Halo structure of $^6$He from $\textit{ab initio}$ two-nucleon spatial correlations

Technical Summary: Halo structure of 6He from ab initio two-nucleon spatial correlations

Problem Statement
Atomic nuclei near the "drip lines" often exhibit clustering phenomena, where a stable core is surrounded by loosely bound nucleons forming a "halo." While neutron halos (e.g., in $^{11}$Li and $^{6}$He) are characterized by large interaction cross-sections and narrow momentum distributions, extracting detailed geometric structures from ab initio calculations remains challenging. Standard one-body density profiles often fail to fully reveal the geometrical structure of multi-neutron halos, particularly in Borromean systems like $^{6}$He (where the $n$-$n$ and $n$-$\alpha$ subsystems are unbound). The authors aim to characterize the structure of $^{4}$He and $^{6}$He by moving beyond one-body probabilities to utilize two-body spatial correlations, specifically focusing on the relative distances between nucleon pairs to probe the halo formation mechanism.

Methodology
The study employs ab initio no-core shell model (NCSM) calculations using the MFDn code. The nuclear Hamiltonian includes the Daejeon16 nucleon-nucleon (NN) interaction (derived from chiral effective field theory via the similarity renormalization group) and the Coulomb interaction. Calculations are performed for $^{4}$He and $^{6}$He with a basis space cutoff of $N_{max} = 16$ and a harmonic oscillator energy spacing of $\hbar\Omega = 10$ MeV.

To analyze the structure, the authors define and compute expectation values for two specific coordinate-space two-body operators projected onto coupled-$J$ basis states with defined two-body spin ($S$) and isospin $z$-projections ($t_{z}$):

1. The Pair-Number Operator ($r^0$): Defined as $r^0[S, t_{za}, t_{zb}] \equiv P_S P_{t_{za}, t_{zb}}$, where $P$ represents projection operators. Its expectation value yields the number of nucleon pairs in specific spin-isospin configurations.
2. The Square-Separation Operator ($r^2$): Defined as $r^2[S, t_{za}, t_{zb}] \equiv r^2 P_S P_{t_{za}, t_{zb}}$, where $r$ is the relative distance between nucleons. Its expectation value provides the mean square separation for specific pair types.

The matrix elements for these operators are constructed in the harmonic oscillator relative motion basis and transformed to the single-particle basis using Moshinsky transformations. The authors analyze the results by decomposing the $^{6}$He neutrons into "core" ($n_\alpha$, occupying $0s_{1/2}$) and "halo" ($n_h$, occupying $0p_{3/2}$) components to construct a binary model of the nucleus.

Key Results

• Spin Configurations: The analysis of $\langle r^0 \rangle$ reveals that the two valence neutrons in the ground state of $^{6}$He predominantly form a spin-singlet configuration. This is consistent with the dominance of $s$-states for the neutron-neutron pairs and the preservation of the $^{4}$He core's spin structure.
• Spatial Separations:
  • The root-mean-square (rms) separation between core nucleons and halo neutrons in $^{6}$He is approximately 80% larger than the pair separations within the "swollen and off-centered $\alpha$ core."
  • The rms separation for proton-proton ($pp$) pairs increases by only ~6% from $^{4}$He to $^{6}$He, whereas the point-proton radius ($r_p$) increases by 25%.
  • The rms separations for neutron-proton ($pn$) and neutron-neutron ($nn$) pairs increase by 40–50% and 66%, respectively.
• Geometric Interpretation:
  • $^{4}$He Structure: The calculated radii and separations suggest a classical tetrahedral structure, where the ratio of the nucleon-nucleon separation to the point-nucleon radius matches the geometric ratio of a tetrahedron's side length to its center-to-vertex distance.
  • $^{6}$He Structure: A binary model analysis indicates that the $\alpha$ core in $^{6}$He is only moderately swollen (consistent with the small $pp$ separation increase). The significant increase in the point-proton radius is primarily attributed to the off-centering of the $\alpha$ core relative to the system's center of mass, rather than a massive expansion of the core itself. The halo neutrons reside at significantly larger distances from the core.

Significance and Claims
The paper claims that two-nucleon spatial correlations, specifically the pair-number operator ($r^0$) and the square-separation operator ($r^2$), encode critical details of halo structures that one-body densities may obscure. The authors demonstrate that these operators can:

1. Identify the dominant spin-singlet configuration of the valence neutrons in $^{6}$He, supporting a "dineutron" halo picture over a "cigar" configuration.
2. Quantitatively distinguish between the swelling of the core and the displacement of the core relative to the center of mass.
3. Provide a route to relate quantum many-body results to classical geometric pictures (e.g., the tetrahedral $^{4}$He).

The authors note that while they have demonstrated the application of this method to $^{6}$He, fully converged ab initio NCSM results for these two-body correlations are still being developed, and suitable extrapolation methods for these operators are deferred to future work. The study serves as a proof of concept for using two-body operators to extract structural information from ab initio wave functions.