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Skyrme-Hartree-Fock-Bogoliubov mass models on a 3D mesh: IV. Improved description of the isospin dependence of pairing

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Pith's one-line read A revised Skyrme-HFB mass model with a new isospin-dependent pairing interpolation improves neutron separation energies and beta-decay energies while retaining state-of-the-art mass accuracy.

arxiv 2411.08007 v2 pith:IIRF5KP7 submitted 2024-11-12 nucl-th astro-ph.HE

Guilherme Grams , Nikolai N. Shchechilin , Adrian Sanchez-Fernandez , Wouter Ryssens , Nicolas Chamel , Stephane Goriely This is my paper · ORCID
classification nucl-thastro-ph.HE
keywords nuclearmatterpropertiesatomicpairingastrophysicalasymmetricbarriers
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

Atomic nuclei are held together by the strong force, and a subtle effect called pairing makes pairs of neutrons or protons bind a little more tightly. This pairing changes nuclear masses, neutron separation energies, and beta-decay energies, which in turn shape the rapid neutron capture process (r-process) that makes heavy elements, and also controls superfluidity inside neutron stars.

The Brussels group has a series of models, called BSkG, that compute nuclear properties across the whole chart of nuclides using an energy density functional. The new model, BSkG4, changes only the pairing channel: instead of using a simple ad hoc interpolation between symmetric nuclear matter and pure neutron matter, it uses a formula motivated by the idea that the isospin dependence of pairing is driven by the splitting of neutron and proton effective masses. The formula is anchored at the two known limits and has no new free parameters.

The result is a model that reproduces known atomic masses with an rms deviation of 0.633 MeV, essentially unchanged from before, but improves neutron separation energies and Qbeta values. It also predicts smaller neutron pairing gaps in neutron-rich matter, which brings proton pairing gaps in neutron star matter closer to several advanced many-body calculations. An r-process simulation shows that the new masses change local abundance predictions by up to a factor of two, though they do not systematically improve the match to the solar r-abundance pattern.

Extended reading notes

Core claim

BSkG4 improves the description of the isospin dependence of 1S0 pairing through the interpolation Delta_q = Delta_NM(rho_q) [Delta_SM(rho) / Delta_NM(rho/2)]^(1 +/- delta) (Eq. 6), with the lower sign for neutrons and upper sign for protons, reducing the rms deviation of neutron separation energies from 0.442 to 0.402 MeV and of Qbeta from 0.534 to 0.493 MeV, while keeping the mass rms at 0.633 MeV.

Load-bearing premise

The interpolation is anchored only at the symmetric matter (SM) and neutron matter (NM) limits computed with EBHF by Cao et al. [37]; its improved behavior at intermediate asymmetry is validated against Zhang et al. [31], which is a BCS calculation with the Argonne AV18 potential that does not include the polarization and self-energy corrections that the EBHF reference includes. The paper explicitly notes the lack of ab initio data across the full asymmetry range, so the central improvement rests on the assumption that the BCS asymmetry trend is a reliable guide for the EBHF-inspired EDF. Location: Sec. 2 and Conclusions.

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Editorial analysis

A structured set of objections, weighed in public.

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

Assumptions & free parameters 30 free parameters · 8 assumptions · 0 invented entities

The central claim rests on 29 fitted BSkG4 parameters, 10 inherited gap-parameterization constants, and several domain assumptions about the reliability of EBHF and BCS reference calculations and the validity of the EDF framework. No new particles, forces, or entities are introduced.

free parameters (30)
  • t0 = -2325.45 MeV fm^3
    Fitted to masses, radii, fission barriers, and INM properties in the BSkG4 protocol.
  • t1 = 731.84 MeV fm^5
    Fitted to masses, radii, fission barriers, and INM properties in the BSkG4 protocol.
  • t2 = 0.01 MeV fm^5
    Fitted to masses, radii, fission barriers, and INM properties in the BSkG4 protocol.
  • t3 = 14092.79 MeV fm^(3+3alpha)
    Fitted to masses, radii, fission barriers, and INM properties in the BSkG4 protocol.
  • t4 = -476.32 MeV fm^(5+3beta)
    Fitted to masses, radii, fission barriers, and INM properties in the BSkG4 protocol.
  • t5 = 271.19 MeV fm^(5+3gamma)
    Fitted to masses, radii, fission barriers, and INM properties in the BSkG4 protocol.
  • x0 = 0.549106
    Fitted to masses, radii, fission barriers, and INM properties in the BSkG4 protocol.
  • x1 = 2.97317
    Fitted to masses, radii, fission barriers, and INM properties in the BSkG4 protocol.
  • x2t2 = -431.435904 MeV fm^5
    Fitted to masses, radii, fission barriers, and INM properties in the BSkG4 protocol.
  • x3 = 0.618431
    Fitted to masses, radii, fission barriers, and INM properties in the BSkG4 protocol.
  • x4 = 5.87636
    Fitted to masses, radii, fission barriers, and INM properties in the BSkG4 protocol.
  • x5 = 0.353345
    Fitted to masses, radii, fission barriers, and INM properties in the BSkG4 protocol.
  • W0 = 122.206 MeV fm^5
    Fitted to masses, radii, fission barriers, and INM properties in the BSkG4 protocol.
  • W_prime_0 = 79.840 MeV fm^5
    Fitted to masses, radii, fission barriers, and INM properties in the BSkG4 protocol.
  • alpha = 1/5
    Exponent of density dependence, fixed at BSkG3 value, part of the EDF ansatz.
  • beta = 1/12
    Exponent of density dependence, fixed at BSkG3 value, part of the EDF ansatz.
  • gamma = 1/4
    Exponent of density dependence, fixed at BSkG3 value, part of the EDF ansatz.
  • kappa_n = 123.20 fm^8
    Pairing functional parameter, fitted in the BSkG4 protocol.
  • kappa_p = 129.07 fm^8
    Pairing functional parameter, fitted in the BSkG4 protocol.
  • Ecut = 7.919 MeV
    Pairing energy cutoff, fitted in the BSkG4 protocol.
  • b = 0.905
    Correction energy parameter, fitted in the BSkG4 protocol.
  • c = 6.764
    Correction energy parameter, fitted in the BSkG4 protocol.
  • d = 0.234
    Correction energy parameter, fitted in the BSkG4 protocol.
  • l = 1.787
    Correction energy parameter, fitted in the BSkG4 protocol.
  • beta_vib = 0.866
    Correction energy parameter, fitted in the BSkG4 protocol.
  • V_W = -1.411 MeV
    Correction energy parameter, fitted in the BSkG4 protocol.
  • lambda = 560.00
    Correction energy parameter, fitted in the BSkG4 protocol.
  • V_prime_W = 0.531 MeV
    Correction energy parameter, fitted in the BSkG4 protocol.
  • A0 = 38.174
    Correction energy parameter, fitted in the BSkG4 protocol.
  • SM/NM gap parameterization (Appendix B, Table 5) = SM: Delta0=11.5586 MeV, k1=0.489932 fm^-1, k2=1.31420 fm^-1, k3=0.906146 fm^-1, km=1.31 fm^-1; NM: Delta0=3.37968 MeV…
    Fitted to EBHF pairing gaps of Cao et al. [37] for BSk30-31-32; inherited as input by BSkG4.
assumptions (8)
  • domain assumption Isospin symmetry of pairing gaps: Delta_n(rho_n,rho_p) = Delta_p(rho_p,rho_n)
    Sec. 2 states 'assuming isospin symmetry'; used to reduce the interpolation problem to one species.
  • ad hoc to paper Exponential effective-mass ansatz for the pairing gap, Eq. (4)
    Assumes Delta_q = Delta_NM(rho_q) exp(C1/m*_q + C2), motivated by the argument that isospin dependence mainly comes from effective mass splitting. The form is inspired by Ref. [31] but not derived.
  • domain assumption Local-density connection between EDF pairing strength and INM pairing gaps
    Eq. (1) uses V_q(rho_n,rho_p) determined by requiring local matching to INM pairing gaps via the procedure of Ref. [34]; standard EDF construction.
  • domain assumption EBHF gaps of Cao et al. (Ref. [37]) are reliable references for NM and SM
    BSkG4 adopts parameterizations of these gaps (Appendix B) as anchor points for the interpolation; the paper does not independently verify them.
  • domain assumption Zhang et al. BCS gaps (Ref. [31]) are acceptable guidance for intermediate asymmetries
    Sec. 2 uses these results to judge the interpolation, despite noting they lack polarization and self-energy corrections.
  • domain assumption HFB mean-field with extended Skyrme EDF is adequate for global nuclear properties
    The entire fitting protocol and mass/radius/fission predictions rely on this framework.
  • domain assumption Belyaev MOI multiplied by 1.32 approximates Thouless-Valatin MOI
    Footnote 3 in Sec. 3.1; used for the MOI comparison in Fig. 6.
  • domain assumption The correction energy parameterization of Ref. [7] remains valid
    Nine Ecorr parameters are taken from the BSkG formalism without re-derivation.

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Pith. "Pith review of Skyrme-Hartree-Fock-Bogoliubov mass models on a 3D mesh: IV. Improved description of the isospin dependence of pairing." pith.science (2026). https://pith.science/paper/IIRF5KP7

@misc{pith2026241108007,
  author       = {Pith},
  title        = {Pith review of: Skyrme-Hartree-Fock-Bogoliubov mass models on a 3D mesh: IV. Improved description of the isospin dependence of pairing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IIRF5KP7}},
  note         = {Machine review of arXiv:2411.08007}
}
abstract

Providing reliable data on the properties of atomic nuclei and infinite nuclear matter to astrophysical applications remains extremely challenging, especially when treating both properties coherently within the same framework. Methods based on energy density functionals (EDFs) enable manageable calculations of nuclear structure throughout the entire nuclear chart and of the properties of infinite nuclear matter across a wide range of densities and asymmetries. To address these challenges, we present BSkG4, the latest Brussels-Skyrme-on-a-Grid model. It is based on an EDF of the extended Skyrme type with terms that are both momentum and density-dependent, and refines the treatment of $^1S_0$ nucleon pairing gaps in asymmetric nuclear matter as inspired by more advanced many-body calculations. The newest model maintains the accuracy of earlier BSkGs for known atomic masses, radii and fission barriers with rms deviations of 0.633 MeV w.r.t. 2457 atomic masses, 0.0246 fm w.r.t. 810 charge radii, and 0.36 MeV w.r.t 45 primary fission barriers of actinides. It also improves some specific pairing-related properties, such as the $^1S_0$ pairing gaps in asymmetric nuclear matter, neutron separation energies, $Q_\beta$ values, and moments of inertia of finite nuclei. This improvement is particularly relevant for describing the $r$-process nucleosynthesis as well as various astrophysical phenomena related to the rotational evolution of neutron stars, their oscillations, and their cooling.

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