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L-shell Photoionisation Cross Sections in the S^{+}, S^{2+}, S^{3+} Isonuclear Sequence

T0 review · 3 major / 7 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Absolute L-shell photoionisation cross sections for S+, S2+ and S3+ are obtained by summing measured single, double and triple ionisation channels, providing benchmarks for MCDF and R-matrix calculations.

desk verdict New absolute L-shell measurements for three S ions, but the S+ and S2+ theory validation is partly fitted via metastable fractions and energy shifts, so the benchmark value is real but weaker than the abstract claims. read the letter →

arxiv 2501.18497 v2 pith:BLYDPWPI submitted 2025-01-30 physics.atom-ph

J.-P. Mosnier , E.T. Kennedy , D. Cubaynes , J.-M. Bizau , S. Guilbaud , C. Blancard , B.M. McLaughlin , M.F. Hasoglu
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T.W. Gorczyca
This is my paper · ORCID
classification physics.atom-ph PACS 32.80.Fb
keywords absolutephotoionisationcrosssectionsL-shellsulphurionsmerged-beamexperimentsynchrotronradiationmulticonfigurationalDirac-FockR-matrixcalculationsisonuclearsequence
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 reports absolute L-shell photoionisation cross sections for the S+, S2+ and S3+ ions in the photon energy range 175–230 eV, where a 2p or 2s electron is excited or ionised. The values are obtained by measuring the single, double and triple ionisation yields in a merged-beam photon–ion experiment and summing them, with a quoted relative uncertainty generally within 15 percent. The paper argues that these data provide new absolute benchmarks for three sulphur isonuclear ions and support the accompanying multiconfigurational Dirac-Fock (MCDF) and Breit-Pauli or Dirac R-matrix calculations as useful interpretation tools. If correct, the results give astrophysical plasma models and X-ray spectroscopy of sulfur-bearing materials a firmer experimental footing in this energy range.

What carries the argument

The absolute scale is set by a merged-beam measurement: a counter-propagating photon beam and ion beam interact over a known length, and the cross section follows from the photoion count rate, the photodiode current, the ion current, the detector efficiencies, and a measured beam-overlap form factor via their Eq. (1). The theoretical interpretation rests on two independent computational approaches: multiconfigurational Dirac-Fock (MCDF), which builds photoabsorption cross sections from variationally optimised eigenstates of the Dirac Hamiltonian, and R-matrix methods, which solve the coupled-channel scattering problem and include autoionising resonances; a Breit-Pauli formulation is used for S2+ and S3+, and a Dirac formulation for S+. The theoretical spectra are convolved with the experimental bandpass, corrected for metastable-state populations, and, where needed, shifted in photon energy to match the measured resonance positions.

What would settle it

Re-measure the S+, S2+ and S3+ total photoionisation cross sections in the 176–187 eV 2p→3d resonance region with an independent merged-beam apparatus and an independent photon-flux calibration, and compare the absolute values. Agreement within the combined ~15 percent uncertainties would confirm the present absolute scale; a disagreement larger than the combined uncertainties would point to an error in the overlap, efficiency, or background subtraction.

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Extended reading notes

Core claim

The central claim is that the absolute L-shell photoionisation cross sections of S+, S2+ and S3+ can be measured by summing all ionisation channels, and that the resulting spectra are rich, resonance-laden, and well reproduced by theory. The measured single-, double- and triple-ionisation cross sections are combined into total photoionisation cross sections; the spectra show narrow (typically ≤100 meV) resonances from 2p→nd excitations below and up to the 2p thresholds, and broad (~1 eV) Rydberg series from 2s→np excitations above the thresholds. The paper reports close experiment–theory agreement in resonance energies, relative intensities, and integrated oscillator strengths for both the MCDF and R-matrix frameworks, after applying small systematic energy shifts and weighting by the estimated ground-state and metastable-state populations in the ion beam. Along the S+, S2+, S3+, S4+ isonuclear series, the integrated 2p→3d oscillator strength increases linearly with ionic charge, with a slope of about 0.25 per unit charge, and the paper notes this trend must peak before the hydrogen-like S15+ value.

Load-bearing premise

The absolute values stand or fall on the calibration chain in their Eq. (1): if the measured beam-overlap integral, the photodiode and channel-plate efficiencies, or the background count subtraction are systematically off, every reported cross section scales with that error.

Editorial extensions

If this is right

  • The absolute cross sections give astrophysical plasma modellers direct input for sulphur ion photoionisation in the L-shell region, where only valence-shell data were previously available for most of these ions.
  • The close experiment–theory comparisons provide a benchmark against which improved MCDF and R-matrix calculations can be tested.
  • The measured double-to-single ionisation intensity ratios quantify the role of shake-off and correlation effects in the decay of 2p and 2s vacancies.
  • The linear increase of the 2p→3d oscillator strength with ionic charge offers a compact prediction that can be extended and tested for neighbouring members of the isonuclear sequence.
  • The resonance energies, quoted with a calibrated accuracy of about 40 meV, can serve as reference data for X-ray absorption spectroscopy of sulfur compounds in low oxidation states.

Reading between the lines

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

  • An extension not drawn in the paper: measuring the missing L-shell spectra of S4+ and S5+ under quieter beam conditions would test whether the 2p→3d oscillator-strength trend continues linearly or bends over near the neon-like S6+ maximum.
  • Because the quoted cross sections include a significant metastable-state fraction, future merged-beam experiments on other ions from an electron cyclotron resonance ion source will need to report the ground-state/metastable beam mixture explicitly before their absolute values can be directly compared with these results.
  • The same merged-beam absolute normalisation approach could be applied to other astrophysically relevant low-Z ions, such as phosphorus, chlorine, or argon in neighbouring stages, to produce a dataset where the systematic calibration errors are common rather than random.
  • The broad 2s→np resonances above the 2p thresholds are natural sources of Fano profiles; fitting those profiles with the parametrisation used here could yield q parameters that connect to dielectronic recombination strengths, a connection the paper mentions but does not pursue quantitatively.
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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 / 7 minor

Summary. The manuscript reports absolute L-shell photoionisation cross sections for S+, S2+, and S3+ in the photon-energy region 175–230 eV, measured with the MAIA merged-beam apparatus at SOLEIL. Single, double, and triple ionisation channels are measured separately and summed to obtain total cross sections, which are then compared with MCDF, Breit-Pauli R-matrix, and Dirac R-matrix (DARC) calculations. The paper also tabulates resonance energies, strengths, and natural widths for the strongest features and presents an isonuclear comparison that includes S4+ and S6+. The absolute scale is obtained directly from Eq. (1) using measured beam currents, overlap integrals, and detector efficiencies.

Significance. The measurements are potentially valuable: absolute L-shell photoionisation data for these three sulphur ions are scarce, and the paper provides them with documented checks on O+ contamination and energy calibration. The internal consistency of the integrated cross sections across channels and the explicit isonuclear trend are strengths. However, the validation of theory for S+ and S2+ is weakened because the metastable-state fractions used to construct the synthetic theoretical spectra are fitted to the same experimental data being benchmarked. The measured absolute cross sections are therefore beam-mixture averages, not pure ground-state cross sections, and the theory-experiment agreement is partly constructed. These issues are addressable in revision, but they affect the force of the benchmark claims.

major comments (3)
  1. [Section IV A, IV B, and III E] The metastable fractions used to generate the theoretical curves are fitted to the experimental spectra being compared. For S+ the DARC comparison uses 90% 4S + 8% 2D + 2% 2P (Fig. 2), while the MCDF comparison uses weights 0.70/0.11/0.17/0.02/0.03; for S2+ weights of approximately 0.74/0.26/0.07 are used. Because these weights are chosen to match the same data, the subsequent agreement in resonance intensities and integrated strengths (e.g., 36.7 vs 35.0 vs 30.7 Mb eV for S+) is partly constructed rather than an independent test. Moreover, the published absolute cross sections are averages over a beam mixture whose composition is source-dependent (Ref. [45]) and not independently measured; they are not pure ground-state cross sections. The authors should either measure or constrain the metastable fractions independently, or explicitly restrict the benchmark claim to beam-average cross sections and provide a sensitivity analysis of the inferred fractions.
  2. [Section IV A, Figs. 2 and 4(c), Table IV] There is an unresolved inconsistency in the DARC comparison. Section IV A states that the theory curves in Fig. 4 are 'energy-unshifted,' and Table IV lists DARC resonance energies about 2.8 eV above the experimental values, yet Fig. 2 shows the DARC spectrum shifted by -2.5 eV to match the data. The integrated-intensity comparison in the same section (30.7 Mb eV for DARC) depends on which energy-shifted version is used. Please state explicitly which DARC results enter Fig. 4(c) and the quoted integrated values, and justify the shift.
  3. [Section IV A and IV B, weight normalization] The quoted metastable weights are not properly normalized: for S2+, 0.74 + 0.26 + 0.07 = 1.07, and for S+ MCDF, 0.70 + 0.11 + 0.17 + 0.02 + 0.03 = 1.03. If these are approximate values, say so; if they are meant to be fractions, they must sum to unity. As written, the synthetic spectra in Figs. 4(b) and 6(b) have an ill-defined absolute normalization, which directly affects the claimed agreement in integrated cross sections.
minor comments (7)
  1. [Fig. 4 caption] The caption contains a duplicated word: 'multiconfigurational Dirac-Fock (MCDF) theoretical theoretical values.'
  2. [Section IV B] The word 'oberved' appears in 'the rich resonance pattern oberved in Fig. 5(b)'; this should be 'observed.'
  3. [Section IV B, sentence preceding Fig. 6] The text says 'The MCDF and R-Matrix theoretical cross sections of Fig. 6(a) and (b) were convolved' but the comparison panels are Figs. 6(b) and 6(c); please correct the cross-reference.
  4. [Table II, footnote b] The footnote to the S3+ metastable state says 'This state is not metastable,' which is contradictory; please clarify whether 2p63s3p2 4P at 8.83 eV is a bound excited state rather than a metastable state.
  5. [Tables V and VI captions] The captions state that theoretical strengths are not corrected for contributions from initial metastable-level populations, while the text in Sections IV B and IV A says that metastable population factors are applied to the theoretical spectra; please reconcile these statements.
  6. [Section IV D] The f=1.53 value for S6+ is derived from a 2p53d 1P1 -> 2p6 1S0 radiative transition at 206.09 eV, which is not the same kind of ground-state 2p -> 3d excitation used for the lower ions; please clarify the comparison in the isonuclear sequence plot.
  7. [Figures 3–9] The paper claims a relative uncertainty of generally 15% but does not show error bars on the cross-section spectra; please add representative error bars or state that they are omitted for clarity.

Circularity Check

2 steps flagged · score 4.0 of 10

Absolute cross sections rest on direct measurement (Eq. 1), but the claimed validation of MCDF/DARC for S+ and S2+ is partly constructed because metastable fractions are fitted to the same spectra that are then used to assert agreement.

  1. fitted input called prediction [Section III E and Section IV A (S+; Figs. 2 and 4)]
    "From a direct comparison of the DARC theoretical cross sections with the experimental photoionization data, we estimated a best fit beam mixture of 90% 4So 3/2, 8% 2Do and 2% 2Po."

    The DARC metastable weights are fit to the measured S+ spectrum, and the same measured spectrum is then used to claim "reasonable agreement" with DARC (integrated 36.7 Mb eV expt vs 30.7 DARC). The MCDF comparison uses a different fitted weight set (0.70, 0.11, 0.17, 0.02, 0.03 from Sec. IV A). Because each theory is allowed its own fitted beam mixture, the integrated-intensity and line-shape agreement is partly manufactured; the data do not independently constrain the theories at the level claimed.

  2. fitted input called prediction [Section IV B and Fig. 7 caption (S2+)]
    "The MCDF and R-Matrix theoretical cross sections were convolved with normalised Gaussian functions of FWHM 50 meV, respectively, and scaled with appropriate metastable population coefficients (see text), to best mimic the experimental spectrum of FIG. 7(a)."

    For S2+, the theoretical spectra are scaled with metastable weights of approximately 0.74 (3P), 0.26 (1D2) and 0.07 (1S0) chosen explicitly "to best mimic" the measured spectrum. The text then reports "a good agreement" and compares integrated strengths (66.5 expt vs 67.1 MCDF vs 60.1 R-Matrix). Since the scale factors are fitted to the same data, the agreement in absolute strength is partly constructed; only the un-scaled resonance-energy pattern and the S3+ ab initio comparison supply independent validation.

full rationale

The central result, the absolute L-shell photoionisation cross sections, is obtained directly from the merged-beam equation sigma(E)=S(E)e^2 eta v q/(I J epsilon integral ...), with calibrated photodiode/channel-plate efficiencies, measured beam overlap, and background subtraction; no theoretical cross section enters this calibration. Thus the main data are not circular. The circularity burden lies in the theory-validation loop: for S+ and S2+, metastable beam fractions are estimated by fitting theoretical spectra to the experimental spectra (90/8/2 for DARC S+; 0.70/0.11/0.17/0.02/0.03 for MCDF S+; 0.74/0.26/0.07 for S2+), and the same fitted spectra are then cited as evidence of "good agreement". Because each theory gets its own fitted weights, the comparison cannot fully validate either theory. The paper does contain independent anchors: S3+ is compared ab initio with no energy shifts or population factors; NIST level energies are used as external checks; and the ASTRID data of Kristensen et al. are overlaid on the S+ SI/DI spectra. These independent elements keep the circularity from dominating the paper's central measurement, so the score is moderate rather than high.

Assumptions & free parameters 11 free parameters · 6 assumptions · 0 invented entities

The central experimental result rests on the merged-beam absolute calibration (Eq. 1), the assumption of a fixed ECRIS metastable mixture, and NIST energy benchmarks; the theoretical comparisons additionally rely on fitted metastable fractions, applied energy shifts, and uniform Lorentzian widths. No new entities are introduced.

free parameters (11)
  • S+ metastable fraction (4S3/2) = 0.90
    Estimated by best-fit comparison of DARC theoretical cross sections to the measured S+ spectrum; Section III E and Figure 2.
  • S+ metastable fraction (2D) = 0.08
    Part of the same best-fit beam mixture for S+, Section III E.
  • S+ metastable fraction (2P) = 0.02
    Part of the same best-fit beam mixture for S+, Section III E.
  • S2+ metastable mixture weights = 0.74 (3P), 0.26 (1D2), 0.07 (1S0)
    Derived to scale theoretical spectra to match measured S2+ total cross sections; Section IV B.
  • DARC energy shift for S+ = -2.5 eV
    Applied to align DARC Model A spectrum with experiment in Figure 2.
  • MCDF energy shift for S2+ high-resolution spectrum = +0.84 eV
    Applied to align MCDF resonances with the measured 181.61 eV peak in Figure 7.
  • R-Matrix energy shift for S2+ high-resolution spectrum = -1.55 eV
    Applied to align R-Matrix resonances with the measured 181.61 eV peak in Figure 7.
  • MCDF Lorentzian width for S+ 2p53s23p33d resonances = 0.069 eV
    Uniform width set to the largest computed spectator Auger rate of the 2p53s23p33d configuration; Section IV A.
  • MCDF Lorentzian widths for S2+ = 26 meV below 2p threshold, 838 meV above
    Uniform widths from largest computed autoionisation and Coster-Kronig rates; Section IV B.
  • MCDF Lorentzian widths for S3+ = 7 meV below 2p threshold, 582 meV above
    Uniform widths from largest computed rates; Section IV C.
  • Gaussian convolution widths = 100 meV, 160 meV, 50 meV depending on scan
    Used to simulate experimental bandpass broadening in synthetic theoretical spectra.
assumptions (6)
  • domain assumption Single-photon, independent-particle interaction in merged beams; cross section given by Eq. (1) with beam-overlap form factor.
    Equation (1) in Section II assumes the measured photoion count is proportional to the overlap integral of photon and ion beams; systematic errors in the form factor directly scale all absolute values.
  • domain assumption ECRIS beam population is a fixed linear mixture of ground and metastable states with time-independent fractions.
    Section II states metastable populations can be produced in variable concentrations; Sections III E and IV B estimate fractions by fitting, assuming they are constant across the energy scan.
  • domain assumption Theoretical resonance energies from MCDF and R-matrix are variational estimates that may require systematic shifts.
    Section V explains that incomplete basis descriptions cause over- or under-estimated photon energies; shifts of -2.5 to +0.84 eV are applied.
  • ad hoc to paper A single representative Lorentzian width (the largest computed Auger width) can be applied uniformly to all resonances in a given region.
    Sections IV A to C dress MCDF spectra with one width below and one above threshold; this ignores n-dependent participator widths which scale as 1/n^3.
  • domain assumption O+ contamination of the S2+ beam can be quantified from oxygen resonances at 528 to 534 eV and scaled to the L-shell region.
    Section II describes the contamination check; assumes no other contaminants and a linear scaling of the O+ contribution.
  • standard math NIST energy levels provide the benchmark for ground and metastable state energies.
    Table II cites Ref. [46]; used for ionisation energies and level identification.

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Cite this review

Pith. "Pith review of L-shell Photoionisation Cross Sections in the S^{+}, S^{2+}, S^{3+} Isonuclear Sequence." pith.science (2026). https://pith.science/paper/BLYDPWPI

@misc{pith2026250118497,
  author       = {Pith},
  title        = {Pith review of: L-shell Photoionisation Cross Sections in the S^+, S^2+, S^3+ Isonuclear Sequence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BLYDPWPI}},
  note         = {Machine review of arXiv:2501.18497}
}
read the original abstract

We present absolute L-shell photoionisation cross sections for the S+, S2+, S3+ions. The cross sections were obtained using the monochromatised photon beam delivered by the SOLEIL synchrotron source coupled with an ion beam extracted from an electron cyclotron resonance source (ECRIS) in the merged dual-beam configuration. The cross sections for single, double and triple ionisation were measured and combined to generate total photoionisation cross sections. For each of the S+, S2+, S3+ ions, the photon energy regions corresponding to the excitation and ionisation of a 2p or a 2s electron (175-230 eV) were investigated. The experimental results are interpreted with the help of multiconfigurational Dirac-Fock (MCDF) and Breit-Pauli R-Matrix (BPRM) or Dirac R-Matrix (DARC) theoretical calculations. The former generates photoabsorption cross sections from eigenenergies and eigenfunctions obtained by solving variationally the multiconfiguration Dirac Hamiltonian while the latter calculate cross sections for photon scattering by atoms. The cross sectional spectra feature rich resonance structures with narrow natural widths (typically less than 100 meV) due to 2p to nd excitations below and up to the 2p thresholds. This behaviour is consistent with the large number of inner-shell states based on correlation and spin-orbit mixed configurations having three open subshells. Strong and wide (typically 1 eV) Rydberg series of resonances due to 2s to np excitations dominate above the 2p threshold.

Figures

Figures reproduced from arXiv: 2501.18497 by the authors.

Figure 1
Figure 1. FIG. 1: Separation of Configuration Space in the [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Comparison of the DARC results from model A [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Absolute photoionisation cross sections of [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (6 more)
Figure 5
Figure 5. Figure 5: FIG. 5: Absolute photoionisation cross sections of [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 4
Figure 4. Figure 4: FIG. 4: Total photoionisation cross sections of singly [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Photoionisation cross sections of doubly-ionised [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Absolute photoionisation cross sections of triply [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Total photoionisation cross sections of triply [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Total experimental photoionisation cross [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]

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