Systematic chart-wide assessment of triaxial-deformation effects on nuclear halo formation in neutron-rich nuclei

The 2021 Rapid Communication by Uzawa, Hagino, and Yoshida demonstrated, with triaxially deformed Woods–Saxon calculations at neutron numbers $N=13$ and $N=43$ , that non-axial (triaxial) deformation mixes different $\Omega$ components (the projection $\Omega$ of the single-particle angular momentum on the nuclear symmetry axis) in a single-particle wave function and can admix low angular momentum (s- and p-wave) components into orbitals that cannot form a halo under axial symmetry — so the region of halo nuclei on the nuclear chart should enlarge if triaxiality is allowed. The paper explicitly left open the systematic study of deformed halo nuclei over a wide range of the neutron-rich nuclear chart with non-axial degrees of freedom. Later work developed a self-consistent triaxial relativistic Hartree–Bogoliubov theory in continuum (TRHBc) that treats triaxial deformation, pairing, and continuum coupling simultaneously, confirmed the mechanism along the aluminum chain (triaxiality-driven halos in $^{40}$ Al and $^{42}$ Al), and extended the idea to proton halos in model calculations. However, the chart-wide survey itself — the prevalence of triaxial-induced halos in medium-heavy and heavy neutron-rich nuclei and the quantitative boundaries of the enlarged halo region — has not been carried out.

Background

A halo nucleus is a weakly bound nucleus in which one or two valence nucleons occupy orbitals of low orbital angular momentum ( $l=0$ or $l=1$ ); the absence of a centrifugal barrier lets their density tail extend far beyond the nuclear core, first observed in $^{11}$ Li in 1985. Because a weakly bound $d$ -, $f$ -, or higher- $l$ orbital is held in by its centrifugal barrier, such orbitals normally do not produce halos. Nuclear deformation changes this: in a deformed potential the orbital angular momentum $l$ is no longer a good quantum number, and components with different $l$ (including $s$ and $p$ waves) are admixed into a single-particle wave function. Under axial symmetry — deformation characterized by a quadrupole parameter $\beta$ with axial shape — the projection $\Omega$ of the single-particle angular momentum on the symmetry axis is conserved, and only orbitals with $\Omega^\pi=1/2^+$ , $1/2^-$ , and $3/2^-$ can contain enough $s$ - or $p$ -wave strength to form a halo ( $\pi$ denotes parity). This deformation-induced halo mechanism is the standard explanation for the observed one-neutron halos in $^{31}$ Ne and $^{37}$ Mg, whose valence neutrons would occupy the $f_{7/2}$ orbital in a spherical nucleus. All of these deformed-halo studies assumed axial symmetry. Uzawa, Hagino, and Yoshida (Phys. Rev. C 104, L011303 (2021)) considered non-axially symmetric (triaxial) deformation, parameterized by $\beta$ and the triaxiality angle $\gamma$ ( $\gamma=0^\circ$ prolate, $\gamma=60^\circ$ oblate, intermediate values triaxial; the paper's Eq. (3) uses the $a_{2\mu}$ expansion of the surface in spherical harmonics). With triaxiality, $\Omega$ itself is no longer a good quantum number: several $\Omega$ components mix, so that, for example, an $\Omega^\pi=5/2^+$ orbital can admix an $\Omega^\pi=1/2^+$ component containing $s_{1/2}$ strength and develop a halo even though no axial deformation could produce one. Using a triaxially deformed Woods–Saxon potential (a phenomenological shell-model potential with spin–orbit coupling) solved on a three-dimensional mesh, the authors showed for the 43rd neutron (the $N=43$ region, e.g. around Zr/Rb/Sr neutron-rich isotopes) that as $\gamma$ moves away from $0^\circ$ and $60^\circ$ the $s$ -wave fraction $P_s$ of the valence orbital grows markedly as the single-particle energy approaches zero, with the rms radius of the valence neutron increasing accordingly — an effect absent for any axial shape. For $N=13$ ( $^{19}$ C, an established one-neutron halo), triaxiality with $\gamma=20^{\circ}$ – $50^{\circ}$ simultaneously reproduced both the measured matter radius and the tiny one-neutron separation energy $S_n=160\pm110$ keV, which the prolate and oblate extremes could not. Their calculations imposed parity and z-signature symmetries (signature is the quantum number associated with invariance under a rotation by $180^\circ$ about the symmetry axis) and a finite box (mesh spacing $\Delta r=0.9$ fm, 24 points per direction) with vanishing boundary conditions; $P_s$ never reached 100%, partly because of the box boundary condition, so numerical convergence for $s$ -wave-dominated weakly bound states ( $S_n\to 0$ ) was flagged as limited. Two facts make the question consequential: finite-range droplet model (FRDM) mass calculations indicate that medium-heavy nuclei near the drip lines (the boundaries of the nuclear chart beyond which an additional neutron is no longer bound) deform triaxially over wide regions of the chart, and symmetry-unrestricted Hartree–Fock plus BCS (Bardeen–Cooper–Schrieffer) pairing calculations find nuclei near the drip lines with $N=60$ –80 and $N=110$ –120 tending to triaxial shapes. The demonstration nevertheless rested on a phenomenological potential without pairing or self-consistency and covered only two neutron numbers, and the authors explicitly identified a systematic study of deformed halo nuclei over a wide range of the nuclear chart, including triaxial deformations, as outstanding future work. Subsequent literature has built the required machinery: a triaxial relativistic Hartree–Bogoliubov theory in continuum (TRHBc) — a self-consistent covariant density functional method in a deformed Woods–Saxon basis that treats triaxial deformation, pairing correlations, and continuum coupling simultaneously — was established by Zhang, Zhang, and Meng and applied systematically to the aluminum chain, where $^{40}$ Al and $^{42}$ Al emerge as triaxial one-neutron halo candidates (the $^{40}$ Al halo appears only when triaxiality is allowed, confirming the mechanism self-consistently); triaxial effects on proton halos were analyzed in a core-plus-proton Woods–Saxon model; and reaction-observable predictions (cross sections and one-neutron-removal momentum distributions computed with the Glauber model, the standard semiclassical description of high-energy nucleus–nucleus scattering) were made for $^{40,42}$ Al. No study has yet extended the triaxial treatment across the nuclear chart.

Problem Statement

Carry out the systematic study of deformed halo nuclei over a wide range of the neutron-rich nuclear chart including the non-axial (triaxial, $\gamma$ ) degrees of freedom, which Uzawa, Hagino, and Yoshida (2021) demonstrated on two neutron numbers ( $N=13$ and $N=43$ ) with a triaxial Woods–Saxon potential and explicitly identified as outstanding. The common scientific objective is a halo/non-halo map of the neutron-rich nuclear chart with triaxial degrees of freedom, against which the effect of triaxiality can be read off nucleus by nucleus. Concretely, a complete answer should: (i) compute ground-state and single-particle structure for neutron-rich nuclei across the chart — at minimum covering the sd-shell halos already established (the neutron shell between the magic numbers $N=8$ and $N=20$ , which contains the established one-neutron halo $^{19}$ C) and extending through the medium-heavy and heavy regions where FRDM and symmetry-unrestricted HF+BCS calculations predict triaxial ground states near the drip lines (roughly $N\approx60$ –80 and $N\approx110$ –120, and neighboring neutron-rich regions) — with the non-axial ( $\gamma$ ) degrees of freedom included and the weak-binding regime treated reliably, by whatever theoretical approach achieves this (for example a chart-wide extension of the triaxially deformed Woods–Saxon method used in the 2021 demonstration, or a self-consistent mean-field approach of the TRHBc type that also includes pairing correlations and continuum coupling); (ii) identify, by direct comparison with matched axially restricted calculations, where allowing triaxiality changes the halo verdict — i.e., which nuclei host triaxial-induced halos that the axial treatment misses — and state the quantitative boundaries of the halo region on the chart with and without the triaxial degree of freedom; and (iii) establish the numerical convergence and asymptotic treatment of weakly bound states ( $S_n\to0$ ) underlying these verdicts, including $s$ -wave-dominated cases where box or basis artifacts matter. The problem is solved when a domain expert can read off from the published results which neutron-rich nuclei are predicted to be halos only because of triaxial deformation, in which mass and neutron-number regions triaxial-induced halos are prevalent, and where the boundary of the halo region lies relative to the axially restricted chart.

References

K. Uzawa, K. Hagino, and K. Yoshida, 'Role of triaxiality in deformed halo nuclei', Phys. Rev. C 104, L011303 (2021). DOI: 10.1103/PhysRevC.104.L011303, https://doi.org/10.1103/PhysRevC.104.L011303 (preprint arXiv:2104.02946, https://arxiv.org/abs/2104.02946)
K. Y. Zhang, S. Q. Zhang, and J. Meng, 'Possible neutron halo in the triaxial nucleus 42Al', Phys. Rev. C 108, L041301 (2023). DOI: 10.1103/PhysRevC.108.L041301, https://doi.org/10.1103/PhysRevC.108.L041301 (preprint arXiv:2212.05703, https://arxiv.org/abs/2212.05703)
K. Y. Zhang, S. Q. Zhang, and J. Meng, 'Triaxial relativistic Hartree-Bogoliubov theory in continuum for exotic nuclei', Phys. Rev. C 112, 044308 (2025). DOI: 10.1103/gk2w-cblb, https://doi.org/10.1103/gk2w-cblb (preprint arXiv:2508.10420, https://arxiv.org/abs/2508.10420)
H. Wang, Z. Wang, C. F. Chen, and Z. Ren, 'Impact of triaxial deformation on proton halos', Phys. Rev. C 112, 024313 (2025). DOI: 10.1103/1r7c-4155, https://doi.org/10.1103/1r7c-4155
J.-L. An, S.-S. Zhang, and K. Zhang, 'One neutron triaxial halo candidates in aluminum isotopes from reaction observables', J. Phys. G: Nucl. Part. Phys. 53, 075105 (2026). DOI: 10.1088/1361-6471/ae84f0, https://doi.org/10.1088/1361-6471/ae84f0 (preprint arXiv:2604.00834, https://arxiv.org/abs/2604.00834)
Y. Xiang, Q. Luo, S. Yang, and K. Zhang, 'Spherical, Axial, and Triaxial Symmetries in the Study of Halo Nuclei with Covariant Density Functional Theory', Symmetry 15, 1420 (2023). DOI: 10.3390/sym15071420, https://doi.org/10.3390/sym15071420
K. Zhang, C. Pan, S. Chen, Q. Luo, K. Wu, and Y. Xiang, 'Recent Progress on Halo Nuclei in Relativistic Density Functional Theory', Nucl. Phys. Rev. 41, 191 (2024). DOI: 10.11804/NuclPhysRev.41.2023CNPC28, https://doi.org/10.11804/NuclPhysRev.41.2023CNPC28
H. Sagawa and K. Hagino, 'Theoretical models for exotic nuclei (2nd edition)', Eur. Phys. J. A 61, 192 (2025). DOI: 10.1140/epja/s10050-025-01641-2, https://doi.org/10.1140/epja/s10050-025-01641-2
L.-Y. Wang, K. Zhang, J.-L. An, and S.-S. Zhang, 'Toward a unified description of the one-neutron halo nuclei 15C and 19C from structure to reaction', Eur. Phys. J. A 60, 251 (2024). DOI: 10.1140/epja/s10050-024-01464-7, https://doi.org/10.1140/epja/s10050-024-01464-7
C. Pan, J.-L. An, P. Ring, X.-H. Wu, P. Papakonstantinou, M.-H. Mun, Y. Kim, S.-S. Zhang, and K.-Y. Zhang, 'Prediction of deformed halo nuclei 43,45Si from multiple criteria based on structure and reaction analyses', Phys. Lett. B 877, 140487 (2026). DOI: 10.1016/j.physletb.2026.140487, https://doi.org/10.1016/j.physletb.2026.140487 (preprint arXiv:2512.14545, https://arxiv.org/abs/2512.14545)
K. Yoshida, 'Cranked Skyrme-Hartree-Fock-Bogoliubov approach for a mean-field description of nuclear rotations near the drip line', Phys. Rev. C 105, 024313 (2022). DOI: 10.1103/PhysRevC.105.024313, https://doi.org/10.1103/PhysRevC.105.024313 (preprint arXiv:2109.08328, https://arxiv.org/abs/2109.08328)