Unsettled structure of the dripline two-neutron halo nuclei: binding energies, orbital content, and radii of 22C and 19B, with extension to the heaviest case 29F

The neutron-dripline nuclei $^{22}$C, $^{19}$B and $^{29}$F are Borromean two-neutron halo systems (a core plus two neutrons, with no two-body subsystem bound) whose binding energies, orbital content, and halo radii are still not settled. The three-body model of Suzuki and Otsuka (2021), which fixes the neutron-neutron interaction at its low-energy limit and represents core correlations through a Pauli-blocking energy shift, reproduces the measured halo radii but leaves a precise set of open questions: the two-neutron separation energies $S_{2n}$ of $^{22}$C and $^{19}$B are not known precisely enough to select among the predicted bound-state branches; the first $2^+$ energy of $^{22}$C, proposed as the diagnostic of the $2s_{1/2}$ occupancy in the correlated $^{20}$C core, is unmeasured; for $^{19}$B, matter-radius analyses favoring $d$-wave dominance remain in conflict with Coulomb-dissociation $B(E1)$ data favoring a substantial $s$-wave admixture, with one or two near-threshold configurations possible; the sensitivity of the predictions to explicit three-body forces and to the adopted neutron-neutron parameters is unquantified; and the configuration content of the $^{29}$F halo remains model-dependent. As of 2026 all five items remain open: no precision mass, no $2^+$ energy, and no dipole-strength measurement exist for $^{22}$C; the $^{19}$B conflict persists and theoretical estimates of its $s$-wave fraction span roughly 20-55%; and uncertainty-quantified three-body and effective-field-theory analyses constrain but do not close the questions.

Background

A two-neutron halo is an extended, low-density cloud of two weakly bound neutrons surrounding an ordinary nuclear core; it occurs only at the neutron dripline (the limit of neutron excess beyond which nuclei are unbound) where the two-neutron separation energy $S_{2n}$ (the energy needed to remove both neutrons) is of order a few hundred keV. All known two-neutron halos, including $^{6}$He, $^{11}$Li, $^{14}$Be, $^{17}$B, $^{19}$B, $^{22}$C and $^{29}$F, are Borromean: the core-plus-one-neutron and neutron-neutron subsystems are individually unbound, so binding is a genuinely three-body effect.

The relevant observables are: (i) the matter radius (the rms radius of the nuclear matter distribution), extracted from measured interaction or reaction cross sections using Glauber-type reaction theory (a high-energy multiple-scattering model), which for $^{22}$C gives $3.44\pm0.08$ fm (Togano et al. 2016); (ii) $S_{2n}$, obtained from direct mass measurements; and (iii) the electric dipole strength $B(E1)$ near threshold (the soft dipole mode: an oscillation of the halo neutrons against the core just above the breakup threshold), measured by Coulomb dissociation, in which the halo is excited and broken up in the Coulomb field of a heavy target. For $^{19}$B, Coulomb dissociation on lead at 220 MeV/nucleon yields a summed soft $B(E1)=1.64\pm0.06\pm0.12$ e$^2$fm$^2$ (Cook et al. 2020), while the matter-radius and mass analyses indicate that halo formation is hindered by the $1d_{5/2}$ character of the valence neutrons (in the standard orbital notation $nl_j$, where $n$ is the principal quantum number, $l$ the orbital angular momentum written with the spectroscopic letters $s,p,d,f$, and $j$ the total angular momentum; a $d$-wave neutron carries a centrifugal barrier that inhibits halo formation) (Gaudefroy et al. 2012).

Suzuki and Otsuka (2021) studied these nuclei with a core$+n+n$ three-body model in which the neutron-neutron interaction is a single-range Gaussian whose parameters are fixed at the low-energy scattering limit (they reproduce the measured neutron-neutron scattering length, the strength parameter of a low-energy two-body interaction, and effective range), the core-neutron interaction is a Woods-Saxon potential (the standard phenomenological well-shaped mean-field potential of nuclear physics), and core correlations are represented by a Pauli-blocking energy shift $\Delta$ (an energy offset enforcing that the halo neutrons avoid orbitals already occupied in the correlated core, here $2s_{1/2}$ strength inside $^{20}$C). The model derives a unique monotonic relation between $S_{2n}$ and the neutron halo radius of $^{22}$C (neutron rms $\approx6.75^{+0.41}_{-0.51}$ fm with $\Delta\simeq1$ MeV, consistent with the measured matter radius), reproduces the $^{29}$F halo with a dominant $2p_{3/2}$ configuration (halo radius 5.80 fm at $S_{2n}=1.44$ MeV, corroborated by a large-space shell-model calculation), and, for the boron isotopes with mixed $2s_{1/2}^2$/$1d_{5/2}^2$ configurations, finds $^{17}$B $d$-wave dominated ($P(1d_{5/2}^2)=0.83\pm0.02$, consistent with the $9(2)$% $s$-orbital spectroscopic factor (the probability of finding a valence neutron in that orbital) measured via quasifree $(p,pn)$ knockout, a reaction that removes one neutron and whose cross section maps orbital occupancy; Yang et al. 2021) while $^{19}$B admits two incompatible solution branches: a pure $1d_{5/2}^2$ branch with small radii agreeing with matter-radius data, and a mixed branch with $S_{2n}\approx0.52$ MeV, $P(2s_{1/2}^2)\approx40$% and a neutron rms radius of 7.2 fm that quantitatively reproduces the Coulomb-dissociation $B(E1)$ and core-$2n$ distance. The authors' publications identify what is missing: precise $S_{2n}$ values for $^{22}$C and $^{19}$B, the first $2^+$ energy of $^{22}$C (to pin down the core $2s_{1/2}$ occupancy and hence $\Delta$), a resolution of the $^{19}$B radius-versus-$B(E1)$ conflict, a quantification of the sensitivity to explicit three-body forces and to alternative neutron-neutron parameters, and extensions beyond the restricted two-component configuration spaces, particularly for $^{29}$F where orbitals such as $1f_{7/2}$ and $1d_{3/2}$ were omitted. Scaling analyses in the unitary limit (the limit of infinite scattering length, in which the details of the interaction drop out; Yamashita et al. 2011) and halo effective field theory (a low-energy expansion treating the core and the two neutrons as the only active degrees of freedom; Hongo and Son 2022) provide model-independent relations among $S_{2n}$, radii and dipole strength that hold when $S_{2n}$ and the neutron-neutron virtual energy are the smallest scales, and that already bound $S_{2n}(^{22}C)\lesssim0.35\!-\!0.4$ MeV.

Problem Statement

Determine the near-threshold structure of the dripline two-neutron halo nuclei - in the first instance $^{22}$C and $^{19}$B, and with the same question posed for the heaviest case $^{29}$F - at the level needed to discriminate the competing structure solutions left open by the core$+n+n$ three-body model of Suzuki and Otsuka (2021), and to establish whether that minimal model (neutron-neutron interaction fixed at the low-energy limit, core correlations represented by a Pauli-blocking shift) survives the confrontation. Concretely, the problem comprises five linked requests; all five are required for a complete resolution, and addressing only a subset settles the corresponding sub-question but does not resolve the problem:

1. Binding: determine $S_{2n}$ of $^{22}$C and $^{19}$B, which are not known precisely enough to select among the predicted bound-state branches, by experiment or theory - for $^{22}$C, to replace the present one-sided upper bound (of a few tenths of an MeV) with a two-sided determination, and for $^{19}$B, at a precision that separates the near-threshold mixed branch from the pure $1d_{5/2}^2$ alternative.
2. Core correlations: determine the $2s_{1/2}$ occupancy of the correlated $^{20}$C core - equivalently the Pauli-blocking energy shift $\Delta$ - for which the first $2^+$ excitation energy of $^{22}$C, unmeasured to date, was proposed by the authors as the diagnostic, with the dipole response of $^{22}$C (whose strength function is predicted to carry the needed configuration information) as a further route to the same occupancy; the request is resolved by an independent determination of that occupancy, not by restating the model's own fitted shift $\Delta\simeq1$ MeV.
3. $^{19}$B orbital content: resolve the conflict between matter-radius analyses favoring $1d_{5/2}^2$ dominance and Coulomb-dissociation $B(E1)$ data favoring a substantial $2s_{1/2}^2$ admixture, and establish whether one or two bound configurations exist near the two-neutron threshold and which is realized (the candidate near-threshold mixed branch lies at $S_{2n}\approx0.52$ MeV); showing that the conflict is an artifact of the spherical core$+2n$ ansatz (e.g. core deformation or additional neutrons participating in the halo) and determining the correct degrees of freedom resolves this request.
4. Model systematics: quantify the sensitivity of the predicted binding energies, radii and configuration probabilities to the neglected higher-order physics - explicit three-body (core$+n+n$) interactions beyond the low-energy-limited Gaussian neutron-neutron term, the prescription for enforcing Pauli blocking, and alternative neutron-neutron low-energy parameters (including effective-range corrections and the extremely large negative scattering lengths used in some studies); all three named items must be covered.
5. Extension to medium mass: determine the configuration content of the $^{29}$F two-neutron halo (relative roles of $2p_{3/2}$, $1f_{7/2}$, $2s_{1/2}$ and $1d_{3/2}$ neutrons and of deformation) using configuration spaces larger than the pure or two-component ansätze, with shell-model cross-checks.

References

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