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Filling fractions for the formation of nuclear pasta in neutron stars: semiclassical vs liquid-drop predictions

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Nuclear pasta in neutron stars appears when clusters fill about one-eighth of their cell, and the liquid-drop curvature correction is why.

desk verdict Solid ETF result on quasi-universal pasta filling fractions, but the CLDM curvature explanation is not yet causal because of unquantified extrapolation of surface tensions to extreme isospin. read the letter →

arxiv 2505.18309 v1 pith:FLNKVGC5 submitted 2025-05-23 astro-ph.HE nucl-th

Nikolai N. Shchechilin , Nicolas Chamel , Andrey I. Chugunov This is my paper · ORCID
classification astro-ph.HEnucl-th
keywords neutronstarinnercrustnuclearpastafillingfractionextendedThomas-FermicompressibleliquiddropmodelcurvaturecorrectiongeneralizedSkyrmefunctionalssymmetryenergy
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Deep in a neutron star's inner crust, nuclei are packed so tightly that theory predicts they deform into rods, slabs, tubes, and bubbles known collectively as nuclear pasta. This paper tries to establish that the filling fraction at which spheres first give way to pasta is quasi-universal, around $u_\mathrm{sp}\approx0.13$–$0.15$, across three precision-fitted generalized Skyrme functionals, and that the low value is caused by the curvature correction to the nuclear surface energy in a liquid-drop description. The result matters because pasta formation becomes a simple geometric condition: once the local filling fraction crosses about one-eighth, the spherical shape should be abandoned, and a cheap algebraic criterion can estimate how much pasta a neutron star contains. The ETF calculations place the onset at $u_\mathrm{sp}\approx0.13$–$0.14$, while the curvature-corrected liquid-drop stability analysis gives $u_\mathrm{sp}\approx0.14$–$0.15$, with later transitions near 0.26–0.27, 0.48–0.51, and 0.66–0.69.

What carries the argument

The carrying machinery is the shape-instability comparison inside the compressible liquid-drop model: for a reference shape A and a competing shape B at the same mean density, the paper forms the ratio $\lambda(\zeta)$ of the two energy densities after each cell size is optimized, and declares A unstable when $\lambda(\zeta)<1$. The surface, curvature, and Coulomb terms enter through the geometric functions $g_s(u)$, $g_c(u)$, and $w(u)$, with the curvature entering through $X_\sigma=g_c\sigma_c/(g_s\sigma_s a)$. This generalizes earlier criteria by solving the cell-size equilibrium condition, Eq. (7), without a perturbative expansion in the curvature. On the ETF side, the mechanism is the extended Thomas-Fermi energy functional with soft-damping nucleon density profiles and the proton-density definition of $u$ given by Eqs. (1)–(2). Using the same functionals in both calculations is what allows the low quasi-universal $u_\mathrm{sp}$ to be attributed to the curvature correction rather than to the nuclear interaction.

What would settle it

A decisive check would be to compute the sphere-to-pasta onset for the same three functionals with a method that does not rely on the liquid-drop geometry or the fitted density-profile ansatz—for instance a full quantum band-structure or molecular-dynamics calculation at inner-crust densities—and see whether the onset still sits at $u\approx0.13$–$0.15$; alternatively, computing $\sigma_c/\sigma_s$ from first principles for semi-infinite neutron-rich matter and feeding it into Eqs. (12)–(13) would test whether the curvature explanation, rather than the calibration, is responsible.

Watch

Extended reading notes

Core claim

The central claim is that the shape transitions of nuclear pasta occur at nearly the same filling fractions for the BSk22, BSk24, and BSk25 generalized Skyrme functionals, even though the densities at which those transitions occur are strongly functional-dependent because they are set by the symmetry energy. Spheres lose to spaghetti at $u\approx0.13$–$0.14$; spaghetti give way to lasagna near 0.26–0.27, lasagna to bucatini near 0.48–0.51, and bucatini to Swiss cheese near 0.66–0.69. The paper then shows that a compressible liquid-drop stability criterion that includes both curvature and neutron-skin corrections reproduces the low onset, $u_\mathrm{sp}\approx0.14$–$0.15$, provided the surface and curvature tensions are calibrated to very neutron-rich systems, as in fits to ETF mass tables. Without curvature, the same criterion yields the older universal thresholds near 0.19–0.215 and misses the pasta onset seen in the ETF calculations; with curvature, the thresholds shift down. The paper concludes that the curvature correction is the reason pasta appears near the long-quoted filling fraction of 1/8, and proposes the algebraic criterion as a fast estimator of pasta abundance and as a guide for future ETF searches. For the SLy4 functional, which in older liquid-drop models produced no pasta, the same criterion gives $u_\mathrm{sp}\approx0.14$, consistent with the ETF result.

Load-bearing premise

The load-bearing premise is that the plane-surface and curvature tensions taken from mass-table fits remain valid at the extreme neutron excesses and high densities of the deepest crustal layers; the authors themselves state that these tensions are very uncertain there, and if the calibration fails, the curvature-based explanation of $u_\mathrm{sp}$ would break down even though the direct ETF result would be less affected.

Editorial extensions

If this is right

  • Pasta onset can be located from geometry alone: once the local filling fraction reaches about 0.13–0.15, a neutron-star crust model should switch from spheres to non-spherical shapes regardless of which of the tested functionals supplies the equation of state.
  • The density thickness of each pasta layer is controlled by the symmetry energy, so different functionals will disagree on where pasta sits but should agree on the filling-fraction ladder 0.13–0.15, 0.26–0.27, 0.48–0.51, 0.66–0.69.
  • Equations (12)–(13) provide an inexpensive way to estimate pasta abundance without full ETF minimization, making large surveys of nuclear models tractable.
  • The same criterion can direct an ETF search: candidate non-spherical shapes need only be evaluated near the predicted filling-fraction thresholds, reducing the cost of the realistic calculation.
  • For the SLy4 functional, the older liquid-drop finding of no pasta is not reproduced; with curvature included the onset appears near 0.14, consistent with the ETF result.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • [Editorial inference] The quasi-universal ladder suggests the pasta-phase sequence is primarily a geometric phenomenon; one could precompute the four threshold filling fractions once and apply them to any equation of state that provides the proton filling fraction, rather than re-running shape searches for every model.
  • [Editorial inference] The criterion could be extended to finite temperature for supernova and merger remnants, where surface and curvature tensions vary; the ratio $\sigma_c/\sigma_s$ rather than their absolute values would be the controlling input.
  • [Editorial inference] The explanation implies a testable ordering: models with larger $\sigma_c/\sigma_s$ at the relevant proton fractions should show pasta at smaller $u$, and this correlation is sharp enough to distinguish curvature physics from other surface effects.
  • [Editorial inference] Because the CLDM thresholds depend on whether the tensions are fitted to experimental masses or to ETF mass tables, improved data or many-body calculations for very neutron-rich nuclei would directly tighten the pasta-onset prediction.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper investigates the conditions for nuclear pasta formation in neutron-star inner crusts. Using the second-order extended Thomas-Fermi (ETF) method with three Brussels-Montreal Skyrme functionals (BSk22, BSk24, BSk25) and a soft-damping density-profile parametrization, it computes the equilibrium pasta sequence and finds that the filling fractions at shape transitions are quasi-universal across functionals, with the sphere-to-pasta onset at u_sp≈0.13–0.14. The paper then derives a compressible liquid-drop model (CLDM) shape-instability criterion that includes curvature and neutron-skin corrections, and shows that with curvature tensions fitted either to ETF mass tables or to experimental atomic masses, the CLDM criterion predicts u_sp≈0.14–0.15, close to the ETF result. The paper proposes this criterion as a fast estimator of pasta abundance and as a guide for ETF searches.

Significance. If the claims hold, the paper provides a simple, falsifiable prediction that pasta appears near a filling fraction of 1/8 in neutron-star crusts, and an analytic criterion to estimate pasta abundance without costly ETF calculations. The ETF results are transparent and cover several functionals, and the use of experimental-mass-based surface tensions for BSk24 and SLy4 provides a partial check against circularity. However, the explanatory power of the CLDM analysis depends on the extrapolation of surface and curvature tensions to extreme isospin asymmetries, which the authors themselves flag as uncertain, and on a consistent definition of the filling fraction between the ETF and CLDM frameworks. The paper does not ship code or deposited data, but the numerical results are described in enough detail to be reproduced in principle.

major comments (2)
  1. [Sec. 3.3, Eq. (17), and Conclusions] The central explanatory claim that the curvature correction shifts u_sp from about 0.215 to about 0.14–0.15 relies entirely on the ratio sigma_c/sigma_s evaluated from Eq. (17) at y_p ≈ 0.03–0.15. This is an extrapolation from fits to finite nuclei, either from ETF mass tables or experimental atomic masses, into a regime that the authors themselves describe in the Conclusions as 'very uncertain.' The paper neither propagates fit uncertainties into Fig. 6 nor compares with direct semi-infinite-matter calculations of sigma_s and sigma_c for the same functionals at the relevant isospin asymmetries. Because the curvature shift scales linearly with sigma_c/sigma_s (Eq. 11), a factor-of-two error in the extrapolated ratio at y_p ≈ 0.05 could move u_sp by roughly 0.05 and potentially erase the agreement. Please quantify this sensitivity and provide an independent check.
  2. [Sec. 2, Eq. (1), versus Sec. 3.1, Table 1] The ETF filling fraction u is defined from the proton density profile in Eq. (1), whereas the CLDM instability criterion used in Sec. 3.2 assumes u is the geometric volume fraction of the dense total-nucleon phase, as implied by the bulk term E_bulk(ni, yp, non, nop, u) and the geometry functions in Table 1. In neutron-rich crustal matter with y_p ≈ 0.03–0.15, the neutron skin makes these two definitions appreciably different. If the ETF proton-based u is inserted into Eqs. (12)–(13), the comparison in Fig. 6 may mix two different quantities. Please justify that the proton-based u is the appropriate variable for the CLDM geometry functions, or repeat the comparison using a total-baryon-density filling fraction.
minor comments (4)
  1. [Table 1] The row for lasagna reads '2 0 πu^2(1−u)^2/6', which is ambiguous; the entries for g_s, g_c, and w should be clearly separated, presumably g_s = 2, g_c = 0, and w = πu^2(1−u)^2/6.
  2. [Fig. 5] The yellow band labeled 'pasta' does not state the corresponding y_p interval; please give the numerical range, e.g., y_p ≈ 0.03–0.15, in the caption.
  3. [Abstract and Secs. 2, 3.3] The abstract quotes u_sp ≈ 0.13–0.15 while Sec. 2 reports 0.13–0.14 for the ETF calculations and Sec. 3.3 reports 0.14–0.15 for the CLDM criterion; stating the separate ranges explicitly in the abstract would avoid confusion.
  4. [Declarations, Data availability] The data availability statement first says 'This manuscript has no associated data' and then says the data are available from N.N.S. upon reasonable request; please harmonize these two statements.

Circularity Check

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No circularity: the CLDM curvature criterion is calibrated to independent mass data and checked against external CLDM calculations, not fitted to ETF pasta thresholds.

full rationale

The paper's central comparison is between two independent calculations: direct ETF minimization of WS-cell energies for five shapes (Sec. 2) and a CLDM shape-instability criterion (Eqs. 12-13) whose only functional input is the ratio sigma_c/sigma_s. This ratio is taken from published fits to isolated-nucleus mass tables, either ETF mass tables (Ref. [36], sharing the underlying BSk functionals) or experimental atomic masses (Ref. [33]), not from pasta thresholds or filling fractions. The CLDM criterion is a parameter-free geometric instability condition whose no-curvature limit reproduces the known universal sequence ut=[0.215,0.355,0.645,0.785] and whose curvature version is validated against independent CLDM calculations (Ref. [38]). Because the BSk ETF mass-table fit and the experimental-mass fit produce similar sigma_c/sigma_s in the pasta y_p band, and both give u_sp ~ 0.14-0.15, the agreement with the ETF result u_sp ~ 0.13-0.14 is a cross-check of the simplified model, not an identity. The paper's own caveat that the surface and curvature tensions 'remain very uncertain at the extreme isospin asymmetries prevailing in the deepest layers of neutron-star crusts' is an extrapolation concern, not evidence of circularity. No equation defines the predicted u_sp in terms of the fit, and no load-bearing premise is justified only by self-citation: the experimental-mass fit by a different group provides external support for the central claim.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

The paper's central claims rest on a chain of model assumptions: the cold-catalyzed crust composition, the WS cell approximation, the second-order gradient expansion, the specific density-profile parametrization, and the extrapolation of surface/curvature tensions to extreme isospin. The only numbers that enter are taken from prior fits to mass tables or from the chosen functionals; none are fitted to the pasta thresholds in this paper.

free parameters (4)
  • Skyrme functional parameters (BSk22, BSk24, BSk25) = from fits to nuclear masses (Ref. [20])
    The three functionals differ in symmetry energy J=32, 30, 29 MeV and are precision-fitted to essentially all nuclear masses; the pasta sequences depend on these inputs.
  • Surface tension parameters (σ0,s, σ0,c, β, α=5.5) = from fits to ETF mass tables (Ref. [36]) and experimental atomic masses (Ref. [33])
    Eq. (17) parameterizes σc(yp) with these coefficients; the CLDM curvature predictions depend on the ratio σc/σs. These come from prior fits, not from the pasta thresholds.
  • Soft-damping density profile parameters = not listed in this paper (Ref. [23])
    The ETF results use the soft-damping parametrization of nucleon density profiles from Ref. [23]; the filling fractions u computed via Eq. (1) depend on this functional form.
  • Symmetry energy coefficients J = 32, 30, 29 MeV for BSk22, BSk24, BSk25
    Fixed inputs defining the functionals; they determine the density ranges of pasta but not the quasi-universal filling fractions.
assumptions (7)
  • domain assumption Cold-catalyzed matter hypothesis: at a given mean baryon density, only one type of nuclear cluster is present.
    Stated in Sec. 1; the ETF minimization compares five shapes at each density, ignoring mixed phases.
  • domain assumption Wigner-Seitz cell approximation: each cell contains one cluster or hole, with specific geometries (sphere, cylinder, slab).
    Used in Sec. 2 and Sec. 3.1; the Coulomb energies w(u) are computed in this approximation, with bcc/hex lattice corrections shown to shift thresholds.
  • domain assumption Second-order semiclassical ETF expansion in gradients of the nucleon densities is sufficient.
    Sec. 2: 'we truncate the semiclassical expansion ... at the second order [24]'. Higher-order terms and proton shell/pairing corrections are neglected.
  • domain assumption The soft-damping parametrization of density profiles is a faithful representation of the equilibrium ETF profiles.
    Sec. 2: 'we employ the soft-damping parametrization ... introduced in Ref. [23]. This new parametrization was shown to be more realistic for the dense crustal layers'.
  • domain assumption In the CLDM shape comparison, the bulk energy and adsorbed-neutron energy densities are the same for both shapes at fixed nbar, ni, non, nop, yp, so they cancel.
    Sec. 3.2: 'the bulk energy ... drops out. This is also the case for the energy density Ns μn / a^3 of adsorbed neutrons since we compare configurations at the same density'.
  • domain assumption Surface and curvature tensions fitted to nuclear mass tables (ETF or experimental) remain valid at the extreme isospin asymmetries of neutron-star crust matter.
    Sec. 3.3 and Conclusions: the authors state these tensions 'remain very uncertain at the extreme isospin asymmetries prevailing in the deepest layers', yet the CLDM explanation relies on them.
  • standard math The virial theorem Eq. (7) (Ωs,pl + 2Ωs,curv = 2Ecoul) gives the equilibrium cell size for shape A.
    Derived in Sec. 3.2 as a generalization of the Baym-Bethe-Pethick virial theorem including curvature; used to eliminate the cell size a.

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Cite this review

Pith. "Pith review of Filling fractions for the formation of nuclear pasta in neutron stars: semiclassical vs liquid-drop predictions." pith.science (2026). https://pith.science/paper/FLNKVGC5

@misc{pith2026250518309,
  author       = {Pith},
  title        = {Pith review of: Filling fractions for the formation of nuclear pasta in neutron stars: semiclassical vs liquid-drop predictions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FLNKVGC5}},
  note         = {Machine review of arXiv:2505.18309}
}
abstract

Historically, a sequence of nuclear pasta shapes was predicted to appear in the deepest region of the inner crust of a neutron star within the compressible liquid-drop picture, when the filling fraction $u$ exceeds some threshold values. However, later calculations showed that these values depend on the details of the liquid-drop model. Here we investigate the existence of pasta in neutron stars within the semiclassical extended Thomas-Fermi approach using various generalized Skyrme functionals. The filling fractions for the different transitions are found to be quasi-universal, unlike the pasta density ranges governed by the symmetry energy at relevant densities. In particular, pasta emerge at $u_\mathrm{sp}\approx0.13-0.15$. By applying a simplified stability criterion within the liquid-drop framework, we show that these values of $u_\mathrm{sp}$ can be explained by the nuclear curvature correction. In this way, the abundance of pasta can be easily estimated. This criterion can also be used to optimize the search of pasta within the more realistic extended Thomas-Fermi approach.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.