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 so-called 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 SO-originated magic number (subshell closure) at the proton number six in 13–20C obtained from systematic analysis of point-proton distribution radii, electromagnetic transition rates and atomic masses of light nuclei. Performing ab initio calculations on 14,15C, we show that the observed proton distribution radii and subshell closure can be explained by the state-of-the-art nuclear theory with chiral nucleon–nucleon and three-nucleon forces, which are rooted in the quantum chromodynamics.

We have measured for the first time the charge-changing cross sections (σCC) of 12−16 C on a 12 C target at energies below 100A MeV. To analyze these low-energy data, we have developed a finiterange Glauber model with a global parameter set within the optical-limit approximation which is applicable to reaction cross section (σR) and σCC measurements at incident energies from 10A to 2100A MeV. Adopting the proton-density distribution of 12 C known from the electron-scattering data, as well as the bare total nucleon-nucleon cross sections, and the real-to-imaginary-part ratios of the forward proton-proton elastic scattering amplitude available in the literatures, we determine the energy-dependent slope parameter βpn of the proton-neutron elastic differential cross section so as to reproduce the existing σR and interaction-cross-section data for 12 C+ 12 C over a wide range of incident energies. The Glauber model thus formulated is applied to calculate the σR's of 12 C on a 9 Be and 27 Al targets at various incident energies. Our calculations show excellent agreement with the experimental data. Applying our model to the σR and σCC for the "neutron-skin"16 C nucleus, we reconfirm the importance of measurements at incident energies below 100A MeV. The proton root-mean-square radii of 12−16 C are extracted using the measured σCC's and the existing σR data. The results for 12−14 C are consistent with the values from the electron scatterings, demonstrating the feasibility, usefulness of the σCC measurement and the present Glauber model.

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