Rotation-induced proton binding at the drip line: pairing confirmation of spin thresholds, particle-decay widths of quasi-bound rotational bands, and functional robustness

Cranked relativistic mean field (CRMF) calculations without pairing predict that in very proton-rich nuclei near the drip line, rotational bands that are proton quasi-bound at low spin become proton bound above a threshold spin, that rotation-induced giant proton halos form in nuclei with $Z\leq 20$, and that the nuclear landscape extends substantially beyond the spin-zero proton drip line. The source work itself flags these predictions as model-limited: static pairing was neglected, so the lowest spins at which proton-bound configurations appear — notably their lowering below the pairing-disappearance spin in nuclei such as $^{16,17}$Ne — await confirmation by cranked relativistic Hartree-Bogoliubov calculations with pairing and blocking; the particle-decay widths that decide the observability of the quasi-bound segments of the bands were not computed; and the sensitivity of the thresholds, halo formation, and landscape extension to the covariant energy density functional was checked only in a single nucleus. The problem posed here is to put these predicted rotation-induced proton-binding phenomena on a quantitatively reliable footing by resolving these three theory-side gaps, with experimental spectroscopy as the ultimate test.

Background

The chart of nuclides ends at the proton and neutron drip lines: the boundaries beyond which an additional proton or neutron is no longer bound. Near the proton drip line the highest occupied proton single-particle states can sit at positive energy, trapped for a finite time behind the combined Coulomb and centrifugal barrier. Such states are called quasi-bound (or resonance) states; they decay by proton emission, in contrast to bound states at negative energy. The present question concerns what happens to these states when the nucleus rotates rapidly.

Collective rotation is described microscopically in cranking models: the nucleus is constrained to rotate about a fixed axis at angular frequency $\Omega$, and the single-particle energies in the rotating frame (routhians) shift with $\Omega$ in proportion to the angular momentum carried by each orbital — the Coriolis interaction. In covariant density functional theory (CDFT), where the nucleus is described by meson fields coupled to nucleons, the rotating problem is solved either at the mean-field level without pairing (cranked relativistic mean field, CRMF) or with pairing correlations included through Bogoliubov quasiparticles built on a pairing interaction (cranked relativistic Hartree-Bogoliubov, CRHB). CRHB is the more complete framework at low and moderate spin; static pairing is known to disappear at high spin, with the disappearance spin $I_{unpair}$ roughly $5\hbar$ for $Z\leq 10$, rising to $10\hbar$ for $12\leq Z\leq 20$ and $15\hbar$ for $22\leq Z\leq 36$ along the yrast line (the lowest-energy configuration at each spin), and pairing is further reduced by blocking — the suppression of pairing when one signature partner of an orbital pair is occupied (signature is the quantum number of an orbital under a rotation by 180° about the rotation axis), as happens in particle-hole excited configurations.

Earlier work on rotation embedded in the particle continuum concentrated on the neutron-rich side. Cranked Skyrme-Hartree-Fock calculations without pairing for weakly bound neutron-rich neon and magnesium isotopes (Nazarewicz, Dobaczewski, Matev, Mizutori, Satula, 2001) revealed the role of weakly bound high-$j$ intruder orbitals (orbitals carrying large single-particle angular momentum $j$ that intrude from a higher major shell), and a particle-plus-core coupled-channel model using the Berggren single-particle ensemble — which treats bound, resonance, and scattering states on an equal footing — computed the neutron-decay widths of rotational states embedded in the continuum in $^{11}$Be and unbound deformed $^{39}$Mg (Fossez et al., 2016). CRMF calculations at the neutron drip line then showed the birth and death of particle-bound rotational bands with changing spin and a rotation-induced extension of the nuclear landscape beyond the spin-zero neutron drip line (Afanasjev, Itagaki, Ray, 2019). Rotational bands based on proton-emitting configurations have been observed experimentally only in $^{59}$Cu and $^{141}$Ho; the rotational excitations of the extremely proton-rich nuclei at and beyond the proton drip line that the source work considers have not been experimentally studied so far.

The source work (Afanasjev, Teeti, Taninah, Physica Scripta 99, 065313 (2024); arXiv:2210.03833) extended this physics to the proton side through systematic CRMF calculations with the NL3* covariant energy density functional (CEDF — the specific functional specifying the meson-nucleon couplings; other examples are DD-MEX and NL1). Its central findings are: (i) strong Coriolis interaction acting on occupied high-$j$ or strongly mixed M intruder orbitals (orbitals with large mixing of harmonic-oscillator shells differing by $\Delta N=2$, hence large angular momentum content) drives the highest occupied proton single-particle states from positive into negative energy, so rotational bands that are proton quasi-bound at zero or low spin become proton bound above a threshold spin; (ii) occupation of these M orbitals triggers a rotation-induced giant proton halo — an extended, low-density proton distribution formed by a mechanism entirely different from the $s$/$p$-orbital halos of non-rotating nuclei — with calculated examples in $Z\leq 20$ nuclei; (iii) the region of proton-bound nuclei extends substantially beyond the spin-zero proton drip line, with the transition spin increasing along an isotopic chain with proton excess; and (iv) the kinematic moment of inertia $J^{(1)}$ and the transition quadrupole moment $Q_t$ differ strongly between normal and giant-halo bands (e.g., $Q_t\approx 0.55$ eb versus $\approx 1.4$ eb), providing experimental fingerprints. All of this was computed without static pairing. In nuclei close to the drip line, such as $^{16,17}$Ne, the lowest spin $I_{lowest}$ at which proton-bound configurations appear lies below $I_{unpair}$, and the authors state explicitly that CRHB calculations with pairing are needed to confirm or reject this lowering. The quasi-bound segments of the bands were characterized only through barrier arguments; no proton-decay widths were computed, the width physics being borrowed from the neutron-rich coupled-channel studies. Functional sensitivity was spot-checked only in $^{14}$Ne with DD-MEX and NL1.

Problem Statement

Put the predicted rotation-induced proton-binding phenomena at the proton drip line on a quantitatively reliable footing by resolving the three theory-side gaps that the source CRMF calculations (performed without static pairing) leave open. The common objective is a model-controlled, falsifiable account of when and where collective rotation makes very proton-rich nuclei proton bound — the prerequisites for experimental tests of the predicted birth of proton-bound rotational bands, rotation-induced giant proton halos, and the extension of the nuclear landscape beyond the spin-zero proton drip line. Specifically:

1. Pairing confirmation of the spin thresholds. Do pairing-inclusive cranked density-functional calculations — cranked relativistic Hartree-Bogoliubov (CRHB) with a stated pairing interaction, blocking of unpaired nucleons (the suppression of pairing correlations in the orbitals occupied by unpaired particles), and the Coriolis anti-pairing effect (the rotation-induced suppression of pairing correlations) treated — confirm or reject the lowering of the lowest spin $I_{lowest}$, at which rotation-induced proton-bound configurations appear, below the pairing-disappearance spin $I_{unpair}$ (the spin above which static pairing is expected to vanish) found in the unpaired CRMF calculations for nuclei near the proton drip line (for example $^{16,17}$Ne), and what are the true lowest spins at which rotation-induced proton-bound configurations occur across the predicted even-$Z$ isotopic chains with $4\leq Z\leq 36$?

2. Decay widths and observability. What are the quantitative proton-decay widths of the quasi-bound (resonance) parts of these rotational bands, computed with explicit coupling to the continuum, and for which borderline states do these widths allow proton emission to compete with — or be overwhelmed by — electromagnetic decay, i.e. which quasi-bound rotational states are in principle observable?

3. Functional robustness. How sensitive are the predicted spin thresholds, the formation of rotation-induced giant proton halos, and the rotation-induced extension of the nuclear landscape to the choice of covariant energy density functional, beyond the spot check in $^{14}$Ne with DD-MEX and NL1, and what uncertainty attaches to these predictions?

Together, these three components — thresholds with pairing, widths for observability, and functional uncertainty bounds — would convert the unpaired mean-field predictions into falsifiable targets (spin thresholds, widths, and the $J^{(1)}$ (kinematic moment of inertia) and $Q_t$ (transition quadrupole moment) fingerprints) for spectroscopy of very proton-rich nuclei.

References

1. A. V. Afanasjev, S. Teeti, A. Taninah, 'Fast rotation of nuclei with extreme isospin in the vicinity of neutron and proton drip lines', Physica Scripta 99, 065313 (2024). DOI: 10.1088/1402-4896/ad4e18, https://doi.org/10.1088/1402-4896/ad4e18 (arXiv:2210.03833, https://arxiv.org/abs/2210.03833)
2. A. V. Afanasjev, N. Itagaki, D. Ray, 'Rotational excitations in near neutron-drip line nuclei: The birth and death of particle-bound rotational bands and the extension of nuclear landscape beyond spin zero neutron drip line', Physics Letters B 794, 7 (2019). DOI: 10.1016/j.physletb.2019.05.021, https://doi.org/10.1016/j.physletb.2019.05.021 (arXiv:1905.11500, https://arxiv.org/abs/1905.11500)
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