Introduction

Atomic nuclei—the finite quantum many-body 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 magic-number nuclei. The nuclear magic numbers, as we know in stable and naturally-occurring nuclei, consist of two different series of numbers. The first series—2, 8, 20—is attributed to the harmonic-oscillator (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 so-called j–j coupling scheme that helped explain1,2 completely the magic numbers, and won Goeppert-Mayer 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 theories3,4,5 with two- (NN) and three-nucleon forces (3NFs). Although the theoretical study6 of the SO splitting of the 1p1/2 and 1p3/2 single-particle states in 15N has suggested possible roles of two-body SO and tensor forces, as well as three-body forces, the discovery of a prevalent SO-type magic number 6 is expected to offer unprecedented opportunities to understand its origins.

Fig. 1
figure 1

Nuclear shell structure. The left diagram is the shell structure for a harmonic-oscillator potential plus a small orbital angular momentum (l2) term. The right diagram shows the splitting of the single-particle orbitals by an additional spin–orbit coupling force

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In her Nobel lecture, Goeppert-Mayer had mentioned the magic numbers 6 and 14—which she described as hardly noticeable—but surmised that the energy gap between the 1p1/2 and 1p3/2 orbitals due to the SO force is fairly small7. That the j–j coupling scheme appears to fail in the p-shell light nuclei was discussed and attributed to their tendency to form clusters of nucleons8. 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 HO-type magic number of the opposite type of nucleons. For instance, possible subshell closures have been suggested in 8He9,10,11, 14O12 and 14C12,13,14. Whether the subshell closure at the proton number Z = 6 is predominantly driven by the shell closure at the neutron number N = 8 in 14C or persists in other carbon isotopes is of fundamental importance.

The isotopic chain of carbon—with six protons and consisting of thirteen particle-bound nuclei—provides an important platform to study the SO splitting of the 1p1/2 and 1p3/2 orbitals. Like other lighter isotopes, the isotopes of carbon are known to exhibit both clustering15,16,17 and single-particle behaviours. Although the second excited Jπ = 0+ state in 12C—the famous Hoyle state and important doorway state that helps produce 12C in stars—is well understood as a triple-alpha state, it seems that the effect of the alpha-cluster-breaking 1p3/2 subshell closure is important to reproduce the ground-state binding energy18. For even–even neutron-rich carbon isotopes, theoretical calculations using the anti-symmetrized molecular dynamics (AMD)19, shell model20,21, as well as the ab initio no-core shell model calculation22 with NN + 3NFs have predicted near-constant proton distributions, a widening gap between proton 1p1/2 and 1p3/2 single-particle orbits, and a remarkably low proton occupancy in the 1p1/2 orbit, respectively. Gupta et al. 23, on the other hand, have suggested the possible existence of closed-shell core nuclei in 15,17,19C on the basis of potential energy surfaces employing the cluster-core model. Experimentally, small B(E2) values comparable to that of 16O were reported from the lifetime measurements of the first excited 2+ \(\left( {2_1^ + } \right)\) states in 16,18C24,25,26. The small B(E2) values indicate small proton contributions to the transitions, and together with the theoretical predictions may imply the existence of a proton-subshell closure.

Although still not well established, the size of a nucleus, which can be defined as the root-mean-square (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 reported27, and suggested as a possible counter-evidence for the double shell closure in 52Ca28. Attempts to identify any emergence of non-traditional magic numbers based on the analysis of the systematics of the experimental proton radii have been reported12,29. For the 4 < Z < 10 region, the lack of experimental data on the proton radii of neutron-rich nuclei due to the experimental and theoretical limitations of the isotope-shift 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 neutron-rich nuclei from the charge-changing cross-section measurements.

Here we present experimental evidence for a prevalent Z = 6 subshell closure in 13–20C, based on a systematic study of proton radii obtained from our recent experiments as well as the existing nuclear charge radii12, electric quadrupole transition rates B(E2) between the \(2_1^ +\) and ground \(\left( {0_{{\mathrm{gs}}}^ + } \right)\) states of even–even nuclei30, and atomic-mass data31. We show, by performing coupled-cluster 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 charge-changing cross section (denoted as σCC) 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 14Be32, 12–17B33 and 12–19C34,35 from the σCC measurements at GSI, Darmstadt, using secondary beams at around 900 MeV per nucleon. In addition, we have also measured σCC’s for 12–18C on a 12C target with secondary beams at around 45 MeV per nucleon at the exotic nuclei (EN) beam line36 at RCNP, Osaka University. To extract proton radii from both low-energy data and high-energy data, we have devised a global parameter set for use in the Glauber-model calculations. The Glauber model thus formulated was applied to the σCC data at both energies to determine the proton radii. A summary on the experiment at RCNP and the Glauber-model analysis is given in Methods. More details can be found in ref. 37

Charge-changing cross sections and proton radii

For simplicity, we show only the results for 17–19C in Table 1; for results on 12–16C, see ref. 37. Rp’s are the proton radii extracted using the Glauber model formulated in ref. 37 The values for 17,18C are the weighted mean extracted using σCC’s at the two energies, while the one for 19C was extracted using the high-energy data. In determining the proton radii, we have assumed harmonic-oscillator (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.

Table 1 Cross sections and proton radii
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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 red-filled and black-filled circles are the data for 12–19C, beryllium and boron isotopes extracted in this and our previous work32,33,34,37. For comparison, the proton radii determined with the electron-scattering and isotope-shift methods12 are also shown in Fig. 2 (open diamonds). Our Rp’s for 12–14C are in good agreement with the electron-scattering 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 AMD19 and relativistic mean field (RMF)38 calculations, respectively. The blue-solid and blue-dash-dotted lines are the results (taken from ref.34) of the ab initio coupled-cluster (CC) calculations with NNLOsat39 and the NN-only interaction NNLOopt40, 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 12C. Overall, the CC calculations with the NNLOsat interactions reproduce the proton radii for 13–18C very well. The calculations without 3NFs underestimate the radii by about 10%, thus suggesting the importance of 3NFs.

Fig. 2
figure 2

Proton radii. Results are shown for carbon, beryllium, boron and oxygen isotopes. The red-filled and black-filled circles are, respectively, the proton radii from this and our recent work32,33,34,37. The open diamonds are the data from electron-scattering and isotope-shift methods12. The error bars for the red-filled circles include the statistical and experimental systematic uncertainties, as well as the uncertainties due to the choice of density distributions. The error bars for other experimental data are taken from the literature. The small symbols connected with dashed and dotted lines are the predictions from the AMD19 and RMF38 models, respectively. The small blue symbols with solid and dash-dotted lines are the results from the ab initio coupled-cluster calculations with NNLOsat39 and the NN-only interaction NNLOopt40

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It is interesting to note that Rp’s are almost constant throughout the isotopic chain from 12C to 19C, fluctuating by less than 5%. Whereas this trend is similar to the one observed/predicted in the proton-closed 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 Rp’s for 12–19C observed in the present work may indicate an inert proton core, that is 1p3/2 proton-subshell 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 out12,29 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 data32,33,34,37, as well as the data from ref. 12, we plot the experimental Rp’s against proton number. To eliminate the smooth mass number dependence of the proton rms radii, we normalised all Rp’s by the following mass-dependent rms radii41:

$$R_{\mathrm{p}}^{{\mathrm{cal}}} = \sqrt {3{\mathrm{/}}5} \left( {1.15 + 1.80A^{ - 2/3} - 1.20A^{ - 4/3}} \right)A^{1/3}{\mathrm{fm}}.$$

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 proton-rich 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 8229, 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.

Fig. 3
figure 3

Systematics of nuclear observables. Evolution of a \(R_{\mathrm{p}}{\mathrm{/}}R_{\mathrm{p}}^{{\mathrm{cal}}}\), b B(E2) and c ep − ep+1 with proton number up to Z = 22 and for isotonic chains up to N = 28. Vertical dotted and thin-dashed lines denote positions of the traditional proton magic numbers and Z = 6, respectively. The error bars for data in a are evaluated using the errors, while the ones in b are the errors from the literature. For clarity, the error bars in c, some of which are slightly larger than the symbols, are not shown. d Two-dimensional lego plot of c

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The possible emergence of a proton-subshell closure at Z = 6 in neutron-rich even–even carbon isotopes is also supported by the small B(E2) values observed in 14–20C25,26,30. 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 follows42:

$${\mathrm{\Delta }}_{\mathrm{p}}^{(3)}(N,Z) \equiv ( - 1)^Z\left[ {S_{\mathrm{p}}(N,Z) - S_{\mathrm{p}}(N,Z + 1)} \right]{\mathrm{/}}2,$$
(1)

where Sp(N, Z) is the one-proton separation energy. In the absence of many-body correlations such as pairings, Sp(N, Z) resembles the single-particle energy, and \(2{\mathrm{\Delta }}_{\mathrm{p}}^{(3)}(N,Z)\) yields the proton single-particle energy-level spacing or shell gap between the last occupied (ep) and first unoccupied proton orbitals (ep+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: Δp = 12A–1/2 MeV. Figure 3c shows the systematics of ep − ep+1 (=\(2{\mathrm{\Delta }}_{\mathrm{p}}^{(3)}\) (N, Z) − 2Δp) for even-Z nuclides. All data were evaluated from the experimental binding energies31. Here, we have omitted odd-Z nuclides to avoid odd–even staggering effects. The cusps observed at Z = N for all isotonic chains are due to the Wigner effect43. 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 two-dimensional 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 proton-subshell closure at Z = 6 in 13–20C. Although the empirical \(2{\mathrm{\Delta }}_{\mathrm{p}}^{(3)}\) for 12C 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 many-body correlations such as cluster correlations. We note that 12C is known to be an intermediate-coupling nucleus lying in the middle of the j–j coupling and L–S coupling limits45. The core is largely broken with only about 40% of the nominal (1p3/2)8 closed-shell component, and the occupation number of nucleons in the 1p1/2 shell is as much as 1.5 from shell model calculations using the Cohen–Kurath interactions46.

Discussion

It is surprising that the systematics of the proton radii, B(E2) values and the empirical proton-subshell gaps for most of the carbon isotopes are comparable to those for proton-closed shell oxygen isotopes. To understand the observed ground-state properties, that are the proton radii and subshell gap of the carbon isotopes, we performed ab initio CC calculations on 14,15C using various state-of-the-art chiral interactions. We employed the CC method in the singles-and-doubles approximation with perturbative triples corrections [Λ-CCSD(T)]47 to compute the ground-state binding energies and proton radii for the closed-(sub)shell 14C. To compute 15C (1/2+), we used the particle-attached equation-of-motion CC (EOM-CC) method48, and included up to three-particle-two-hole (3p–2h) and two-particle-three-hole (2p–3h) corrections as recently developed in ref. 49 Figure 4 shows the binding energies as functions of the proton radii for (a) 14C and (b) 15C. The coloured bands are the experimental values; the binding energies (red-horizontal lines) are taken from ref. 31, while proton radii are from ref. 37 (orange bands) and the electron-scattering data12 (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 downward-pointing triangle), 1.8/2.0 (EM)(black circle), 2.2/2.0 (EM)(black diamond), 2.8/2.0 (EM)(black triangle), and NNLOsat39 (black star). Here, the NN interactions are the next-to-next-to-next-to-leading order (N3LO) chiral interaction from ref. 51, evolved to lower cutoffs (1.8/2.0/2.2/2.8 fm–1) via the similarity-renormalisation-group (SRG) method52, while the 3NF is taken at NNLO with a cutoff of 2.0 fm–1 and adjusted to the triton binding energy and 4He 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 NN-only 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 NN-only part of the chiral interaction NNLOsat (white downward-pointing 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 NNLOsat 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 14C (15C). These results highlight the importance of comparing both experimental observables to examine the employed interactions.

Fig. 4
figure 4

Binding energies versus proton radii. The results for a 14C and b 15C are shown. The coloured bands and red-horizontal lines are the experimental values. The green band represents the proton radius from the electron scattering. The filled black symbols are the CC predictions with SRG-evolved NN + 3NF chiral effective interactions at different NN/3NF cutoffs and NNLOsat, whereas the open symbols are the predictions with the NN-only EM and NNLOsat interactions. The error bars are the estimated theoretical uncertainties due to truncations of the employed method and model space53. See text for details

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The importance of the Fujita–Miyazawa type54 or the chiral NNLO 3NFs55,56 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 one-proton separation energies for carbon and oxygen isotopes. In Fig. 5, the horizontal bars represent the experimental one-proton addition (\(\epsilon _{ + {\mathrm{p}}}\)) and removal (\(\epsilon _{ - {\mathrm{p}}}\)) energies for (a) carbon and (b) oxygen isotopes deduced from one-proton 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 19,21N, which have been tentatively assigned as 3/2−57. Other experimental data are taken from refs. 31,58,59. For comparison, we show the one-proton addition and removal energies (blue symbols) calculated using the shell model with the YSOX interaction60, which was constructed from a monopole-based universal interaction (VMU). Because the phenomenological effective two-body interactions were determined by fitting experimental data, they are expected to partially include the three-nucleon effect and thus can reproduce relatively well the ground-state 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.

Fig. 5
figure 5

Shell evolution. Empirical one-proton addition (\(\epsilon _{ + {\mathrm{p}}}\)) and removal (\(\epsilon _{ - {\mathrm{p}}}\)) energies (horizontal bars) for a carbon, and b oxygen isotopes deduced from one-proton separation energies 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 19,21N, which have been tentatively assigned as 3/2−57. Other experimental data are taken from refs. 31,58,59. The blue symbols are the shell model calculations using the YSOX interactions60. Results of the CC calculations with and without 3NFs are shown by the red-solid and red-dashed lines, respectively

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As mentioned earlier, in the absence of many-body correlations, \(\epsilon _{ \pm {\mathrm{p}}}\) resemble the proton single-particle energies, and the gap between them can be taken as the (sub)shell gap. In the following, we consider 14,15C and the closed-shell 14,16,22O isotopes in more detail. We computed their ground-state binding energies and those of their neighbouring isotones 13,14B, 13,15,16,21N and 15,17,23F. We applied the Λ-CCSD(T) and the particle-attached/removed EOM-CC methods to compute the binding energies for the closed-(sub)shell and open-shell nuclei, respectively. The ground-state binding energies of 14B (2−) and 16N (2−) were computed using the EOM-CC method with reference to 14C and 16O employing the charge-exchange EOM-CC technique61. Results of the CC calculations on 14,15C and 14,16,22O with and without 3NFs are shown by the red-solid and red-dashed lines, respectively. Here, we have opted for EM(1.8/2.0 fm−1), which yield the smallest chi-square value for the calculated and experimental binding energies considered, as the NN+3NF interactions. For the NN-only interaction, we show the calculations with EM(2.8 fm−1). The calculated \(\epsilon _{ - {\mathrm{p}}}\)(3/2−) for 22O with EM(2.8 fm−1) (and other NN-only interactions) has an unrealistic positive value, and is thus omitted. We found the norms of the wave functions for the one-particle (1p) 1/2− and one-hole (1h) 3/2− states of 14C, and the two corresponding 1p and 1h states of 15C (2− states in 14B and 16N) to be almost 90%. The calculations suggest that these states can be accurately interpreted by having dominant single-particle structure, and that the gaps between these 1p–1h states resemble the proton-subshell gaps. It is obvious from the figure that the calculations with the NN + 3NF interactions reproduce the experimental \(\epsilon _{ \pm {\mathrm{p}}}\) for 14,15C and 14,16,22O very well. Overall, the calculations without 3NFs predict overbound proton states, and in the case of 14,15C, much reduced subshell gaps. These results suggest that 14C is a doubly-magic nucleus, and 15C a proton-closed 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 SO-originated magic numbers such as 28. Although we have shown only results for 14,15C, 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 neutron-number dependence of the SO splitting and its origin. It will be interesting to understand also the origins of the diverse structures in 12C.

Finally, we would like to point out that an inert 14C core, built on the N = 8 closed shell, has been postulated to explain several experimental data for 15,16C. For instance, a 14C + n model was successfully applied62 to explain the consistency between the measured g-factor and the single-particle-model prediction (the Schmidt value) of the excited 5/2+ state in 15C. Wiedeking et al. 25, on the other hand, have explained the small B(E2) value in 16C assuming a 14C + n + n model in the shell-model calculation. In terms of spectroscopy studies using transfer reactions, the results from the 14C(d, p)15C63 and 15C(d, p)16C64 measurements are also consistent with the picture of a stable 14C core. On the proton side, a possible consolidation of the 1p3/2 proton-subshell closure when moving from 12C to 14C was reported decades ago from the measurements of the proton pick-up (d,3He) reaction on 12,13,14C targets65, consistent with shell model predictions. An attempt to study the ground-state configurations with protons outside the 1p3/2 orbital in 14,15C has also been reported66 very recently. To further investigate the proton-subshell closure in the neutron-rich carbon isotopes, more experiments using one-proton 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 12–18C beams were produced, in separate runs, by projectile fragmentation of 22Ne10+ ions at 80 MeV per nucleon incident on a 9Be (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 450-mg cm−2-thick natural carbon (reaction) target. The incident beam was identified by the measurements of energy loss in a 320-mm-thick 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 100-μm-thick plastic scintillation detector placed before the reaction target and the radio-frequency 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 7-cm-thick NaI(Tl) scintillation detector. The outgoing particles were identified using the energy-loss and total-energy information obtained with the MUSIC and NaI(Tl) detectors. Data acquisition was performed using the software package babirlDAQ67. The charge-changing 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 Glauber-model analysis

The point-proton root-mean-square radius is defined as follows:

$$R_{\mathrm{p}} \equiv \left\langle {r_{\mathrm{p}}^2} \right\rangle ^{1/2} = \left( {{\int} {\rho _{\mathrm{p}}({\bf{r}})r^2\mathrm{d}{\bf{r}}} } \right)^{1/2},$$
(2)

where ρp(r) is the proton density distribution, r is the radial vector, and r is the radius. To extract the proton radii from the measured charge-changing cross sections, we performed reaction calculations using the recently formulated Glauber model37 within the optical-limit approximation. We assumed that charge-changing cross section depends only on the proton density distribution in the carbon projectile. By adopting a simple one-parameter HO or a two-parameter Woods–Saxon (WS) density distribution for the protons, we determined the parameter(s) so as to reproduce the experimental data. Rp is then calculated by substituting the obtained proton density distribution into Eq. (2). The difference (about 0.5%) between the Rp values determined with different functional forms was taken as the systematic uncertainty. The HO-type and WS-type density distributions are given by:

$$\begin{array}{l}\rho _{\mathrm{p}}^{{\mathrm{HO}}}\left( {Z,R_{{\mathrm{HO}}},r} \right) = \rho _{\mathrm{0}}^{{\mathrm{HO}}}{\kern 1pt} {\mathrm{exp}}\left[ { - \left( {\frac{r}{{R_{{\mathrm{HO}}}}}} \right)^2} \right]\left[ {1 + \frac{{Z - 2}}{3}\left( {\frac{r}{{R_{{\mathrm{HO}}}}}} \right)^2} \right],\\ \rho _{\mathrm{p}}^{{\mathrm{WS}}}\left( {R_{{\mathrm{WS}}},a,r} \right) = \rho _{\mathrm{0}}^{{\mathrm{WS}}}\left[ {1 + {\mathrm{exp}}\frac{{\left( {r - R_{{\mathrm{WS}}}} \right)}}{a}} \right]^{ - 1}\end{array}$$

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). RHO is the HO width parameter, while the parameters RWS and a are the half-density radius and diffuseness, respectively.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.