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Extended Skyrme effective interactions with higher-order momentum-dependence for transport models and neutron stars

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Pith's one-line read This paper shows that a Skyrme pseudopotential extended to N5LO, with momentum dependence up to $p^{10}$, describes the empirical nucleon optical potential up to 2 GeV and, in the same framework, produces interactions consistent with…

desk verdict Useful formal extension of the Skyrme pseudopotential to N5LO, but the 2 GeV optical-potential claim is fit to an extrapolation, not to data. read the letter →

arxiv 2412.09393 v2 pith:ASK3S3O5 submitted 2024-12-12 nucl-th astro-ph.HEhep-phnucl-ex

Si-Pei Wang , Xin Li , Rui Wang , Jun-Ting Ye , Lie-Wen Chen This is my paper · ORCID
classification nucl-thastro-ph.HEhep-phnucl-ex
keywords Skyrmepseudopotentialnuclearequationofstatesymmetryenergysingle-nucleonpotentialnucleonopticalheavy-iontransportneutronstarseffectivemasssplitting
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

This paper extends the Skyrme pseudopotential, a quasilocal expansion of the nuclear mean field, to arbitrary order NnLO by adding derivative terms up to $2n$th order to its central term. The authors derive the Hamiltonian density and single-nucleon potential under nonequilibrium Hartree-Fock conditions and find that the N5LO version, with momentum dependence up to $p^{10}$, reproduces the empirical nucleon optical potential through 2 GeV of kinetic energy while lower orders fail above 1 GeV. They also extend the density-dependent terms through a Fermi-momentum expansion, giving the equation of state and symmetry energy enough flexibility to satisfy nuclear-matter, microscopic neutron-matter, and neutron-star constraints at the same time. The paper constructs 24 interaction parameter sets spanning three interaction orders and eight isospin-splitting scenarios, and a lattice-Hamiltonian transport benchmark of Au+Au collisions at 1.23 GeV per nucleon reproduces the measured proton collective flows.

What carries the argument

The load-bearing object is the NnLO Skyrme central pseudopotential, obtained by assuming that the $2n$th-order derivative term is proportional to the binomial expansion $(\hat A+\hat B)^n$, where $\hat A$ is the even relative-momentum operator and $\hat B$ the odd one. The split into even and odd powers of $\hat B$ fixes two coupling families, and after Hartree-Fock averaging it generates momentum-dependent and gradient terms in the Hamiltonian density. In the single-nucleon potential the momentum dependence is a polynomial in $p^2$ whose coefficients $a_n$ and $b_n$ can be fitted directly to the optical potential; the polynomial form factors the $\vec p$-dependence out of the phase-space integral, which is what keeps high-order terms computationally affordable in transport simulations. The density-dependent part follows a Fermi-momentum expansion $\rho^{(2n-1)/3}$, giving independent control of the saturation and symmetry-energy expansion coefficients up to fifth order.

What would settle it

Measure or reliably extract the real part of the nucleon optical potential in symmetric nuclear matter at saturation density for nucleon kinetic energies between 1 and 2 GeV, for example from high-quality proton-nucleus elastic-scattering data; if the extracted potential differs from the N5LO prediction by more than the fitting tolerance across that range, the central claim fails.

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Extended reading notes

Core claim

The central claim is that a Skyrme pseudopotential whose central term is built from the binomial expansion $(\hat A+\hat B)^n$ produces a single-nucleon potential with polynomial momentum dependence up to $p^{2n}$, and that at N5LO ($p^{10}$) this polynomial stays close to the empirical nucleon optical potential up to about 2 GeV kinetic energy, including the saturated behavior above 1 GeV. The same construction at N3LO and N4LO reproduces the optical potential only up to about 1 GeV and 1.5 GeV, respectively, with rapid growth beyond those energies. With the Fermi-momentum-expanded density-dependent terms, the interactions give a flexible description of symmetric nuclear matter, the symmetry energy (through $E_{\rm sym}$, $L$, $K_{\rm sym}$, $J_{\rm sym}$, $I_{\rm sym}$, $H_{\rm sym}$), pure neutron matter, and neutron-star masses and radii compatible with current multimessenger constraints. The paper also reports that the N5LO interaction, used in a lattice-Hamiltonian transport solver, reproduces proton directed, elliptic, triangular, and quadrangular flow data from Au+Au collisions at 1.23 GeV per nucleon.

Load-bearing premise

The load-bearing premise is that the extrapolation of the empirical optical potential above 1 GeV, used as fitting data for the $a_8$ and $a_{10}$ coefficients, faithfully represents the true single-nucleon potential between 1 and 2 GeV; if it does not, the 2 GeV claim has no experimental footing.

Editorial extensions

If this is right

  • Transport simulations for heavy-ion collisions at incident energies up to about 2 GeV per nucleon can use the N5LO mean field without the fast potential rise that limited the N3LO interaction to energies below 1 GeV.
  • The N5LO interactions provide six independent characteristic parameters for symmetric nuclear matter and six for the symmetry energy, so future data can constrain the equation of state at suprasaturation densities rather than only around saturation.
  • All 24 constructed interactions satisfy current flow constraints on the symmetric-matter pressure, microscopic pure-neutron-matter predictions, pulsar mass-radius measurements, and the gravitational-wave upper limit on the 1.4-solar-mass tidal deformability.
  • The proton elliptic flow $v_2$ responds to both the isoscalar effective mass and the high-momentum slope of the single-nucleon potential: a larger effective mass weakens $v_2$, while a rapidly rising potential strengthens it.
  • The benchmark Au+Au simulation at 1.23 GeV per nucleon reproduces the measured proton $v_1$, $v_2$, $v_3$, and $v_4$, establishing that the extended interactions are usable transport inputs above 1 GeV per nucleon.

Reading between the lines

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

  • Because the high-order coefficients $a_8$ and $a_{10}$ contribute negligibly to bulk nuclear-matter properties near saturation but dominate the single-nucleon potential above 1.5 GeV/c, the optical-potential fit and the equation-of-state fit are almost decoupled; this separation could let future analyses constrain the high-momentum mean field and the dense-matter EOS independently from the same co
  • If direct 1-2 GeV elastic-scattering data confirm the N5LO potential, the same binomial construction could be extended to N6LO and beyond, although each added coefficient increases the extrapolation risk.
  • The symmetry-potential family is generated by a single scaling constant through a cosine-like Taylor expansion, so the eight effective-mass-splitting scenarios do not exhaust the possible high-momentum symmetry potentials; testing other shapes of the higher coefficients would clarify how much of the flow agreement depends on that choice.
  • A fully relativistic mean-field treatment is the natural stress test of the claimed 2 GeV applicability, since the transport kinematics are relativistic while the Skyrme potential itself is nonrelativistic.
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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

3 major / 5 minor

Summary. This paper generalizes the previously constructed N3LO Skyrme pseudopotential to general NnLO order by including central-term derivative operators up to 2n-th order, and derives the corresponding Hamiltonian density and single-nucleon potential in the Hartree-Fock approximation under nonequilibrium conditions. The authors construct 24 new interactions from three models (N3LO, N4LO, N5LO) and eight symmetry potentials characterized by linear isospin-splitting coefficients Δm*_1(ρ0), study their predictions for nuclear matter, neutron stars, and the symmetry energy, and perform lattice BUU simulations of Au+Au collisions at 1.23 GeV/nucleon, comparing proton flow harmonics with HADES data. The central quantitative claim is that the N5LO single-nucleon potential, with momentum dependence up to p^10, gives a good description of the empirical nucleon optical potential up to 2 GeV kinetic energy.

Significance. If accepted at face value, the framework would be a useful step for transport simulations of heavy-ion collisions in the 1-2 GeV/nucleon regime: the polynomial form of the mean-field potential retains a computational advantage (the p-dependence is factored out of the test-particle sums, as shown in Appendix A), and the extended density-dependent terms provide flexible equations of state that can simultaneously accommodate microscopic neutron-matter calculations, neutron-star radii from NICER, and the tidal deformability constraint from GW170817. The paper is also strong in its systematic formal derivations and in the transparency of the parameter tables and fitting procedure. The HADES benchmark, although limited in scope, is a constructive addition that demonstrates the potential applicability of the constructed interactions.

major comments (3)
  1. [Section III, Eq. (57), Table II] The headline claim that the N5LO single-nucleon potential 'can give a nice description for the empirical nucleon optical potential up to energy of 2 GeV' is not established by the fit described in Eq. (57). The fitting functional explicitly includes 'its extrapolation above 1 GeV' with equal weights and no practical errors, and the N5LO fit extends to 2.5 GeV/c, approximately corresponding to 2 GeV kinetic energy. The coefficients a8 and a10 in Table II are therefore fixed by the analytic continuation of the Hama et al. parametrization, not by measured proton-nucleus scattering data in the 1-2 GeV range. If the true high-energy potential differs from that continuation, the fitted high-order coefficients and the claimed saturated behavior above 1 GeV would change. The abstract, Section IV.B, and Section VI should be reworded to state that the N5LO potential reproduces the Hama et al. parametrization including its extrapolation, or the authors should add a quantitative robustness test against alternative high-energy extrapolations and restrict the empirical claim accordingly.
  2. [Section IV.B, Eq. (59), Fig. 7] The agreement between the predicted symmetry potential and the global optical-model analyses is partly preselected by construction. The coefficients bn are generated by the cosine-like expansion of Eq. (59) with the single scale A tuned to produce the chosen Δm*_1(ρ0) values; the observation that the Δm*_1 = 0.3 and 0.5 curves are consistent with the optical-model value (0.41 ± 0.15)δ is therefore a consequence of the parametrization, not an independent measurement. The text should state explicitly that Eq. (59) is an assumed functional form and that the comparison in Fig. 7 is a consistency check, not a determination of the isospin splitting of the effective mass. A sensitivity study varying the assumed shape of the symmetry-potential momentum dependence would materially strengthen the paper.
  3. [Section V.B, Figs. 12-13] The HADES benchmark is presented as providing 'good predictions' of proton collective flows, but it does not isolate the 1-2 GeV single-nucleon potential. The simulations use only four interactions with Δm*_1(ρ0) = 0.3 fixed, a single beam energy 1.23 GeV/nucleon, a fixed impact parameter b = 7.4 fm, and a transport model containing many additional ingredients including the collision term, nucleon resonances, Δ single-particle potentials, and the gradient parameter E[2] fitted to the 197Au binding energy. Agreement with the flow data is therefore a global consistency check of the whole LBUU setup rather than a direct validation of the high-momentum behavior of the mean field. The claims in Section VI should be softened accordingly, and the text should state that the flow comparison provides at most qualitative support for the constructed interactions.
minor comments (5)
  1. [Section III, Eq. (59) and Table III] The dimensional status of Eq. (59) is unclear: since bn has units MeV fm^n, the factor (π/10)^n and the scale A need their units specified; as written, a single A value cannot generate coefficients with different fm^n units for n = 2, 4, 6, 8, 10.
  2. [Section IV.B, Fig. 6(a)] The figure and caption should distinguish more clearly between the Hama et al. data points and their extrapolated curve, for example by using different symbols and by stating in the text that the portion above 1 GeV is the extrapolation rather than the measured optical potential.
  3. [Section III, Eq. (57)] The quantity in Eq. (57) is called a weighted squared difference χ2, but the text immediately states that there are no practical errors σ_i and that equal weights are assigned. This is not a statistical chi-square; please describe it as an unweighted least-squares measure and report residuals or a goodness-of-fit metric.
  4. [Section VI] There is a grammatical error in 'we have constructed a parameter sets'; this should read 'a parameter set' or 'parameter sets'. Similar small language issues appear in a few other places, e.g., 'mainly due to its polynomial structure' in the introduction could be made more precise.
  5. [Table II] Several entries contain stray spaces in the numeric values, for example '8 .133 × 10−4'; please correct the formatting throughout the tables.

Circularity Check

2 steps flagged · score 6.0 of 10

The headline 2 GeV optical-potential agreement is a fit to the very curve it claims to describe, and the neutron-star compatibility is inherited from a self-cited fitted parameter set.

  1. fitted input called prediction [Section III, Eq. (57); Section IV.B, Fig. 6(a)]
    "Next, we use GEKKO optimization suite [124] to minimize the weighted squared difference χ2 between U0 in Eq. (37) and the nucleon optical potential data Uopt [70, 71] and its extrapolation above 1 GeV: χ2 = Σ (U0,i − Uopt,i/σi)^2, (57) ... Since there are actually no practical errors σi here, we assign equal weights to each data point within the range of the nucleon momentum up to 1.5 GeV/c, 2.0 GeV/c, and 2.5 GeV/c (approximately corresponding to nucleon kinetic energy of 1 GeV, 1.5 GeV and 2 GeV) for N3LO, N4LO and N5LO models, respectively."

    The N5LO coefficients a8 and a10 are the fitted parameters of this χ2 minimization, with the fit range deliberately extended to 2.5 GeV/c (about 2 GeV kinetic energy). The paper then reports in Sec. IV.B that 'through the optimization process we have performed in Sec. III, U0(ρ,p) conforms rather well to the empirical nucleon optical potential (and its extrapolation) for nucleon kinetic energy up to ... 2 GeV, with SP6, SP8, and SP10 models.' That statement is a description of the fit quality, not an independent validation. Above 1 GeV the fitted target is Hama et al.'s extrapolated curve, not measured proton-nucleus data, so calling the agreement 'empirical' up to 2 GeV also overstates the content of the check.

  2. fitted input called prediction [Section III (parameter choice from Ref. [88]); Section IV.C, Fig. 10]
    "We set Esym(ρ0), L, Ksym, and Jsym to their values in parameter set 'SP6L45' from Ref. [88], namely Esym(ρ0) = 30 MeV, L = 45 MeV, Ksym = −110 MeV, and Jsym = 700 MeV, to simultaneously satisfy the theoretical prediction of the EOS of PNM and various neutron star observations. ... As shown in Fig. 10, all of the 24 interactions are compatible with the constraint for PSR J0030+0451, PSR J0437-4715 and PSR J0740+6620, falling within the 68.3% CI."

    The SP6L45 parameter set was constructed in Ref. [88] by fitting to pure-neutron-matter microscopic calculations and neutron-star observations. The present paper fixes the lower-order density dependence of the symmetry energy to exactly those fitted values and sets the higher-order SNM/symmetry-energy coefficients to the corresponding N3LO values. The subsequent agreement of the mass-radius relations and tidal deformabilities with NICER and GW170817 is therefore inherited from the earlier fit rather than produced by the new N4LO/N5LO framework. Ref. [88] is supplying a fitted input that is then presented as independent confirmation, so the neutron-star compatibility claim is partially circular.

full rationale

The paper contains a genuine and largely self-contained formal development: the NnLO generalization of the Skyrme pseudopotential, the Hartree-Fock Hamiltonian density, the single-nucleon potential, and the lattice-BUU implementation are derived in detail, and the HADES flow comparison is an independent benchmark because the interactions were not tuned to those flow data. No uniqueness theorem or unsupported external authority is invoked for the formal steps. However, the central 2 GeV optical-potential claim reduces to a fit: Eq. (57) minimizes χ2 between U0 and Hama's Uopt including its extrapolation above 1 GeV, so the close agreement shown in Fig. 6(a) is a property of the optimization, not a prediction. Similarly, the neutron-star compatibility is largely inherited from the self-cited SP6L45 set of Ref. [88], whose symmetry-energy parameters were already fitted to PNM and neutron-star observations. These two reductions make the main empirical claims partially circular, while the formal framework and the HADES benchmark retain independent content. Overall circularity score: 6.

Assumptions & free parameters 7 free parameters · 8 assumptions · 0 invented entities

The construction relies on the standard Skyrme/Hartree-Fock machinery, the Fermi-momentum expansion for density dependence, and several ad hoc choices specific to this paper: the binomial ansatz for higher-order derivative terms, the cosine-like symmetry-potential shape, and the use of an extrapolated optical potential as data above 1 GeV. The large number of adjustable characteristic quantities is partially offset by the equal number of model parameters, but most of the empirical content enters through prior fits and by-hand choices rather than through independent predictions.

free parameters (7)
  • a_n coefficients of U0(rho0,p) for n=0,2,4,6,8,10 = N5LO: -64.97, 7.104, -0.1628, 1.731e-3, -8.614e-6, 1.621e-8 in MeV fm^{2n}
    Fitted with GEKKO to the Hama optical potential plus its extrapolation; these coefficients directly set the high-momentum shape of the single-nucleon potential.
  • A (symmetry-potential scale in Eq. (59)) = Values from 30.17 to -269.7 depending on Delta m*_1 (Table III)
    A single adjustable scale chosen to reproduce each preselected value of the linear isospin splitting coefficient Delta m*_1(rho0); the momentum shape itself is fixed by a cosine-like Taylor ansatz.
  • Delta m*_1(rho0) linear isospin splitting coefficient = ±0.1, ±0.3, ±0.5, ±0.7
    Eight values chosen by hand to span the uncertainty in the symmetry potential; optical-model data favor roughly 0.3 to 0.5, but the values are not fitted to data.
  • J0 (SNM skewness coefficient) = -383 MeV
    Set to the maximum allowed value from heavy-ion flow data; this choice affects the density behavior of the symmetric matter EOS in all constructed interactions.
  • I0 and H0 (SNM kurtosis and hyper-skewness) = 1818.9 MeV and -12065 MeV
    For N4LO and N5LO these are adjustable in principle but are fixed to the N3LO model values to isolate the effect of the new momentum dependence.
  • Esym(rho0), L, Ksym, Jsym = 30 MeV, 45 MeV, -110 MeV, 700 MeV
    Taken from the SP6L45 parameter set of Ref. [88], which was previously constructed to satisfy microscopic PNM calculations and neutron-star observations.
  • Gradient parameter E[2] in BUU simulations = -305, -300, -310, -310 MeV fm^5 for SP6, SP8, SP10, SP6Ms83
    Adjusted per interaction to reproduce the experimental binding energy of 197Au; higher-order gradient parameters are omitted.
assumptions (8)
  • ad hoc to paper The 2n-th order central term of the pseudopotential is built solely from the binomial expansion of (A+B)^n, with [A,B]=0 and the specific parity forms of Eq. (A13).
    This structural assumption is not derived from a more fundamental theory and determines all higher-order momentum and gradient terms in the NnLO pseudopotential. It is introduced in Appendix A, Eqs. (A11)-(A13).
  • domain assumption The Hartree-Fock approximation with spin-averaged quantities is valid, and spin-orbit and tensor terms do not contribute to the studied observables.
    Used throughout Section II and Appendix A to derive the Hamiltonian density and single-nucleon potential from the pseudopotential.
  • domain assumption The real part of the empirical nucleon-nucleus optical potential (Schroedinger equivalent potential) represents the single-nucleon potential in symmetric nuclear matter at saturation density.
    This identification is the basis for fitting the a_n coefficients to the Hama et al. optical potential in Section III.
  • domain assumption The density-dependent terms follow the Fermi-momentum expansion proposed in Ref. [88].
    The form of V_DD in Eq. (6) is imported from the authors' prior work and is assumed to provide a sufficiently flexible density dependence for the EOS.
  • ad hoc to paper The symmetry potential momentum dependence is parametrized by a cosine-like Taylor expansion, Eq. (59), with a single scale A.
    This ansatz is introduced to keep the symmetry potential well behaved up to 2 GeV/c and to reduce the number of free parameters; it has no microscopic derivation.
  • domain assumption Neutron-star cores are composed of npe mu matter without phase transitions, hyperons, or other exotic degrees of freedom.
    Used in Section IV.C when solving the TOV equation and comparing with NICER and GW170817 constraints.
  • domain assumption A non-relativistic mean-field potential combined with relativistic kinematics is adequate for transport simulations up to about 2 GeV per nucleon.
    The authors acknowledge this point in Section V.B and argue that the successful HADES comparison supports the approximation.
  • ad hoc to paper The extrapolation of the Hama optical potential above 1 GeV is a valid fit target for the N4LO and N5LO models.
    No direct empirical optical-potential data above 1 GeV are presented; the claim of describing the empirical potential up to 2 GeV depends on this extrapolation.

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Pith. "Pith review of Extended Skyrme effective interactions with higher-order momentum-dependence for transport models and neutron stars." pith.science (2026). https://pith.science/paper/ASK3S3O5

@misc{pith2026241209393,
  author       = {Pith},
  title        = {Pith review of: Extended Skyrme effective interactions with higher-order momentum-dependence for transport models and neutron stars},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ASK3S3O5}},
  note         = {Machine review of arXiv:2412.09393}
}
abstract

The recently developed extended Skyrme effective interaction based on the so-called N3LO Skyrme pseudopotential is generalized to the general N$n$LO case by incorporating the derivative terms up to 2$n$th-order into the central term of the pseudopotential. The corresponding expressions of Hamiltonian density and single-nucleon potential are derived within the Hartree-Fock approximation under general nonequilibrium conditions. The inclusion of the higher-order derivative terms provides additional higher-order momentum dependence for the single-nucleon potential, and in particular, we find that the N5LO single-nucleon potential with momentum dependent terms up to $p^{10}$ can give a nice description for the empirical nucleon optical potential up to energy of $2$ GeV. At the same time, the density-dependent terms in the extended Skyrme effective interaction are extended correspondingly in the spirit of the Fermi momentum expansion, which allows highly flexible variation of density behavior for both the symmetric nuclear matter equation of state and the symmetry energy. Based on the Skyrme pseudopotential up to N3LO, N4LO and N5LO, we construct a series of interactions with the nucleon optical potential having different high-momentum behaviors and with the symmetry potentials featuring different linear isospin-splitting coefficients for nucleon effective mass, by which we study the properties of nuclear matter and neutron stars. Furthermore, within the lattice BUU transport model, some benchmark simulations with selected interactions are performed for the Au+Au collisions at a beam energy of $1.23$ GeV/nucleon, and the predicted collective flows for protons are found to nicely agree with the data measured by HADES collaboration.

Figures

Figures reproduced from arXiv: 2412.09393 by the authors.

Figure 1
Figure 1. FIG. 1. The pressure of SNM [ [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The EOS of PNM predicted by different interactions, categorized according to different models. The band represents [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The energy dependence of the symmetry energy predicted by different interactions, categorized according to different [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The EOS of SNM as a function of nucleon density [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The EOS of PNM as a function of nucleon density [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The energy dependence of the single-nucleon poten [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The momentum dependence of the single-nucleon [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. The momentum dependence of the symmetry poten [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. M-R relation for static neutron stars from different interactions, categorized according to different models (see text [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. The squared sound speed ( [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Directed ( [PITH_FULL_IMAGE:figures/full_fig_p020_13.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Directed ( [PITH_FULL_IMAGE:figures/full_fig_p020_12.png]

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