Establishing the deformation-driven two-neutron halo mechanism at the fluorine neutron drip line: $^{29}$F and $^{31}$F

The nucleus $^{29}$F (Z=9, N=20) is the heaviest known two-neutron Borromean halo nucleus, and $^{31}$F (N=22) is the last bound fluorine isotope. Hamamoto (2021) interpreted the $^{29}$F halo in a phenomenological deformed Woods-Saxon picture as evidence for a prolate ground-state deformation that mixes the near-degenerate neutron $2p_{3/2}$ and $1f_{7/2}$ resonant orbitals, and posed the open issues of two-neutron correlations beyond the independent-particle picture, confirmation in microscopic models, whether $^{31}$F has an observable halo, and the untested tensor-force alternative for the N=22 drip-line extension, noting in addition the missing experimental inputs (deformation, spin-parity assignments). The open problem is to establish this mechanism quantitatively: reproduce the halo observables of $^{29}$F and determine those of $^{31}$F in microscopic treatments that go beyond the independent-particle picture; and quantify the relative roles of quadrupole deformation, proton-neutron tensor-force shell evolution and - as stressed by later self-consistent studies - continuum coupling in extending the fluorine drip line to N=22. The discriminating experimental constraints (matter radius of $^{31}$F, deformation and orbital occupancy of $^{29}$F) remain missing and would distinguish the competing interpretations.

Background

A neutron halo forms when one or two valence neutrons occupy orbitals with separation energies of only a few hundred keV, so that their wave functions extend far beyond the nuclear core; the signature is a matter radius and a reaction cross section far above the trend set by the mass number $A$. A two-neutron 'Borromean' halo is a three-body system (core + n + n) that is bound even though each of its two-body subsystems is unbound. The neutron drip line is the boundary of particle stability on the neutron-rich side of the nuclear chart; for fluorine (Z=9) it falls at $^{31}$F (N=22), whose particle stability was first shown by Sakurai et al. (1999) and whose position as the last bound fluorine isotope was fixed by Ahn et al. (2019), while $^{28}$F and $^{30}$F are neutron-unbound.

In the conventional shell model, N=20 is a magic number: a gap separates the $sd$ shell from the $pf$ shell, favoring spherical closed-shell shapes. Nuclei with Z~10-12 near N=20 form the 'island of inversion', where intruder $pf$-shell configurations dominate the ground states. 'Shell evolution' denotes the systematic change of single-particle energies with proton and neutron number, driven by the monopole (angular-momentum-averaged) part of the nucleon-nucleon interaction; the proton-neutron tensor force is the canonical mechanism, reviewed by Otsuka et al. (2020). $^{29}$F sits at the low-Z border of the island of inversion, and how the N=20 closure erodes there is precisely what is at stake in this problem.

Experimentally, Bagchi et al. (2020) measured the reaction cross sections of $^{27,29}$F on carbon at RIKEN and found an unexpectedly large cross section for $^{29}$F, identifying it as the heaviest two-neutron Borromean halo known, with a halo root-mean-square radius of about 6.6 fm; the interpretation assigns the halo neutrons to the $2p_{3/2}$ orbital (vanishing N=20 shell closure). Large-scale shell-model calculations reproduce the data, while coupled-cluster calculations (an ab initio method that solves the many-body problem from internucleon forces by an exponential expansion of correlated particle-hole excitations) based on chiral effective field theory (nuclear forces derived as a systematic low-energy expansion of quantum chromodynamics) describe $^{27}$F but are 'challenged' for $^{29}$F. Quasi-free neutron-removal (knockout) momentum distributions (Revel et al. 2020) show that the valence neutrons in the $^{29}$F ground state have predominantly $\ell=1$ (p-wave) character, about 79%. A bound excited state at 1.08 MeV in $^{29}$F was observed by Doornenbal et al. (2017). The first observation of unbound $^{30}$F (Kahlbow et al. 2024) shows that the N=20 gap is not restored toward $^{28}$O: the neutron $d$ and $fp$ orbitals are nearly degenerate, with $1p_{3/2}$ and $1p_{1/2}$ below $0f_{7/2}$, making both $^{29}$F and $^{31}$F good two-neutron halo candidates.

Hamamoto (2021) analyzed $^{29}$F with a phenomenological one-body Woods-Saxon potential (a mean nuclear potential with a diffuse surface) as a function of the quadrupole deformation parameter $\beta_2$. Because the neutron resonant $1f_{7/2}$ and $2p_{3/2}$ levels are nearly degenerate at $\beta_2=0$ and the proton number Z=9 favors a prolate shape, the nucleus is predicted to be prolate ($\beta_2\approx0.38$); the last neutrons then occupy the weakly bound Nilsson orbit (a single-particle orbit of a deformed potential, labeled by the projection $\Omega$ of the nucleon angular momentum on the nuclear symmetry axis together with the parity $\pi$) of $\Omega^\pi=1/2^-$ with mixed $2p_{3/2}$/$1f_{7/2}$ parentage, producing the halo. Hamamoto concluded that the halo is evidence for prolate deformation of $^{29}$F, noted that for $^{31}$F the halo depends subtly on the deformation magnitude and which orbitals the last two neutrons fill, questioned whether proton-neutron tensor-force shifts of neutron levels (as invoked for the $^{25}$F core by Tang et al. 2020) can explain a drip line extended to N=22 with acceptable tensor strength, and asked for confirmation by advanced microscopic models.

Subsequent theory falls into three classes, none final. (i) Phenomenological three-body models (core + n + n with adjusted core-neutron potentials) with explicit two-neutron correlations: for $^{29}$F by J. Singh et al. (2020) and Casal et al. (2020, electric dipole response); for $^{31}$F by Michel et al. (2020) in the Gamow shell model (a shell model whose basis spans bound states and continuum resonances), by Masui et al. (2020, 2022), and by G. Singh et al. (2022). These find the halo compatible with correlated two-neutron dynamics and identify a 'pairing anti-halo' effect - pairing energy gain that shrinks weakly bound systems - which however reduces but does not eliminate the $^{31}$F halo in most configurations. (ii) Self-consistent mean-field: Luo et al. (2021) in a relativistic mean field solved in the complex-momentum (Berggren) representation, a basis in which bound states and continuum resonances are treated on an equal footing, showing that modest quadrupole deformation plus continuum coupling produce the observed structure in $^{28,29,31}$F while stating explicitly that the relative responsibility of deformation, tensor-force shell evolution, and continuum coupling remains unclear; Zhang et al. (2026) in the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc, a self-consistent mean-field method with pairing and exact continuum) reproduce a deformed two-neutron halo in $^{29}$F at $\beta_2\approx0.4$ and obtain for $^{31}$F a well-deformed ground state ($\beta_2\approx0.24$) with a more pronounced halo. (iii) Reviews in 2025-2026 (Kahlbow; Barman et al.; Fortunato) concur that $^{29}$F is a moderately extended two-neutron halo consistent with the data, that $^{31}$F remains an unmeasured, model-dependent 'speculative' halo candidate, and that the mechanism question is open.

Problem Statement

Establish, quantitatively and with controlled uncertainties, whether the two-neutron halo of $^{29}$F and the structure of the drip-line nucleus $^{31}$F are produced by the deformation-driven mechanism proposed by Hamamoto - prolate ground-state deformation mixing the near-degenerate neutron $2p_{3/2}$ and $1f_{7/2}$ orbitals - and, more generally, how quadrupole deformation, proton-neutron tensor-force shell evolution, and coupling to the continuum combine to extend the fluorine neutron drip line to N=22. A complete answer must: (a) provide an advanced microscopic model treatment of $^{29}$F beyond the independent-particle picture - incorporating whatever pairing, continuum coupling, and two-neutron correlations are needed to describe the halo - that quantitatively reproduces the measured halo observables, the reaction cross sections and matter radius (Bagchi et al. 2020) and the approximately 79% $\ell=1$ valence-neutron character (Revel et al. 2020), and state whether the deformation-driven halo mechanism survives at this level; (b) determine whether $^{31}$F possesses an observable two-neutron halo, resolving the sensitivity to the deformation magnitude and orbital occupancy - including the pairing anti-halo effect - over the experimentally allowed range of its two-neutron separation energy; and (c) quantify the relative roles of quadrupole deformation and proton-neutron tensor-force monopole shifts of neutron levels in extending the fluorine drip line to N=22 - deciding whether realistic tensor-force strengths can account for bound $^{31}$F without deformation and how tensor and deformation effects interplay - and, within the same account, clarify the role of continuum coupling stressed by later self-consistent calculations (Luo et al. 2021). Components (a)-(c) are jointly linked: they all test the single deformation-driven-halo interpretation of the fluorine drip line. The problem is complete when (a) and (b) are answered and the mechanism decomposition of (c) is quantified - or when the (a)-(c) evidence jointly refutes the deformation-driven picture in favor of an identified alternative mechanism. The discriminating experimental inputs - the matter radius or reaction cross section of $^{31}$F, and the deformation, orbital occupancy, and spin-parity assignments of $^{29}$F - remain missing; their absence is a known limitation of the field rather than a deliverable required of a solution.

References

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