Halo Phenomena in Light to Medium Mass Nuclei with Three-Body Models

1. Introduction

The year 2025 marks the 40th year anniversary of the discovery of halo nuclei to which this conference in Beijing is dedicated. The experimental verification by Isao Tanihata and colleagues at RIKEN (Japan) [6], that the radius of the heaviest lithium isotopes significantly deviates from the simple $A^{1/3}$ Fermi rule has challenged our views of basic nuclear physics. This initial accomplishment is an important cornerstone and must be celebrated because it has generated a sort of butterfly effect in nuclear physics, with far-reaching consequences that were unimaginable only 40 years ago. It has spurred new developments in quantum mechanics and the rise of new theoretical ideas, the first of which is the interpretations of halos as extended nucleon orbitals by Jonson and Geissel [7] (see Figure 1). One- and two-neutron halos have been found in many isotopes of the nuclear chart, especially far from stability line, where the imbalance of neutrons over protons makes it easier to populate low-ℓ near-threshold energy levels that are more prone to give rise to extended halos and a wealth of new phenomena. This, in turn, has demanded for new data and the experiments have pushed the limits of our reach far beyond the valley of stability venturing deep into the region of the nuclear chart where unstable systems little by little fade toward the drip-line.

The field is now too big to attempt a balanced and complete review, therefore I will concentrate on the contribution given by the Padova group, mainly initiated by the keen interest of the late Prof. Vitturi in this matter, and including the work of all the post-docs, collaborators and friends that have participated to this adventure.

While ab initio theories are theoretically superior to empirical few-body models, they still struggle to reproduce the physics of halo and cluster nuclei. Consequently, there remains a significant opportunity for our current approach to provide meaningful new results, particularly for medium-mass nuclei.

In the next sections, we will discuss very briefly some of the most successful works that we have published in recent years on the topic of halo phenomena, cluster nuclei, three-body models and pairing interactions. In particular, the results of the three-body model based on the hyper-spherical formalism for ²⁹F and ³¹F will be described, highlighting the relations with the island of inversion. Then, another work dealing with proton-neutron correlations will be discussed for the interesting case of ¹⁰²Sb, where an effective weakening of the p-n interaction is in place.

2. New Insights on the Structure of ²⁹F at the Border of the Island of Inversion

In 2020, the results of two new experiments have been published. Ref. [8] has provided various measurements of reaction cross sections for beams of ²⁹F on carbon, essentially confirming the large matter radius and the two-neutron halo structure of this isotope. In Ref. [9], new spectroscopic information on the sub-system ²⁸F was measured that proved crucial for our interpretation of these systems. The central point is the role played by the $2p_{3/2}$ orbital in determining the structure and the halo nature of ²⁹F. About the same time, we had reported the results of a three-body approach on ²⁹F = ²⁷F $+2n$ where we analyzed various hypotheses for the ordering of the single-particle levels, that we called standard, intruder, degenerate and inverted scenarios in Ref. [3]. It is not uncommon that, moving to the drip-lines, large energy gaps that are associated to shell gaps, close and rearrange to such an extent that some states of the $N=3$ pf-shell might come lower than some states in the $N=2$ sd-shell, giving a shell inversion, as shown in Figure 2. The region between ³⁰Ne and ³⁴Mg, where this phenomenon occurs, is thus called island of inversion. We believe that ²⁹F must also be part of this island, extending its southern shore.

The details of the hyper-spherical approach that we used to model this system are summarized in Ref.[3] and more can be found in Ref. [10,11]. The core-neutron interaction is a Woods-Saxon plus spin-orbit adjusted to the available experimental information in ²⁸F, while the neutron-neutron potential is chosen as the Gogny-Pires-Tourreil (GPT) potential. A small phenomenological Gaussian three-body potential is added to the hamiltonian. The resulting phase shifts for the ²⁷F $+n$ system are shown in Figure 3 for the various scenarios.

Subsequently, the ²⁹F (²⁷F+n + n) wave functions are built within the hyper-spherical harmonics expansion formalism (HHE) and the three-body hamiltonian is then diagonalized in a Transformed Harmonic Oscillator (THO) basis, that allows some control over the density of states in the continuum region. This is necessary because we want to calculate the ground and excited states, but we also want to calculate low-lying continuum states, in order to study the electric dipole excitations to the continuum.

Soon after the publication of the new experimental data [9], we refined our model coming up in Ref. [1] with the scenario that we believe is the correct one for ²⁹F, i.e. an inverted scenario (D), but not too far from the degenerate one. We have also calculated the dipole strength distribution (with a total cumulative strength of about $\sim 2e^{2}f{m}^2$), see Figure 4, and predicted the Relativistic Coulomb Excitation cross-section, suggesting how this could lead to a further confirmation of these results. These two quantities are proportional at low energy and in certain regimes.

In a subsequent in-depth analysis, [2], we have studied the convergence of the HHE method with the hyper-angular momentum for the dipole distribution. We employed continuum states in a pseudostate approach using THO basis and we calculated form factors that are used in continuum-discretized coupled-channels (CDCC) calculations to describe low-energy scattering. Our predictions show the typical low-lying enhancement of the E1 response expected for halo nuclei and the relevance of dipole couplings for low-energy reactions on heavy targets.

3. Extensions to ³¹F: radii, B(E1) and Halo Structure

The next interesting system in this chain of fluorine isotopes is ³¹F: this nucleus is studied here as a core of ²⁹F plus two neutrons. Admittedly, this is not ideal, because ²⁹F is already seen as a ²⁷F core plus two neutrons, so that a more appropriate set-up would consider a five-body model with 4 valence neutrons. As this was beyond our reach, we tried with the three-body model, relying on the fact that the core has been found to posses only a moderate halo.

If, as in Figure 2, the $1d_{3/2}$ and $p{f}_{3/2}$ orbitals are switched, ²⁹F has (ideally) 4 neutrons in the p state and the next isotope, ³¹F might have a very similar structure with the crucial p shell filled and some neutrons in the next shell. Therefor, a delicate balance is in place between the more tightening effect of two extra neutrons in a high angular momentum state (not leading to halo) and an even lower binding energy (that leads to halo formation). It might turn out that ³¹F is less of a pronounced halo that its predecessor!

Again, we have built several different scenarios, stretching the models’ parameters to the limits in all possible ways, and we have examined the consequences. The radii obtained in all the scenarios indicate, without much spread, that ³¹F has an extended halo structure, as can be seen in Figure 5 and far from the $A^{1/3}$ behaviour.

Without repeating the analysis in Ref. [4], our results indicate that the crucial factor is the dominance of pf-shell configurations, with various degrees of p-wave components. In all cases they point to increased radius and larger B(E1), the hallmarks of a halo formation. We predicted a total integrated B(E1) strengths of in excess of $\gtrsim 2.6e^{2}f{m}^2$, compatible with the formation of a halo.

This nucleus still defies a proper experimental verification due to the extreme neutron-to-proton ratio and thus low experimental beam intensity.

4. p-n Correlations in the External Orbitals of ¹⁰²Sb

We move now, from neutron-neutron correlations in neutron-rich nuclei to proton-neutron correlations in proton-rich nuclei.

The ¹⁰⁰Sn, located at the crossing of the $N=50$ and $Z=50$ magic number lines, is a bound system. The ¹⁰¹Sn system, i.e. a ¹⁰⁰Sn +n, is also bound, while the mirror ¹⁰¹Sb = ¹⁰⁰Sn +p is not bound. Therefore, we are looking at a critical proton-drip line region, where the addition of a single proton is sufficient to change the game. The 1⁺ ground state of ¹⁰²Sb = ¹⁰⁰Sn +p+n is an interesting nucleus, because it is interpreted as an effective deuteron built out of the orbitals surrounding the ¹⁰⁰Sn core. In Ref. [5] it is suggested that it might be a proton emitter due to a weakening effect of the proton-neutron (p-n) interaction with respect to a bare deuteron. A time-dependent three-body calculation (similar to the approach adopted by Oishi in Ref.[12]) is performed, based on the quantum evolution operator in which the hamiltonian (with center of mass correction) is modeled as in Figure 6.

The effective proton-neutron interaction is corrected by a factor $f_{pn}{v}_{pn}({\overrightarrow{r}}_p,{\overrightarrow{r}}_n)$, such that $f_{pn}=1$ gives back the bare deuteron interaction.

The 1⁺ ground state of ¹⁰²Sb is suggested as a possible proton emitter unbound resonance. This conclusion is reached by assuming a weakening effect on the proton-neutron (pn) interaction with respect to a bare deuteron. An analogous, but less dramatic, reduction is necessary to reproduce the empirical binding energies of similar systems with lower mass: ⁴²Sc(1⁺) and ¹⁸F_gs. The results are reported in Figure 7.

We expect that realistic calculation give a coefficient f that is way smaller than 1 and in this case $f=0.92$ is the threshold for binding. Thus, it is very likely that ¹⁰²Sb is an unbound nucleus, prone to proton emission.

If one tries a simple power-law fit of the $f_{pn}$ coefficient from Figure 7 with masses, including bare deuteron, fluorine and scandium, one gets for tin a value of about $f_{pn}\sim 0.33$, that is much lower than the threshold. Another observation is that the coefficients lowers with increasing ℓ, because forming a deuteron becomes progressively harder for orbitals with higher angular momentum. Therefore, going from f to g, one expects a lowering.

This system might prove very important in order to further constraint the p-n interaction, therefore experiments should be planned to search for a low-lying 1⁺ resonance in the spectrum of ¹⁰²Sb.

5. Conclusions

The long quest for the halo nuclei, initiated 40 years ago is not anywhere concluded. Experimental physicists are conceiving new accelerators to push the limits of the asymmetry and the intensity frontiers and new detectors to enhance our discovering potential and enlarge the number of measured observables. Theorists are inventing new models, comparing with experimental data, adjusting the theories and repeating the process, spiraling towards a comprehensive description of nuclear physics far from stability. Forty years later, the excitement for this endeavor remains as strong as ever.