First observation of 28O

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 neutron-to-proton (N/Z) ratios than in stable nuclei. Light, neutron-rich 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 short-lived resonances (about 10−21 s), provide the most stringent tests of modern nuclear-structure theories. Here we report on the first observation of 28O and 27O through their decay into 24O and four and three neutrons, respectively. The 28O nucleus is of particular interest as, with the Z = 8 and N = 20 magic numbers1,2, it is expected in the standard shell-model picture of nuclear structure to be one of a relatively small number of so-called ‘doubly magic’ nuclei. Both 27O and 28O were found to exist as narrow, low-lying resonances and their decay energies are compared here to the results of sophisticated theoretical modelling, including a large-scale shell-model 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 cross-section for the production of 28O from a 29F beam is consistent with it not exhibiting a closed N = 20 shell structure.

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 neutron-to-proton (N/Z) ratios than in stable nuclei.Light, neutron-rich 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 short-lived resonances (about 10 −21 s), provide the most stringent tests of modern nuclear-structure 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 shell-model picture of nuclear structure to be one of a relatively small number of so-called 'doubly magic' nuclei.Both 27 O and 28 O were found to exist as narrow, low-lying resonances and their decay energies are compared here to the results of sophisticated theoretical modelling, including a large-scale shell-model 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 cross-section for the production of 28 O from a 29 F beam is consistent with it not exhibiting a closed N = 20 shell structure.
One of the most active areas of present-day 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 initio-type approaches whereby the underlying interactions between the constituent nucleons are constructed from first-principles 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, neutron-rich nuclei (Fig. 1a).Here, beyond the limits of nuclear binding-the so-called neutron drip line-nuclei can exist as very-short-lived (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 Article calculations to be made than attempting to model an A-body (A = Z + N) nucleus from the full ensemble of nucleons.Indeed, this approach has been a fundamental premise of the shell-model 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 neutron-rich 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, two-neutron unbound nucleus 10 He (Z = 2, N = 8; N/Z = 4), has been observed as a well-defined 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 neutron-rich 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 sd-shell and pf-shell orbitals, rather than being well pronounced (Fig. 1b), is weakened or even vanishes, and configurations with neutrons excited into the pf-shell 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 low-lying 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 particle-bound oxygen isotope, 24 O, has been found to be doubly magic, with a new closed shell forming at N = 16 (refs.19-23).As such, the structural character of the more neutron-rich 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 single-neutron and two-neutron 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 high-energy 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 neutron-drip-line beams coupled to a thick, active liquidhydrogen target system and a high-performance multineutron detection array.

Experiment
The neutron-unbound 27,28 O were produced through proton-induced 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) liquid-hydrogen 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 decay-energy resolution.
The forward-focused beam-velocity reaction products-charged fragments and fast neutrons-were detected and their momenta determined using the SAMURAI spectrometer 29 , including the three large-area segmented plastic scintillator walls of the NeuLAND 30 and NEBULA arrays.An overall detection efficiency for the three-neutron and four-neutron 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 invariant-mass 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 two-body 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 decay-energy 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 decay-energy 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 three-body ( 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 four-body decay-energy (E 0123 ) spectrum of Fig. 2b.As confirmed by the simulations, which are able to simultaneously describe the 24 O+3n and 4n decay-energy spectra, the well-populated peak-like 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 two-proton 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 higher-lying 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  0.02(syst) MeV was found, with an upper limit of the width of the resonance of 0.7 MeV (68% confidence interval).The cross-section for single-proton removal from 29 F populating the resonance was deduced to be 1.36 (stat) ±0.13(syst) mb −0.14 +0.16 . The systematic uncertainties for the decay energy and the width were dominated by the precise conditions used in the neutron-crosstalk-rejection procedures, whereas the principal contribution to that for the cross-section arose from the uncertainty in the neutron-detection 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.

Article
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-0.18MeV (68% confidence interval)-through a fit of a gated E 012 spectrum for the much higher statistics 24 O and two-neutron 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 ground-state 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 large-scale shell-model calculations 9,37 , including those with continuum effects 38,39 .We focus on large-scale shell-model and coupled-cluster calculations, in which the latter is augmented with a new statistical method.Both techniques include explicitly three-nucleon forces, which are known to play a key role in describing the structure of neutron-rich nuclei, including the oxygen isotopes and the location of the Z = 8 neutron drip line at 24 O (refs. 40-42).
The large-scale shell-model 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 pf-shell 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 low-lying 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 three-nucleon 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 pf-shell 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 pf-shell orbitals, as expected for nuclei in the IoI.This is supported by the measured cross-section as discussed below.
First-principles calculations were performed using the coupledcluster (CC) method guided by history matching (HM) [44][45][46] to explore the parameter space of the 17 low-energy constants (LECs) in the χEFT description of the two-nucleon and three-nucleon interactions.HM identifies the region of parameter space for which the emulated CC method generates non-implausible results (see Methods).A reliable, low-statistic sample of 121 different LEC parameterizations was extracted, for which the CC posterior predictive distribution (ppd) was computed for the ground-state 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  , 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 neutron-rich 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 sd-shell 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 drip-line and unbound nuclei.The shell-model calculation using the SDPF-M interaction 37 includes the pf-shell 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 SDPF-M interaction, predict a J π = 3/2 + 27 O gs .In the case of the SDPF-M, 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 valence-space in-medium similarity renormalization group (VS-IMSRG) 33 uses the 1.8/2.0(EM) EFT potential 43 .The results for the self-consistent Green's function (SCGF) approach are shown for the NNLO sat (ref.47) and NN+3N(lnl) potentials 35 .The coupled-cluster calculation (Λ-CCSD(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 cross-section for single-proton 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 pf-shell 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 distorted-wave  27,28 O are the present results and those for 25,26 O are taken from the atomic mass evaluation 54 .The experimental uncertainties are smaller than the symbol size.Comparison is made with predictions of shell-model calculations using the EEdf3 (refs.31,32), USDB 9 and SDPF-M 37 (see text for 27 O) interactions, the coupled-cluster method with the statistical approach (CC) and shell-model calculations incorporating continuum effects (CSM 38 and GSM 39 ).Also shown are the predictions of ab initio approaches (VS-IMSRG 33 , SCGF 35 and Λ-CCSD(T) 36 ).The vertical bars for CC denote 68% credible intervals.The shaded band for GSM shows the uncertainties owing to pf-continuum couplings.
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.5,31,32,48-50), 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 single-particle cross-section provides for an experimentally deduced spectroscopic factor of C S = 0.48 (stat) ±0.05(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 shell-model calculations (in which the centre-of-mass 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 well-known reduction factor observed in (p, 2p) and (e, e′p) reactions 52 .Notably, the EEdf3 calculations predict admixtures of the ground-state wavefunction of 29 F with sd-closed-shell 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 pf-shell 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 pf-shell 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 positive-parity and negative-parity states of 27 O as seen in 28 F (ref. 5).

Conclusions
We have reported here on the first observation of the extremely neutron-rich oxygen isotopes 27,28 O.Both nuclei were found to exist as relatively low-lying resonances.These observations were made possible using a state-of-the-art setup that permitted the direct detection of three and four neutrons.From an experimental point of view, the multineutron-decay spectroscopy demonstrated here opens up new perspectives in the investigation of other extremely neutron-rich 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 coupled-cluster 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 shell-model picture to be a doubly magic nucleus (Z = 8 and N = 20), the single-proton removal cross-section 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.

Online content
Any methods, additional references, Nature Portfolio reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at https://doi.org/10.1038/s41586-023-06352-6.

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 345-MeV-per-nucleon 48 Ca beam on a 15-mm-thick 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 two-proton 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 cross-section 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 three-body decay-energy (E 012 ) spectrum gated by E 01 < 0.08 MeV, corresponding to selection of the 26 O ground-state 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).

Invariant-mass 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 p , 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 decay-energy 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 decay-energy 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 non-active material), as it reproduces well the experimentally determined single-neutron 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 decay-energy spectra
The energies, widths and amplitudes of the resonances, as modelled by intrinsic line shapes with a Breit-Wigner form with energy-dependent widths, were obtained through fits of the corresponding decay-energy spectra using the maximum-likelihood 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 one-neutron and two-neutron 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 single-neutron 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 non-resonant component is not included in the fitting as it is small, if not negligible, as in the cases of 25,26 O produced in one-proton-removal 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 non-resonant component-modelled with a line shape given by p E pE exp(− ) 0 0123 1 0123 , in which p 0 and p 1 are fitting parameters-has been examined.This gives 8% reduction in the 28 O cross-section with a very limited impact on the energies and widths of the 27,28 O resonances.

Neutron crosstalk
A single beam-velocity 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 so-called 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 single-neutron beams.
In the case of the four-neutron 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 peak-like structure in the E 01234 decay-energy 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 two-body 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 neutron-decay threshold) in the two-body partial decay energy E 01 and three-body 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 two-neutron emission to the 26 O ground state, which-in turn-decays by two-neutron emission to the 24 O ground state, which describes well the E 012 and E 034 spectra.By comparison, a simulation assuming five-body phase-space 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 one-neutron and three-neutron emission are energetically forbidden.The two-neutron decay to the 26 O ground state and the four-neutron decay to 24 O are allowed with nearly equal decay energies.The former decay should be favoured as the effective few-body 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 single-particle 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 s-wave, p-wave, d-wave and f-wave 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 d-wave or f-wave 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 best-fit normalization of the theoretical distribution obtained by the distorting potential with the Dirac phenomenology (microscopic folding-model potential) through a χ 2 minimization gives reduced-χ 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 shell-model 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 next-to-next-to-next-to-leading-order (N 3 LO) in the χEFT.It was then renormalized using the V low-k approach 64,65 with a cutoff of Λ =2.0 fm , to obtain a low-momentum interaction decoupled from high-momentum phenomena.The EKK method [66][67][68] was then used to obtain the effective NN interaction for the sd-pf shells, by including the so-called Q ˆ-box, which incorporates unfolded effects coming from outside the model space 69 , up to the third order and its folded diagrams.As to the single-particle basis vectors, the eigenfunctions of the three-dimensional 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 pion-exchange processes and includes the effects of Δ-hole 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 closed-shell 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 density-dependent 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 single-particle 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 pf-shell orbitals by 0.78 MeV.

Coupled-cluster 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 three-nucleon 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 coupled-cluster method [75][76][77][78][79][80][81] with singles-doubles and perturbative triples excitations, known as the CCSDT-3 approximation 82,83 , to compute the ground-state energy of 28 O, and the particleremoved equation-of-motion (EOM) coupled-cluster method from refs.84,85 for the ground-state energy of 27 O.The coupled-cluster 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 three-nucleon force is limited to three-body energies up to E 3max = 14ħω.For energy differences, the effects of model-space truncations and coupling to the scattering continuum are small and were neglected in the history-matching analysis.
The LECs of this interaction are constrained by a history-matching approach using high-precision emulators enabled by eigenvector continuation 86 .These tools mimic the results of actual coupled-cluster 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 17-dimensional 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 space-filling lattice hypercube design for which we perform coupled-cluster computations of ground-state 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 coupled-cluster eigenstates as the low-energy 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 coupled-cluster eigenstates is referred to as the subspace-projected coupled-cluster (SP-CC) method.In this work, we extended the SP-CC method of ref. 88 to excited states and increased the precision by including triples excitations by means of the CCSDT-3 and EOM-CCSDT-3 methods, respectively.Our SP-CC 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 coupled-cluster 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 coupled-cluster computations.The use of emulation hence represents a critical advance, which facilitates a far deeper analysis of the coupled-cluster method that was previously infeasible owing to the substantial computational expense of the coupled-cluster calculations.Hence, these techniques overcome a substantial barrier to the use of such coupled-cluster methods.

Coupled-cluster 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 method model exp In this relation, we consider experimental uncertainties, ϵ exp , as well as method approximation errors, ϵ method .The latter represent, for example, model-space truncations and other approximations in the ab initio many-body solvers and are estimated from method-convergence 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 EFT-convergence 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 86-88 provides us with an efficient approximation,  M θ ( ), of the model.This approach entails an emulator error ϵ emulator such that M θ M θ ( ) = ( ) + 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.

Coupled-cluster 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 z ( ) Q 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  M θ ( ) i and the observation z i for all observables i in the target set 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 three-sigma 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  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 Q j using a space-filling design of sample values for the parameter inputs θ.Choose a rejection strategy based on implausibility measures for a set j Z of informative observables.2. Construct or refine emulators for the model predictions across the current non-implausible volume Q j .3. The implausibility measures are then calculated over j Q , using the emulators, and implausibility cutoffs are imposed.This defines a new, smaller NI volume j+1 Q that should satisfy Q Q ⊂ j j +1 .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 j+1 Z 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 Q 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 short-ranged 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 few-body and many-body 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 lower-left triangle of Extended Data Fig. 6.In all four waves, the input volume for c 1,2,3,4 is a four-dimensional 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 two-nucleon 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 P-wave, we therefore ignored this subset of the inputs and compensated by slightly enlarging the corresponding method errors.This is a well-known 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, wave-by-wave, 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 lower-left 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.

Coupled-cluster 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(θ|A = 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 upper-right 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 R ˆ− 1 < 10 −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 t-distribution (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 low-statistics 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(θ|A = 2-24) for the LECs.This complements the statistical analysis of the ab initio model with important information content from an odd and neutron-rich oxygen isotope.Using the pdf p(θ|A = 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 ground-state 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 few-nucleon 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 proton-induced 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 proton-induced knockout reactions; in ref. 52, it was shown that the spectroscopic factors deduced from (p, 2p) reactions for the single-particle levels near the Fermi surface of several nuclei are consistent with those extracted from electron-induced (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 bound-state 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 cross-section.The non-locality 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) cross-section is given by F kin C 2 S|T p2p | 2 , with F kin being a kinetic factor and C 2 S the spectroscopic factor.
In this study, the cross-section 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 single-particle potential 108 to compute ϕ p .We have used two types of the one-body distorting potential to obtain χ-specifically, the Dirac phenomenology (set EDAD2 (ref.109)) and a microscopic folding model potential based on the Melbourne G-matrix interaction 110 and one-body nuclear densities calculated with the Bohr-Mottelson single-particle model 108 .It was found that the difference in the (p, 2p) cross-sections calculated with the two sets of distorting potentials was at most 7.5%.Also, they give almost identical shapes for the momentum distributions.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 five-body phase-space decay, respectively.b, Same as a but for the three-body decay energy E 034 .c, The filled grey histogram is the partial decay-energy 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 four-body 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 (phase-space) decay.d, Same as c but for the two-body decay energy E 03 .e, Decay-energy 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 near-zero decay energy is interpreted as arising from direct population of the 26 O ground state from 29 Ne.f, Decay-energy 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.

Fig. 1 |
Fig. 1 | Nuclear chart and shell structure.a, Nuclear chart up to Z = 18 showing the stable and short-lived (β-decaying) nuclei.The experimentally established neutron drip line is shown by the thick blue line.Known doubly magic nuclei are also indicated.b, Schematic illustration of the neutron configuration for a nucleus with a closed N = 20 shell.c, The neutron configuration involving particle-hole excitations across a quenched N = 20 shell gap.

Fig. 2 |
Fig. 2 | Decay-energy spectra and decay scheme.a, Five-body decay-energy (E 01234 ) spectrum for 24 O+4n events.The solid red histogram shows the best-fit result taking into account the experimental response function.The dotted red histogram shows the contribution arising from residual crosstalk that survives the rejection procedures (see Methods).b, Four-body decay-energy (E 0123 ) spectrum for 24 O+3n events.c, Same as b but gated by the partial decay energy E 012 < 0.08 MeV.The dashed red histograms represent the contributions from 28 O and 27 O events and the solid red histogram shows the sum.d, Definition of the partial decay energies.e, Decay scheme of the unbound oxygen isotopes.The newly observed resonances and their decays are shown in red.

Fig. 3 |
Fig.3| Ground-state energies with respect to24 O. Experiment is shown by the black circles, in which the values for27,28 O are the present results and those for25,26 O are taken from the atomic mass evaluation 54 .The experimental uncertainties are smaller than the symbol size.Comparison is made with predictions of shell-model calculations using the EEdf3 (refs.31,32),USDB9  and SDPF-M37 (see text for27 O) interactions, the coupled-cluster method with the statistical approach (CC) and shell-model calculations incorporating continuum effects (CSM38 and GSM39 ).Also shown are the predictions of ab initio approaches (VS-IMSRG33 , SCGF35 and Λ-CCSD(T)36 ).The vertical bars for CC denote 68% credible intervals.The shaded band for GSM shows the uncertainties owing to pf-continuum couplings.

. 1 |. 2 |
Schematic view of the experimental setup.The insets show the overall efficiency as a function of decay energy for detecting24 O and four and three neutrons.Partial decay-energy spectra.a, The filled grey histogram is the three-body decay energy E 012 gated on the total decay energy E 01234 < 1 MeV for the

. 3 |. 4 |Extended Data Fig. 5 |
Probability distribution of the calculated energy differences.Survived non-implausible 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 one-dimensional probability density distributions.The values given by the other theories are plotted as squares: green, USDB, GSM and CSM; red, SDPF-M and EEdf3; purple, VS-IMSRG.Transverse momentum distribution of the24 O+3n system in the rest frame of the 29 F beam.Events corresponding to the population of the28 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.Cross-validation of emulators.Upper-left panel, total energies of24 O computed with the coupled-cluster method in the CCSDT-3 approximation versus the SP-CC emulator for a validation set of 100 parameter samples.Upper-right panel, distribution of residuals in percent.Lower-left panel, 2 + excitation energies of24 O computed with the coupled-cluster method in the EOM-CCSDT-3 approximation versus the SP-CC emulator for a validation set of 40 parameter samples.Lower-right panel, distribution of residuals in percent.Extended Data Fig.6| History-matching waves and Bayesian posterior sampling.Lower-left 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/dash-dotted, blue/dashed and black/solid rectangles, respectively.The optical depths of non-implausible samples in the final wave are shown in red, with darker regions corresponding to a denser distribution of non-implausible samples.Upper-right triangle, parameter posterior pdf from MCMC sampling with the non-implausible samples of the history-matching 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 .
2428O and 28 O-24O energy differences: 47.Ekström, A. etal.Accurate nuclear radii and binding energies from a chiral interaction.Phys.Rev. C 91, 051301(R) (2015).48.Utsuno, Y. et al.Shape transitions in exotic Si and S isotopes and tensor-force-driven Jahn-Teller effect.Phys.Rev. C 86, 051301(R) (2012).49.Caurier, E., Nowacki, F. & Poves, A. Merging of the islands of inversion at N = 20 and N = 28.Phys.Rev. C 90, 014302 (2014).50.Fossez, K. & Rotureau, J. Density matrix renormalization group description of the island of inversion isotopes 28-33 F. Phys.Rev. C 106, 034312 (2022).51.Dieperink, A. E. L. & de Forest, T. Center-of-mass effects in single-nucleon knock-out reactions.Phys.Rev. C 10, 543-549 (1974).52.Wakasa, T., Ogata, K. & Noro, T. Proton-induced knockout reactions with polarized and unpolarized beams.Prog.Part.Nucl.Phys.96, 32-87 (2017).53.Macchiavelli, A. O. et al.Structure of 29 F in the rotation-aligned coupling scheme of the particle-rotor model.Phys.Lett.B 775, 160-162 (2017).54.Wang, M., Huang, W. J., Kondev, F. G., Audi, G. & Naimi, S. The AME 2020 atomic mass evaluation (II).Tables, graphs and references.Chin.Phys.C 45, 030003 (2021).Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material.If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.