A Halo: The Trigger to a New Era of Nuclear Correlations

In this contribution to the Halo-40 Proceedings, we discuss two topics regarding halo phenomena: The first is the pairing anti-halo effect on the neutron radius of halo nuclei and its restoration due to the coupling to the continuum; the second is the soft dipole excitation of deformed halo nuclei. We demonstrate the importance of Hartree-Fock-Bogoliubov and the relativistic Hartree-Bogoliubov theory in continuum for properly taking into account the halo nature of extended wave functions in calculations of neutron radii, as well as the soft dipole excitations of halo nuclei. It was shown that the anti-halo effect is very sensitive to the continuum coupling induced by Bogoliubov-type quasi-particles, which largely cancels the anti-halo effect on the neutron radius. The soft dipole excitations of deformed halo nuclei Ne-31 and Mg-37 are discussed within the deformed Woods-Saxon model. We point out that the sharp peak just above the threshold in the dipole response is created by the halo effect, and its strength can be used to identify the magnitude of deformation and the halo configuration in the Nilsson level scheme.

💡 Research Summary

In this contribution to the Halo‑40 Proceedings, the authors address two intertwined aspects of halo phenomena in exotic nuclei: (i) the pairing anti‑halo effect on neutron radii and its partial restoration through coupling to the particle continuum, and (ii) the soft dipole (E1) response of deformed halo systems, exemplified by 31Ne and 37Mg. The work emphasizes that a proper theoretical description of halo nuclei requires self‑consistent frameworks that treat pairing correlations and continuum coupling on an equal footing, namely the Hartree‑Fock‑Bogoliubov (HFB) and Relativistic Hartree‑Bogoliubov (RHB) theories formulated in coordinate space (DRHBc).

The paper begins with a concise historical overview, noting the discovery of the anomalously large matter radius of 11Li and the subsequent identification of a variety of one‑ and two‑nucleon halo systems across the nuclear chart. It lists the broad set of exotic phenomena associated with halos—skin formation, shell evolution, di‑nucleon correlations, BEC‑BCS crossover, deformation decoupling, soft dipole modes, anti‑halo effects, and novel decay channels—setting the stage for the two focal topics.

Pairing anti‑halo effect. In a pure mean‑field (Hartree‑Fock) picture, a weakly bound s‑ or p‑wave neutron leads to a wave function that decays as exp(−αr) with α∝√|ε|, causing the rms radius to diverge as the separation energy ε→0. When static pairing is introduced (HFB or RHB), the lower component v(r) of the quasiparticle wave function acquires an exponential tail governed by the pairing gap Δ rather than the single‑particle energy. If Δ remains finite as ε→0, the radius saturates at a value proportional to 1/Δ, which is the essence of the anti‑halo effect. The authors illustrate this with a schematic calculation for a 3s1/2 state (ε=−0.5 MeV, λ=−0.15 MeV), showing that the HFB v‑component is markedly more compact than the corresponding HF+BCS result.

However, the same pairing interaction also scatters nucleon pairs into the continuum. For quasiparticle energies above the Fermi surface (E_i>|λ|), the upper component u_i(r) becomes non‑localized, behaving as sin(βr)/r, and the pairing field Δ(r) acquires a longer range. This continuum coupling counteracts the anti‑halo suppression, effectively “restoring” part of the halo extension. The authors quantify this competition by varying the depth of a Woods‑Saxon potential for the 2p3/2 orbital in 31Ne, comparing rms radii obtained with HF and HFB canonical wave functions. The HFB curve shows a pronounced reduction at shallow binding, but the inclusion of continuum states in a full DRHBc calculation leads to a partial recovery of the radius and an enhanced dipole strength.

Soft dipole excitations in deformed halos. The second part of the work focuses on the low‑energy E1 response of deformed halo nuclei. Using a deformed Woods‑Saxon model, the authors calculate the dipole strength function for 31Ne and 37Mg, identifying a sharp peak just above the neutron emission threshold. This peak originates from the oscillation of the weakly bound halo neutrons against the deformed core and is highly sensitive to the Nilsson configuration of the valence neutron. By employing the DRHBc theory with the PC‑PK1 density functional, they demonstrate that the same peak appears in fully self‑consistent relativistic calculations, and its magnitude varies with the deformation parameter β_2 and the occupation of specific Nilsson orbitals (e.g.,