The triaxial relativistic Hartree-Bogoliubov theory in continuum (TRHBc) combined with the Glauber reaction model predicts that the very neutron-rich isotopes 40^{40}Al and 42^{42}Al are the first triaxially deformed one-neutron pp-wave halo candidates: their reaction cross sections σR\sigma_R on carbon at 240 and 900 MeV per nucleon are predicted to rise pronouncedly above the systematic trend of the lighter aluminum isotopes, and the longitudinal momentum distributions of their one-neutron-removal residues to be roughly 30% and 50% narrower than those of 38^{38}Al and 36^{36}Al, reflecting about 60% pp-wave valence-neutron occupation. Neither isotope has ever been studied with any reaction-observable measurement — 42^{42}Al was first observed only in 2007 as rare projectile-fragmentation events — so these quantitative predictions are completely untested. The open problem is their experimental test: to design and quantitatively assess a measurement of σR\sigma_R and the inclusive momentum distributions dσ/dp∥d\sigma/dp_{\parallel} for 40,42^{40,42}Al on carbon at intermediate and high energies that could establish the empirical presence and magnitude of the halo signatures — or their absence — and thereby confirm or refute the triaxial one-neutron pp-wave halo interpretation.

Background

A neutron halo forms near the neutron dripline (the limit of neutron excess beyond which nuclei are unbound) when a valence neutron binds with a separation energy of only a few hundred keV: its wave function then extends far beyond the nuclear core, giving the nucleus an anomalously large matter radius. Such nuclei live only milliseconds and cannot be made into targets, so their structure is probed in inverse kinematics through reaction observables: the reaction cross section σR\sigma_R (the probability of any inelastic event in a collision with a light target such as carbon, measured at beam energies of hundreds of MeV per nucleon, a technique introduced by Tanihata and collaborators in 1985), and the longitudinal momentum distribution dσ/dp∥d\sigma/dp_{\parallel} of the residue left after one-neutron removal — by the uncertainty principle, a spatially extended valence neutron produces a narrow momentum distribution, and the width and shape of that distribution encode the orbital angular momentum of the valence neutron (a pp-wave neutron gives a distinctly narrower distribution than an ff-wave one). The standard tool connecting nuclear structure to these observables is the Glauber (eikonal) model, a high-energy scattering approximation fed with microscopic nuclear densities and valence-neutron wave functions.

Halos were long associated with nearly spherical, weakly bound nuclei, but deformation changes the single-particle level order near the Fermi surface and can push a low-angular-momentum orbital close to the continuum, so a weakly bound valence neutron occupies a spatially extended pp-wave state even in a deformed system. This mechanism was established for the deformed halos 31^{31}Ne and 37^{37}Mg; 37^{37}Mg is the heaviest experimentally confirmed halo nucleus, and its halo status rests precisely on the two observables at issue here — an enhanced reaction cross section on carbon (Takechi et al., 2014) and a pp-wave one-neutron-removal signature (Kobayashi et al., 2014). Fewer than twenty halo nuclei are confirmed in total, so extending the halo landscape to heavier masses and to triaxial deformation (a shape with three unequal principal axes, beyond the axial prolate/oblate cases) is a major goal of the field. The theoretical framework is covariant density functional theory in the continuum — the relativistic Hartree-Bogoliubov theory, a mean-field method with pairing correlations solved in a deformed Woods-Saxon basis so that weakly bound and continuum states are treated properly. Its triaxial extension (TRHBc) was developed by Zhang, Zhang, and Meng (2025) and applied to the aluminum chain (Z=13Z=13), where it predicted that 42^{42}Al, the last bound odd-neutron aluminum isotope, has a triaxially deformed halo (Zhang, Zhang, Meng, 2023). Experimentally, 42^{42}Al was first observed in 2007, together with 40^{40}Mg, as rare fragmentation products (Baumann et al., Nature 2007). In Ref. [1] (An, Zhang, Zhang, 2026), TRHBc densities and valence-neutron wave functions were coupled for the first time to the Glauber model, yielding quantitative, falsifiable predictions: σR\sigma_R of 40,42^{40,42}Al + 12^{12}C at 240 and 900 MeV per nucleon rises pronouncedly above the systematic trend of 31^{31}–39^{39}Al at both energies; the full widths at half maximum of the one-neutron-removal momentum distributions are approximately 30% (40,42^{40,42}Al vs 38^{38}Al) and 50% (vs 36^{36}Al) smaller; and the valence neutron has about 60% pp-wave occupation (2p1/22p_{1/2} plus 2p3/22p_{3/2}) in both isotopes, against ff-wave dominance in 36,38^{36,38}Al. On this basis 40,42^{40,42}Al were identified as the first candidate triaxially deformed one-neutron pp-wave halos. Whether these predicted signatures are real is entirely untested: no reaction cross section or one-neutron-removal measurement exists for either isotope.

Problem Statement

Design and quantitatively assess the experimental test of whether 40^{40}Al and 42^{42}Al possess the predicted one-neutron pp-wave halo structure, using the reaction observables that the TRHBc + Glauber framework identifies as the only currently available probe of that structure. Since neither isotope has yet been measured and both lie beyond the reach of current beam intensities, the resolving deliverable is the design and feasibility analysis of this test rather than the measured data themselves: it must specify the beam production and target requirements, the detector setup, the expected yields and statistical precision, and the systematic uncertainties, and must state an explicit falsification criterion for the predicted signatures. Specifically: (1) assess the reaction cross sections σR\sigma_R of 40,42^{40,42}Al + 12^{12}C at intermediate and high energies (predicted at 240 and 900 MeV per nucleon) and establish whether the predicted pronounced, energy-independent enhancement above the systematic trend of the lighter aluminum isotopes would be distinguishable at the achievable precision; and (2) assess the inclusive longitudinal momentum distributions dσ/dp∥d\sigma/dp_{\parallel} of the residues after one-neutron removal of 40,42^{40,42}Al on carbon, establishing whether their full widths at half maximum — predicted to be approximately 30% and 50% below those of 38^{38}Al and 36^{36}Al — and their shapes — predicted to reflect a dominant pp-wave valence-neutron occupation of about 60%, against ff-wave dominance in 36,38^{36,38}Al — would be resolvable. The outcome should establish whether the halo signatures could be empirically detected at their predicted magnitude — or would be excluded — and thereby test the identification of 40,42^{40,42}Al as triaxially deformed one-neutron pp-wave halo nuclei within the TRHBc + Glauber framework. Since σR\sigma_R and dσ/dp∥d\sigma/dp_{\parallel} are angle-averaged probes computed from angle-averaged densities, they test the halo extension and its orbital character; they do not by themselves discriminate the triaxial deformation, so the triaxial attribution must be reported as an inference of the TRHBc + Glauber calculation unless a complementary shape-sensitive probe — for example a shape-sensitive Coulomb-excitation or breakup measurement — is included.

References

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