In the time-odd deformed relativistic Hartree-Bogoliubov theory in continuum (TODRHBc), Pan, Zhang, and Zhang [Phys. Lett. B 855 (2024) 138792] found that in the one-neutron halo nucleus 31^{31}Ne the weakly bound valence neutron orbital (dominant p3/2p_{3/2} component, ϵ=−0.087\epsilon=-0.087 MeV) becomes unbound (ϵ=+0.049\epsilon=+0.049 MeV) when the time-odd (magnetic) mean fields of the covariant density functional are neglected — nuclear magnetism converts an unbound orbital into a bound one and stabilizes the nucleus against one-neutron emission. The authors themselves flagged this conclusion as unestablished: it rests on a single functional (PC-PK1), an unpublished detailed formalism, a cranking-approximation rotational correction, and no systematic survey. This problem asks to establish the role of nuclear magnetism in near-threshold structure of weakly bound, deformed odd-A nuclei: verify the unbound-to-bound mechanism robustly across density functionals, blocking configurations, and numerical implementations; quantify the effect of beyond-cranking rotational correlations on the neutron separation energy and deformation energetics; complete and fully document the TODRHBc formalism so the calculations are independently reproducible; and determine systematically how time-odd fields shift near-threshold spectra, drip-line locations, nuclear-chart shape systematics, and magnetic moments of exotic nuclei. No published study has yet extended the time-odd treatment beyond PC-PK1 or documented the full TODRHBc formalism, and subsequent odd-A exotic-nucleus studies by the same community still use the time-even equal-filling approximation.

Background

Halo nuclei are weakly bound systems in which one or two valence nucleons tunnel far beyond the nuclear core, producing an extended low-density tail; 31^{31}Ne (10 protons, 21 neutrons) is the heaviest established one-neutron halo, with spin-parity Jπ=3/2−J^{\pi}=3/2^{-} and an experimental one-neutron separation energy Sn=0.15−0.10+0.16S_n=0.15^{+0.16}_{-0.10} MeV, i.e., a neutron bound by only about 0.15 MeV. The theoretical framework of record for such nuclei is relativistic (covariant) density functional theory, in which nucleons move in self-consistent scalar and vector mean fields derived from an energy density functional; pairing correlations are added by the Hartree-Bogoliubov (HB) formalism, which mixes orbitals across the Fermi surface. Near the drip lines, where separation energies approach zero, the coupling of weakly bound orbitals to the particle continuum (unbound scattering states) must be treated explicitly; this is done in the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc), which solves the HB equations in a Dirac Woods-Saxon basis that represents the continuum, for axially deformed shapes. For odd-mass nuclei, the unpaired nucleon blocks a valence orbital; practical DRHBc applications use the equal-filling approximation (EFA), which averages over the two blocked time-reversed partners. Because the EFA is time-reversal symmetric, it cancels the vector current of the blocked nucleon and eliminates the nuclear magnetic potential. In covariant density functionals the functional generates, besides the standard time-even fields (densities), also time-odd mean fields — vector fields coupled to the nucleon current and spin densities — collectively called nuclear magnetism; they break time-reversal symmetry, lift the Kramers degeneracy of time-reversed single-particle pairs (in deformed nuclei by roughly 0–0.2 MeV in 31^{31}Ne), and were previously known to matter for magnetic moments and rotating nuclei, but had never been included self-consistently together with deformation, pairing, and continuum coupling for a halo candidate. Pan, Zhang, and Zhang developed the time-odd extension TODRHBc (based on the point-coupling functional PC-PK1) and applied it to 31^{31}Ne: nuclear magnetism adds about 0.09 MeV binding, splits canonical single-particle spectra by 0–0.2 MeV, and — the striking result — the blocked p3/2p_{3/2}-dominant valence level is bound at ϵ=−0.087\epsilon=-0.087 MeV with magnetism but unbound at ϵ=+0.049\epsilon=+0.049 MeV without it, so time-odd fields alone decide whether 31^{31}Ne is stable against one-neutron emission at the single-particle level. After a rotational correction of 0.93 MeV computed in the cranking approximation — in which the intrinsic mean field is placed in self-consistent uniform rotation at fixed angular velocity to estimate rotational energies — Sn=0.56S_n=0.56 MeV, somewhat above experiment. The authors state that the detailed TODRHBc formalism will be presented in a future work, that the cranking approximation is unsuitable for spherical nuclei and a collective-Hamiltonian treatment is expected to improve the results, that the unbound-to-bound conclusion may depend on numerical conditions (the orbital is bound by less than 0.1 MeV), and that systematic studies of nuclear magnetism's influence on the shape of the nuclear chart, drip-line locations, and magnetic moments of exotic nuclei are in progress.

Problem Statement

Establish the role of nuclear magnetism — the time-odd mean fields of covariant density functionals — in the near-threshold structure of weakly bound, deformed odd-A nuclei, with the halo nucleus 31^{31}Ne as the prototype. The single overarching objective is to determine whether, and under what conditions, time-odd fields genuinely convert unbound valence orbitals into bound ones and thereby change the stability, structure, and predicted limits of exotic nuclei. A complete answer must jointly deliver all four components below (a partial result on any one of them, including the overarching objective, is not sufficient): (1) a fully documented, self-consistent time-odd deformed relativistic Hartree-Bogoliubov treatment in the continuum (the complete TODRHBc formalism, documented so that independent groups can reproduce the implementation), and its application to 31^{31}Ne reproducing or correcting the published benchmark: blocked p3/2p_{3/2}-dominant valence level at ϵ=−0.087\epsilon=-0.087 MeV with nuclear magnetism versus ϵ=+0.049\epsilon=+0.049 MeV without, and the ~0.09 MeV binding-energy contribution; (2) a robustness determination for the unbound-to-bound mechanism — whether the bound/unbound status of the blocked orbital survives variation of the density functional (PC-PK1 plus at least one alternative with a controlled treatment of the time-odd channels), of the blocking configuration, and of the numerical implementation (box size, Dirac Woods-Saxon basis truncation, continuum discretization), given that the orbital is bound by less than 0.1 MeV, with the minimum set of variations tested on each of these axes and the numerical tolerance within which the status counts as having survived stated explicitly (for example, at least two values each of box size and Dirac Woods-Saxon basis truncation and at least one alternative blocking configuration); (3) the effect of rotational correlations beyond the cranking approximation (using any documented beyond-cranking treatment, of which a collective-Hamiltonian treatment is one example) on the neutron separation energy and deformation energetics of 31^{31}Ne, replacing the present 0.93 MeV cranking correction that yields Sn=0.56S_n=0.56 MeV; and (4) a systematic determination, over a finite, explicitly stated survey of at least three representative isotopic chains toward the drip lines, rather than the entire nuclear chart, with the chains chosen and the basis on which they are deemed representative stated, of how nuclear magnetism shifts near-threshold single-particle spectra, one- and two-nucleon separation energies, predicted drip-line locations, shape systematics of the nuclear chart, and magnetic moments of odd-A exotic nuclei — establishing whether the 31^{31}Ne mechanism is an isolated fine-structure effect or a generic correction that covariant continuum density functional theory must include for odd-A nuclei.

References

  1. Cong Pan, Kaiyuan Zhang, Shuangquan Zhang, "Nuclear magnetism in the deformed halo nucleus 31^{31}Ne", Physics Letters B 855, 138792 (2024). DOI: 10.1016/j.physletb.2024.138792; arXiv:2403.03713, https://arxiv.org/abs/2403.03713
  2. K. Y. Zhang, X. X. Lu, "Microscopic description of the proton halo in 12^{12}N", Physics Letters B 871, 139989 (2025). DOI: 10.1016/j.physletb.2025.139989; arXiv:2510.11038, https://arxiv.org/abs/2510.11038
  3. C. Pan, J. L. An, P. Ring, X. H. Wu, P. Papakonstantinou, M.-H. Mun, Y. Kim, S. S. Zhang, K. Y. Zhang, "Prediction of deformed halo nuclei 43,45^{43,45}Si from multiple criteria based on structure and reaction analyses", Physics Letters B 877, 140487 (2026). DOI: 10.1016/j.physletb.2026.140487; arXiv:2512.14545, https://arxiv.org/abs/2512.14545
  4. Panagiota Papakonstantinou, Myeonghwan Mun, Cong Pan, Kaiyuan Zhang, "Proton halo structures and 22^{22}Al", Physical Review C 112, 044301 (2025). DOI: 10.1103/yt6n-bp9f; arXiv:2501.07759, https://arxiv.org/abs/2501.07759
  5. Cong Pan, Yaochen Yang, Xiaofei Jiang, Xin-Hui Wu, "Exploratory study on the masses of odd-Z nuclei and r-process simulation based on the deformed relativistic Hartree-Bogoliubov theory in continuum", The European Physical Journal A 61, 275 (2025). DOI: 10.1140/epja/s10050-025-01756-6; arXiv:2503.09324, https://arxiv.org/abs/2503.09324
  6. Kaiyuan Zhang, Ling Li, Jie Yan, Guixiang Ye, Xinhui Wu, Jia-Lin An, Shi-Sheng Zhang, Cong Pan, Xiang-Xiang Sun, "Description of 29^{29}F and 31^{31}F Within the Deformed Relativistic Hartree-Bogoliubov Theory in Continuum", Particles 9, 36 (2026). DOI: 10.3390/particles9020036, https://www.mdpi.com/2542-4286/9/2/36
  7. K. Y. Zhang, S. Q. Zhang, J. Meng, "Triaxial relativistic Hartree-Bogoliubov theory in continuum for exotic nuclei", Physical Review C 112, 044308 (2025). DOI: 10.1103/gk2w-cblb; arXiv:2508.10420, https://arxiv.org/abs/2508.10420
  8. Xuwei Sun, Jacek Dobaczewski, Markus Kortelainen, Jhilam Sadhukhan, Adrián Sánchez-Fernández, Herlik Wibowo, "Moments of inertia of rare-earth nuclei and the nuclear time-odd mean fields within exact solutions of the adiabatic theory", Physics Letters B 868, 139685 (2025). DOI: 10.1016/j.physletb.2025.139685; arXiv:2502.07433, https://arxiv.org/abs/2502.07433
  9. Yun Dong Wang, Hui Hui Xie, Tian Shuai Shang, Peng Xiang Du, Jian Li, Haozhao Liang, Kaiyuan Zhang, "Microscopic study of charge properties in halo nuclei", Physical Review C 114, 024315 (2026). DOI: 10.1103/vnmp-n6vf; arXiv:2607.11156, https://arxiv.org/abs/2607.11156
  10. Heesung Kwon, Kyoungsu Heo, Seonghyun Kim, Eunja Ha, Myung-Ki Cheoun, "Phenomenological Criteria of Halo Nuclei in Ne Isotopes via Diffuseness and Helm-Model Approaches with Reaction Cross Sections" (2026). arXiv:2604.03901, https://arxiv.org/abs/2604.03901