Microscopic description of the proton halo in $^{12}$N

TL;DR

Proton halos are challenging to describe due to the Coulomb barrier; this work extends the deformed relativistic Hartree-Bogoliubov in continuum (DRHBc) framework to the proton-halo regime using $^{12}$N as a benchmark. By solving the DRHBc equations with a Dirac Woods-Saxon basis and incorporating axial deformation, the study reproduces available $S_p$ and matter rms radii for the $N=5$ isotones and reveals a dramatic halo in $^{12}$N. The halo originates from occupation of a weakly bound $(1/2)_2^-$ proton orbital with dominant $p$-wave content, yielding an $r_{ m rms}$ over $3.2$ fm and contributing about $90\\%$ of the proton density at large $r$, alongside a pronounced core–halo shape decoupling (prolate core with $eta_2 \\approx 0.20$ vs. oblate halo with $eta_2 \\approx -0.33$). These results demonstrate DRHBc’s ability to provide a microscopic, self-consistent description of proton halos and highlight avenues for future extensions, such as angular momentum projection to access electromagnetic moments and spectra.

Abstract

The year 2025 marks the 40th anniversary of the discovery of halo nuclei and the 15th anniversary of the development of the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc). In this work, we present the first DRHBc description of the proton halo phenomenon. The available experimental proton separation energies and empirical matter root-mean-square (rms) radii are reasonably well reproduced for the $N=5$ isotones, ranging from the stable nucleus $^{9}$Be to the drip-line nucleus $^{12}$N. In particular, the DRHBc theory captures the abrupt increase in rms radii at $^{12}$N, unambiguously corroborating its proton halo structure. The formation of this halo is attributed to the occupation of a weakly bound orbital with dominant $1p$ components by the valence proton, which contributes approximately $90\%$ to the diffused proton density of $^{12}$N at large distances from the nuclear center. A shape decoupling between the prolate core and the oblate halo in $^{12}$N is predicted.

Figures (5)

• Figure 1: (a) Proton separation energies, (b) matter rms radii, and (c) quadrupole deformation parameters for the $N=5$ isotones from $Z=4$ ($^9$Be) to $Z=7$ ($^{12}$N). In order to quantify theoretical uncertainties, in the DRHBc calculations the density functionals NL3$^*$Lalazissis2009PLB, NL3 Lalazissis1997PRC, NLSH Sharma1993PLB, and PK1 Long2004PRC are employed, with the pairing strength varied from $-340$ to $-380$ MeV fm$^3$. For comparison, the experimental separation energies from AME20 AME2020(3) are shown in (a), and the empirical matter radii estimated by Warner et al. Warner2006PRC and Ahmad et al. Ahmad2017PRC are shown in (b).
• Figure 2: (a) Proton rms radii for the $N=5$ isotones from $Z=4$ ($^9$Be) to $Z=7$ ($^{12}$N), and (b--e) their proton density distributions in the $xz$ plane with $z$ being the symmetry axis.
• Figure 3: Rms radii versus single-particle energies for the proton orbitals in the canonical basis for $^{12}$N. Each orbital, labeled by $\Omega^\pi_i$, is represented by a vertical bar, with different colors indicating the contributions from its main components.
• Figure 4: (a) Angle-averaged densities of individual single-proton orbitals in $^{12}$N, along with the total proton density, plotted as functions of the radial coordinate $r$; (b) Corresponding contributions of each orbital to the total proton density as functions of $r$.
• Figure 5: Same as Fig. \ref{['fig2']}(e), but shown separately for (a) the core and (b) the halo of $^{12}$N.