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REVIEW 3 major objections 3 minor 50 references

The d-electron contribution to the stopping power of transition metals

T0 review · 3 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The low-energy stopping of protons in six transition metals is reproduced by the atomic d-subshell momentum distribution, not by a free-electron gas.

desk verdict A genuinely new d-electron stopping model for transition metals; the atomic-to-solid momentum distribution assumption is the key thing a referee should test. read the letter →

arxiv 2411.12810 v1 pith:OJ7ZBE4Y submitted 2024-11-19 physics.atom-ph cond-mat.mtrl-sci

J. P. Peralta , A. M. P. Mendez , D. M. Mitnik , C. C. Montanari This is my paper · ORCID
classification physics.atom-phcond-mat.mtrl-sci PACS 34.50.Bw
keywords stoppingpowertransitionmetalsd-electronsinhomogeneousmomentumdistributionnon-perturbativemodellow-energyionplasmonfrequencyelectroniccrosssection
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 proposes a way to compute the d-electron contribution to the stopping power of the later transition metals (Ni, Pd, Pt, Cu, Ag, and Au) without perturbation theory: treat the partially filled d-subshell as a gas of electrons with the momentum distribution of the free atom, $f_{nl}(p)=(2\pi)^{3/2}|\Phi_{nl}(p)|^2$, rather than as part of the homogeneous free-electron gas. Only the number of electrons actually promoted to the conduction band is taken from the measured plasmon frequency; the rest stay bound and respond to the ion through an effective collision cross section built from a velocity-dependent screened potential. The model is claimed to reproduce the low-energy stopping cross sections with good agreement with experiment and with time-dependent density functional theory (TDDFT), and, when combined with free-electron-gas and inner-shell contributions, to give coherent total stopping cross sections from 0.1 keV to 100 MeV. The d-electron contribution appears as a smooth nonlinearity in the velocity dependence, with a maximum near the mean d-electron velocity, rather than as the sharp break of linearity discussed in older experiments. The authors do not rule out a small overestimation at the lowest velocities because the model contains no explicit band gap.

What carries the argument

The load-bearing object is the inhomogeneous momentum distribution $f_{nl}(p)=(2\pi)^{3/2}|\Phi_{nl}(p)|^2$: the squared Fourier transform of the atomic d-subshell wavefunction, normalized to the number $N_d$ of bound d electrons. It replaces the Fermi-sphere step function of the free-electron-gas model, so d-electrons of all momenta contribute, and it is what carries the low-energy d-electron response. The stopping cross section is then computed by folding this distribution with the transport cross section $\sigma_{\mathrm{tr}}(v_r)$ (the momentum-transfer cross section) generated by a velocity-dependent screened potential that satisfies the cusp condition; the Fourier transforms are evaluated analytically after expanding the wavefunctions in exponential-type basis functions. The model becomes a full stopping theory by adding free-electron-gas and inner-shell contributions, covering 0.1 keV to 100 MeV.

What would settle it

Measure proton stopping in a single crystal of Ni or Au along a well-characterized off-channelling direction at velocities below about 0.3 atomic units with a few-percent uncertainty and compare with the model curve: a deviation larger than the stated few-percent agreement, or the appearance of a sharp slope break that the model excludes, would falsify the free-atom d-electron description. A less costly test is to recompute $f_{3d}(p)$ using a band-structure d-projected wavefunction and check whether the low-velocity total stopping shifts by more than the model's current scatter.

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Extended reading notes

Core claim

The paper's central discovery is that the stopping of protons in the later transition metals is not governed by how the d-electrons sit in the Fermi sphere but by their full atomic momentum profile. Concretely, the d-subshell stopping cross section is obtained by replacing the Fermi step function with $f_{nl}(p) = (2\pi)^{3/2}|\Phi_{nl}(p)|^2$ in the transport-cross-section integral, with $\Phi_{nl}(p)$ the Fourier transform of the atomic d wavefunction normalized to the number $N_d$ of d electrons that remain bound. The integer $N_{\mathrm{FEG}}$ of electrons in the free-electron gas is taken as the integer closest to the value implied by the measured plasmon frequency, so the only solid-state input is that frequency. Adding the resulting d-curve to a free-electron-gas curve and to inner-shell curves yields total stopping cross sections that match recent low-energy experiments closely for Ni, Pd, and Pt, lie between the conflicting data sets for Cu, Ag, and Au, and agree with available TDDFT results. The authors interpret the low-energy 'break of linearity' seen in group 11 measurements as a soft nonlinearity generated by the inhomogeneous momentum distribution, with the d-curve peaking near the mean d-electron velocity $v_d$.

Load-bearing premise

The load-bearing premise is that the d electrons that stay bound in the solid respond like free-atom d electrons, with the number of such electrons fixed by rounding the conduction-electron count from the measured plasmon frequency to an integer; if solid-state band effects or that electron-count assignment are wrong, the low-energy d contribution changes.

Editorial extensions

If this is right

  • For Ni, Pd, and Pt the calculated total stopping cross sections fall within a few percent of recent low-energy measurements, so the model provides a quantitative account of the group 10 low-energy response without a slope break.
  • For Cu, Ag, and Au the calculated curves sit between the two conflicting families of low-energy data, clarifying that the historical 'break of linearity' is not reproduced as a sharp kink but as a smooth d-electron-driven rise.
  • The d-subshell contribution to the stopping cross section is largest when the projectile speed is close to the mean d-electron speed $v_d$; this places the d-curve maximum at a distinct impact energy for each metal (about 350 keV/amu for Ni, about 100 keV for Pt, with the other metals in between).
  • Combining the d-model with free-electron-gas and inner-shell models yields a fully theoretical stopping-power curve over 0.1 keV to 100 MeV, with the only electron-count parameter fixed by the measured plasmon frequency.

Reading between the lines

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

  • Beyond the paper: the same inhomogeneous-momentum construction could be tested on other d-block metals (for example, group 9 or group 12), where low-energy stopping data are sparser; the only inputs needed are a d-subshell wavefunction and a measured or computed plasmon frequency.
  • Beyond the paper: replacing the free-atom d wavefunctions in Eq. (7) with solid-state d-projected wavefunctions would isolate the size of band-structure and band-gap effects the authors identify as a possible source of low-velocity overestimation.
  • Beyond the paper: a systematic comparison of d-only stopping curves (not total stopping) between this model and TDDFT across all six metals would test whether the atomic momentum profile captures the same physics as explicit time-dependent electron dynamics.
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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 / 3 minor

Summary. The paper proposes a non-perturbative model for the d-electron contribution to the electronic stopping cross section of the late transition metals Ni, Pd, Pt, Cu, Ag, and Au. The stopping integral of Eq. (1) is evaluated with an atomic "inhomogeneous" momentum distribution f_nl(p) built from Hartree-Fock or relativistic wave functions (Eq. (7)), and this d contribution is combined with a free-electron-gas description of the valence electrons and a shellwise local approximation for inner shells. The authors compare their low-velocity results with experimental data from the IAEA database and with available TDDFT calculations, and present extended-energy curves from 0.1 keV to 100 MeV. The central claim is that the d-electron stopping response can be captured by atomic momentum distributions rather than by a free-electron gas or full TDDFT, and that this yields good agreement with experiment and with TDDFT where available.

Significance. If validated, the model is attractive because it is simple and analytic, it requires no fitting to stopping data, and the electron-count inputs (N_FEG, N_d) are inferred from experimental plasmon frequencies. The paper's strengths include a transparent derivation, extensive comparison with a large experimental database, and direct comparison with TDDFT results for several targets. The main obstacle is that the central assumption — that free-atom d-orbital momentum distributions represent the solid-state d-electron response — is not tested against any solid-state calculation, and the sensitivity of the results to the N_FEG/N_d partition is not quantified. These issues are addressable, but they are load-bearing for the claimed agreement at low velocities.

major comments (3)
  1. [Sec. II, Eq. (7); Sec. III A] The central assumption is that the d-electron response in the solid is described by the free-atom momentum distribution f_nl(p). In the low-velocity limit, Eqs. (5)-(6) weight the distribution at p ~ v, so the low-energy stopping power is controlled by the low-momentum tail of f_nd(p). In a transition metal this tail is affected by band structure, s-d hybridization, and d-band occupation, none of which is captured by an isolated-atom orbital. The authors' caveat at the end of Sec. III A that a band-gap effect may cause overestimation at the lowest velocities is a symptom of this atomic-versus-solid gap, but it does not test the shape of f_nd(p) itself. I request a quantitative comparison with a DFT-based d-projected momentum density (or equivalent solid-state calculation) for at least one Group-10 and one Group-11 target, with a discussion of how hybridization changes the low-p weight and how that would affect the computed stopping curves.
  2. [Table I; Sec. III] The partition of outer electrons into N_FEG and N_d is based on integer rounding of the experimental plasmon frequency through the free-electron relation, together with the free-atom configuration; it is not validated by a solid-state occupation calculation. Because the d contribution is proportional to N_d through the normalization in Eq. (7), the low-velocity results depend directly on this assignment. The authors state that N_FEG and N_d 'may be the subject of discussion' and compare with an N_FEG=10 variant for Pt in Figs. 6 and 12, but this is not a systematic sensitivity analysis for the present model. I would like to see either a quantitative sensitivity study over the admissible integer choices of N_FEG/N_d for all six targets, or an independent estimate of the solid d-occupation from DFT projected densities of states.
  3. [Eqs. (6), (A11)-(A13)] As printed, the integration limits in I'(v_r), g5, and g7 are reversed. After exchanging the order of integration in Eqs. (3)-(4), the p-integral should run from |v_r - v| to v_r + v. The printed limits |v_r + v| to |v_r - v| decay over the integration interval and would give the negative of the intended value. This is likely typographical, but it appears in the central derivation and in the analytical expressions of Appendix A, so it must be corrected and the numerical implementation must be confirmed to use the correct limits.
minor comments (3)
  1. [Eq. (9)] The Heaviside function is written as Θ(p - p_F), but the text says the step function lies 'within the Fermi sphere'; this should be Θ(p_F - p). In addition, the numerical constant (2π)^{3/2} on the right-hand side is inconsistent with the Fermi-sphere volume factor: with p_F = (3π^2 n_e)^{1/3}, one has ∫ d^3p = 4π^3 n_e, not (2π)^{3/2} n_e. Please clarify the normalization convention and reconcile Eq. (9) with Eq. (8).
  2. [Table I caption] The phrase 'the dump value γ_p^exp' appears to be a typo for 'damping value'; the text around Table I should be corrected.
  3. [Sec. III A, Figs. 2-7] The comparison with TDDFT for Au is complicated by the fact that the cited TDDFT results are channelling calculations for Au<100>; the authors note this, but the figure caption could make the distinction clearer to avoid the impression of a direct off-channelling comparison.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the d-electron stopping is computed from atomic wave functions and independently inferred electron counts, and then compared with external experimental and TDDFT benchmarks.

full rationale

The paper's derivation chain is a forward calculation, not a fit to the target data. Equation (7) defines the d-electron momentum distribution from Hartree-Fock or relativistic atomic wave functions, and Eqs. (5)-(6) then compute the stopping cross section by integrating that distribution against a transport cross section. No stopping-power measurement is used to set any parameter in this chain. The electron counts NFEG and Nd are inferred from experimental plasmon frequencies quoted from Werner et al. [40], an independent measurement, with NFEG taken as the nearest integer and Nd as the complement; this affects the magnitude of the d contribution but is not tuned to the stopping data. The velocity-dependent screened potential is taken from the authors' prior work [4], but it is a fixed input model that is itself being tested by the present comparisons rather than a result derived from the target claim. The validation step, where the computed curves are compared with IAEA data and with TDDFT, is external to the derivation. Even the acknowledged low-velocity caveat about the missing band gap is a stated limitation of the atomic approximation, not evidence that the prediction is equivalent to an input. The only parts of the paper that involve self-citations are the use of the authors' earlier screened-potential and shell-model ingredients and the comparison with their own earlier FEG-only calculation; these are normal building blocks and do not make the central claim reduce to its inputs. The central claim has independent content: it predicts quantitative stopping curves from independent atomic-structure and plasmon-frequency inputs, and those predictions are then checked against experiment and TDDFT.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The model introduces no new particles, forces, or conserved quantities. Its main adjustable inputs are the per-element electron counts NFEG and Nd, which are chosen from experimental plasmon frequencies rather than derived from first principles. The remaining uncertainty comes from using atomic wave functions and a self-cited screened potential, together with the absence of an explicit band gap.

free parameters (2)
  • NFEG (integer number of electrons in the free-electron gas) = Ni=3, Cu=3, Pd=7, Pt=7, Ag=3, Au=7
    Set to the integer closest to the experimental plasmon frequency from reflection EELS data in Ref. [40]. NFEG determines Nd and therefore the amplitude of the d-electron stopping contribution.
  • Nd (number of d-bound electrons) = Ni=7, Cu=8, Pd=3, Pt=3, Ag=8, Au=4
    Derived by subtracting NFEG from the nominal d+s valence configuration. This number scales the d-electron momentum distribution and is chosen rather than derived from first principles.
assumptions (4)
  • standard math The standard kinetic-theory formula, Eq. (1), with transport cross-section and phase shifts, gives the stopping cross-section from a momentum distribution.
    Invoked as the starting point from Nagy and Bergara (1996) and Wang, Nagy, and Echenique (1998). The paper assumes this framework is valid.
  • domain assumption The velocity-dependent screened potential of Ref. [4] verifies the cusp condition and correctly describes the projectile-target interaction.
    The phase shifts in Eq. (2) are computed with this potential. If the potential is inaccurate, the d-electron stopping cross-sections would change.
  • domain assumption Free-atom Hartree-Fock or relativistic atomic wave functions represent the bound d-electrons in the solid target.
    The authors use atomic wave functions for Ni, Cu, Pd, Ag, Pt, and Au and explicitly acknowledge the atomic-versus-solid distinction in Section III. Solid-state band structure and the d-band energy gap are not included.
  • domain assumption Independent-shell approximation: total electronic stopping is the sum of d-electron, free-electron-gas, and inner-shell contributions.
    The paper combines the d model with Mermin-Lindhard FEG results and SLPA-LM inner-shell contributions, assuming that interference between shells is negligible.

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Pith. "Pith review of The d-electron contribution to the stopping power of transition metals." pith.science (2026). https://pith.science/paper/OJ7ZBE4Y

@misc{pith2026241112810,
  author       = {Pith},
  title        = {Pith review of: The d-electron contribution to the stopping power of transition metals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OJ7ZBE4Y}},
  note         = {Machine review of arXiv:2411.12810}
}
abstract

We present a new non-perturbative model to describe the stopping power by ionization of the $d$-electrons of transition metals. These metals are characterized by the filling of the d-subshell and the promotion of part of the electrons to the conduction band. The contribution of d-electrons at low-impact energies has been noted experimentally in the past as a break of the linear dependence of the stopping power with the ion velocity. In this contribution, we describe the response of these electrons considering the atomic "inhomogeneous" momentum distribution. We focus on the transition metals of Groups 10 and 11 in the periodic table: Ni, Pd, Pt, Cu, Ag, and Au. Results describe the low energy-stopping power, with good agreement with the experimental data and available TDDFT results. By combining the present non-perturbative model for the $d$-subshell contribution with other approaches for the valence electrons and for the inner shells, we provide a coherent theoretical method capable of describing the stopping power of these transition metals from the very low to the high energy region.

Figures

Figures reproduced from arXiv: 2411.12810 by the authors.

Figure 1
Figure 1. FIG. 1. (Color online) Distribution functions of the [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 4
Figure 4. FIG. 4. (Color online) Low-energy stopping cross-section of [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. FIG. 5. (Color online) Low-energy stopping cross-section of [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figures from the paper (7 more)
Figure 6
Figure 6. Figure 6: FIG. 6. (Color online) Low-energy stopping cross-section of [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Stopping cross-section of Ni for H as a function [PITH_FULL_IMAGE:figures/full_fig_p006_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. (Color online) Stopping cross-section of Cu for H [PITH_FULL_IMAGE:figures/full_fig_p007_9.png]
Figure 11
Figure 11. Figure 11: FIG. 11. (Color online) Stopping cross-section of Ag for H [PITH_FULL_IMAGE:figures/full_fig_p007_11.png]
Figure 10
Figure 10. Figure 10: FIG. 10. (Color online) Stopping cross-section of Pd for [PITH_FULL_IMAGE:figures/full_fig_p007_10.png]
Figure 12
Figure 12. Figure 12: FIG. 12. (Color online) Stopping cross-section of Pt for H [PITH_FULL_IMAGE:figures/full_fig_p008_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. (Color online) Stopping cross-section of Au for H [PITH_FULL_IMAGE:figures/full_fig_p008_13.png]

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    F. F. Selau, H. Trombini, G. G. Marmitt, A. M. De An- drade, J. Morais, P. L. Grande, I. Alencar, M. Vos, and R. Heller, Physical Review A 102, 032812 (2020). Appendix A: Momentum distribution calculations In the present model, we propose an inhomoge- neous momentum distributi...

    2020

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.