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arxiv: 2410.01997 · v2 · pith:YAXYFOEPnew · submitted 2024-10-02 · ⚛️ nucl-th · astro-ph.HE

Role of neutron pairing with density-gradient dependence in the semi-microscopic treatment of the inner crust of neutron stars

Nicolas Chamel , John-Michael Pearson , Nikolay N. Shchechilin This is my paper

Pith reviewed 2026-05-23 19:52 UTC · model grok-4.3

classification ⚛️ nucl-th astro-ph.HE
keywords neutron star crustpairing interactionequation of statesuperfluiditySkyrme functionalThomas-Fermi approximationnuclear clusters
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The pith

The proton number Z in neutron-star inner crust clusters remains 40 whether neutron pairing is included.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper applies the BSk31 functional, which features an improved pairing interaction combining microscopic homogeneous matter results and an empirical density-gradient term, to model the inner crust of neutron stars via the extended Thomas-Fermi method. It establishes that the equilibrium proton number Z is constantly 40 across the density range, unaffected by neutron pairing, and that the resulting clusters do not permit penetration by the neutron superfluid. This yields an equation of state and composition similar to earlier models but with distinct pairing fields, making BSk31 more suitable for superfluidity studies. The findings matter because the superfluid dynamics influence observable phenomena such as pulsar timing irregularities.

Core claim

Calculations with the fourth-order extended Thomas-Fermi method including Strutinsky-integral corrections show that the equilibrium proton number Z equals 40 throughout the considered densities in the inner crust, with or without neutron pairing. The clusters formed are impermeable to the neutron superfluid. The equation of state and composition closely match those from the BSk24 functional, though the neutron pairing fields differ substantially from previous predictions.

What carries the argument

The BSk31 Skyrme energy-density functional incorporating a pairing interaction with a term fitted to microscopic homogeneous nuclear matter calculations and an empirical density-gradient dependent term.

If this is right

  • The equation of state remains largely unchanged from prior models.
  • Neutron superfluid dynamics will reflect the impermeability of clusters to the superfluid.
  • The functional is better suited for investigations of neutron superfluidity due to its more realistic pairing.
  • Proton pairing is treated in BCS theory while neutron pairing uses local density approximation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This impermeability may reduce the effective superfluid fraction in the crust compared to models allowing penetration.
  • It could alter predictions for the entrainment between the superfluid and the lattice.
  • Future work might test this by performing full Hartree-Fock-Bogoliubov calculations in the inhomogeneous crust environment.

Load-bearing premise

The empirical density-gradient dependent term in the pairing functional remains valid in the inhomogeneous density environment of the neutron star crust.

What would settle it

A microscopic calculation or observation indicating that the equilibrium proton number deviates from 40 or that the neutron pairing gap is non-zero inside the clusters would falsify the central results.

Figures

Figures reproduced from arXiv: 2410.01997 by John-Michael Pearson, Nicolas Chamel, Nikolay N. Shchechilin.

Figure 1
Figure 1. Figure 1: FIG. 1: The [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Energy per nucleon (in MeV) in the inner crust of a neutron star as function of mean density ¯n [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Pressure (in MeV fm [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Therefore the changes in the EoS of the inner crust with BSk31 compared with BSk24 are mainly caused by [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Difference in energy per nucleon (in MeV) for BSk31 calculated with and without neutron pairing, as function of mean [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Relative difference in pressure (in MeV fm [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Energy per nucleon (in MeV) as a function of proton number [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: As for Fig. 6 at density ¯n [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: As for Fig. 6 at density ¯n [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Proton fraction [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Symmetry energies (in MeV) of BSk31 and BSk30 as a function of density (in fm [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: (left panel) Nucleon density distributions in the inner crust of a neutron star at mean density ¯n [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: As for Fig. 11 at density ¯n [PITH_FULL_IMAGE:figures/full_fig_p011_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: As for Fig. 11 at density ¯n [PITH_FULL_IMAGE:figures/full_fig_p012_13.png] view at source ↗
read the original abstract

Using the fourth-order extended Thomas-Fermi method with Strutinsky-integral shell and pairing corrections, we calculate the inner crust of neutron stars with the BSk31 functional, whose pairing has two terms: i) a term that is fitted to the results of microscopic calculations on homogeneous nuclear matter (accounting for both medium polarization and self-energy effects) that are more realistic than those of our earlier functionals; ii) an empirical term that is dependent on the density gradient, which permits an excellent fit to nuclear masses. Both proton and neutron pairing are taken into account, the former in the BCS theory and the latter in the local density approximation. We found that the equilibrium value of the proton number $Z$ remains 40 over the entire density range considered, whether or not neutron pairing is included. The new equation of state and the composition are very similar to those of our previously preferred functional, BSk24. However, the predicted neutron pairing fields are quite different. In particular, clusters are found to be impermeable to the neutron superfluid. The implications for the neutron superfluid dynamics are briefly discussed. Since the new pairing is more realistic, the functional BSk31 is better suited for investigating neutron superfluidity in neutron-star crusts.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript calculates the inner crust of neutron stars using the fourth-order extended Thomas-Fermi method with Strutinsky-integral shell and pairing corrections and the BSk31 functional. The pairing includes a term fitted to microscopic homogeneous-matter calculations plus an empirical density-gradient term fitted to nuclear masses; proton pairing is treated in BCS and neutron pairing in LDA. The central results are that the equilibrium proton number Z remains fixed at 40 over the full density range (with or without neutron pairing), the EOS and composition are similar to those from BSk24, and the clusters are impermeable to the neutron superfluid, with implications for superfluid dynamics.

Significance. If the results hold, the work supplies an updated crust model with a pairing functional that incorporates more realistic homogeneous-matter input than earlier BSk functionals, potentially refining predictions for neutron superfluid entrainment and dynamics in the inner crust.

major comments (2)
  1. [Abstract (description of pairing terms) and results on neutron pairing fields] The impermeability conclusion (clusters impermeable to the neutron superfluid) is obtained by applying the empirical density-gradient term of the BSk31 pairing functional via LDA to the strongly inhomogeneous crust profiles; this term was fitted only to nuclear masses, and no benchmark, sensitivity test, or microscopic validation of its accuracy under crust conditions (high neutron excess, strong density gradients) is reported.
  2. [Results section on equilibrium Z values] The headline claim that Z remains exactly 40 whether or not neutron pairing is included is presented as robust, yet the underlying calculations rely on the same unvalidated empirical pairing term for the neutron sector; a direct comparison isolating the effect of that term on the proton shell corrections would be needed to confirm the claim is independent of the fitting assumptions.
minor comments (2)
  1. [Abstract] The abstract states that calculations support the Z=40 and impermeability results but supplies no error bars, convergence tests with respect to ETF order, or tabulated comparison data against BSk24; adding these would strengthen the presentation.
  2. [Methods] Notation for the two pairing terms (microscopic vs. empirical) should be defined explicitly with equations when first introduced to avoid ambiguity in later discussion of their separate contributions.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and the detailed comments on our manuscript. We respond point by point to the major comments, addressing the technical concerns raised regarding the pairing functional and its implications.

read point-by-point responses

  1. Referee: [Abstract (description of pairing terms) and results on neutron pairing fields] The impermeability conclusion (clusters impermeable to the neutron superfluid) is obtained by applying the empirical density-gradient term of the BSk31 pairing functional via LDA to the strongly inhomogeneous crust profiles; this term was fitted only to nuclear masses, and no benchmark, sensitivity test, or microscopic validation of its accuracy under crust conditions (high neutron excess, strong density gradients) is reported.

    Authors: The density-gradient term is an empirical component of the BSk31 pairing functional, introduced to optimize the fit to nuclear masses after the primary term (fitted to microscopic homogeneous-matter calculations) is fixed. Its application via LDA to the crust follows the identical methodology used for BSk24 and earlier functionals. The impermeability result is a direct output of the functional form, which suppresses the pairing field where density gradients are large. We agree that dedicated microscopic benchmarks for this term under crust-specific conditions (extreme neutron excess and gradients) are not reported and would require separate ab initio calculations outside the scope of the present work; the manuscript relies on the functional's established performance for finite nuclei. revision: no

  2. Referee: [Results section on equilibrium Z values] The headline claim that Z remains exactly 40 whether or not neutron pairing is included is presented as robust, yet the underlying calculations rely on the same unvalidated empirical pairing term for the neutron sector; a direct comparison isolating the effect of that term on the proton shell corrections would be needed to confirm the claim is independent of the fitting assumptions.

    Authors: The equilibrium Z is determined by minimization of the total energy, with proton shell corrections obtained independently via the Strutinsky-integral method. These proton corrections are unaffected by the form of the neutron pairing functional. Explicit calculations were performed both with the full neutron pairing (including the gradient term) and without any neutron pairing; Z remains 40 in both cases. The neutron pairing contribution modifies the total energy but does not shift the energy minimum away from Z=40. An additional isolation of only the gradient term's indirect influence is not required to support the reported result, as the full neutron pairing already leaves Z unchanged. revision: no

standing simulated objections not resolved
  • Microscopic validation or sensitivity tests of the empirical density-gradient pairing term specifically under inner-crust conditions

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper applies the pre-existing BSk31 functional (with its two pairing terms) as an input model to perform a new fourth-order ETF + Strutinsky calculation of crust equilibrium. The reported outputs (Z fixed at 40, impermeability of clusters) are obtained by minimizing the total energy functional over the inhomogeneous density profiles; they are not identical to the nuclear-mass fit by construction, nor do they reduce to any self-citation chain. The functional's gradient term is an external input whose validity in the crust is an assumption, not a definitional tautology. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on the validity of the BSk31 pairing functional (microscopic term plus empirical gradient term) and on the applicability of the local-density approximation for neutron pairing in the crust.

free parameters (1)
  • coefficients of the empirical density-gradient dependent pairing term
    Fitted to nuclear masses to achieve excellent agreement; required for the functional to be used in the crust calculation.
axioms (2)
  • domain assumption Local density approximation accurately captures neutron pairing in the inhomogeneous crust
    Explicitly stated for neutron pairing treatment.
  • domain assumption Fourth-order extended Thomas-Fermi method plus Strutinsky-integral corrections suffice for the inner-crust structure
    Method used for all reported results.

pith-pipeline@v0.9.0 · 5767 in / 1342 out tokens · 21139 ms · 2026-05-23T19:52:01.316389+00:00 · methodology

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Reference graph

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