Abstract
The nuclear shell structure, which originates in the nearly independent motion of nucleons in an average potential, provides an important guide for our understanding of nuclear structure and the underlying nuclear forces. Its most remarkable fingerprint is the existence of the socalled magic numbers of protons and neutrons associated with extra stability. Although the introduction of a phenomenological spin–orbit (SO) coupling force in 1949 helped in explaining the magic numbers, its origins are still open questions. Here, we present experimental evidence for the smallest SOoriginated magic number (subshell closure) at the proton number six in ^{13–20}C obtained from systematic analysis of pointproton distribution radii, electromagnetic transition rates and atomic masses of light nuclei. Performing ab initio calculations on ^{14,15}C, we show that the observed proton distribution radii and subshell closure can be explained by the stateoftheart nuclear theory with chiral nucleon–nucleon and threenucleon forces, which are rooted in the quantum chromodynamics.
Introduction
Atomic nuclei—the finite quantum manybody systems consisting of protons and neutrons (known collectively as nucleons)—exhibit shell structure, in analogy to the electronic shell structure of atoms. Atoms with filled electron shells—known as the noble gases—are particularly stable chemically. The filling of the nuclear shells, on the other hand, leads to the magicnumber nuclei. The nuclear magic numbers, as we know in stable and naturallyoccurring nuclei, consist of two different series of numbers. The first series—2, 8, 20—is attributed to the harmonicoscillator (HO) potential, while the second one—28, 50, 82 and 126—is due to the spin–orbit (SO) coupling force (see Fig. 1). It was the introduction of this phenomenological SO force—a force that depends on the intrinsic spin of a nucleon and its orbital angular momentum, and the socalled j–j coupling scheme that helped explain^{1,2} completely the magic numbers, and won GoeppertMayer and Jensen the Nobel Prize. However, the microscopic origins of the SO splitting have remained unresolved due to the difficulty to describe the structure of atomic nuclei from ab initio nuclear theories^{3,4,5} with two (NN) and threenucleon forces (3NFs). Although the theoretical study^{6} of the SO splitting of the 1p_{1/2} and 1p_{3/2} singleparticle states in ^{15}N has suggested possible roles of twobody SO and tensor forces, as well as threebody forces, the discovery of a prevalent SOtype magic number 6 is expected to offer unprecedented opportunities to understand its origins.
In her Nobel lecture, GoeppertMayer had mentioned the magic numbers 6 and 14—which she described as hardly noticeable—but surmised that the energy gap between the 1p_{1/2} and 1p_{3/2} orbitals due to the SO force is fairly small^{7}. That the j–j coupling scheme appears to fail in the pshell light nuclei was discussed and attributed to their tendency to form clusters of nucleons^{8}. Experimental and theoretical studies in recent decades, however, have hinted at the possible existence of the magic number 6 in some semimagic unstable nuclei, each of which has a HOtype magic number of the opposite type of nucleons. For instance, possible subshell closures have been suggested in ^{8}He^{9,10,11}, ^{14}O^{12} and ^{14}C^{12,13,14}. Whether the subshell closure at the proton number Z = 6 is predominantly driven by the shell closure at the neutron number N = 8 in ^{14}C or persists in other carbon isotopes is of fundamental importance.
The isotopic chain of carbon—with six protons and consisting of thirteen particlebound nuclei—provides an important platform to study the SO splitting of the 1p_{1/2} and 1p_{3/2} orbitals. Like other lighter isotopes, the isotopes of carbon are known to exhibit both clustering^{15,16,17} and singleparticle behaviours. Although the second excited J^{π} = 0^{+} state in ^{12}C—the famous Hoyle state and important doorway state that helps produce ^{12}C in stars—is well understood as a triplealpha state, it seems that the effect of the alphaclusterbreaking 1p_{3/2} subshell closure is important to reproduce the groundstate binding energy^{18}. For even–even neutronrich carbon isotopes, theoretical calculations using the antisymmetrized molecular dynamics (AMD)^{19}, shell model^{20,21}, as well as the ab initio nocore shell model calculation^{22} with NN + 3NFs have predicted nearconstant proton distributions, a widening gap between proton 1p_{1/2} and 1p_{3/2} singleparticle orbits, and a remarkably low proton occupancy in the 1p_{1/2} orbit, respectively. Gupta et al. ^{23}, on the other hand, have suggested the possible existence of closedshell core nuclei in ^{15,17,19}C on the basis of potential energy surfaces employing the clustercore model. Experimentally, small B(E2) values comparable to that of ^{16}O were reported from the lifetime measurements of the first excited 2^{+} \(\left( {2_1^ + } \right)\) states in ^{16,18}C^{24,25,26}. The small B(E2) values indicate small proton contributions to the transitions, and together with the theoretical predictions may imply the existence of a protonsubshell closure.
Although still not well established, the size of a nucleus, which can be defined as the rootmeansquare (rms) radius of its nucleon distribution, is expected to provide important insights on the evolution of the magic numbers. Recently, an unexpectedly large proton rms radius (denoted simply as proton radius hereafter) was reported^{27}, and suggested as a possible counterevidence for the double shell closure in ^{52}Ca^{28}. Attempts to identify any emergence of nontraditional magic numbers based on the analysis of the systematics of the experimental proton radii have been reported^{12,29}. For the 4 < Z < 10 region, the lack of experimental data on the proton radii of neutronrich nuclei due to the experimental and theoretical limitations of the isotopeshift method has hindered systematic analysis of the radii behaviour. Such systematic analysis has become possible very recently owing to the development of an alternative method to extract the proton radii of neutronrich nuclei from the chargechanging crosssection measurements.
Here we present experimental evidence for a prevalent Z = 6 subshell closure in ^{13–20}C, based on a systematic study of proton radii obtained from our recent experiments as well as the existing nuclear charge radii^{12}, electric quadrupole transition rates B(E2) between the \(2_1^ +\) and ground \(\left( {0_{{\mathrm{gs}}}^ + } \right)\) states of even–even nuclei^{30}, and atomicmass data^{31}. We show, by performing coupledcluster calculations, that the observations are supported by the ab initio nuclear model that employs the nuclear forces derived from the effective field theory of the quantum chromodynamics.
Results
Experimental details
The chargechanging cross section (denoted as σ_{CC}) of a projectile nucleus on a nuclear/proton target is defined as the total cross section of all processes that change the proton number of the projectile nucleus. Applying this method, we have determined the proton radii of ^{14}Be^{32}, ^{12–17}B^{33} and ^{12–19}C^{34,35} from the σ_{CC} measurements at GSI, Darmstadt, using secondary beams at around 900 MeV per nucleon. In addition, we have also measured σ_{CC}’s for ^{12–18}C on a ^{12}C target with secondary beams at around 45 MeV per nucleon at the exotic nuclei (EN) beam line^{36} at RCNP, Osaka University. To extract proton radii from both lowenergy data and highenergy data, we have devised a global parameter set for use in the Glaubermodel calculations. The Glauber model thus formulated was applied to the σ_{CC} data at both energies to determine the proton radii. A summary on the experiment at RCNP and the Glaubermodel analysis is given in Methods. More details can be found in ref. ^{37}
Chargechanging cross sections and proton radii
For simplicity, we show only the results for ^{17–19}C in Table 1; for results on ^{12–16}C, see ref. ^{37}. R_{p}’s are the proton radii extracted using the Glauber model formulated in ref. ^{37} The values for ^{17,18}C are the weighted mean extracted using σ_{CC}’s at the two energies, while the one for ^{19}C was extracted using the highenergy data. In determining the proton radii, we have assumed harmonicoscillator (HO)type distributions for the protons in the Glauber calculations. The uncertainties shown in the brackets include the statistical uncertainties, the experimental systematic uncertainties, and the uncertainties attributed to the choice of functional shapes, that is HO or Woods–Saxon, assumed in the calculations.
To get an overview of the isotopic dependence, we compare the proton radii of the carbon isotopes with those of the neighbouring beryllium, boron and oxygen isotopes. Figure 2 shows the proton radii for carbon, beryllium, boron and oxygen isotopes. The redfilled and blackfilled circles are the data for ^{12–19}C, beryllium and boron isotopes extracted in this and our previous work^{32,33,34,37}. For comparison, the proton radii determined with the electronscattering and isotopeshift methods^{12} are also shown in Fig. 2 (open diamonds). Our R_{p}’s for ^{12–14}C are in good agreement with the electronscattering data. In addition, we performed theoretical calculations. The small symbols connected with dashed and dotted lines shown in the figure are the results from the AMD^{19} and relativistic mean field (RMF)^{38} calculations, respectively. The bluesolid and bluedashdotted lines are the results (taken from ref.^{34}) of the ab initio coupledcluster (CC) calculations with NNLO_{sat}^{39} and the NNonly interaction NNLO_{opt}^{40}, respectively. The AMD calculations reproduce the trends of all isotopes qualitatively but overestimate the proton radii for carbon and beryllium isotopes. The RMF calculations, on the other hand, reproduce most of the proton radii of carbon and oxygen isotopes but underestimate the one of ^{12}C. Overall, the CC calculations with the NNLO_{sat} interactions reproduce the proton radii for ^{13–18}C very well. The calculations without 3NFs underestimate the radii by about 10%, thus suggesting the importance of 3NFs.
It is interesting to note that R_{p}’s are almost constant throughout the isotopic chain from ^{12}C to ^{19}C, fluctuating by less than 5%. Whereas this trend is similar to the one observed/predicted in the protonclosed shell oxygen isotopes, it is in contrast to those in the beryllium and boron isotopic chains, where the proton radii change by as much as 10% (for berylliums) or more (for borons). It is also worth noting that most theoretical calculations shown predict almost constant proton radii in carbon and oxygen isotopes. The large fluctuations observed in Be and B isotopes can be attributed to the development of cluster structures, whereas the almost constant R_{p}’s for ^{12–19}C observed in the present work may indicate an inert proton core, that is 1p_{3/2} protonsubshell closure.
Systematics of nuclear observables
Examining the Z dependence of the proton rms radii along the N = 8 isotonic chain, Angeli et al. have pointed out^{12,29} a characteristic change of slope (existence of a kink), a feature closely associated with shell closure, at Z = 6. Here, by combining our data with the recent data^{32,33,34,37}, as well as the data from ref. ^{12}, we plot the experimental R_{p}’s against proton number. To eliminate the smooth mass number dependence of the proton rms radii, we normalised all R_{p}’s by the following massdependent rms radii^{41}:
Figure 3a shows the evolution of \(R_{\mathrm{p}}{\mathrm{/}}R_{\mathrm{p}}^{{\mathrm{cal}}}\) with proton number up to Z = 22 and for isotonic chains up to N = 28. Each isotonic chain is connected by a solid line. For simplicity, only the symbols for N = 3–16 are shown in the legend in Fig. 3c; the data for N = 6–13 isotones are displayed in colours for clarity. For nuclides with more than one experimental value, we have adopted the weighted mean values. The discontinuities observed at Z = 10 and Z = 18 are due to the lack of experimental data in the protonrich region. Note the increase/change in the slope at the traditional magic numbers Z = 8 and 20. Marked kinks, similar to those observed at Z = 20, 28, 50 and 82^{29}, are observed at Z = 6 for isotonic chains from N = 7 to N = 13, indicating a possible major structural change, for example emergence of a subshell closure, at Z = 6.
The possible emergence of a protonsubshell closure at Z = 6 in neutronrich even–even carbon isotopes is also supported by the small B(E2) values observed in ^{14–20}C^{25,26,30}. Figure 3b shows the systematics of B(E2) values in Weisskopf unit (W.u.) for even–even nuclei up to Z = 22. All data are available in ref. ^{30} Nuclei with shell closures manifest themselves as minima. Besides the traditional magic number Z = 8, clear minima with B(E2) values smaller than 3 W.u. are observed at Z = 6 for N = 8, 10, 12 and 14 isotones.
To further examine the possible subshell closure at Z = 6, we consider the second derivative of binding energies defined as follows^{42}:
where S_{p}(N, Z) is the oneproton separation energy. In the absence of manybody correlations such as pairings, S_{p}(N, Z) resembles the singleparticle energy, and \(2{\mathrm{\Delta }}_{\mathrm{p}}^{(3)}(N,Z)\) yields the proton singleparticle energylevel spacing or shell gap between the last occupied (e_{p}) and first unoccupied proton orbitals (e_{p+1}) in the nucleus with Z protons (and N neutrons). To eliminate the effect of proton–proton (p–p) pairing, we subtract out the p–p pairing energies using the empirical formula: Δ_{p} = 12A^{–1/2} MeV. Figure 3c shows the systematics of e_{p} − e_{p+1} (=\(2{\mathrm{\Delta }}_{\mathrm{p}}^{(3)}\) (N, Z) − 2Δ_{p}) for evenZ nuclides. All data were evaluated from the experimental binding energies^{31}. Here, we have omitted oddZ nuclides to avoid odd–even staggering effects. The cusps observed at Z = N for all isotonic chains are due to the Wigner effect^{43}. Apart from the Z = N nuclides, sizable gaps (>2 MeV) are also observed at Z = 6 for N = 7–14, and at Z = 8 for N = 8–10 and 12–16. For clarity, we show the corresponding twodimensional lego plot in Fig. 3d.
By requiring a magic nucleus to fulfil at least two signatures in Fig. 3a–c, we conclude that we have observed a prominent protonsubshell closure at Z = 6 in ^{13–20}C. Although the empirical \(2{\mathrm{\Delta }}_{\mathrm{p}}^{(3)}\) for ^{12}C is large (~14 MeV), applying the prescription from ref. ^{44}, we obtain about 10.7 MeV for the total p–p and p–n pairing energy. This estimated large pairing energy indicates possible significant manybody correlations such as cluster correlations. We note that ^{12}C is known to be an intermediatecoupling nucleus lying in the middle of the j–j coupling and L–S coupling limits^{45}. The core is largely broken with only about 40% of the nominal (1p_{3/2})^{8} closedshell component, and the occupation number of nucleons in the 1p_{1/2} shell is as much as 1.5 from shell model calculations using the Cohen–Kurath interactions^{46}.
Discussion
It is surprising that the systematics of the proton radii, B(E2) values and the empirical protonsubshell gaps for most of the carbon isotopes are comparable to those for protonclosed shell oxygen isotopes. To understand the observed groundstate properties, that are the proton radii and subshell gap of the carbon isotopes, we performed ab initio CC calculations on ^{14,15}C using various stateoftheart chiral interactions. We employed the CC method in the singlesanddoubles approximation with perturbative triples corrections [ΛCCSD(T)]^{47} to compute the groundstate binding energies and proton radii for the closed(sub)shell ^{14}C. To compute ^{15}C (1/2^{+}), we used the particleattached equationofmotion CC (EOMCC) method^{48}, and included up to threeparticletwohole (3p–2h) and twoparticlethreehole (2p–3h) corrections as recently developed in ref. ^{49} Figure 4 shows the binding energies as functions of the proton radii for (a) ^{14}C and (b) ^{15}C. The coloured bands are the experimental values; the binding energies (redhorizontal lines) are taken from ref. ^{31}, while proton radii are from ref. ^{37} (orange bands) and the electronscattering data^{12} (green band). The filled black symbols are CC predictions with the NN + 3NF chiral interactions from ref. ^{50} labelled 2.0/2.0 (EM)(black square), 2.0/2.0 (PWA)(black downwardpointing triangle), 1.8/2.0 (EM)(black circle), 2.2/2.0 (EM)(black diamond), 2.8/2.0 (EM)(black triangle), and NNLO_{sat}^{39} (black star). Here, the NN interactions are the nexttonexttonexttoleading order (N^{3}LO) chiral interaction from ref. ^{51}, evolved to lower cutoffs (1.8/2.0/2.2/2.8 fm^{–1}) via the similarityrenormalisationgroup (SRG) method^{52}, while the 3NF is taken at NNLO with a cutoff of 2.0 fm^{–1} and adjusted to the triton binding energy and ^{4}He charge radius. The error bars are the estimated theoretical uncertainties due to truncations of the employed method and model space. For details on the CC method and error estimation, see refs. ^{5,53}. Note that the error bars for the binding energies are smaller than the symbols. Depending on the NN cutoff, the calculated binding energy correlates strongly with the calculated proton radius. In addition, we performed the CC calculations with chiral effective interactions without 3NFs, that are the NNonly EM interactions with NN cutoffs at 1.8 (white circle), 2.0 (white square), 2.2 (white diamond) and 2.8 fm^{–1} (white triangle), and the NNonly part of the chiral interaction NNLO_{sat} (white downwardpointing triangle). Overall, most calculations that include 3NFs reproduce the experimental proton radii well. For the binding energies, the calculations with the EM(1.8/2.0) and NNLO_{sat} interactions reproduce both data very well. It is important to note that without 3NFs the calculated proton radii are about 9–15% (18%) smaller, while the ground states are overbound by as much as about 24% (26%) for ^{14}C (^{15}C). These results highlight the importance of comparing both experimental observables to examine the employed interactions.
The importance of the Fujita–Miyazawa type^{54} or the chiral NNLO 3NFs^{55,56} in reproducing the binding energies and the drip lines of nitrogen and oxygen isotopes have been suggested in recent theoretical studies. Here, to shed light on the role of 3NFs on the observed subshell gap, that is the SO splitting in the carbon isotopes, we investigate the evolution of oneproton separation energies for carbon and oxygen isotopes. In Fig. 5, the horizontal bars represent the experimental oneproton addition (\(\epsilon _{ + {\mathrm{p}}}\)) and removal (\(\epsilon _{  {\mathrm{p}}}\)) energies for (a) carbon and (b) oxygen isotopes deduced from oneproton separation energies, that are binding energies of boron to fluorine isotopes, and the excitation energies of the lowest 3/2^{−} states in the odd–even nitrogen isotopes. The dotted bars indicate the adopted values for the observed excited states in ^{19,21}N, which have been tentatively assigned as 3/2^{−}^{57}. Other experimental data are taken from refs. ^{31,58,59}. For comparison, we show the oneproton addition and removal energies (blue symbols) calculated using the shell model with the YSOX interaction^{60}, which was constructed from a monopolebased universal interaction (V_{MU}). Because the phenomenological effective twobody interactions were determined by fitting experimental data, they are expected to partially include the threenucleon effect and thus can reproduce relatively well the groundstate energies, drip lines, energy levels, as well as the electric and spin properties of carbon and oxygen isotopes. As shown in Fig. 5, the shell model calculations reproduce the \(\epsilon _{ \pm {\mathrm{p}}}\)’s for carbon and oxygen isotopes very well.
As mentioned earlier, in the absence of manybody correlations, \(\epsilon _{ \pm {\mathrm{p}}}\) resemble the proton singleparticle energies, and the gap between them can be taken as the (sub)shell gap. In the following, we consider ^{14,15}C and the closedshell ^{14,16,22}O isotopes in more detail. We computed their groundstate binding energies and those of their neighbouring isotones ^{13,14}B, ^{13,15,16,21}N and ^{15,17,23}F. We applied the ΛCCSD(T) and the particleattached/removed EOMCC methods to compute the binding energies for the closed(sub)shell and openshell nuclei, respectively. The groundstate binding energies of ^{14}B (2^{−}) and ^{16}N (2^{−}) were computed using the EOMCC method with reference to ^{14}C and ^{16}O employing the chargeexchange EOMCC technique^{61}. Results of the CC calculations on ^{14,15}C and ^{14,16,22}O with and without 3NFs are shown by the redsolid and reddashed lines, respectively. Here, we have opted for EM(1.8/2.0 fm^{−1}), which yield the smallest chisquare value for the calculated and experimental binding energies considered, as the NN+3NF interactions. For the NNonly interaction, we show the calculations with EM(2.8 fm^{−1}). The calculated \(\epsilon _{  {\mathrm{p}}}\)(3/2^{−}) for ^{22}O with EM(2.8 fm^{−1}) (and other NNonly interactions) has an unrealistic positive value, and is thus omitted. We found the norms of the wave functions for the oneparticle (1p) 1/2^{−} and onehole (1h) 3/2^{−} states of ^{14}C, and the two corresponding 1p and 1h states of ^{15}C (2^{−} states in ^{14}B and ^{16}N) to be almost 90%. The calculations suggest that these states can be accurately interpreted by having dominant singleparticle structure, and that the gaps between these 1p–1h states resemble the protonsubshell gaps. It is obvious from the figure that the calculations with the NN + 3NF interactions reproduce the experimental \(\epsilon _{ \pm {\mathrm{p}}}\) for ^{14,15}C and ^{14,16,22}O very well. Overall, the calculations without 3NFs predict overbound proton states, and in the case of ^{14,15}C, much reduced subshell gaps. These results suggest that ^{14}C is a doublymagic nucleus, and ^{15}C a protonclosed shell nucleus.
Our results show that the phenomenon of large spin–orbit splitting is indeed universal in atomic nuclei, and the magic number 6 is as prominent as other classical SOoriginated magic numbers such as 28. Although we have shown only results for ^{14,15}C, we expect further systematic and detailed theoretical analyses on other carbon isotopes, in particular ab initio calculations using realistic and/or chiral interactions, to provide quantitative insights on the neutronnumber dependence of the SO splitting and its origin. It will be interesting to understand also the origins of the diverse structures in ^{12}C.
Finally, we would like to point out that an inert ^{14}C core, built on the N = 8 closed shell, has been postulated to explain several experimental data for ^{15,16}C. For instance, a ^{14}C + n model was successfully applied^{62} to explain the consistency between the measured gfactor and the singleparticlemodel prediction (the Schmidt value) of the excited 5/2^{+} state in ^{15}C. Wiedeking et al. ^{25}, on the other hand, have explained the small B(E2) value in ^{16}C assuming a ^{14}C + n + n model in the shellmodel calculation. In terms of spectroscopy studies using transfer reactions, the results from the ^{14}C(d, p)^{15}C^{63} and ^{15}C(d, p)^{16}C^{64} measurements are also consistent with the picture of a stable ^{14}C core. On the proton side, a possible consolidation of the 1p_{3/2} protonsubshell closure when moving from ^{12}C to ^{14}C was reported decades ago from the measurements of the proton pickup (d,^{3}He) reaction on ^{12,13,14}C targets^{65}, consistent with shell model predictions. An attempt to study the groundstate configurations with protons outside the 1p_{3/2} orbital in ^{14,15}C has also been reported^{66} very recently. To further investigate the protonsubshell closure in the neutronrich carbon isotopes, more experiments using oneproton transfer and/or knockout reactions induced by radioactive boron, carbon and nitrogen beams at facilities such as ATLAS, FAIR, FRIB, RCNP, RIBF and SPIRAL2 are anticipated.
Methods
Experiment and data analysis
Secondary ^{12–18}C beams were produced, in separate runs, by projectile fragmentation of ^{22}Ne^{10+} ions at 80 MeV per nucleon incident on a ^{9}Be (production) target with thickness ranging from 1.0 to 5.0 mm. The carbon beam of interest was selected by setting the appropriate particle magnetic rigidities using the RCNP EN fragment separator. The carbon beam thus produced was transported to the experimental area, and directed onto a 450mg cm^{−2}thick natural carbon (reaction) target. The incident beam was identified by the measurements of energy loss in a 320mmthick silicon detector, and the time of flight (TOF) between the production and reaction targets. The TOF was determined from the timing information obtained with a 100μmthick plastic scintillation detector placed before the reaction target and the radiofrequency signal from the accelerator. Particles exiting the reaction target were detected by a multisampling ionisation chamber (MUSIC), consisting of eight anodes and nine cathodes, before being stopped in a 7cmthick NaI(Tl) scintillation detector. The outgoing particles were identified using the energyloss and totalenergy information obtained with the MUSIC and NaI(Tl) detectors. Data acquisition was performed using the software package babirlDAQ^{67}. The chargechanging cross sections were measured using the transmission method taking into account the geometrical acceptance of the MUSIC and NaI(Tl) detectors. In the present transmission method, the numbers of incident carbon beam and outgoing carbon particles, including lighter carbon isotopes, were identified and counted.
Proton radii and Glaubermodel analysis
The pointproton rootmeansquare radius is defined as follows:
where ρ_{p}(r) is the proton density distribution, r is the radial vector, and r is the radius. To extract the proton radii from the measured chargechanging cross sections, we performed reaction calculations using the recently formulated Glauber model^{37} within the opticallimit approximation. We assumed that chargechanging cross section depends only on the proton density distribution in the carbon projectile. By adopting a simple oneparameter HO or a twoparameter Woods–Saxon (WS) density distribution for the protons, we determined the parameter(s) so as to reproduce the experimental data. R_{p} is then calculated by substituting the obtained proton density distribution into Eq. (2). The difference (about 0.5%) between the R_{p} values determined with different functional forms was taken as the systematic uncertainty. The HOtype and WStype density distributions are given by:
where \(\rho _{\mathrm{0}}^{{\mathrm{HO}}}\) and \(\rho _{\mathrm{0}}^{{\mathrm{WS}}}\) are the central densities, which are uniquely determined by the conservation of proton number (Z). R_{HO} is the HO width parameter, while the parameters R_{WS} and a are the halfdensity radius and diffuseness, respectively.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank T. Shima, H. Toki, K. Ogata and H. Horiuchi for discussion, K. Hebeler for providing matrix elements in Jacobi coordinates for 3NFs at NNLO, and C. Yuan for comments. H.J.O. and I.T. thank A. Tohsaki and his spouse, D.T.T. and T.T.N. acknowledge RCNP Visiting Young Scientist Support Program, D.T.T. and T.H.H. thank Nishimura International Scholarship Foundation and Matsuda Yosahichi Memorial Foreign Student Scholarship, respectively, for support. This work was supported in part by Hirose International Scholarship Foundation, the JSPSVAST Bilateral Joint Research Project, GrandinAid for Scientific Research Nos. 20244030, 20740163 and 23224008 from Japan Monbukagakusho, the Office of Nuclear Physics, U.S. Department of Energy, under grants DEFG0296ER40963, DESC0008499 (NUCLEI SciDAC collaboration), the Field Work Proposal ERKBP57 at Oak Ridge National Laboratory (ORNL), and the Vietnam government under the Program of Development in Physics by 2020. Computer time was provided by the Innovative and Novel Computational Impact on Theory and Experiment (INCITE) program. This research used resources of RCNP Accelerator Facility and the Oak Ridge Leadership Computing Facility located at ORNL, which is supported by the Office of Science of the Department of Energy under Contract No. DEAC0500OR22725.
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H.J.O. initiated the project, performed systematic analysis, prepared the figures and wrote the manuscript. D.T.T. performed data analysis and Glaubermodel calculations. G.H., T.D.M. and G.R.J. performed the CC calculations. T.S. and T.O. performed the shell model calculations. Y.K.E. and L.S.G. performed the AMD and RMF calculations, respectively. D.T.T., H.J.O., N.A., S.T., I.T., T.T.N., Y.A., P.Y.C., M.F., H.G., M.N.H., T.H., T.H.H., E.I., A.I., R.K., T.K., L.H.K., W.P.L., K.M., M.M., S.M., D.N., N.D.N., D.N., A.O., P.P.R., H.S., C.S., J.T., M.T., R.W. and T.Y. performed the experiments. All authors discussed and commented on the manuscript.
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Tran, D.T., Ong, H.J., Hagen, G. et al. Evidence for prevalent Z = 6 magic number in neutronrich carbon isotopes. Nat Commun 9, 1594 (2018). https://doi.org/10.1038/s4146701804024y
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