Constraining the shell-versus-$\alpha$-cluster composition of the $^{12}$C and $^{16}$O ground states through density-profile-sensitive elastic scattering

Whether the ground states of $^{12}$C and $^{16}$O are dominated by single-particle shell-model motion or by $\alpha$-cluster ($^{4}$He subunit) correlations remains contested. Horiuchi and Itagaki [Phys. Rev. C 107, L021304 (2023)] showed, with radius-constrained trial states of the antisymmetrized quasi-cluster model, that $\alpha$ clustering leaves measurable imprints in one-body density profiles — enhanced elastic proton-scattering cross sections and electron-scattering charge form factors near the first diffraction peak — but that the internal density differences are only probed at momentum transfers beyond the second diffraction minimum, where existing data provide no constraint for $^{16}$O at $q\gtrsim 3\ \mathrm{fm}^{-1}$. The open problem is to settle this competition quantitatively: determine the $\alpha$-cluster admixture and the internal density-profile differences of the $^{12}$C and $^{16}$O ground states from elastic electron and proton scattering including the high-momentum-transfer region. The diagnostic could also be extended to heavier $4N$ nuclei and neutron-rich unstable systems, including unstable-beam measurements via inverse kinematics, but this extension is an outlook rather than part of the posed question.

Background

Two competing pictures describe the structure of light nuclei. In the single-particle shell model, protons and neutrons move independently in a self-consistent mean field and fill orbitals, with angular momenta coupled in the so-called $jj$-coupling scheme (each nucleon's orbital and spin angular momenta first combine into a total angular momentum $j$, and these $j$'s then couple to the nuclear spin). In the cluster picture, the large binding energy of $^{4}$He causes two protons and two neutrons to form tightly bound $\alpha$ particles that behave as subunits of the nucleus. These pictures predict visibly different one-body density profiles $\rho(r)$: a developed $\alpha$-cluster configuration has a sharper nuclear surface and a more depressed central density than a shell-model configuration of the same charge radius (the root-mean-square radius of the proton distribution, a well-measured ground-state observable). For $^{12}$C this competition is a long-standing theme of nuclear-structure physics: the excited Hoyle state — the famous $0^{+}$ state of $^{12}$C at 7.65 MeV that governs carbon production in stars — is known to have prominent $\alpha$-cluster character (Chernykh et al., Phys. Rev. Lett. 98, 032501 (2007)), and the question of how much $\alpha$ clustering is present already in the ground states of $^{12}$C and $^{16}$O is unresolved. The (p,p$\alpha$) knockout reaction recently indicated unexpectedly enhanced $\alpha$-particle preformation even in the medium-mass nucleus $^{48}$Ti (Taniguchi et al., Phys. Rev. C 103, L031305 (2021)), sharpening interest in model-independent measures of cluster content. Density profiles are probed by elastic scattering. Elastic electron scattering measures the charge form factor $F(q)$, the Fourier transform of the charge density, where $q$ is the momentum transfer; $|F(q)|^{2}$ shows diffraction minima whose positions and depths encode the density distribution, with progressively larger $q$ probing progressively more internal regions of the nucleus. Elastic proton-nucleus scattering at ~1 GeV, described by the Glauber model (a high-energy multiple-scattering theory in which the projectile scatters eikonal-style from the target density), is mainly sensitive to the surface region around the half-density radius. Hamiltonian-based calculations of such observables exist for $^{12}$C — e.g., the quantum Monte Carlo charge form factor of Lovato et al., Phys. Rev. Lett. 111, 092501 (2013) — but have not been combined with a controlled shell-versus-cluster decomposition. Horiuchi and Itagaki [Phys. Rev. C 107, L021304 (2023)] introduced a simple diagnostic: trial states generated by the antisymmetrized quasi-cluster model (AQCM), a framework in which both $jj$-coupling shell-model and $\alpha$-cluster configurations — and mixtures of them — are written as intrinsic Slater determinants in a single scheme, with parameters fixed to reproduce the experimental charge radii. Computing the proton-nucleus elastic cross sections and the elastic charge form factors of these states and comparing with data (the electron data of Sick and McCarthy, Nucl. Phys. A 150, 631 (1970), and proton data at 1 GeV), they found that a significant $\alpha$-cluster component is present in both nuclei: the observables near the first peak are enhanced when clustering is developed, and comparisons with the data support mixed shell-cluster content — for the proton cross sections the mixed configuration fits $^{12}$C best, while for $^{16}$O, where only the pure four-$\alpha$ cluster configuration was tested, that cluster configuration fits best. However, the internal density differences between configurations only separate at momentum transfers beyond the second diffraction minimum, and there the comparison is unconstrained — for $^{16}$O the existing data give no constraint at $q\gtrsim 3\ \mathrm{fm}^{-1}$. The authors also note that the proton probe can be extended to unstable nuclei via inverse kinematics (detecting the recoil proton in a fixed-target experiment with a radioactive beam), and that electron scattering of unstable nuclei is becoming available at the SCRIT facility (an electron-storage-ring setup at RIKEN that traps ions in an electron beam and detects elastically scattered electrons), which has demonstrated elastic electron scattering from the unstable isotope $^{132}$Xe and, most recently, from an online-produced $^{137}$Cs target — though so far only at low momentum transfer. The same class of density-profile diagnostic has since been applied to the surface densities of $^{44,52}$Ti (Phys. Rev. C 106, 044330 (2022)) and to $^{20}$Ne (Phys. Rev. C 108, 014322 (2023)), $^{8}$He (Phys. Rev. C 108, L011304 (2023)), and $^{48}$Ti (Phys. Rev. C 109, 054324 (2024)), but these works do not resolve the internal-density question for $^{12}$C and $^{16}$O or the quantification of the cluster admixture.

Problem Statement

Settle, quantitatively, the competition between the $jj$-coupling shell-model and $\alpha$-cluster pictures in the ground states of $^{12}$C and $^{16}$O through their one-body density profiles as probed by elastic scattering. Concretely, the following linked questions remain open: (1) What are the internal (central-region) density differences between the shell-model, $\alpha$-cluster, and mixed configurations of the $^{12}$C and $^{16}$O ground states, and can they be constrained experimentally — in particular by elastic electron-scattering charge form factors at momentum transfers beyond the second diffraction minimum, where for $^{16}$O existing data provide no constraint at $q\gtrsim 3\ \mathrm{fm}^{-1}$, and by proton-nucleus elastic scattering extended to comparable momentum transfer? (2) How large is the $\alpha$-cluster admixture in these ground states? Beyond the posed questions, proton scattering can be more easily applied to unstable nuclei using inverse kinematics, electron scattering of unstable nuclei is becoming available, and applications of the method to heavier $4N$ systems (nuclei with mass number a multiple of four, where $\alpha$ clustering is most natural) as well as neutron-rich unstable nuclei may help toward a more universal understanding of the shell-cluster competition; these generalizations are suggested directions rather than part of the posed open problem. A complete answer must supply all three of the following, jointly, for both $^{12}$C and $^{16}$O. Together they resolve questions (1) and (2). (a) The internal (central-region) density differences among the shell-model, $\alpha$-cluster, and mixed configurations, as quantitative profile values rather than only a qualitative comparison. (b) The size of the $\alpha$-cluster admixture, reported as the best-fit value of the AQCM mixing (cluster-breaking) parameter $\Lambda$ of the mixed trial state ($\Lambda=1$: pure $jj$-coupling shell-model limit; $\Lambda=0$: pure $\alpha$-cluster limit). (c) A statement, which may be qualitative, of which observables discriminate these configurations in the internal density region and what measurement would decide the question. Because no mixed configuration is available for $^{16}$O, one must be constructed to address parts (a) and (b) for that nucleus. Where the existing data leave a quantity unconstrained — for $^{16}$O at $q\gtrsim 3\ \mathrm{fm}^{-1}$ in particular — the answer must state explicitly what the current data do and do not determine.

References

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