The discovery of 39^{39}Na at RIKEN (2022) and the subsequent deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc) study by Zhang et al. (Phys. Rev. C 107, L041303 (2023)) established that 39^{39}Na may be the first nucleus combining a collapsed N=28N=28 shell closure, a Borromean two-neutron halo, and shape decoupling between a prolate core and an oblate halo. That work left a precise, interlinked set of open questions: (i) the location of the neutron dripline in the sodium chain (predicted as 39^{39}Na by some covariant density functionals and 41^{41}Na by others) is undetermined; (ii) the experimental observables that most directly diagnose core-halo shape decoupling have not been identified; (iii) finite-range pairing forces have not been implemented in DRHBc, so their impact on weak binding and halo properties is unquantified; and (iv) the Borromean three-body dynamics of 39^{39}Na has not been treated microscopically beyond mean field or connected to measurable signatures. Recent work provides three-body halo models, Glauber reaction-observable predictions, and soft-dipole response calculations, but every component remains open: no mass or heavier-isotope search has fixed the Na dripline, no measurement exists for 39^{39}Na, no shape-decoupling diagnostic has been demonstrated against confounds, and finite-range pairing is still absent from deformed continuum calculations.

Background

The neutron dripline is the boundary of the chart of nuclides beyond which nuclei are no longer bound against neutron emission; mapping it is a major goal of nuclear physics because it directly constrains the nuclear force and nucleosynthesis models. Experimentally the neutron dripline is known only up to neon (Z=10Z=10): Ahn et al. fixed it at 31^{31}F and 34^{34}Ne (Phys. Rev. Lett. 123, 212501 (2019)), and the discovery of 39^{39}Na in nine events (Ahn et al., Phys. Rev. Lett. 129, 212502 (2022)) left the sodium dripline undetermined, since the existence of heavier sodium isotopes was not excluded. 39^{39}Na has neutron number N=28N=28, a conventional magic (closed-shell) number; the disappearance of such shell closures in very neutron-rich nuclei, notably in the so-called island of inversion around N≈20N\approx20 to N≈28N\approx28, is a central theme of exotic-nuclear-structure research. A neutron halo is an extended, low-density neutron distribution arising when weakly bound neutrons occupy low orbital-angular-momentum (ss or pp) orbitals near threshold; a Borromean nucleus is a bound three-body system none of whose two-body subsystems is bound (39^{39}Na would be 37^{37}Na+n+n+n+n, since 38^{38}Na is calculated to be one-neutron unbound). A deformed halo is a halo around a deformed core; shape decoupling means the halo and the core adopt different quadrupole shapes-here an oblate neutron halo forming around a prolate core. Zhang et al. (Phys. Rev. C 107, L041303 (2023); arXiv:2305.02636) studied 39^{39}Na in the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc)-a covariant density functional treatment that combines axial deformation with a proper continuum treatment via a Dirac Woods-Saxon basis (single-particle states of a deformed relativistic Woods-Saxon potential, including resonant and non-resonant continuum components) and pairing correlations (in their case a density-dependent zero-range surface pairing of strength V0=−325 MeV fm3V_0=-325\ \mathrm{MeV\ fm^3} in a 100 MeV pairing window, the standard choice of the DRHBc mass-table effort). They found that the lowering of 2p1/2,3/22p_{1/2,3/2} orbitals collapses the N=28N=28 shell closure, producing a well-deformed ground state (quadrupole deformation parameter β2≈0.45\beta_2\approx0.45) with two-neutron separation energy S2n≲1S_{2n}\lesssim1 MeV, partially occupied weakly bound or continuum pp-wave orbitals (about 30% of the valence neutrons), and an oblate halo around the prolate core in 39,41^{39,41}Na. However, their dripline prediction is functional-dependent: PC-PK1, NL3*, and PK1 give 41^{41}Na as the last bound isotope, while PC-F1 and NL-SH give 39^{39}Na. Their paper explicitly leaves open: the Na dripline location (to be settled by experimental dripline/mass determination and better-constrained functionals and pairing interactions); the identification of the observables most closely tied to shape decoupling in deformed halos; the implementation of finite-range pairing forces (Gogny or separable type) in DRHBc and their quantitative effect on weak binding, occupations, and halo properties; and the microscopic three-body treatment of the Borromean character of 39^{39}Na through reaction observables and the low-lying electric-dipole response. Prior and subsequent work frames these questions: Skyrme Hartree-Fock-Bogoliubov (HFB) studies (Chai et al., Phys. Rev. C 102, 014312 (2020)) already inferred deformed-halo character for 39^{39}Na and dripline constraints, and complex-momentum relativistic mean-field calculations (Luo et al., Chin. Phys. C 48, 044103 (2024)) show the halo predictions depend on the interaction and on the resonant states included; finite-range Gogny pairing has existed only in spherical relativistic Hartree-Bogoliubov theory (Gonzalez-Llarena et al., Phys. Lett. B 379, 13 (1996)), never in the deformed continuum framework.

Problem Statement

Convert the DRHBc prediction that 39^{39}Na combines a quenched N=28N=28 shell closure, a Borromean two-neutron halo, and core-halo shape decoupling into validated knowledge, and fix the neutron dripline of the sodium chain. This single objective comprises four linked components, exactly as posed by Zhang et al. (Phys. Rev. C 107, L041303 (2023)): (1) Dripline location: determine, by experimental mass/separation-energy measurements and/or production limits on 40,41^{40,41}Na together with better-constrained density functionals and pairing interactions, whether 39^{39}Na, 41^{41}Na, or another isotope is the last bound sodium isotope. (2) Three-body structure: establish by microscopic beyond-mean-field calculation (three-body core+nn+nn treatment including the deformed core) and/or experiment whether 39^{39}Na is indeed a Borromean two-neutron deformed halo, through its predicted reaction observables and low-lying electric-dipole response. (3) Shape-decoupling diagnostics: identify which experimental signatures-reaction cross sections, core momentum distributions, Coulomb dissociation spectra, low-lying E1E1 strength, or scattering of hadronic probes-correlate most directly with core-halo shape decoupling (an oblate halo around a prolate core) as opposed to halo diluteness or core deformation alone, and demonstrate their discriminating power within a consistent model framework, since no 39^{39}Na measurement exists. (4) Pairing systematics: implement finite-range pairing forces (Gogny or separable finite-range) in the deformed continuum Hartree-Bogoliubov framework and quantify their impact on weak binding, occupation probabilities, and halo properties in the sodium chain. The components are jointly supporting: (4) and (2) test whether the mean-field prediction survives pairing and beyond-mean-field corrections; (3) supplies the observables by which (1) and (2) are decided experimentally. The problem is complete when the sodium dripline is fixed or experimentally bounded (component (1)), and the deformed-halo/shape-decoupling picture of 39^{39}Na is validated or refuted through identified observables (components (2)-(3)) with quantified model dependence, where the quantified model dependence covers the beyond-mean-field three-body treatment of component (2) and the finite-range pairing systematics of component (4). The four components are jointly mandatory, and because no 39^{39}Na measurement exists, completion does not require experimental 39^{39}Na data: a theory-level determination that fixes or bounds the sodium dripline (component (1)) and validates or refutes the shape-decoupled deformed-halo picture of 39^{39}Na through identified observables (components (2)-(3)), with the model dependence of components (2) and (4) quantified, satisfies the completion condition.

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