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arxiv: 2605.22442 · v1 · pith:TLFPSMGKnew · submitted 2026-05-21 · ⚛️ physics.atom-ph · physics.app-ph

Inelastic collisions of fast charged particles with atoms. Relativistic plane-wave Born approximation

Francesc Salvat This is my paper

Pith reviewed 2026-05-22 01:53 UTC · model grok-4.3

classification ⚛️ physics.atom-ph physics.app-ph
keywords relativistic Born approximationinelastic collisionsgeneralized oscillator strengthsstopping poweratomic cross sectionsDirac-Hartree-Fock-Slaterenergy-loss spectra
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The pith

The double-differential cross section for inelastic collisions separates into longitudinal and transverse generalized oscillator strengths each multiplied by kinematic factors.

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

This paper gives a full relativistic plane-wave Born approximation for fast charged particles losing energy in atoms and ions. Target electrons are treated in an independent-electron model whose orbitals come from the Dirac-Hartree-Fock-Slater potential, and the interaction is written in the Coulomb gauge. The cross section is expressed as the sum of a longitudinal term driven by the instantaneous Coulomb field and a transverse term arising from exchange of virtual photons, each term being a kinematic prefactor times the corresponding generalized oscillator strength. Closed analytic expressions for both oscillator strengths are obtained in terms of angular-momentum coupling coefficients and radial integrals, permitting a systematic numerical database and the derivation of high-energy asymptotic formulas including shell corrections.

Core claim

Within the relativistic plane-wave Born approximation the double-differential cross section is written as the sum of two products, each consisting of purely kinematic factors and a generalized oscillator strength; the longitudinal GOS accounts for transitions induced by the instantaneous Coulomb interaction while the transverse GOS accounts for transitions caused by the transverse electromagnetic field, and both GOSs are given in closed form by vector coupling coefficients and radial integrals evaluated with Dirac-Hartree-Fock-Slater orbitals.

What carries the argument

The longitudinal and transverse generalized oscillator strengths, which isolate the atomic response from the kinematic factors in the cross section and are evaluated from the independent-electron Dirac orbitals.

If this is right

  • A complete numerical database of longitudinal and transverse GOSs now exists for every subshell of the ground-state configuration of neutral atoms from hydrogen through einsteinium.
  • Asymptotic high-energy formulas follow directly for the total inelastic cross section, the stopping cross section, and the energy-straggling cross section.
  • Shell corrections to the asymptotic stopping cross section are obtained by subtracting the computed numerical values from the high-energy limit.
  • The same expressions yield energy-loss spectra and integrated quantities for any atomic number without further approximation.

Where Pith is reading between the lines

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

  • The clean separation between longitudinal and transverse channels may make it easier to isolate relativistic magnetic and retardation corrections in measured spectra.
  • The tabulated GOS database supplies a ready input for Monte Carlo codes that track fast particles through matter.
  • The same formal structure can be reused for positive ions or for excited target states once the corresponding orbitals are supplied.
  • Systematic comparison of the relativistic results with the non-relativistic limit quantifies the energy range where relativistic kinematics become essential.

Load-bearing premise

Wave functions obtained from a central-field independent-electron model with the Dirac-Hartree-Fock-Slater self-consistent potential are accurate enough to give reliable generalized oscillator strengths for the collision kinematics of interest.

What would settle it

Direct experimental measurement of the double-differential energy-loss cross section for, say, 100 MeV protons on neon or argon, compared against the numerical values generated from the derived GOS database.

Figures

Figures reproduced from arXiv: 2605.22442 by Francesc Salvat.

Figure 10
Figure 10. Figure 10: + 2|κa| ⟨ψa |Knr| ψa⟩ 6(Q + mec 2 ) 5 [PITH_FULL_IMAGE:figures/full_fig_p064_10.png] view at source ↗
read the original abstract

A detailed formulation of the relativistic plane-wave Born approximation for inelastic collisions of charged particles with free atoms and positive ions is presented. The wave functions of the target atom or ion are calculated from a central-field independent-electron model with the Dirac-Hartree-Fock-Slater self-consistent potential, and the electromagnetic field is expressed in the Coulomb gauge. The double-differential cross section, depending on the energy loss and the recoil energy, is given as a sum of two terms which are products of purely kinematic factors and the generalized oscillator strengths (GOSs). Transitions induced by the instantaneous Coulomb interaction between the projectile and the active target electron are described by the longitudinal GOS. Transitions caused by the transverse interaction (exchange of virtual photons) are accounted for by a transverse GOS. We derive closed expressions for the longitudinal and transverse GOSs in terms of vector coupling coefficients and radial integrals. A set of Fortran programs have been written to compute the GOSs, the energy-loss differential cross section, and integrals of the latter. A complete numerical database of GOSs has been calculated for all the subshells of the ground-state configuration of neutral atoms of the elements with atomic numbers from 1 (hydrogen) to 99 (einsteinium). A systematic derivation of asymptotic formulas for the total cross section, the stopping cross section and the energy-straggling cross section is presented. The shell correction to the asymptotic formula for the stopping cross section of protons is obtained from the difference between computed numerical values and the predictions of that formula.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript presents a detailed relativistic plane-wave Born approximation (PWBA) formulation for inelastic collisions of fast charged particles with free atoms and positive ions. Target wave functions are obtained from a central-field independent-electron model using the Dirac-Hartree-Fock-Slater potential in the Coulomb gauge. The double-differential cross section is expressed as a sum of two terms, each a product of kinematic factors and generalized oscillator strengths (GOSs): the longitudinal GOS for instantaneous Coulomb interactions and the transverse GOS for virtual-photon exchange. Closed expressions for both GOSs are derived in terms of vector coupling coefficients and radial integrals. A numerical database of GOSs is computed for all subshells of neutral atoms with Z from 1 to 99, Fortran programs are provided for GOSs and cross sections, and asymptotic formulas for total, stopping, and straggling cross sections are derived, with the shell correction to the stopping cross section obtained from the difference between numerical results and the asymptotic formula.

Significance. If the central results hold, the work supplies a systematic relativistic framework, closed-form GOS expressions, and an extensive numerical database that can serve as a reference for high-energy stopping-power calculations in radiation physics and particle transport. The separation into longitudinal and transverse contributions, the derivation of asymptotics, and the extraction of shell corrections from direct numerical comparison represent practical strengths that extend beyond purely numerical tabulations.

minor comments (3)
  1. [Introduction] The manuscript should specify the energy range and Z values over which the PWBA remains valid, including any quantitative criteria for the Born approximation (e.g., projectile velocity relative to orbital velocities).
  2. [Numerical implementation] A brief description of the Fortran programs' input/output formats, compilation requirements, and example usage would improve reproducibility of the reported GOS database and cross-section integrals.
  3. [Derivation of GOSs] The definition and normalization of the transverse GOS could be cross-referenced more explicitly to the longitudinal GOS to clarify the kinematic prefactors in the double-differential cross section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the detailed and accurate summary of our manuscript, for highlighting its practical strengths in providing a systematic relativistic PWBA framework, closed-form GOS expressions, an extensive numerical database, and asymptotic formulas with shell corrections, and for recommending minor revision. We are pleased that the separation into longitudinal and transverse contributions and the extraction of shell corrections from numerical comparison were noted as useful extensions beyond tabulations.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper derives the double-differential cross section as kinematic prefactors times longitudinal and transverse GOSs from the relativistic PWBA in Coulomb gauge, obtains closed expressions for the GOSs via standard angular-momentum recoupling of Dirac matrix elements, computes a numerical database using the DHFS central-field model as an explicit input assumption, and derives asymptotic formulas for total, stopping, and straggling cross sections directly from those expressions. The shell correction is then obtained by subtracting the asymptotic predictions from the full numerical results, which is a post-processing difference rather than a fitted parameter renamed as a prediction. No step reduces by construction to its own inputs, no load-bearing self-citation chain is present, and the derivation remains self-contained against external benchmarks and the stated atomic model.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The formulation rests on the independent-electron approximation and the specific self-consistent potential; no new particles or forces are introduced.

axioms (2)
  • domain assumption Central-field independent-electron model with Dirac-Hartree-Fock-Slater potential yields adequate single-particle wave functions for GOS calculations.
    Stated in the abstract as the basis for target wave functions.
  • domain assumption Plane-wave Born approximation is valid for the fast-particle kinematics considered.
    Implicit in the title and formulation of the cross section.

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