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Mirror displacement energies from four measured mirror pairs constrain mainly one effective charge-symmetry-breaking combination, and the same calibrated finite-density functional predicts the charge radii of 40Ti, 42Ti, 46Cr, and 50Fe—so m

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2026-08-03 01:51 UTC pith:ISVG4OUW

load-bearing objection The paper identifies a useful effective CSB combination and makes testable radius predictions, but the predictions rest on an untested linear-response extrapolation and the 'validation' is in-sample; worth a serious referee with revisions. the 4 major comments →

arxiv 2607.18621 v2 pith:ISVG4OUW submitted 2026-07-21 nucl-th

A common finite-density charge-symmetry-breaking response in mirror displacement energies and charge radii

Myeong-Hwan Mun , Kyoungsu Heo , Jubin Park , Myung-Ki Cheoun , H. Sagawa This is my paper
classification nucl-th
keywords mirror nucleicharge radiicharge-symmetry breakingmirror displacement energyneutron skinenergy-density functionalisospin symmetry breakingsymmetry energy
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper argues that the binding-energy differences between mirror nuclei (mirror displacement energies, MDEs) and their charge-radius differences are governed by the same finite-density charge-symmetry-breaking (CSB) effect, described by a volume term and a surface-gradient term. The measured MDEs pin down almost entirely one effective combination of the two coupling strengths, not each coupling separately. Adding measured charge radii of the same four calibration pairs selects the point on that weakly constrained direction, and the resulting functional predicts charge radii for four unmeasured proton-rich nuclei. If the predictions hold, mirror charge-radius differences must be corrected for finite-density CSB before being interpreted as neutron-skin or symmetry-energy probes. The work matters because it supplies concrete, testable radii and a method for separating isospin-symmetry-breaking effects from the physics of neutron skins.

Core claim

The central claim is that a class-III charge-symmetry-breaking functional—volume term (1/2)t0^III ρ0ρ1 plus surface-gradient term C_Δ^III (ρ0Δρ1+ρ1Δρ0)—added to a Coulomb-only Skyrme energy-density functional simultaneously describes the residual MDEs and mirror charge-radius differences of the four pairs 34Ar–34S, 36Ca–36S, 38Ca–38Ar, and 54Ni–54Fe. The response is surface-gradient-dominated: the sensitivity ratio λ_B = -S_Δ^B/S_0^B is nearly the same for all four anchors (1.32±0.04 fm^-2 with SLy4, 1.20±0.05 fm^-2 with SkM*), so the MDEs determine chiefly the effective combination t_eff^III = t0^III - λ_cl C_Δ^III ≈ -6.4 MeV fm^3, rather than t0^III and C_Δ^III separately. Measured anchor

What carries the argument

The central objects are the volume term (1/2)t0^III ρ0ρ1, which probes the bulk-weighted neutron–proton density imbalance, and the surface-gradient term C_Δ^III(ρ0Δρ1+ρ1Δρ0), which after integration by parts acts as -2C_Δ^III[(∇ρ_n)^2-(∇ρ_p)^2] and shifts the radial region where CSB acts. Their MDE responses are captured by self-consistent local sensitivity coefficients S_0^B and S_Δ^B, whose ratio λ_B=-S_Δ^B/S_0^B is nearly constant across the four mirror pairs. This near-degeneracy means the MDE data constrain mostly the combination t_eff^III=t0^III-λ_cl C_Δ^III, while the complementary direction t_⊥^III is weakly constrained and is fixed by the measured mirror charge radii in the joint fi

Load-bearing premise

The calibration assumes that the experiment–theory difference left after the adopted Coulomb-only calculation is entirely due to the two-term class-III CSB functional; if omitted electromagnetic, surface, shell, pairing, or deformation effects contribute at the 0.1 MeV or 0.005 fm level, the extracted couplings and predicted radii are biased.

What would settle it

Measure the charge radius of 42Ti (or 50Fe) with an uncertainty below roughly 0.006 fm and compare with the predicted 3.5780(54) fm (or 3.7110(82) fm); a deviation beyond the combined experimental error and the two-functional spread would falsify the calibrated CSB correction. On the energy side, an ab initio calculation of the four anchor-pair MDE residuals that reproduces the residual without any surface-gradient term would falsify the claim that a common surface-gradient-dominated class-III response is required.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Mirror displacement energies do not separately determine the volume and surface-gradient CSB couplings; only the combination t_eff^III is pinned down by the energy data alone.
  • If the calibration is right, 42Ti and 50Fe are the cleanest near-term tests, with SLy4–SkM* spreads of about 0.002 fm in the predicted radii.
  • 46Cr is the strongest test of model dependence: its predicted radius differs by 0.0212 fm between the two functionals.
  • Calibrated CSB corrections shift proton-rich mirror neutron skins at the 10^-2 fm level, so mirror charge-radius differences used as neutron-skin or symmetry-energy probes require the same CSB subtraction.
  • The surface-gradient-dominated response pattern is more robust across the two functionals than the absolute radius correction, which remains energy-density-functional dependent.

Where Pith is reading between the lines

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

  • A precise measurement of R_ch(42Ti) or R_ch(50Fe), with uncertainty comparable to the quoted parenthetical errors, would directly test whether the calibrated CSB functional captures the real spatial content of the MDE residual.
  • The λ_B-ratio method could be applied to other isospin-breaking observables, such as triplet displacement energies, to identify which combinations of class-II and class-III couplings are actually constrained before interpreting individual coupling strengths.
  • Extending the same volume-plus-surface-gradient CSB calibration to a relativistic mean-field framework with ω-ρ meson mixing would test whether the surface-gradient-dominated pattern survives a different many-body expansion.
  • If the predicted radii are confirmed, the same calibration can be reused to assign CSB corrections to other proton-rich mirror pairs, effectively converting mirror-radius measurements into probes of isospin-symmetry breaking rather than direct neutron-skin measurements.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper calibrates a finite-density class-III charge-symmetry-breaking (CSB) functional, consisting of a volume term t0^III ρ0ρ1 and a surface-gradient term C_Δ^III (ρ0Δρ1 + ρ1Δρ0), against four measured mirror pairs (34Ar–34S, 36Ca–36S, 38Ca–38Ar, 54Ni–54Fe) using both mirror displacement energies (MDEs) and mirror charge-radius differences. Starting from Coulomb-only SLy4 and SkM* Skyrme EDF baselines, the authors compute local sensitivity coefficients and find that the MDEs constrain mainly the effective combination t_eff^III = t0^III − λ_cl C_Δ^III, with λ_cl ≈ 1.32 fm^{-2} (SLy4) and 1.20 fm^{-2} (SkM*). A joint mass–radius fit is then used to select a point on the MDE-compatible covariance band, and linear-response extrapolation yields predictions for the charge radii of 40Ti, 42Ti, 46Cr, and 50Fe, with claimed uncertainties of ~0.005 fm plus a two-EDF spread up to 0.021 fm. The paper argues that finite-density CSB corrections must be included before mirror charge-radius differences are used as clean neutron-skin or symmetry-energy probes.

Significance. If the central extraction is correct, the paper provides a useful framework for connecting MDE data to charge-radius predictions: it explicitly separates the energy-selected CSB direction from the radius-selected point on that direction, and it makes four concrete, falsifiable predictions for proton-rich charge radii. The identification of the effective combination and the compact λ_cl band across two EDFs is a genuinely useful diagnostic. The paper also honestly flags that the extracted couplings are reference-dependent effective parameters and that the absolute radius correction is EDF dependent. However, the strength of the quantitative claims — especially the ~0.005 fm precision of the predicted radii — currently rests on an untested linear-response extrapolation and on an in-sample 'validation' of the radius residuals, so the significance is conditional on resolving those issues.

major comments (4)
  1. [Supplementary Eqs. (S2)–(S4), (S14); Table 3] The predicted radii and the Table 2 validation residuals are linear-response extrapolations O_i(t0,CΔ) ≈ O_i^(C) + S0 t0 + SΔ CΔ, where the slopes are evaluated at t0=CΔ=0 with finite-difference steps h0=5.6 MeV fm3 and hΔ=1.0 MeV fm5. The joint-fit couplings are t0 ≈ −25.8 MeV fm3 (SLy4) and CΔ ≈ −14.7 MeV fm5, i.e., about 4.6×h0 and 14.7×hΔ. The quoted nonlinearity test (even finite-step remainder ≤0.026 MeV in the MDE) is performed at the small steps, not at the operating point, and does not cover the radius channel where the two terms add rather than cancel. A 10% nonlinearity in the individual radius responses would shift the predictions by more than the quoted 0.005 fm floor. A direct HFB calculation at the fitted couplings is needed to verify the reported radii and residuals.
  2. [Eq. (16) and Table 2] The joint calibration in Eq. (16) uses η_R=1, so the measured anchor charge radii enter the χ² directly. The subsequent presentation of Table 2 as 'pair-by-pair validation' is therefore in-sample: the residuals shown are reduced not because the CSB functional was independently confirmed by radii, but because those same radii were used to select the couplings. This is a legitimate calibration procedure, but it should be called a reproduction, not a validation. The genuinely out-of-sample statements are the target predictions in Table 3, and these rest on the linear-response issue noted above.
  3. [Table 1, SkM* row] The SkM* joint fit gives RMS_R = 0.0136 fm and χ²/dof = 3.9, with the radius residual barely improved from the MDE-only fit (0.0174 fm → 0.0136 fm) while the MDE RMS becomes worse than the SLy4 joint fit. This indicates that the same two-parameter functional cannot simultaneously reproduce the MDE and radius residuals in SkM* at the adopted error floors. The paper's claim of a 'common finite-density CSB response' is thus only partially supported; the SkM* result is more naturally read as a model tension. The authors should either lower the claim to 'SLy4 supports the common response; SkM* does not', or quantify why the SkM* tension does not undermine the target predictions.
  4. [Sec. 1, Eq. (5)] The entire calibration treats the residual R_B,i = ΔB_exp − ΔB^(C) as if it were dominated by the two adopted class-III CSB terms. The paper itself states that this residual 'may absorb omitted electromagnetic, surface, shell, pairing, and deformation effects.' Only a small pp contact term is varied as a check, and the check does not cover deformation, pairing, or beyond-mean-field effects. Given that the residuals are at the 0.1–2 MeV level and the target radii are claimed to 0.005 fm, the authors should provide a more explicit estimate of how large such omitted effects would have to be to change the target predictions, or restrict the conclusions to the level allowed by this ambiguity.
minor comments (5)
  1. [Table 1 caption] The fit labels 'mass' in Table 1 are confusing; these are MDE-only fits, not fits to nuclear masses. Please rename to 'MDE' or 'MDE-only'.
  2. [Fig. 2 caption / Table 2 caption] Table 2 refers to 'More detailed numbers are illustrated in Fig. 2', but the figure is numbered Fig. 2 in the main text while the reference in Sec. 2.3 says 'illustrated in Fig. 1'. Please harmonize the cross-references.
  3. [Eq. (3)] The symbol Rp is used for the point-proton rms radius, which is standard, but the text should explicitly distinguish Rp from R_p^ch to avoid confusion in Eq. (19) where R_p^ch denotes the predicted charge radius.
  4. [Table 3 and Sec. 3] The parenthetical errors in Table 3 explicitly exclude coupling-covariance and λ_B-band assignment uncertainties. This should be stated directly in the main text (it is only in the Supplementary Material) so that readers do not mistake the parentheses for total theoretical uncertainties.
  5. [General] The data availability statement says data are available 'upon reasonable request'. Given that the paper is built on a specific numerical procedure, consider providing the fitted couplings, slopes, and HFBTHO inputs in a small data file for full reproducibility.

Circularity Check

1 steps flagged

Radius 'validation' is in-sample by construction; the Table 3 target predictions are genuinely out-of-sample.

specific steps
  1. fitted input called prediction [Sec. 2.2-2.3, Eq. (16), Table 2; Introduction: 'the measured radii validate its spatial content']
    "The fitted parameters minimize χ2 = Σ_i (R^{(C+CSB)}_{B,i}/σ_{B,i})^2 + η_R Σ_i (R^{(C+CSB)}_{R,i}/σ_{R,i})^2 ... We use η_R = 0 for MDE-only fits and η_R = 1 for joint mass-radius fits. ... The pair-by-pair validation is shown in Table 2. ... A reduction of |RR,i| after an MDE-compatible CSB correction indicates that the energy-selected CSB correction has the appropriate radial structure."

    The radius residuals R^{(C+CSB)}_{R,i} in Table 2 are exactly the residuals entering the χ2 objective in Eq. (16) with η_R=1. The joint-fit couplings are chosen to minimize them, so their reduction is a fitting result, not an independent validation. Calling this 'validation' of the 'spatial content' is in-sample. The Table 3 predictions for 40Ti, 42Ti, 46Cr, and 50Fe do not enter the calibration and remain out-of-sample; the circularity is confined to the validation claim, not the central extrapolation.

full rationale

The main predictive derivation is self-contained: the MDE and target-radius residuals are computed from an EDF plus a two-term class-III CSB functional, and the target predictions (Eqs. 18-19/S14-S15) use only the calibrated couplings and EDF-computed slopes; no target MDE or target proton-rich radius enters. The t_eff parametrization (Eqs. 11-14) is a mathematical re-expression of the two-coupling local model, not a circular redefinition. The linear-response extrapolation (Supp. Eq. S4) at couplings 5-15 times the finite-difference steps is a robustness assumption, not circularity; the paper quotes the even finite-step remainder and notes covariance errors are not included. The one genuine circular step is the radius 'validation' in Sec. 2.3/Table 2, because the same radius residuals are minimized in Eq. (16); however, this does not affect the out-of-sample target predictions. No load-bearing self-citation or imported uniqueness theorem was found.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

No new physical entities (particles, forces, dimensions) are introduced. The CSB volume and surface-gradient terms are effective EDF terms borrowed from earlier Skyrme-ISB literature and RMF ω-ρ0 mixing references, not newly invented objects. The main load-bearing choices are the residual-attribution assumption, the two-term truncation, the linear-response approximation, and the specific error floors.

free parameters (3)
  • t0^III (volume class-III CSB coupling) = -25.82(7.90) MeV fm^3 (SLy4 joint); -24.73(5.57) MeV fm^3 (SkM* joint)
    Fitted to MDE and anchor-radius residuals in Eq. (16). Strongly covariant with C_Δ^III; only the combination t_eff is well determined.
  • C_Δ^III (surface-gradient class-III CSB coupling) = -14.71(6.00) MeV fm^5 (SLy4 joint); -15.25(4.69) MeV fm^5 (SkM* joint)
    Fitted alongside t0^III; individual value is poorly constrained due to compensation with t0^III along the MDE-selected direction.
  • Adopted error floors σ_B=0.10 MeV and σ_R=0.005 fm = 0.10 MeV, 0.005 fm
    Chosen by hand to weight MDE vs radius residuals in the χ2; they are not derived from data and influence the balance of the joint fit.
axioms (5)
  • ad hoc to paper The experiment-theory residual after the Coulomb-only EDF is dominated by the two adopted class-III CSB terms (volume and surface-gradient).
    Sec. 1 admits the residual 'may absorb omitted electromagnetic, surface, shell, pairing, and deformation effects'; only a small pp contact term is tested, so the completeness of the decomposition is assumed.
  • domain assumption The local linear-response extrapolation (Supplementary Eq. S4) is valid at the fitted couplings, including induced density polarization.
    The finite-step non-odd remainder is checked to be below 0.05 MeV, but no direct verification is provided at the large fitted t0^III and C_Δ^III values.
  • domain assumption SLy4 and SkM* are adequate representative reference functionals.
    Only two EDFs are used; the authors explicitly state that their difference is a 'limited diagnostic' of model dependence, not a statistical uncertainty.
  • domain assumption Coulomb exchange in the Slater approximation and neglect of the spin-orbit radius term are covered by the 0.005 fm radius-error floor.
    Eq. (3) sets R_so^2=0 and folds expected residual radius-model effects into the adopted 0.005 fm floor.
  • ad hoc to paper Target selection by requiring λB compatibility with the calibration band does not bias the predictions.
    Supplementary Sec. 3 excludes low-λB pairs such as 30S-30Si and 32Ar-32Si from the target set; the selection is based on calculated sensitivities, not on measured target radii, but it is nonetheless a post-hoc restriction.

pith-pipeline@v1.3.0-alltime-deepseek · 13783 in / 15162 out tokens · 156262 ms · 2026-08-03T01:51:15.244432+00:00 · methodology

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read the original abstract

Mirror charge radii can constrain neutron skins only if nuclear isospin-symmetry breaking beyond the Coulomb interaction is controlled. We present a differential Skyrme energy-density-functional analysis in which mirror displacement energies (MDEs) and mirror charge-radius differences are generated by the same class-III charge-symmetry-breaking (CSB) functional. Starting from self-consistent Coulomb-only SLy4 and SkM* baselines, we map Coulomb-subtracted MDE residuals onto volume and surface-gradient CSB terms. The MDEs determine the nearly degenerate volume-surface direction, while the measured anchor radii select the point on that direction used for target predictions, without introducing radius-specific parameters. Measured pairs $^{34}$Ar-$^{34}$S, $^{36}$Ca-$^{36}$S, $^{38}$Ca-$^{38}$Ar, and $^{54}$Ni-$^{54}$Fe form compact surface-gradient-like response classes, with $\lambda_B=1.32\pm0.04~\mathrm{fm}^{-2}$ for SLy4 and $1.20\pm0.05~\mathrm{fm}^{-2}$ for SkM*. MDEs therefore constrain mainly the effective combination $t_0^{\mathrm{III}}-\lambda_{\mathrm{cl}}C_\Delta^{\mathrm{III}}$, rather than the two couplings separately. A joint volume-plus-surface-gradient fit gives MDE and $\Delta R_{\rm ch}^{\rm mirr}$ RMS residuals of $0.0529$ MeV and $0.0031$ fm in SLy4; SkM* recovers the response class but gives a larger $\Delta R_{\rm ch}^{\rm mirr}$ residual. Calibrated response changes proton-rich mirror skins at the $10^{-2}$ fm level and yields charge radius predictions for $^{40}$Ti, $^{42}$Ti, $^{46}$Cr, and $^{50}$Fe, with an SLy4 - SkM* spread of up to $0.021$ fm. Surface-gradient-sensitive CSB corrections must therefore be quantified before mirror charge-radius differences are used as clean neutron-skin or symmetry-energy probes. The agreement of SLy4 and SkM* in the response-class assignment is more robust than their absolute radius corrections, which remain EDF dependent.

Figures

Figures reproduced from arXiv: 2607.18621 by H. Sagawa, Jubin Park, Kyoungsu Heo, Myeong-Hwan Mun, Myung-Ki Cheoun.

Figure 1
Figure 1. Figure 1: MDE residuals and mirror charge-radius di [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: Schematic connection between mirror charge radii and neutron skin thickness. For a neutron-rich nucleus [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: MDE residuals and mirror charge-radius residuals for the four measured calibration pairs. Bars correspond to the reference calculation, the volume-CSB [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

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Works this paper leans on

26 extracted references · 1 canonical work pages

  1. [2]

    Chabanat, P

    E. Chabanat, P. Bonche, P. Haensel, J. Meyer, R. Schaeffer, A Skyrme parametrization from subnuclear to neutron star densities. Part II. Nuclei far from stabilities, Nucl. Phys. A 635 (1998) 231--256, doi:10.1016/S0375-9474(98)00180-8

    doi:10.1016/s0375-9474(98)00180-8 1998

  2. [3]

    Bartel, P

    J. Bartel, P. Quentin, M. Brack, C. Guet, H.-B. H kansson, Towards a better parametrisation of Skyrme-like effective forces: A critical study of the SkM force, Nucl. Phys. A 386 (1982) 79--100, doi:10.1016/0375-9474(82)90403-1

    doi:10.1016/0375-9474(82)90403-1 1982

  3. [4]

    Okamoto, Coulomb energy of ^ 3 He and possible charge asymmetry of nuclear forces, Phys

    K. Okamoto, Coulomb energy of ^ 3 He and possible charge asymmetry of nuclear forces, Phys. Lett. 11 (1964) 150--151, doi:10.1016/0031-9163(64)90650-X

    doi:10.1016/0031-9163(64)90650-x 1964

  4. [5]

    J. A. Nolen, J. P. Schiffer, Coulomb energies, Annu. Rev. Nucl. Sci. 19 (1969) 471--526, doi:10.1146/annurev.ns.19.120169.002351

    arXiv 1969

  5. [6]

    B. A. Brown, W. A. Richter, R. Lindsay, Displacement energies with the Skyrme Hartree--Fock method, Phys. Lett. B 483 (2000) 49--54, doi:10.1016/S0370-2693(00)00589-X

    doi:10.1016/s0370-2693(00)00589-x 2000

  6. [7]

    B a czyk, J

    P. B a czyk, J. Dobaczewski, M. Konieczka, W. Satu a, T. Nakatsukasa, K. Sato, Isospin-symmetry breaking in masses of N Z nuclei, Phys. Lett. B 778 (2018) 178--183, doi:10.1016/j.physletb.2017.12.068

    doi:10.1016/j.physletb.2017.12.068 2018

  7. [8]

    Naito, G

    T. Naito, G. Col \`o , H. Liang, X. Roca-Maza, H. Sagawa, Toward ab initio charge-symmetry breaking in nuclear energy density functionals, Phys. Rev. C 105 (2022) L021304, doi:10.1103/PhysRevC.105.L021304

    doi:10.1103/physrevc.105.l021304 2022

  8. [9]

    Sagawa, T

    H. Sagawa, T. Naito, X. Roca-Maza, T. Hatsuda, QCD-based charge symmetry breaking interaction and the Okamoto--Nolen--Schiffer anomaly, Phys. Rev. C 109 (2024) L011302, doi:10.1103/PhysRevC.109.L011302

    doi:10.1103/physrevc.109.l011302 2024

  9. [10]

    B. A. Brown, Mirror charge radii and the neutron equation of state, Phys. Rev. Lett. 119 (2017) 122502, doi:10.1103/PhysRevLett.119.122502

    doi:10.1103/physrevlett.119.122502 2017

  10. [11]

    B. A. Brown, K. Minamisono, J. Piekarewicz, et al., Implications of the ^ 36 Ca-- ^ 36 S and ^ 38 Ca-- ^ 38 Ar difference in mirror charge radii on the neutron matter equation of state, Phys. Rev. Research 2 (2020) 022035(R), doi:10.1103/PhysRevResearch.2.022035

    doi:10.1103/physrevresearch.2.022035 2020

  11. [12]

    o rtersh \

    A. J. Miller, K. Minamisono, A. Klose, D. Garand, C. Kujawa, J. D. Lantis, Y. Liu, B. Maa , P. F. Mantica, W. Nazarewicz, W. N \"o rtersh \"a user, S. V. Pineda, P.-G. Reinhard, D. M. Rossi, F. Sommer, C. Sumithrarachchi, A. Teigelh \"o fer, J. Watkins, Proton superfluidity and charge radii in proton-rich calcium isotopes, Nat. Phys. 15 (2019) 432--436, d...

    doi:10.1038/s41567-019-0416-9 2019

  12. [13]

    o nig, D. M. Rossi, B. A. Brown, A. Incorvati, J. Lantis, K. Minamisono, W. N \

    S. V. Pineda, K. K \"o nig, D. M. Rossi, B. A. Brown, A. Incorvati, J. Lantis, K. Minamisono, W. N \"o rtersh \"a user, J. Piekarewicz, R. Powel, F. Sommer, Charge radius of neutron-deficient ^ 54 Ni and symmetry energy constraints using the difference in mirror pair charge radii, Phys. Rev. Lett. 127 (2021) 182503, doi:10.1103/PhysRevLett.127.182503

    doi:10.1103/physrevlett.127.182503 2021

  13. [14]

    Reinhard, W

    P.-G. Reinhard, W. Nazarewicz, Information content of the differences in the charge radii of mirror nuclei, Phys. Rev. C 105 (2022) L021301, doi:10.1103/PhysRevC.105.L021301

    doi:10.1103/physrevc.105.l021301 2022

  14. [15]

    Naito, X

    T. Naito, X. Roca-Maza, G. Col \`o , H. Liang, H. Sagawa, Isospin symmetry breaking in the charge radius difference of mirror nuclei, Phys. Rev. C 106 (2022) L061306, doi:10.1103/PhysRevC.106.L061306

    doi:10.1103/physrevc.106.l061306 2022

  15. [16]

    S. J. Novario, D. Lonardoni, S. Gandolfi, G. Hagen, Trends of neutron skins and radii of mirror nuclei from first principles, Phys. Rev. Lett. 130 (2023) 032501, doi:10.1103/PhysRevLett.130.032501

    doi:10.1103/physrevlett.130.032501 2023

  16. [17]

    Hu, How do mirror charge radii constrain density dependence of the symmetry energy?, Phys

    B.-S. Hu, How do mirror charge radii constrain density dependence of the symmetry energy?, Phys. Lett. B 857 (2024) 138969, doi:10.1016/j.physletb.2024.138969

    arXiv 2024

  17. [18]

    M. V. Stoitsov, J. Dobaczewski, W. Nazarewicz, P. Ring, Axially deformed solution of the Skyrme--Hartree--Fock--Bogolyubov equations using the transformed harmonic oscillator basis: The program HFBTHO (v1.66p), Comput. Phys. Commun. 167 (2005) 43--63, doi:10.1016/j.cpc.2005.01.001

    doi:10.1016/j.cpc.2005.01.001 2005

  18. [19]

    M. V. Stoitsov, N. Schunck, M. Kortelainen, N. Michel, H. Nam, E. Olsen, J. Sarich, S. Wild, Axially deformed solution of the Skyrme--Hartree--Fock--Bogolyubov equations using the transformed harmonic oscillator basis (II): HFBTHO v2.00d, Comput. Phys. Commun. 184 (2013) 1592--1604, doi:10.1016/j.cpc.2013.01.013

    doi:10.1016/j.cpc.2013.01.013 2013

  19. [20]

    Navarro P \'e rez, N

    R. Navarro P \'e rez, N. Schunck, R.-D. Lasseri, C. Zhang, J. Sarich, Axially deformed solution of the Skyrme--Hartree--Fock--Bogolyubov equations using the transformed harmonic oscillator basis (III): HFBTHO v3.00, Comput. Phys. Commun. 220 (2017) 363--375, doi:10.1016/j.cpc.2017.06.022

    doi:10.1016/j.cpc.2017.06.022 2017

  20. [21]

    Marevi \'c , N

    P. Marevi \'c , N. Schunck, E. M. Ney, R. Navarro P \'e rez, M. Verriere, J. O'Neal, Axially-deformed solution of the Skyrme--Hartree--Fock--Bogoliubov equations using the transformed harmonic oscillator basis (IV): HFBTHO v4.0, Comput. Phys. Commun. 276 (2022) 108367, doi:10.1016/j.cpc.2022.108367

    arXiv 2022

  21. [22]

    M. Wang, W. J. Huang, F. G. Kondev, G. Audi, S. Naimi, The AME 2020 atomic mass evaluation (II). Tables, graphs and references, Chin. Phys. C 45 (2021) 030003, doi:10.1088/1674-1137/abddaf

    doi:10.1088/1674-1137/abddaf 2020

  22. [23]

    Angeli, K

    I. Angeli, K. P. Marinova, Table of experimental nuclear ground state charge radii: An update, At. Data Nucl. Data Tables 99 (2013) 69--95, doi:10.1016/j.adt.2011.12.006

    doi:10.1016/j.adt.2011.12.006 2013

  23. [24]

    B a czyk, W

    P. B a czyk, W. Satu a, J. Dobaczewski, M. Konieczka, Isobaric multiplet mass equation within nuclear density functional theory, J. Phys. G: Nucl. Part. Phys. 46 (2019) 03LT01, doi:10.1088/1361-6471/aaffe4

    doi:10.1088/1361-6471/aaffe4 2019

  24. [25]

    Suzuki, H

    T. Suzuki, H. Sagawa, N. Van Giai, Charge independence and charge symmetry breaking interactions and the Coulomb energy anomaly in isobaric analog states, Phys. Rev. C 47 (1993) R1360--R1363, doi:10.1103/PhysRevC.47.R1360

    doi:10.1103/physrevc.47.r1360 1993

  25. [26]

    K \"o nig, J

    K. K \"o nig, J. C. Berengut, A. Borschevsky, A. Brinson, B. A. Brown, A. Dockery, S. Elhatisari, E. Eliav, R. F. Garcia Ruiz, J. D. Holt, B.-S. Hu, J. Karthein, D. Lee, Y.-Z. Ma, U.-G. Mei ner, K. Minamisono, A. V. Oleynichenko, S. V. Pineda, S. D. Prosnyak, M. L. Reitsma, L. V. Skripnikov, A. Vernon, A. Zaitsevskii, Nuclear charge radii of silicon isoto...

    doi:10.1103/physrevlett.132.162502 2024

  26. [27]

    Tanimura, Y., Naito, T., Sagawa, H. et al. Charge symmetry breaking effects of - _0 mixing in relativistic mean-field model. Eur. Phys. J. A 61, 229 (2025). https://doi.org/10.1140/epja/s10050-025-01699-y

    work page doi:10.1140/epja/s10050-025-01699-y 2025