Quantitative account of continuum contributions to neutron pair correlation at the neutron drip line: resonance-width relation, continuum background, deformation, and the adequacy of HFB for halos

In a coordinate-space Hartree-Fock-Bogoliubov (HFB) study of pair correlation in nuclei near the neutron drip line, Hamamoto and Mottelson (Phys. Rev. C 68, 034312, 2003) found that weakly bound low-angular-momentum neutron orbits decouple from the pair field and become almost unavailable for pairing once the corresponding Hartree-Fock (HF) level is unbound, while high-angular-momentum orbits with narrow one-particle resonances can be treated in the BCS (Bardeen-Cooper-Schrieffer) discrete-level approximation at the resonant energies. They left open four linked issues that together ask how the unbound continuum participates in pairing: (1) the quantitative relation between the HF one-particle resonant width and the degree to which the level contributes to the many-body pair correlation; (2) the interpretation and quantitative accounting of the weak, slowly varying 'continuum background' occupation in particle-number-consistent calculations; (3) the extension to deformed drip-line nuclei, where low-angular-momentum components are spread over many Nilsson orbitals; and (4) the adequacy of the HFB approximation itself for halo correlations in which weakly bound s-wave neutrons are nearly decoupled from the mean pair field. Follow-up literature transformed but did not settle these questions: schematic fixed-pair-potential calculations by the authors confirmed strong decoupling, whereas self-consistent continuum HFB calculations found that broad quasiparticle resonances and non-resonant continuum states still contribute appreciably, tracing the discrepancy to the spatial structure of the self-consistent pair potential. No universal quantitative width-pairing criterion, settled non-resonant-background prescription, systematic self-consistent deformed confirmation, or validated assessment of HFB for halos exists.

Background

Setting. The neutron drip line is the boundary of nuclear existence beyond which additional neutrons are no longer bound. In nuclei approaching it the last neutrons are weakly bound and the neutron Fermi energy $\lambda$ (the chemical potential) lies close to zero, so the pairing interaction — the effective attraction between time-reversed neutron states that produces nuclear superfluidity — couples bound states to the low-energy scattering continuum rather than to discrete levels alone. Nuclear pairing is treated either by the BCS (Bardeen-Cooper-Schrieffer) approximation (Hartree-Fock single-particle energies plus a constant pairing gap $\Delta$) or, more generally, by Hartree-Fock-Bogoliubov (HFB) theory, in which a pair potential $\Delta(r)$ and the mean field are treated on the same footing. HFB diagonalizes the problem in terms of quasiparticles, superpositions of particle and hole states with quasiparticle energy $E_{qp}$; the occupation of a level is carried by the lower component $v(r)$ of the quasiparticle wave function, and an unbound level appears as a resonance — a quasi-bound state embedded in the continuum whose width $\Gamma$ (its inverse lifetime, related to the energy derivative of the scattering phase shift) measures its escape rate. Near the drip line the HFB equation must be solved in coordinate space with the correct asymptotic outgoing-wave boundary conditions, because quasiparticle states with $\lambda+E_{qp}>0$ have continuum character (Grasso, Sandulescu, Van Giai, and Liotta, Phys. Rev. C 64, 064321 (2001), established this exact-continuum methodology). A standard shortcut is to replace a sufficiently narrow one-particle resonance by a discrete level at the resonant energy (the resonant BCS approximation).

The 2003 study. Hamamoto and Mottelson solved such a coordinate-space HFB model — a spherical Woods-Saxon potential (the standard phenomenological nuclear mean field, whose depth is constant in the nuclear interior and falls off smoothly toward the surface) plus a fixed volume- or surface-peaked pair potential, following the $3s_{1/2}$, $2d_{3/2}$, and $1g_{7/2}$ orbits as the potential depth is varied — and found that for a given one-particle energy the occupation probability of low-$\ell$ (low orbital angular momentum $\ell$; especially the $s$ and $p$ waves) orbits is much smaller than that of high-$\ell$ orbits, because the low-$\ell$ wave functions extend far beyond the pair field; once the Hartree-Fock (HF) level becomes unbound the low-$\ell$ orbit becomes almost unavailable for pair correlation, whereas high-$\ell$ orbits supporting narrow one-particle resonances can still be estimated in the BCS approximation (their HFB lower components remaining well bound). In addition to resonant peaks, a weak, slowly varying occupation tail extends over higher quasiparticle energies (the 'continuum background').

The open issues. The paper's conclusions identify four directions left open: (1) the quantitative relation between the HF one-particle resonant width and the degree to which that level contributes to the many-body pair correlation — only the qualitative correlation (narrow widths allow BCS-like treatment) is established; (2) the interpretation and quantitative accounting of the continuum background, which must be included when a fixed particle number is imposed but for which no simple terminology or modeling prescription exists; (3) the extension to deformed nuclei, where low-$\ell$ components are spread over essentially all Nilsson orbitals with small $\Omega$ (Nilsson orbitals are the single-particle eigenstates of an axially deformed potential, labeled by $\Omega^{\pi}$, the projection of the total angular momentum on the symmetry axis, and the parity; every $\Omega^{\pi}=1/2^{+}$ orbital can contain $s_{1/2}$ components), so a small $s_{1/2}$ admixture may make such orbitals unavailable for pairing; (4) the adequacy of the HFB approximation itself for the correlations central to halo phenomena (a halo is the spatially extended tail of very weakly bound neutrons) in which $s_{1/2}$ neutrons are nearly decoupled from the mean pair field.

What followed. The subsequent literature split over the strength of the decoupling effect. The authors' schematic fixed-$\Delta(r)$ calculations (2004-2006) confirmed strong decoupling and small effective pair gaps (the pairing gap experienced by a specific orbit) for weakly bound low-$\ell$ and $\Omega^{\pi}=1/2^{+}$ orbits. In contrast, self-consistent continuum HFB calculations — in which the pair potential is generated by a density-dependent effective interaction (typically a Skyrme energy-density functional, the standard family of effective interactions of self-consistent nuclear mean-field theory) and the continuum is summed exactly with the coordinate-space Green's-function technique (an exact reformulation of the HFB equations that integrates over the continuous quasiparticle spectrum instead of discretizing it in a finite box) — found that broad quasiparticle resonances and non-resonant continuum states still contribute appreciably to pair correlation (Oba and Matsuo, Phys. Rev. C 80, 024301 (2009); Zhang, Matsuo, and Meng, Phys. Rev. C 83, 054301 (2011)), because the self-consistent pair potential is enhanced at the low-density nuclear surface where the extended low-$\ell$ wave functions still overlap it. Whether a quantitative width-to-contribution relation can be stated, in what form, and for which class of pairing functionals, is the crux of the present problem.

Problem Statement

Determine, with quantitative control over the theoretical framework, how unbound neutron states participate in the many-body pair correlation in nuclei near the neutron drip line. As posed by the conclusions of Hamamoto and Mottelson (Phys. Rev. C 68, 034312 (2003)), this comprises four linked questions sharing one objective: (1) establish the quantitative relation between the width of a one-particle resonance (in the Hartree-Fock potential, or of the corresponding HFB quasiparticle resonance) and the degree to which that level contributes to the many-body pair correlation — equivalently, how narrow the resonant width must be for the discrete-level (BCS) replacement at the resonant energy to be a good approximation, and what happens for broad resonances whose widths are comparable to their energies; (2) provide a robust interpretation and quantitative accounting of the 'continuum background' — the weak, slowly varying occupation probability extending over higher quasiparticle energies — including how it is to be included in calculations imposing a fixed particle number; (3) determine the pair correlation in deformed drip-line nuclei, where low-$\ell$ components are spread over many Nilsson orbitals (for example $\Omega^{\pi}=1/2^{+}$ orbitals containing $s_{1/2}$ components), establishing whether and when such orbitals become unavailable for pairing; and (4) assess the adequacy of the HFB approximation for the correlations central to halo phenomena in which weakly bound $s_{1/2}$ neutrons are nearly decoupled from the mean pair field, and construct the more appropriate correlated description if HFB fails. The single objective uniting the four components is a quantitative, model-controlled account of continuum pairing at the neutron drip line; that account is not restricted to the schematic fixed-pair-potential model of the original paper, because the later literature shows the conclusions there depend on the spatial structure and self-consistency of the pair potential. The four components are jointly posed, and no single one of them resolves the question.

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