Density-driven $\alpha$-particle clustering in neutron-rich oxygen isotopes

Technical Summary: Density-driven $\alpha$-particle clustering in neutron-rich oxygen isotopes

Problem Statement
While $\alpha$-clustering is a well-established phenomenon in light, self-conjugate ($N=Z$) nuclei such as $^{12}$C and $^{16}$O, its behavior in neutron-rich systems remains less understood. Specifically, it is unclear how excess valence neutrons influence the "Mott-like" transition where nuclei reorganize from spatially delocalized configurations into localized $\alpha$-cluster structures under conditions of progressive density dilution. Previous studies have established that constrained radial expansion in $^{16}$O drives a transition to a tetrahedral 4$\alpha$ structure; however, the role of valence neutrons in modifying this transition, the stability of the resulting cluster core, and the nature of the molecular-orbital-like bonding in these dilute, neutron-rich environments requires further investigation.

Methodology
The authors employ the multidimensionally constrained relativistic Hartree–Bogoliubov (MDC-RHB) framework using the density-dependent covariant energy-density functional DD-ME2. The study focuses on neutron-rich oxygen isotopes ($^{18}$O, $^{20}$O, and $^{24}$O), using $^{16}$O as a reference system.

Key methodological features include:

• Constrained Expansion: Calculations are performed by constraining the mass root-mean-square (rms) radius ($R_{\text{rms}}$) to progressively larger values (up to 6.0 fm), effectively simulating isotropic nuclear dilution.
• Deformation Constraints: Global quadrupole deformation is suppressed ($\langle Q_{2\mu} \rangle = 0$) to isolate radius-driven structural evolution. However, all other symmetry-allowed multipole moments (including non-axial octupole $\beta_{32}$ and axial hexadecapole $\beta_{40}$) are left unconstrained to allow the spontaneous emergence of complex shapes.
• Basis Optimization: A harmonic oscillator basis with $N_{\text{shell}} = 12$ and $\hbar\omega = 13$ MeV is utilized to minimize truncation artifacts in the dilute regime.
• Analysis Tools: The study analyzes total energy, pairing correlations, deformation parameters, and 3D density distributions (decomposed into core and valence components). Additionally, a phenomenological four-centre Hückel-type mapping is applied to the single-particle spectrum to interpret valence neutron behavior in terms of molecular-orbital-like states.

Key Results

1. Universal Transition to Tetrahedral Clustering: All studied isotopes ($^{16}$O, $^{18}$O, $^{20}$O, $^{24}$O) exhibit a structural transition from a spatially delocalized nucleonic configuration to a localized tetrahedral 4$\alpha$-like structure as $R_{\text{rms}}$ increases. This transition is marked by a sudden onset of non-zero non-axial octupole deformation ($\beta_{32}$) and a plateau in the total energy curve.
2. Critical Dilution Parameter: The transition occurs at a similar relative dilution parameter, $\chi_{\text{PT}} = \rho_{\text{PT}}/\rho_{\text{GS}} \approx 0.28\text{--}0.31$, across all isotopes. While the absolute transition radius ($R_{\text{PT}}$) increases with neutron number, the critical relative dilution required to trigger clustering remains largely isotope-independent.
3. Role of Valence Neutrons: Excess neutrons do not suppress the formation of the tetrahedral 4$\alpha$ core. Instead, they modify the density distribution:
   • The core (protons and $N=8$ neutrons) retains the localized tetrahedral structure.
   • Valence neutrons remain spatially delocalized, increasing density in the central region and the inter-cluster spaces.
   • This "smoothing" effect enhances spatial connectivity between the $\alpha$-like substructures without destroying the underlying core symmetry.
4. Pairing Correlations: In $^{16}$O, pairing correlations vanish throughout the expansion. In neutron-rich isotopes, neutron pairing develops during expansion, showing sharp changes at the transition radius, which signals a rearrangement of the single-particle spectrum rather than the clustering mechanism itself.
5. Hückel-Type Mapping: The effective coupling ($\mu$) between the four cluster centers decreases as the nucleus expands and the substructures separate. Consequently, the occupation-weighted energy shift ($\Delta E_{\text{val}}$) becomes less negative with increasing neutron number. This indicates that while the tetrahedral core persists, the molecular-orbital-like binding energy contribution from valence neutrons relative to the centroid diminishes as the system dilutes.

Significance and Claims
The paper claims to clarify the mechanism of density-driven localization in neutron-rich finite nuclei. It demonstrates that the emergence of $\alpha$-clustering is primarily driven by the overall dilution of the nuclear system, reaching a critical threshold independent of the specific neutron excess. The study establishes that valence neutrons act as a "glue" that modifies the density profile and maintains connectivity between clusters, consistent with molecular-orbital descriptions where valence nucleons occupy states extending over cluster cores.

The authors conclude that their findings provide a theoretical basis for understanding how neutron excess affects clustering in oxygen isotopes. They note that while the current work identifies the structural evolution and density profiles, direct comparison with experimental spectra and decay observables would require calculations beyond the constrained configurations presented here. The results motivate further experimental investigation into neutron-excess effects on clustering in oxygen isotopes.