutron unbound nucleus ^{10}He (Z = 2, N = 8; N/Z = 4), has been observed as a welldefined resonance but its magicity or otherwise has yet to be established (ref. ^{12}, and references therein).
The N = 20 shell closure has long been known, however, to disappear in the neutronrich Ne, Na and Mg (Z = 10–12) isotopes (see, for example, refs. ^{13,14}). This region is referred to as the ‘Island of Inversion’ (IoI)^{15}, whereby the energy gap between the neutron sdshell and pfshell orbitals, rather than being well pronounced (Fig. 1b), is weakened or even vanishes, and configurations with neutrons excited into the pfshell orbitals dominate the ground state (gs) of these nuclei, as shown schematically in Fig. 1c. The IoI nuclei with such configurations are well deformed, rather than spherical, and exhibit lowlying excited states. Very recently, the IoI has been shown to extend to the fluorine isotopes ^{28,29}F (N = 19, 20)^{5,16,17,18} that neighbour ^{28}O. On the other hand, the last particlebound oxygen isotope, ^{24}O, has been found to be doubly magic, with a new closed shell forming at N = 16 (refs. ^{19,20,21,22,23}). As such, the structural character of the more neutronrich oxygen isotopes and, in particular, ^{28}O is an intriguing question. So far, however, only ^{25,26}O (N = 17, 18) have been observed, as singleneutron and twoneutron unbound systems, respectively^{24,25,26,27}, with the latter existing as an extremely narrow, barely unbound resonance.
This investigation focused on the search for ^{27,28}O, produced in highenergy reactions, through the direct detection of their decay products—^{24}O and three or four neutrons. Critical to the success of this work was the capability of the RIKEN RI Beam Factory to produce intense neutrondripline beams coupled to a thick, active liquidhydrogen target system and a highperformance multineutron detection array.
Experiment
The neutronunbound ^{27,28}O were produced through protoninduced nucleon knockout reactions from a 235 MeV per nucleon beam of ^{29}F. As depicted in Extended Data Fig. 1, the ^{29}F ions were characterized and tracked onto a thick (151 mm) liquidhydrogen reaction target using plastic scintillators and multiwire drift chambers. The hydrogen target was surrounded by the MINOS Time Projection Chamber^{28}, which allowed for the determination of the reaction vertex. This combination provided for both the maximum possible luminosity together with the ability to maintain a good ^{27,28}O decayenergy resolution.
The forwardfocused beamvelocity reaction products—charged fragments and fast neutrons—were detected and their momenta determined using the SAMURAI spectrometer^{29}, including the three largearea segmented plastic scintillator walls of the NeuLAND^{30} and NEBULA arrays. An overall detection efficiency for the threeneutron and fourneutron decay of around 2% and 0.4%, respectively, was achieved for decay energies of 0.5 MeV (Extended Data Fig. 1). The decay energies were reconstructed from the measured momenta using the invariantmass technique with a resolution (full width at half maximum, FWHM) of around 0.2 MeV at 0.5 MeV decay energy (see Methods).
Analysis and results
The ^{24}O fragments were identified by the magnetic rigidity, energy loss and time of flight derived from the SAMURAI spectrometer detectors. The neutrons incident on the NeuLAND and NEBULA arrays were identified on the basis of the time of flight and energy deposited in the plastic scintillators. Notably, the multineutron detection required the application of dedicated offline analysis procedures to reject crosstalk (see Methods), that is, events in which a neutron is scattered between and registered in two or more scintillators.
In the analysis, the decay neutrons were denoted n_{1}, n_{2},… by ascending order of the twobody relative energy E_{0i} between ^{24}O and n_{i}, such that E_{01} < E_{02} < E_{03} < E_{04} (Fig. 2d). The ^{28}O decay energy, E_{01234}, reconstructed from the measured momentum vectors of the five decay particles, is shown in Fig. 2a. A narrow peak is clearly observed at about 0.5 MeV and may be assigned to be the ^{28}O ground state. As a small fraction of crosstalk events could not be eliminated by the rejection procedures, care must be taken to understand their contribution to the E_{01234} spectrum. In particular, ^{24}O+3n events, in which one of the neutrons creates crosstalk and is not identified as such in the analysis, can mimic true ^{28}O decay. In this context, to provide a complete and consistent description of all the ^{24}O+xn decayenergy spectra, a full Monte Carlo simulation was constructed (see Methods). As shown in Fig. 2a, the contribution from the residual crosstalk events is found to be rather limited in magnitude in the ^{24}O+4n decayenergy spectrum and, moreover, produces a very broad distribution.
The decay of ^{28}O was investigated by examining the correlations in the ^{24}O plus neutrons subsystems (see Methods). In particular, the threebody (^{24}O+n_{1}+n_{2}) partial decay energy E_{012} (Extended Data Fig. 2a) was reconstructed from the ^{24}O+4n dataset. The corresponding spectrum exhibits a sharp threshold peak arising from ^{26}O_{gs}, which is known to have a decay energy of only 18(5) keV (ref. ^{27}). This observation clearly indicates that ^{28}O sequentially decays through ^{26}O_{gs} as shown by the arrows A and B in Fig. 2e.
We have also observed, in the ^{24}O+3n channel, a ^{27}O resonance for the first time, as may be seen in the fourbody decayenergy (E_{0123}) spectrum of Fig. 2b. As confirmed by the simulations, which are able to simultaneously describe the ^{24}O+3n and 4n decayenergy spectra, the wellpopulated peaklike structure below about 0.5 MeV corresponds to ^{28}O events in which only three of the four emitted neutrons are detected. The peak at E_{0123} ≈ 1 MeV, however, cannot be generated by such events and must arise from a ^{27}O resonance. This was confirmed by the analysis of the data acquired with a ^{29}Ne beam (see Methods and Extended Data Fig. 2e), in which ^{27}O can be produced by twoproton removal but not ^{28}O, as this requires the addition of a neutron. The ^{27}O resonance also decays sequentially through ^{26}O_{gs}, as shown by the arrows B and C in Fig. 2e from the analysis of the partial decay energies (Extended Data Fig. 2c,d).
The decay energies of the ^{27,28}O resonances were derived from a fit of the E_{0123} spectrum with the condition that the partial decay energy satisfies E_{012} < 0.08 MeV (Fig. 2c), that is, decay through the ^{26}O ground state was selected so as to minimize the uncertainties owing to contributions from higherlying ^{28}O resonances that were not identified in the present measurements owing to the limited detection efficiency (Extended Data Fig. 1). The fitting used line shapes that incorporated the effects of the experimental response functions, as derived from the simulations, including the contribution arising from the residual crosstalk (see Methods).
In the case of ^{28}O, a decay energy of \({E}_{01234}={0.46}_{0.04}^{+0.05}({\rm{stat}})\,\pm \)\(0.02({\rm{syst}})\,{\rm{MeV}}\) was found, with an upper limit of the width of the resonance of 0.7 MeV (68% confidence interval). The crosssection for singleproton removal from ^{29}F populating the resonance was deduced to be \({1.36}_{0.14}^{+0.16}({\rm{stat}})\pm 0.13({\rm{syst}})\,{\rm{mb}}\). The systematic uncertainties for the decay energy and the width were dominated by the precise conditions used in the neutroncrosstalkrejection procedures, whereas the principal contribution to that for the crosssection arose from the uncertainty in the neutrondetection efficiency. It may be noted that, if the resonance observed here is an excited state of ^{28}O (presumably the 2^{+} level), then the ground state must lie even closer to threshold and the excitation energy of the former must be less than 0.46 MeV. This, however, is very much lower than theory suggests (2 MeV or more), even when the N = 20 shell closure is absent (see below). As such, it is concluded that the ground state has been observed.
In the case of ^{27}O, a decay energy of E_{0123} = 1.09 ± 0.04(stat) ± 0.02(syst) MeV was found. The width of the resonance was comparable with the estimated experimental resolution of 0.22 MeV (FWHM). Nevertheless, it was possible to obtain an upper limit on the width—0.18 MeV (68% confidence interval)—through a fit of a gated E_{012} spectrum for the much higher statistics ^{24}O and twoneutron coincidence events, as shown in Extended Data Fig. 2f. The spin and parity (J^{π}) of the resonance may be tentatively assigned to be 3/2^{+} or 7/2^{−} based on the upper limit of the width (see Methods).
Comparison with theory
The experimental groundstate energies of the oxygen isotopes ^{25–28}O are summarized in Fig. 3 and compared with theoretical calculations based on chiral effective field theory (χEFT)^{31,32,33,34,35,36} and largescale shellmodel calculations^{9,37}, including those with continuum effects^{38,39}. We focus on largescale shellmodel and coupledcluster calculations, in which the latter is augmented with a new statistical method. Both techniques include explicitly threenucleon forces, which are known to play a key role in describing the structure of neutronrich nuclei, including the oxygen isotopes and the location of the Z = 8 neutron drip line at ^{24}O (refs. ^{40,41,42}).
The largescale shellmodel calculations were undertaken using the new EEdf3 interaction, which was constructed on the basis of χEFT (see Methods). Because the calculations use a model space that includes the pfshell orbitals, the disappearance of the N = 20 shell closure can be naturally described. The EEdf3 interaction is a modified version of EEdf1 (refs. ^{31,32}), which correctly predicts the neutron drip line at F, Ne and Na, as well as a relatively lowlying ^{29}F excited state^{17} and the appreciable occupancy of the neutron 2p_{3/2} orbital^{5,18}. The EEdf3 interaction, which includes the effects of the EFT threenucleon forces^{43}, provides a reasonable description of the trends in the masses of the oxygen isotopes. However, as may be seen in Fig. 3, it predicts slightly higher ^{27,28}O energies (about 1 MeV) than found in the experiment. The calculated sum of the occupation numbers for the neutron pfshell orbitals is 2.5 (1.4) for ^{28}O (^{27}O) and for the 1d_{3/2} orbital 2.0 (2.1), which are consistent with a collapse of the N = 20 shell closure. The EEdf3 calculations show that ^{28}O_{gs} has large admixtures of configurations involving neutron excitations in the pfshell orbitals, as expected for nuclei in the IoI. This is supported by the measured crosssection as discussed below.
Firstprinciples calculations were performed using the coupledcluster (CC) method guided by history matching (HM)^{44,45,46} to explore the parameter space of the 17 lowenergy constants (LECs) in the χEFT description of the twonucleon and threenucleon interactions. HM identifies the region of parameter space for which the emulated CC method generates nonimplausible results (see Methods). A reliable, lowstatistic sample of 121 different LEC parameterizations was extracted, for which the CC posterior predictive distribution (ppd) was computed for the groundstate energies of ^{27,28}O, which are shown in Fig. 3. The predicted ^{27,28}O energies are correlated, as is clearly seen in the plot of energy distributions shown in Extended Data Fig. 3. From this, the median values and 68% credible regions were obtained for the ^{27}O–^{28}O and ^{28}O–^{24}O energy differences: \(\Delta E({}^{27,28}{\rm{O}})={0.11}_{+0.36}^{0.39}\,{\rm{M}}{\rm{e}}{\rm{V}}\) and \(\Delta E{(}^{28,24}{\rm{O}})={2.1}_{+1.2}^{1.3}\,{\rm{M}}{\rm{e}}{\rm{V}}\). The experimental values ΔE(^{27,28}O) = 0.63 ± 0.06(stat) ± 0.03(syst) MeV and \(\Delta E{(}^{28,24}{\rm{O}})={0.46}_{0.04}^{+0.05}({\rm{s}}{\rm{t}}{\rm{a}}{\rm{t}})\pm 0.02({\rm{s}}{\rm{y}}{\rm{s}}{\rm{t}})\,{\rm{M}}{\rm{e}}{\rm{V}}\), located at the edge of the 68% credible region, are consistent with the CC ppd. However, it is far enough away from the maximum to suggest that only a few finely tuned chiral interactions may be able to reproduce the ^{27}O and ^{28}O energies. Also, the obtained credible regions of the ^{27,28}O energies with respect to ^{24}O are relatively large, demonstrating that the measured decay energies of the extremely neutronrich isotopes ^{27,28}O are valuable anchors for theoretical approaches based on χEFT.
In Fig. 3, the predictions of a range of other models are shown. The USDB^{9} effective interaction (constructed within the sd shell) provides for arguably the most reliable predictions of the properties of sdshell nuclei. The continuum shell model (CSM)^{38} and the Gamow shell model (GSM)^{39} include the effects of the continuum, which should be important for dripline and unbound nuclei. The shellmodel calculation using the SDPFM interaction^{37} includes the pfshell orbitals in its model space, which should be important if either or both ^{27,28}O lie within the IoI. All the calculations, except those with the SDPFM interaction, predict a J^{π} = 3/2^{+} ^{27}O_{gs}. In the case of the SDPFM, a 3/2^{−} ground state is found with essentially degenerate 3/2^{+} (energy plotted in Fig. 3) and 7/2^{−} excited states at 0.71 MeV.
The remaining theoretical predictions are based on χEFT interactions. The valencespace inmedium similarity renormalization group (VSIMSRG)^{33} uses the 1.8/2.0 (EM) EFT potential^{43}. The results for the selfconsistent Green’s function (SCGF) approach are shown for the NNLO_{sat} (ref. ^{47}) and NN+3N(lnl) potentials^{35}. The coupledcluster calculation (ΛCCSD(T)^{36}) using NNLO_{sat} is also shown. Except for the results obtained using the GSM, all of the calculations shown predict higher energies than found here for ^{27}O and ^{28}O.
We now turn to the question of whether the N = 20 shell closure occurs in ^{28}O. Specifically, the measured crosssection for singleproton removal from ^{29}F may be used to deduce the corresponding spectroscopic factor (C^{2}S), which is a measure of the degree of overlap between initial and final state wavefunctions. As noted at the start of this paper, the N = 20 shell closure disappears in ^{29}F and the ground state is dominated by neutron pfshell configurations^{5,16,17,18}. As such, if the neutron configuration of ^{28}O is very similar to ^{29}F and the Z = 8 shell closure is rigid, the spectroscopic factor for proton removal will be close to unity. The spectroscopic factor was deduced using the distortedwave impulse approximation (DWIA) approach (see Methods). As recent theoretical calculations predict J^{π} = 5/2^{+} or 1/2^{+} for ^{29}F_{gs} (see, for example, refs. ^{5,31,32,48,49,50}), the momentum distribution has been investigated (Extended Data Fig. 4) and was found to be consistent with proton removal from the 1d_{5/2} orbital (see Methods), leading to a 5/2^{+} assignment. The ratio of the measured to theoretical singleparticle crosssection provides for an experimentally deduced spectroscopic factor of \({C}^{2}S={0.48}_{0.06}^{+0.05}({\rm{stat}})\pm 0.05({\rm{syst}})\). Such an appreciable strength indicates that the ^{28}O neutron configuration resembles that of ^{29}F. This value may be compared with that of 0.68 derived from the EEdf3 shellmodel calculations (in which the centreofmass correction factor^{51} (29/28)^{2} has been applied). The 30% difference between the experimental C^{2}S as compared with theory is in line with the wellknown reduction factor observed in (p, 2p) and (e, e′p) reactions^{52}. Notably, the EEdf3 calculations predict admixtures of the groundstate wavefunction of ^{29}F with sdclosedshell configurations of only 12%. Consequently, even when the neutrons in ^{28}O are confined to the sd shell, a spectroscopic factor of only 0.13 is obtained. As such, it is concluded that, as in ^{29}F, the pfshell neutron configurations play a major role in ^{28}O and that the N = 20 shell closure disappears. Consequently, the IoI extends to ^{28}O and it is not a doubly magic nucleus.
More effort will be required to properly quantify the character of the structure of ^{28}O and the neutron pfshell configurations. In this context, the determination of the excitation energy of the first 2^{+} state is the next step that may be deduced experimentally^{17}. The EEdf3 calculations predict an excitation energy of 2.097 MeV, which is close to that of approximately 2.5 MeV computed by the particle rotor model assuming moderate deformation^{53}. Both predictions are much lower than the energies found in doubly magic nuclei, for example, 6.917 MeV in ^{16}O and 4.7 MeV in ^{24}O (refs. ^{21,23}). A complementary probe of the neutron sd–pf shell gap, which is within experimental reach, is the energy difference between the positiveparity and negativeparity states of ^{27}O as seen in ^{28}F (ref. ^{5}).
Conclusions
We have reported here on the first observation of the extremely neutronrich oxygen isotopes ^{27,28}O. Both nuclei were found to exist as relatively lowlying resonances. These observations were made possible using a stateoftheart setup that permitted the direct detection of three and four neutrons. From an experimental point of view, the multineutrondecay spectroscopy demonstrated here opens up new perspectives in the investigation of other extremely neutronrich systems lying beyond the neutron drip line and the study of multineutron correlations. Comparison of the measured energies of ^{27,28}O with respect to ^{24}O with a broad range of theoretical predictions, including two approaches using nuclear interactions derived from effective field theories of quantum chromodynamics, showed that—in almost all cases—theory underbinds both systems. The statistical coupledcluster calculations indicated that the energies of ^{27,28}O can provide valuable constraints of such ab initio approaches and, in particular, the interactions used. Finally, although ^{28}O is expected in the standard shellmodel picture to be a doubly magic nucleus (Z = 8 and N = 20), the singleproton removal crosssection measured here, when compared with theory, was found to be consistent with it not having a closed neutron shell character. This result suggests that the IoI extends beyond ^{28,29}F into the oxygen isotopes.
Methods
Production of the ^{29}F beam
The beam of ^{29}F ions was provided by the RI Beam Factory operated by the RIKEN Nishina Center and the Center for Nuclear Study, University of Tokyo. It was produced by projectile fragmentation of an intense 345MeVpernucleon ^{48}Ca beam on a 15mmthick beryllium target. The secondary beam, including ^{29}F, was prepared using the BigRIPS^{55} fragment separator operated with aluminium degraders of 15 mm and 7 mm median thicknesses at the first and fifth intermediate focal planes, respectively. The primary ^{48}Ca beam intensity was typically 3 × 10^{12} particles per second. The average intensity of the ^{29}F beam was 90 particles per second.
Measurement with a ^{29}Ne beam
Data were also acquired to measure the direct population of ^{27}O through twoproton removal from ^{29}Ne. The beam was produced in a similar manner to that for ^{29}F and the energy was 228 MeV per nucleon with an average intensity of 8 × 10^{3} particles per second.
Unfortunately, in this measurement, the crosssection for the twoproton removal was much lower than expected and the statistics obtained for ^{24}O+3n coincidence events was too low to be usefully exploited. Nevertheless, the decay of ^{27}O could be identified from the ^{24}O+2n coincidence data. As may be seen in Extended Data Fig. 2e, the threebody decayenergy (E_{012}) spectrum gated by E_{01} < 0.08 MeV, corresponding to selection of the ^{26}O groundstate decay, exhibits a clear peak at around 1 MeV. As the simulations demonstrate, this is consistent with the sequential decay of the ^{27}O resonance observed in the ^{29}F beam data (Fig. 2c).
Invariantmass method
The invariant mass of ^{28}O, M(^{28}O), was reconstructed from the momentum vectors of all the decay particles (^{24}O and 4n) with \(M({}^{28}{\rm{O}})=\sqrt{{(\sum {E}_{i})}^{2}{\sum {{\bf{p}}}_{i}}^{2}}\), in which E_{i} and p_{i} denote the total energy and momentum vector of the decay particles, respectively. The decay energy is then obtained as E_{01234} = M(^{28}O) − M(^{24}O) − 4M_{n}, in which M(^{24}O) and M_{n} are the masses of ^{24}O and the neutron, respectively. The decayenergy resolution is estimated by Monte Carlo simulations. The resolution (FWHM) varies as a function of the decay energy approximately as 0.14(E_{01234} + 0.87)^{0.81} MeV.
Simulations
The experimental response functions, for both the full and partial decayenergy spectra, were derived from a Monte Carlo simulation based on GEANT4 (ref. ^{56}). All relevant characteristics of the setup (geometrical acceptances and detector resolutions) were incorporated, as well as those of the beam, target and reaction effects. The QGSP_INCLXX physics class was used to describe the interactions of the neutrons in the detectors (as well as nonactive material), as it reproduces well the experimentally determined singleneutron detection efficiency as well as the detailed characteristics of neutron crosstalk events^{57,58}. The generated events were treated using the same analysis procedure as for the experimental data. The overall efficiency as a function of decay energy for detecting ^{24}O and three and four neutrons, as estimated by the simulations, is shown by the insets of Extended Data Fig. 1.
Fitting of decayenergy spectra
The energies, widths and amplitudes of the resonances, as modelled by intrinsic line shapes with a Breit–Wigner form with energydependent widths, were obtained through fits of the corresponding decayenergy spectra using the maximumlikelihood method, in which the experimental responses were obtained by the simulations. As the decays of both ^{27}O and ^{28}O proceed through the ^{26}O ground state (18 keV (ref. ^{27})), the width of which is very small, the observed widths will be dominated by the oneneutron and twoneutron decay, respectively, to ^{26}O. We assume an \({E}_{01234}^{2}\) dependence of the width for the 2n emission^{59} to ^{26}O in the case of ^{28}O and an energy dependence for the width of the singleneutron emission^{60} from ^{27}O to ^{26}O. Fits with orbital angular momentum (L) dependent widths (L = 2 and 3) for the latter gave consistent results within the statistical uncertainties.
A nonresonant component is not included in the fitting as it is small, if not negligible, as in the cases of ^{25,26}O produced in oneprotonremoval reactions in previous experiments^{24,25,26,27}. The event selection with E_{012} < 0.08 MeV should further reduce any such contribution. As a quantitative check, a fit with a nonresonant component—modelled with a line shape given by \({p}_{0}\sqrt{{E}_{0123}}\exp \left({p}_{1}{E}_{0123}\right)\), in which p_{0} and p_{1} are fitting parameters—has been examined. This gives 8% reduction in the ^{28}O crosssection with a very limited impact on the energies and widths of the ^{27,28}O resonances.
Neutron crosstalk
A single beamvelocity neutron may scatter between individual plastic scintillator detectors of the three neutron walls of the setup. Such crosstalk events can mimic true multineutron events and present a source of background. By examining the apparent kinematics of such events and applying socalled causality conditions, this background can be almost completely eliminated^{57,58}. Notably, both the rejection techniques and the rate and characteristics of the crosstalk have been benchmarked in and compared with the simulations for dedicated measurements with singleneutron beams.
In the case of the fourneutron detection to identify ^{28}O, only 16% of the events arise from crosstalk that could not be eliminated (Fig. 2a). Most of these residual crosstalk events arise in cases in which one (or occasionally more) of the neutrons emitted in the decay of ^{28}O is subject to crosstalk. A much smaller fraction is also estimated to be produced when one of the three neutrons from the decay of ^{27}O, produced directly by proton and neutron knockout, undergoes crosstalk. Notably, the crosstalk cannot generate a narrow peaklike structure in the E_{01234} decayenergy spectrum.
Partial decay energy of subsystems
The partial decay energies of the ^{24}O+xn subsystems can be used to investigate the manner in which ^{27,28}O decay. In this analysis, the decay neutrons are numbered (n_{1}, n_{2},…) by ascending order of twobody relative energy E_{0i} between ^{24}O and n_{i}, that is, such that, E_{01} < E_{02} < E_{03} < E_{04}. Of particular interest here is the extremely low decay energy of the ^{26}O ground state (18 keV (ref. ^{27})), such that it appears just above zero energy (or the neutrondecay threshold) in the twobody partial decay energy E_{01} and threebody partial decay energy (E_{012}).
Extended Data Fig. 2a,b shows the distributions of the partial decay energies E_{012} and E_{034} for the ^{24}O+4n coincidence events with a total decay energy E_{01234} < 1 MeV. The resulting sharp threshold peak in the E_{012} spectrum is a clear sign of sequential decay through the ^{26}O ground state. This is confirmed quantitatively by a simulation assuming twoneutron emission to the ^{26}O ground state, which—in turn—decays by twoneutron emission to the ^{24}O ground state, which describes well the E_{012} and E_{034} spectra. By comparison, a simulation assuming fivebody phasespace decay fails to reproduce both of these spectra. We thus conclude that the ^{28}O ground state sequentially decays through the ^{26}O ground state as depicted in Fig. 2e.
In a similar vein, the sequential decay of ^{27}O through the ^{26}O ground state was identified from the analysis of the partial decay energies for the ^{24}O+3n coincidence events. Extended Data Fig. 2c,d show the distributions of the partial decay energies E_{012} and E_{03} for events for which 1.0 < E_{0123} < 1.2 MeV. The E_{012} spectrum exhibits a strong enhancement at zero energy indicative of sequential decay through the ^{26}O ground state. This interpretation is confirmed by the comparison shown with a simulation for the sequential decay of ^{27}O including the contribution from the decay of ^{28}O.
Widths of the ^{27,28}O resonances
As the energy of the ^{28}O resonance is lower than those of ^{27}O and ^{25}O (Fig. 2e), both oneneutron and threeneutron emission are energetically forbidden. The twoneutron decay to the ^{26}O ground state and the fourneutron decay to ^{24}O are allowed with nearly equal decay energies. The former decay should be favoured as the effective fewbody centrifugal barrier increases according to the number of emitted particles^{59}. It may be noted that the upper limit of 0.7 MeV observed here for the ^{28}O resonance width is consistent with the theoretical estimates for its sequential decay^{59}.
The upper limit for the ^{27}O width (0.18 MeV) may be compared with the singleparticle widths^{61} for neutron decay. Because the width of ^{26}O is very narrow owing to the extremely small decay energy (18 keV (ref. ^{27})), the ^{27}O width should be dominated by that for the first step ^{27}O → ^{26}O+n. The widths for swave, pwave, dwave and fwave neutron emission are 5, 3, 0.8 and 0.06 MeV, respectively. Assuming that the corresponding spectroscopic factors are not small (≳0.1), this would suggest that the decay occurs through dwave or fwave neutron emission. As such, the spin and parity of the ^{27}O resonance may be tentatively assigned to be 3/2^{+} or 7/2^{−}.
Momentum distribution
Extended Data Fig. 4 shows the transverse momentum (P_{x}) distribution of the ^{24}O+3n system in the rest frame of the ^{29}F beam for events gated by E_{012} < 0.08 MeV and E_{0123}<0.8 MeV, that is, events corresponding to population of the ^{28}O ground state. We note that this analysis used the ^{24}O+3n events, as the limited ^{24}O+4n statistics could not be usefully exploited in distinguishing between the momentum distributions for the proton knockout from different orbitals. Even though the momentum distribution is slightly broadened by the undetected decay neutron, it still reflects directly the character of the knocked out proton.
The experimental P_{x} distribution is compared with DWIA reaction theory calculations (see below) for knockout of a proton from the 1d_{5/2} and 2s_{1/2} orbitals. The theoretical distributions are convoluted with the experimental resolution, as well as the much smaller broadening induced by the undetected neutron (σ = 34 MeV/c). The bestfit normalization of the theoretical distribution obtained by the distorting potential with the Dirac phenomenology (microscopic foldingmodel potential) through a χ^{2} minimization gives reducedχ^{2} values of 2.0 (2.0) for the 1d_{5/2} proton knockout and 3.7 (4.7) for the 2s_{1/2} knockout. The curves in Extended Data Fig. 4 represent the calculations obtained by the distorting potential with the Dirac phenomenology. The better agreement for the 1d_{5/2} proton knockout suggests that the spin and parity of the ^{29}F ground state is 5/2^{+}, as predicted by the shellmodel calculations, including those using the EEdf3 interaction.
EEdf3 calculations
The EEdf3 Hamiltonian^{31} is a variant of the EEdf1 Hamiltonian, which was used in ref. ^{32} for describing F, Ne, Na and Mg isotopes up to the neutron drip line^{31}. The EEdf1 Hamiltonian was derived from χEFT interaction, as described below. The χEFT interaction proposed by Entem and Machleidt^{62,63} was taken with Λ = 500 MeV, as the nuclear force in vacuum, up to the nexttonexttonexttoleadingorder (N^{3}LO) in the χEFT. It was then renormalized using the V_{lowk} approach^{64,65} with a cutoff of \({\Lambda }_{{V}_{{\rm{low}} \mbox{} k}}=2.0\,{{\rm{fm}}}^{1}\), to obtain a lowmomentum interaction decoupled from highmomentum phenomena. The EKK method^{66,67,68} was then used to obtain the effective NN interaction for the sd–pf shells, by including the socalled \(\hat{Q}\)box, which incorporates unfolded effects coming from outside the model space^{69}, up to the third order and its folded diagrams. As to the singleparticle basis vectors, the eigenfunctions of the threedimensional harmonic oscillator potential were taken as usual. Also, the contributions from the Fujita–Miyazawa threenucleon force (3NF)^{70} were added in the form of the effective NN interaction^{40}. The Fujita–Miyazawa force represents the effects of the virtual excitation of a nucleon to a Δ baryon by pionexchange processes and includes the effects of Δhole excitations, but does not include other effects, such as contact (c_{D} and c_{E}) terms.
In this study, we explicitly treat neutrons only, whereas the protons remained confined to the ^{16}O closedshell core. As such, there is no proton–neutron interaction between active nucleons, and the neutron–neutron interaction is weaker. As this increases the relative importance of the effects from 3NF, we use the more modern 3NF of Hebeler et al.^{43}, which is expected to have finer details and improved properties. We obtain effective NN interactions from this 3NF first by deriving densitydependent NN interactions from them^{71} and then by having the density dependence integrated out with the normal density. It was suggested that this 3NF produces results similar to those reported in ref. ^{32} for the F, Ne, Na and Mg isotopes. As a result of this change, the singleparticle energies are shifted for the 1d_{5/2} and 2s_{1/2} orbitals by −0.72 MeV, for the 1d_{3/2} orbital by −0.42 MeV and for the pfshell orbitals by 0.78 MeV.
Coupledcluster calculations and emulators
The starting point for the calculations is the intrinsic Hamiltonian,
Here T_{kin} is the kinetic energy, T_{CoM} the kinetic energy of the centre of mass and V_{NN} and V_{NNN} are nucleon–nucleon and threenucleon potentials from χEFT^{62,72,73} and include Delta isobars^{74}. The momentum space cutoff of this interaction is Λ = 394 MeV/c.
We used the coupledcluster method^{75,76,77,78,79,80,81} with singlesdoubles and perturbative triples excitations, known as the CCSDT3 approximation^{82,83}, to compute the groundstate energy of ^{28}O, and the particleremoved equationofmotion (EOM) coupledcluster method from refs. ^{84,85} for the groundstate energy of ^{27}O. The coupledcluster calculations start from a spherical Hartree–Fock reference of ^{28}O in a model space of 13 major harmonic oscillator shells with an oscillator frequency of ħω = 16 MeV. The threenucleon force is limited to threebody energies up to E_{3max} = 14ħω. For energy differences, the effects of modelspace truncations and coupling to the scattering continuum are small and were neglected in the historymatching analysis.
The LECs of this interaction are constrained by a historymatching approach using highprecision emulators enabled by eigenvector continuation^{86}. These tools mimic the results of actual coupledcluster computations but are several orders of magnitude faster to evaluate, hence facilitating comprehensive exploration of the relevant parameter space. The emulators work as follows. In the 17dimensional space of LECs, the parameterization of the ΔNNLO_{GO}(394) potential^{74} serves as a starting point around which we select emulator training points according to a spacefilling lattice hypercube design for which we perform coupledcluster computations of groundstate energies, radii and excited states of ^{16,22,24}O (see Extended Data Table 1 for details). Keeping track of the variations of the observables and the corresponding coupledcluster eigenstates as the lowenergy constants are varied allows us to construct an emulator that can be used to predict the results for new parameterizations. This emulator strategy is rather general^{87} and possible because the eigenvector trajectory generated by continuous changes of the LECs only explores a relatively small subspace of the Hilbert space. Eigenvector continuation emulation tailored to coupledcluster eigenstates is referred to as the subspaceprojected coupledcluster (SPCC) method. In this work, we extended the SPCC method of ref. ^{88} to excited states and increased the precision by including triples excitations by means of the CCSDT3 and EOMCCSDT3 methods, respectively. Our SPCC emulators use up to 68 training points for each observable of interest and use model spaces consisting of 11 major harmonic oscillator shells. We checked the precision of each emulator by performing emulator diagnostics^{89}: confronting the emulator predictions with the results of actual coupledcluster computations; see Extended Data Fig. 5. Once constructed, the emulators are inexpensive computational tools that can precisely predict the results for virtually arbitrary parameterizations of the EFT potentials. This allows us to explore several hundred million parameterizations with the computational cost of only a few hundred actual coupledcluster computations. The use of emulation hence represents a critical advance, which facilitates a far deeper analysis of the coupledcluster method that was previously infeasible owing to the substantial computational expense of the coupledcluster calculations. Hence, these techniques overcome a substantial barrier to the use of such coupledcluster methods.
Coupledcluster calculations: linking models to reality
We describe the relationship between experimental observations, z, and ab initio model predictions M(θ), in which θ denotes the parameter vector of the theoretical model, as
In this relation, we consider experimental uncertainties, ϵ_{exp}, as well as method approximation errors, ϵ_{method}. The latter represent, for example, modelspace truncations and other approximations in the ab initio manybody solvers and are estimated from methodconvergence studies^{74}. Most notably, we acknowledge the fact that, even if we were to evaluate the model M(θ) at its best possible choice of the parameter vector, θ*, the model output, M(θ*), would still not be in exact quantitative agreement with reality owing to, for example, simplifications and approximations inherent to the model. We describe this difference in terms of a model discrepancy term, ϵ_{model}. The expected EFTconvergence pattern of our model allows us to specify further probabilistic attributes of ϵ_{model} a priori^{90,91,92,93}. We use the model errors defined in ref. ^{94}. The use of emulators based on eigenvector continuation^{86,87,88} provides us with an efficient approximation, \(\widetilde{M}(\theta )\), of the model. This approach entails an emulator error ϵ_{emulator} such that \(M(\theta )=\widetilde{M}(\theta )+{{\epsilon }}_{{\rm{emulator}}}\), as outlined in the previous section.
Obviously, we do not know the exact values of the errors in equation (2), hence we represent them as uncertain quantities and specify reasonable forms for their statistical distributions, in alignment with the Bayesian paradigm. This allows for these uncertainties to be formally incorporated in all subsequent calculations and inferences. We also assume that the errors add independently of each other and the inputs θ.
Coupledcluster calculations: HM
In this work, we use an iterative approach for complex computer models known as HM^{44,45,46}, in which the model, solved at different fidelities, is confronted with experimental data z using the relation in equation (2). The aim of HM is to estimate the set \({\mathcal{Q}}(z)\) of values for θ, for which the evaluation of a model M(θ) yields an acceptable—or at least not implausible (NI)—match to a set of observations z. HM has been used in various studies^{95,96,97} ranging, for example, from effects of climate modelling^{98,99} to systems biology^{46}. This work represents the first application in nuclear physics. We introduce the standard implausibility measure
which is a function over the input parameter space and quantifies the (mis)match between our (emulated) model output \({\widetilde{M}}_{i}(\theta )\) and the observation z_{i} for all observables i in the target set \({\mathcal{Z}}\). This specific definition uses the maximum of the individual implausibility measures (one for each observable) as the restricting quantity. We consider a particular value for θ as implausible if I(θ) > c_{I} ≡ 3.0 appealing to Pukelsheim’s threesigma rule^{100}. In accordance with the assumptions leading to equation (2), the variance in the denominator of equation (3) is a sum of independent squared errors. Generalizations of these assumptions are straightforward if further information on error covariances or possible inaccuracies in our error model would become available. An important strength of the HM approach is that we can proceed iteratively, excluding regions of input space by imposing cutoffs on implausibility measures that can include further observables z_{i} and corresponding model outputs M_{i}, and possibly refined emulators \({\widetilde{M}}_{i}\), as the iterations proceed. The iterative HM proceeds in waves according to a straightforward strategy that can be summarized as follows:

1.
At iteration j: evaluate a set of model runs over the current NI volume \({{\mathcal{Q}}}_{j}\) using a spacefilling design of sample values for the parameter inputs θ. Choose a rejection strategy based on implausibility measures for a set \({{\mathcal{Z}}}_{j}\) of informative observables.

2.
Construct or refine emulators for the model predictions across the current nonimplausible volume \({{\mathcal{Q}}}_{j}\).

3.
The implausibility measures are then calculated over \({{\mathcal{Q}}}_{j}\), using the emulators, and implausibility cutoffs are imposed. This defines a new, smaller NI volume \({{\mathcal{Q}}}_{j+1}\) that should satisfy \({{\mathcal{Q}}}_{j+1}\subset {{\mathcal{Q}}}_{j}\).

4.
Unless (i) the emulator uncertainties for all observables of interest are sufficiently small in comparison with the other sources of uncertainty, (ii) computational resources are exhausted or (iii) all considered points in the parameter space are deemed implausible, we include any further informative observables in the considered set \({{\mathcal{Z}}}_{j+1}\) and return to step 1.

5.
If 4(i) or (ii) is true, we generate a large number of acceptable runs from the final NI volume \({{\mathcal{Q}}}_{{\rm{final}}}\), sampled according to scientific need.
The ab initio model for the observables we consider comprises at most 17 parameters; four subleading pion–nucleon couplings, 11 nucleon–nucleon contact couplings and two shortranged threenucleon couplings. To identify a set of NI parameter samples, we performed iterative HM in four waves using observables and implausibility measures as summarized in Extended Data Table 1. For each wave, we use a sufficiently dense Latin hypercube set of several million candidate parameter samples. For the model evaluations, we used fast computations of neutron–proton (np) scattering phase shifts and efficient emulators for the fewbody and manybody observables listed. See Extended Data Table 2 for the list of included observables and key information for each wave. The input volume for wave 1 included large ranges for the relevant parameters, as indicated by the panel ranges in the lowerleft triangle of Extended Data Fig. 6. In all four waves, the input volume for c_{1,2,3,4} is a fourdimensional hypercube mapped onto the multivariate Gaussian probability density function (pdf) resulting from a Roy–Steiner analysis of πN scattering data^{101}. In wave 1 and wave 2, we sampled all relevant parameter directions for the set of included twonucleon observables. In wave 3, the extra ^{3}H and ^{4}He observables were added. As they are known to be insensitive to the four model parameters acting solely in the Pwave, we therefore ignored this subset of the inputs and compensated by slightly enlarging the corresponding method errors. This is a wellknown emulation procedure called inactive parameter identification^{44}. For the final iteration, that is, wave 4, we considered all 17 model parameters and added a set of observables for the oxygen isotopes ^{16,22,24}O and emulated the model outputs for 5 × 10^{8} parameter samples. Extended Data Fig. 6 summarizes the sequential NI volume reduction, wavebywave, and indicates the set Q_{4} of 634 NI samples after the fourth and final wave. The volume reduction is guided by the implausibility measure in equation (3) and the optical depths (see equations (25) and (26) in ref. ^{46}), in which the latter are illustrated in the lowerleft triangle of Extended Data Fig. 6. The NI samples summarize the parameter region of interest and can directly aid insight about interdependencies between parameters induced by the match to observed data. This region is also that in which we would expect the posterior distribution to reside. We see that the iterative HM process trains a nested series of emulators that become more and more accurate over this posterior region, as the iterations progress.
Coupledcluster calculations: Bayesian posterior sampling
The NI samples in the final HM wave also serve as excellent starting points for extracting the posterior pdf of the parameters θ, that is, p(θA = 2–24). To this end, we assume a normally distributed likelihood, according to equation (2), and a uniform prior corresponding to the initial volume of wave 1. Note that the prior for c_{1,2,3,4} is the multivariate Gaussian resulting from a Roy–Steiner analysis of πN scattering data^{101}. We sample the posterior using the affine invariant Markov chain Monte Carlo (MCMC) ensemble sampler emcee^{102} and the resulting distribution is shown in the upperright triangle of Extended Data Fig. 6. The sampling was performed with four independent ensemble chains, each with 150 walkers, and satisfactory convergence was reached (diagnosed using the Gelman–Rubin test with \( \hat{R}1 < 1{0}^{4}\) in all dimensions). We performed 5 × 10^{5} iterations per walker—after an initial warmup of 5,000 steps—and kept one final sample for every 500 steps. Combining all chains, we therefore end up with 4 × 150 × 1,000 = 6 × 10^{5} final samples. Also, we explored the sensitivity of our results to modifications of the likelihood definition. Specifically, we used a Student’s tdistribution (ν = 5) to see the effects of allowing heavier tails, and we introduced an error covariance matrix to study the effect of correlations (ρ ≈ 0.6) between selected observables. In the end, the differences in the extracted credibility regions were not great and we therefore present only results obtained with the uncorrelated, multivariate normal distribution (see Extended Data Table 3).
A subset of marginal ppds is shown in Extended Data Fig. 7. Clearly, a subset of 100 samples provides an accurate lowstatistics representation of this marginalized ppd. We exploit this feature in our final predictions for ^{27,28}O presented in the main text. Note that the ppd does not include draws from the model discrepancy pdf. To include information about the ^{25}O separation energy with respect to ^{24}O, we perform a straightforward Bayesian update of the posterior pdf p(θA = 2–24) for the LECs. This complements the statistical analysis of the ab initio model with important information content from an odd and neutronrich oxygen isotope. Using the pdf p(θA = 2–24), we draw 500 model predictions for ΔE(^{25,24}O) and account for all independent and normally distributed uncertainties according to Extended Data Table 1. Next, we draw 121 different LEC parameterizations from the revised posterior and use coupled cluster to compute the corresponding groundstate energies of ^{27,28}O. The full bivariate ppd for the ^{28}O–^{24}O and ^{27}O–^{28}O energy differences, ΔE(^{28,24}O) and ΔE(^{27,28}O), with associated credible regions, are shown in Extended Data Fig. 3. The effect of the continuum on the energy difference was estimated to be about 0.5 MeV in ref. ^{36} and was neglected in this work. We note that our ability to examine the full ppd for these expensive ab initio calculations provides welcome further insight, which is a direct consequence of the use of the HM procedure. We note that a sufficiently precise determination of ΔE(^{28,24}O) and ΔE(^{27,28}O) requires wave 4 in the HM and also using the separation energy ΔE(^{24,25}O) for the construction of the pdf. Without input about ^{25}O, the separation energy ΔE(^{27,28}O) becomes too uncertain to be useful. It is in this sense that a sufficiently precise prediction of ΔE(^{27,28}O) is finely tuned and cannot be based only on the properties of light nuclei up to ^{4}He. Changes in the LECs that have small impact in fewnucleon systems are magnified in ^{28}O. Apparently, one needs information about all nuclear shells, including the sd shell, to meaningfully predict this key nucleus.
DWIA calculations
The DWIA^{52,103,104} describes protoninduced proton knockout—(p, 2p)—processes as proton–proton (pp) elastic scattering. This is referred to as the impulse approximation, which is considered to be valid at intermediate energies when both outgoing protons have large momenta with respect to the residual nucleus. The DWIA approach has been successful in describing protoninduced knockout reactions; in ref. ^{52}, it was shown that the spectroscopic factors deduced from (p, 2p) reactions for the singleparticle levels near the Fermi surface of several nuclei are consistent with those extracted from electroninduced (e, e′p) reactions. The transition matrix of (p, 2p) processes within DWIA theory is given by T_{p2p} = ⟨χ_{1}χ_{2}t_{pp}χ_{0}ϕ_{p}⟩, in which χ_{i} are the distorted waves of the incoming proton (0) and the two outgoing protons (1 and 2), whereas ϕ_{p} is the normalized boundstate wavefunction of the proton inside the nucleus. The pp effective interaction is denoted by t_{pp}, the absolute square of which is proportional to the pp elastic crosssection. The nonlocality corrections^{105} to both χ_{i} and ϕ_{p} are taken into account, as well as the Møller factor^{106} for t_{pp} that guarantees the Lorentz invariance of the pp reaction probability. The (p, 2p) crosssection is given by F_{kin}C^{2}ST_{p2p}^{2}, with F_{kin} being a kinetic factor and C^{2}S the spectroscopic factor.
In this study, the crosssection integrated over the allowed kinematics of the outgoing particles was calculated. We used the Franey–Love parameterization^{107} for t_{pp} and the Bohr–Mottelson singleparticle potential^{108} to compute ϕ_{p}. We have used two types of the onebody distorting potential to obtain χ—specifically, the Dirac phenomenology (set EDAD2 (ref. ^{109})) and a microscopic folding model potential based on the Melbourne Gmatrix interaction^{110} and onebody nuclear densities calculated with the Bohr–Mottelson singleparticle model^{108}. It was found that the difference in the (p, 2p) crosssections calculated with the two sets of distorting potentials was at most 7.5%. Also, they give almost identical shapes for the momentum distributions. As such, we have used here the average value of the crosssections for each singleparticle configuration.
Code availability
Our unpublished computer codes used to generate the results reported in this paper are available from the corresponding author on reasonable request.
Change history
07 November 2023
A Correction to this paper has been published: https://doi.org/10.1038/s4158602306815w
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Acknowledgements
We thank the RIKEN Nishina Center and the Center for Nuclear Study, the University of Tokyo accelerator staff for the excellent beam delivery. This work was supported in part by JSPS KAKENHI grant nos. JP18K03672 and JP18H05404. This work was also supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) project ID 279384907  SFB 1245, the GSITU Darmstadt cooperation agreement, the GSI under contract KZILGE1416, the German Federal Ministry for Education and Research (BMBF) under contract nos. 05P15RDFN1 and 05P21PKFN1, the European Research Council (ERC) grant agreement no. 258567 and the ERC under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 758027) and the Swedish Research Council grant nos. 20115324, 201703839 and 201704234, 202005127. Partial support was also supplied by the FrenchJapanese International Associated Laboratory for Nuclear Structure Problems, as well as the French ANR14CE33002202 EXPAND. This work was also supported in part by the Institute for Basic Science (IBSR031D1) in Korea and the US Department of Energy, Office of Science, Office of Nuclear Physics, under award nos. DEFG0296ER40963 and DESC0018223. This work was also supported in part by the National Science Foundation, USA under grant no. PHY1102511. Computer time was provided by the Innovative and Novel Computational Impact on Theory and Experiment (INCITE) programme. This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the US Department of Energy under contract no. DEAC0500OR22725, and resources provided by the Swedish National Infrastructure for Computing (SNIC) at Chalmers Centre for Computational Science and Engineering (C3SE) and the National Supercomputer Centre (NSC) partially funded by the Swedish Research Council through grant agreement no. 201805973. Y.T. acknowledges the support of the JSPS GrantinAid for Scientific Research grant no. JP21H01114. I.G. has been supported by HIC for FAIR and Croatian Science Foundation under project nos. 1257 and 7194. Z.D. and D.S. have been supported by the National Research, Development and Innovation Fund of Hungary through project nos. TKP2021NKTA42 and K128947. T. Otsuka., N.S., N.T., Y.U. and S.Y. acknowledge valuable support from the ‘Priority Issue on PostK computer’ (hp190160), ‘Program for Promoting Researches on the Supercomputer Fugaku’ (JPMXP1020200105, hp200130, hp210165) and KAKENHI grants (JP17K05433, JP20K03981, JP19H05145, JP21H00117). The material presented here is based on work supported in part by the US Department of Energy, Office of Science, Office of Nuclear Physics, under contract no. DEAC0206CH11357 (ANL). T. Nakamura acknowledges the support of the JSPS GrantinAid for Scientific Research grant no. JP21H04465. I.V. gratefully acknowledges UKRI (EP/W011956/1) and Wellcome (218261/Z/19/Z) funding.
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Y.K. designed and proposed the experiment and performed the offline analysis and the Monte Carlo simulations. Y.K., T. Nakamura, T. Otsuka, K.O. and N.A.O. drafted the manuscript. Y.K., T.N., N.L.A., H.A.F., L.A., T.A., H.B., K.B., C.C., D.C., H.C., N.C., A.C., F.D., A.D., Q.D., Z.D., C.A.D., Z.E., I.G., J.M.G., J.G., A.G., M.N.H., A.H., C.R.H., M.H., A.H., Á.H., J.W.H., T.I., J.K., N.K.N., S. Kawase, S. Kim, K.K., T.K., D.K., S.K., I.K., V.L., S.L., F.M.M., S.M., J.M., K.M., T.M., M.N., K.N., N.N., T.N., A.O., F.d.O.S., N.A.O., H.O., T. Ozaki, V.P., S.P., A.R., D.R., A.T.S., T.Y.S., M.S., H. Sato, Y.S., H. Scheit, F.S., P.S., M.S., Y.S., H. Simon, D.S., O.S., L.S., S.T., M. Tanaka, M. Thoennessen, H.T., Y.T., T.T., J. Tscheuschner, J. Tsubota, T.U., H.W., Z.Y., M.Y. and K.Y. took part in the setting up of the experiment and/or monitored the data accumulation and/or maintained the operation of the experiment and detectors. T. Otsuka, N.S., N.T., Y.U. and S.Y. performed the EEdf3 calculations. K.O. performed the DWIA calculations. A.E., C.F., G.H., W.G.J., T.P., Z.H.S. and I.V. performed the coupledcluster calculations and their statistical analysis and wrote the associated sections of the manuscript.
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Extended data figures and tables
Extended Data Fig. 1 Schematic view of the experimental setup.
The insets show the overall efficiency as a function of decay energy for detecting ^{24}O and four and three neutrons.
Extended Data Fig. 2 Partial decayenergy spectra.
a, The filled grey histogram is the threebody decay energy E_{012} gated on the total decay energy E_{01234} < 1 MeV for the ^{24}O+4n coincidence events. The red and blue histograms are the results of simulations of sequential decay through the ^{26}O ground state (A and B in Fig. 2e) and fivebody phasespace decay, respectively. b, Same as a but for the threebody decay energy E_{034}. c, The filled grey histogram is the partial decayenergy spectrum E_{012} gated by 1.0 < E_{0123} < 1.2 MeV for the ^{24}O+3n coincidence events. The red and blue dashed histograms are the results of simulations assuming ^{27}O sequential (B and C in Fig. 2e) and fourbody phasespace decay, respectively. The green hatched histogram represents the contribution from the decay of ^{28}O. The red (blue) solid histogram is the sum of the contributions from ^{28}O and ^{27}O for sequential (phasespace) decay. d, Same as c but for the twobody decay energy E_{03}. e, Decayenergy spectrum of ^{24}O+2n events from the ^{29}Ne beam data. The grey histogram represents events with E_{01} < 0.08 MeV. The red histogram shows the results of the simulation for the decay of the ^{27}O resonance. The excess observed nearzero decay energy is interpreted as arising from direct population of the ^{26}O ground state from ^{29}Ne. f, Decayenergy spectrum of ^{24}O+2n events from the ^{29}F beam. The grey histogram represents events with E_{01} < 0.08 MeV. The red histogram shows the best fit in the region of the peak arising from the decay of the ^{27}O resonance (dashed histogram) and an exponential distribution (dotted curve) arising from all other contributions that come primarily from the decay of ^{28}O.
Extended Data Fig. 3 Probability distribution of the calculated energy differences.
Survived nonimplausible calculations are shown by blue dots as functions of energy differences ΔE(^{28,24}O) and ΔE(^{27,28}O). The black circle shows experiment. The dashed curves indicate 68% and 90% highest probability density regions. The top and right distributions are the onedimensional probability density distributions. The values given by the other theories are plotted as squares: green, USDB, GSM and CSM; red, SDPFM and EEdf3; purple, VSIMSRG.
Extended Data Fig. 4 Transverse momentum distribution of the ^{24}O+3n system in the rest frame of the ^{29}F beam.
Events corresponding to the population of the ^{28}O ground state (E_{012} < 0.08 MeV and E_{0123} < 0.8 MeV) are shown by the data points. The blue and red solid lines represent the DWIA calculations, including the experimental effects for s_{1/2} and d_{5/2} proton knockout, respectively, whereby the distributions have been scaled to best fit experiment.
Extended Data Fig. 5 Crossvalidation of emulators.
Upperleft panel, total energies of ^{24}O computed with the coupledcluster method in the CCSDT3 approximation versus the SPCC emulator for a validation set of 100 parameter samples. Upperright panel, distribution of residuals in percent. Lowerleft panel, 2^{+} excitation energies of ^{24}O computed with the coupledcluster method in the EOMCCSDT3 approximation versus the SPCC emulator for a validation set of 40 parameter samples. Lowerright panel, distribution of residuals in percent.
Extended Data Fig. 6 Historymatching waves and Bayesian posterior sampling.
Lowerleft triangle, the panel limits correspond to the input volume of wave 1. The domain is iteratively reduced and the input volumes of waves 2, 3 and 4 are indicated by the green/dashdotted, blue/dashed and black/solid rectangles, respectively. The optical depths of nonimplausible samples in the final wave are shown in red, with darker regions corresponding to a denser distribution of nonimplausible samples. Upperright triangle, parameter posterior pdf from MCMC sampling with the nonimplausible samples of the historymatching analysis as starting points. We use an uncorrelated, multivariate normal likelihood function and a uniform prior bounded by the first wave initial volume. Note that the relevant posterior regions are small in some directions but larger in others, such as c_{D} and c_{E}.
Extended Data Fig. 7 ppds for ^{16,22,24}O.
MCMC samples of the ppd for selected oxygen observables. The black (maroon) histogram shows results obtained with an uncorrelated, Gaussian likelihood (including a discrete probability p(E_{np.1S0} > 0θ) = 1). The red histogram illustrates a lowstatistics sample. The 68% credible regions and the medians are indicated by dashed lines on the diagonal, whereas the solid, vertical grey (blue) lines show the experimental target (prediction with the ΔNNLO_{GO}(394) interaction).
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Kondo, Y., Achouri, N.L., Falou, H.A. et al. First observation of ^{28}O. Nature 620, 965–970 (2023). https://doi.org/10.1038/s41586023063526
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DOI: https://doi.org/10.1038/s41586023063526
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