Threshold soft $E1$ strength of the deformed halo nuclei $^{31}$Ne and $^{37}$Mg: experimental extraction and microscopic continuum theory for configuration and deformation assignment

The nuclei $^{31}$Ne and $^{37}$Mg, the heaviest established one-neutron halos, owe their halo character to deformation-driven $p$-wave intruder configurations, but their last-neutron Nilsson configurations and ground-state deformations $\beta_2$ remain unsettled. Lu, Sagawa and Zhou (Phys. Rev. C 111, 064320 (2025)) predicted that the soft electric-dipole ($E1$) strength just above the neutron threshold is a sensitive fingerprint of the candidate configurations, differing by factors of several between halo and non-halo assignments in $^{31}$Ne, with the enhancement in $^{37}$Mg about 60% larger than that of the halo $[321]3/2$ configuration in $^{31}$Ne, using single-particle deformed Woods–Saxon wave functions with a plane-wave treatment of the continuum. The problem asks for a quantitative determination of the threshold $E1$ response of both nuclei, that is, the distribution $dB(E1)/dE$, its peak energy, width, partition among $K$ bands, and integrated $B(E1)$, such as by direct extraction from exclusive Coulomb-breakup measurements or by microscopic deformed continuum QRPA/finite-amplitude-method calculations including pairing and blocking, in order to fix the configuration and deformation of each nucleus. The problem is open as of August 2026: the last-neutron configuration and the ground-state deformation of both nuclei remain unsettled.

Background

Far from the valley of stability, in and beyond the 'island of inversion' — the region around the magic neutron number $N=20$ where the conventional $N=20$ shell gap collapses and $pf$-shell intruder orbitals enter the ground-state configuration, with deformation persisting toward heavier neutron numbers for the lightest elements — nuclear deformation and weak binding combine to produce the heaviest known one-neutron halos, $^{31}$Ne ($N=21$) and $^{37}$Mg ($N=25$). A one-neutron halo is an extended neutron density tail that arises when the valence neutron is bound by $S_n\lesssim 1$ MeV ($S_n$ is the one-neutron separation energy) and occupies a state of low orbital angular momentum $\ell\leq 1$, so that no centrifugal barrier confines it. Both nuclei were identified as halos through anomalously large one-neutron-removal and reaction cross sections: inclusive Coulomb breakup of $^{31}$Ne on Pb (Nakamura et al., 2009, 2014), and one-neutron removal from $^{37}$Mg on C and Pb targets together with reaction cross sections (Kobayashi et al., 2014; Takechi et al., 2014). Their empirical binding is extremely weak, $S_n\approx 0.15$ MeV for $^{31}$Ne and $\approx 0.22$ MeV for $^{37}$Mg. In an axially deformed mean field the valence neutron occupies a Nilsson orbital labeled $[N n_z \Lambda]\Omega$, where $N$ is the principal oscillator quantum number, $n_z$ and $\Lambda$ are the projections of the oscillator and orbital quantum numbers on the symmetry axis, and $\Omega$ is the projection of the total single-particle angular momentum; deformation mixes spherical components of both parities with the same $\Omega$, so a nominally $pf$-shell orbital such as $[321]3/2$ can acquire a large halo $p$-wave amplitude. Because a soft electric-dipole ($E1$) excitation, i.e. the transition of the weakly bound neutron into continuum states just above threshold, quantified by the reduced transition-probability distribution $dB(E1)/dE$, is dominated by the halo tail, its magnitude and shape are extremely sensitive to the $p$-wave content and hence to the configuration and to the quadrupole deformation parameter $\beta_2$. In an axially symmetric nucleus the strength further splits among final-state rotational bands labeled by the projection $K$ of the angular momentum on the symmetry axis, and the partition of $B(E1)$ among $K$ components is itself configuration dependent. Coulomb breakup on a heavy target at a few hundred MeV per nucleon proceeds dominantly through absorption of virtual photons, so the measured relative-energy spectrum of the core fragment plus neutron maps the $E1$ strength, making exclusive Coulomb breakup the natural experimental probe. Lu, Sagawa and Zhou (2025) calculated the threshold $E1$ strength of $^{31}$Ne and $^{37}$Mg with single-neutron wave functions from a phenomenological deformed Woods–Saxon potential (a standard phenomenological nuclear mean field) whose depth is tuned to the empirical $S_n$, treating the continuum final states in a plane-wave approximation (PWA), that is, neglecting the residual interaction of the escaping neutron with the deformed core. They find that the halo configurations $[321]3/2$ (at $\beta_2\approx 0.5$) and $[330]1/2$ (at $\beta_2\approx 0.24$) in $^{31}$Ne enhance the threshold peak by factors of several relative to the non-halo $[202]3/2$ configuration (at $\beta_2\approx 0.32$), that the two halo configurations differ by about a factor of two in peak height, and that in $^{37}$Mg the assigned $[321]1/2$ configuration ($\beta_2\approx 0.46$) gives an enhancement about 60% larger than $[321]3/2$ in $^{31}$Ne. The soft $E1$ strength is thus a quantitative fingerprint of configuration and deformation. Two limitations frame the open problem. First, the existing measurements are inclusive and cannot resolve the predicted peak structure or the $K$ partition, so the candidate configurations of both nuclei remain unsettled. Second, the calculation omits collective correlations and a consistent continuum treatment: the random-phase approximation (RPA, the standard small-amplitude response theory; in its quasiparticle version QRPA, built on a Hartree–Fock–Bogoliubov pairing ground state) or the finite-amplitude method (FAM, a numerically efficient solution of the RPA equations by iterative linear response), combined with the deformed relativistic Hartree–Bogoliubov theory in continuum (DRHBc, a covariant density-functional framework that solves the pairing problem in coordinate space with scattering boundary conditions), provides the microscopic framework called for, and it has so far been demonstrated only for even-even nuclei.

Problem Statement

The problem is to establish quantitatively the near-threshold (soft) electric-dipole response of the deformed one-neutron halo nuclei $^{31}$Ne and $^{37}$Mg and to use it to assign the last-neutron Nilsson configuration and the ground-state quadrupole deformation $\beta_2$ of each nucleus. For each nucleus the quantities to be determined are the strength distribution $dB(E1)/dE$ just above the neutron threshold, specifically its peak energy, its width, the partition of the strength among final-state $K$ bands, and the integrated $B(E1)$ over the low-energy region, taken up to $5$ MeV above the neutron threshold following the convention of the source calculation. The quantities may be established along complementary routes, each individually valuable; the following are the natural ones. (1) Experimental: a direct extraction of these quantities from exclusive Coulomb-breakup or equivalent measurements (for example on a Pb target at energies of order a few hundred MeV per nucleon), with reaction-model control of nuclear breakup and higher-order Coulomb effects, at a precision sufficient to discriminate the candidate configurations, $[321]3/2$, $[330]1/2$ and $[202]3/2$ in $^{31}$Ne and $[321]1/2$ in $^{37}$Mg, whose predicted threshold strengths differ by factors of several and whose two halo peaks in $^{31}$Ne differ by about a factor of two. (2) Theoretical: a microscopic calculation of the soft $E1$ response built on deformed Hartree–Fock–Bogoliubov or DRHBc ground states with explicit continuum coupling and pairing, including quasiparticle blocking of the odd neutron (excluding its occupied level from the pairing field), via deformed QRPA or the finite-amplitude method, yielding predictions for the configuration–$\beta_2$ discrimination that experiment can test. A complete answer replaces the present configuration ambiguity of both nuclei by a determined assignment with stated confidence; an answer that establishes either the measured strength distribution or the validated microscopic strength function for either nucleus constitutes substantive partial progress.

References

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