The effect of inversion of $p$ and $f$ orbits on halo formation in heavy sodium isotopes

TL;DR

This work addresses halo formation in neutron-rich Na isotopes near the drip line and investigates how the potential inversion of the $f_{7/2}$ and $p_{3/2}$ neutron orbitals influences ground-state structure. It employs a few-body core+$n$ and core+$2n$ framework with Woods-Saxon core–neutron potentials, a GPT $n$-$n$ interaction, and a phenomenological three-body force, analyzed within a discrete pseudostate basis formed by a transformed harmonic oscillator in hyperspherical coordinates. The main finding is that shell inversion markedly enhances halo features, predicting a possible one-neutron halo in $^{34}$Na and Borromean halos in $^{37}$Na and $^{39}$Na, with larger matter radii and strong low-energy $B(E1)$ strength serving as diagnostic signals. These results offer predictive guidance for experiments probing interaction cross sections, breakup reactions, and dipole responses, and point to extensions that include core excitations and more complete angular-momentum treatments to refine the halo scenario.

Abstract

The role of the inversion of the $p$ and $f$ shell-model orbits in the emergence of halo structures in the ground states of neutron-rich $^{34,37,39}$Na is investigated. Families of two- and three-body models are constructed with effective core-neutron interactions, with parameter choices based on a combination of the available experimental data and systematic trends, as well as the GPT $n$-$n$ interaction and a phenomenological three-body force. Our results indicate a possible one-neutron halo in $^{34}$Na, while $^{37,39}$Na exhibit features of Borromean halos. The halo formation is driven by the weakening of the shell gap and inversion of the $2p_{3/2}$ and $1f_{7/2}$ orbits expected to occur somewhere near these masses. We further show that the electric dipole response provides a clear and sensitive probe of halo structure in these isotopes.

Figures (5)

• Figure 1: Schematic representation of a potential scenario for the evolution of neutron shell-model orbits in the independent-particle shell model and the associated change in filling for sodium isotopes with $A={32, \cdots, 39}$ (energies not to scale). $\Delta_\ell$ (arrows) and $P_\ell$ (boxes) represent the single-particle energy level gaps to the threshold and correlation energies, respectively. An inversion between $p$ and $f$ orbits occurs between $^{33}$Na and $^{34}$Na, then the $f$ orbit gets very close to threshold at $A=36$ and becomes definitely unbound for $A=38$.
• Figure 2: The valence single-particle energies of the neutron $f_{7/2}$ and $p_{3/2}$ orbitals as a function of the central Woods-Saxon depth ($V^{(l)}_0$) for the isotopes $^{34,36,38}\mathrm{Na}$. Negative energies correspond to bound states, while positive energies indicate unbound continuum (resonance) states. Resonance energies are extracted from a phase-shift analysis.
• Figure 3: Matter radii ($R_m$) of Na isotopes as a function of mass number $A$. The relativistic mean-field (RMF-BCS) (circles) and shell-model results (pluses and crosses), are taken from Refs. GENG2004 and Otsuka2022, respectively. The gray points (with error bars) are extracted from the preliminary data analysis in Ref. Suzuki2014. Our results are represented as a band, corresponding to a range of parameters used. Thus, the magenta and green bands correspond to the normal and inverted parameter sets as shown in Fig. \ref{['Fig_2bpot']}, respectively, for different values of $s_{1n}$ ($A=34$) and $s_{2n}$ ($A=37,39$). The solid line inside each band is the result obtained with the central experimental or evaluated value of $s_{1n/2n}$. The coloured bands correspond to the full range between the lower and upper limits of the neutron removal energy; the open bands for $A=37,39$ represent a wider range from the deeply bound state with no three-body force (lower limit) to an almost unbound nucleus (upper limit).
• Figure 4: Filling fractions of the $(f_{7/2})^2$ (dashed blue) and $(p_{3/2})^2$ (solid red) orbits in $^{37}\mathrm{Na}$ and $^{39}\mathrm{Na}$ as a function of two-neutron separation energy $s_{2n}$. Panels (a) and (b) present the results for $^{37}\mathrm{Na}$, while panels (c) and (d) correspond to $^{39}\mathrm{Na}$. The upper row (a, c) displays calculations using the normal parameter set, and the lower row (b, d) shows those obtained with the inverted set. Solid symbols show the central value, with transparently filled symbols show the normal limits. Open symbols show the extreme limits (white boxes in Fig. \ref{['FigmatR']}).
• Figure 5: $B(E1)$ distributions for $^{34}\mathrm{Na}$ (panels a, b), $^{37}\mathrm{Na}$ (panels c, d), and $^{39}\mathrm{Na}$ (panels e, f). The upper panels (a, c, e) show results calculated at the central experimental/evaluated separation energy for both the normal (dashed magenta line) and inverted (solid green line) sets. The lower panels (b, d, f) display results for the inverted set. The shaded band corresponds to the solid light green band in Fig. \ref{['FigmatR']}, for the full range of two-neutron separation energies, and the solid line is for the central value. For details, see main text.