{"id":"da89f1b4-9066-489a-a547-202c0d811dc7","arxiv_id":"2506.21004","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A scattering-matrix 4D STEM reconstruction pipeline, extended to handle spatial incoherence, unknown defocus, and dark-field signal, recovers the SrTiO3 projected potential from experimental data.","lead":"This paper extends scattering-matrix reconstruction of atomic-scale potentials from 4D scanning transmission electron microscopy to experimental conditions, adding corrections for partial spatial coherence, unknown defocus, and dark-field electrons. The authors demonstrate recovery of the SrTiO3 potential from measured data, with results consistent with an independent ptychographic reconstruction.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Experimental success hinges on an unverified Lorentzian effective source (HWHM 0.3 Å) chosen by grid search after Gaussian models failed, while Fig.","rationale":"The reader and I converge on the same weak point: the central experimental claim depends on an unverified, grid-selected Lorentzian effective source model. The analytic developments, particularly the t+2Δf constraint (Eqs. 9-14), are clean and parameter-free given their assumptions, and the simulation studies are extensive and honest about failure modes, including the thermal-scattering model comparison (Fig. 2) and the dose/coherence studies (Figs. 3-5). The ptychographic reconstruction in Fig. 8(b) is genuinely supportive, though not fully independent because it is initialised with the same thickness and defocus values and uses the same pre-processed data. The decisive weakness is that the Lorentzian HWHM 0.3 Å source is selected after Gaussian models failed, using criteria that include agreement of the reconstructed potential with the expected structure; the paper's own Fig. 3 shows the algorithm is fragile to source underestimation and to larger source widths. This is an experimental-validation gap rather than an internal inconsistency, so it does not warrant rejection; it does warrant keeping the verdict conditional until the source model is independently calibrated or an uncertainty analysis is provided. If the proposed source-calibration check passes, the paper would comfortably support its central claim.","tokens_in":23232,"tokens_out":4586,"duration_ms":55083,"concrete_test":"Independently calibrate the effective source distribution for the identical optical configuration (e.g., from a vacuum or amorphous-region defocus series or Ronchigram, or from a known standard) and rerun the reconstruction pipeline of Figs. 7-8 with that measured source in place of the grid-selected Lorentzian; compare the resulting low-order Fourier coefficients and real-space potential against Fig. 8(a). If they shift by more than the estimated noise level, the source assumption is load-bearing and the central quantitative claim needs revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The experimental reconstruction is only as strong as the assumed effective source model, and that model is not independently measured. After Gaussian effective source models failed, the authors adopted a purely Lorentzian source and selected HWHM 0.3 Å by grid search over reconstructions of the same experimental data (Experimental reconstruction, Fig. 7), using qualitative appearance of the potential and Pearson correlation with the 4D STEM data as selection criteria. The paper's own simulations show the method fails when the source HWHM is underestimated by 50% (Fig. 3a,d) and fails for a 0.6 Å HWHM source even when known exactly (Fig. 3e). The experimental success therefore requires both the Lorentzian functional form and the 0.3 Å width to be accurate enough, despite the authors noting that Gaussian models failed and that model mismatch could be absorbed by the reconstruction. No uncertainty estimates accompany the final reconstruction (Fig. 8a), and the absorptive potential is acknowledged to disagree with simulation, underscoring that model mismatch can be present without invalidating the visual fit. If the true effective source is wider or non-Lorentzian, the reconstructed low-order Fourier coefficients could be biased while still matching the data.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript presents three algorithmic modifications to the scattering-matrix (S-matrix) approach for retrieving the projected electrostatic potential from 4D STEM data: a coherence-aware phase retrieval update (Eq. 7), a symmetry-based constraint that estimates the combination t+2Δf (Eqs. 9–14), and inclusion of dark-field detector regions. The methods are tested on simulated data with noise, spatial incoherence, and systematic thickness/defocus grids, then applied to experimental 4D STEM data from monolithic SrTiO3, yielding a reconstruction that the authors describe as in excellent quantitative agreement with the expected structure. The experimental reconstruction is compared with mixed-state inverse multislice ptychography to provide an external consistency check.","tokens_in":23417,"tokens_out":4730,"duration_ms":54239,"significance":"If the experimental result is robust, the paper delivers the first demonstration that a single-defocus 4D STEM dataset from a partially coherent instrument can be inverted to a projected potential under multiple scattering via the S-matrix. The t+2Δf derivation is clean and parameter-free, and the simulation study is unusually thorough, including 100 noise realisations, a dose ladder, and systematic thickness/defocus grids. The significance is tempered by the fact that the experimental demonstration rests on an assumed Lorentzian effective source whose width is chosen by grid search on the same data; the paper's own simulations show strong sensitivity to this parameter, so the central experimental claim is not yet established as robust.","major_comments":[{"comment":"The central experimental claim rests on the assumption of a purely Lorentzian effective source with HWHM 0.3 Å, selected by grid search over reconstructions of the same experimental data (Fig. 7). The paper's own simulations (Fig. 3a,d) show that underestimating the HWHM by 50% causes the reconstruction to fail, and Fig. 3e shows failure for a 0.6 Å HWHM source even when known exactly. Because no independent measurement of the effective source is provided, and because Gaussian source models failed (requiring the Lorentzian form), the experimental success is not yet established as robust. I request either an independent source characterization, a demonstration that the final potential is stable under plausible variation of the source model and width, or a quantitative statement of the resulting uncertainty in the reconstructed potential.","section":"Experimental reconstruction"},{"comment":"The final parameter selection uses qualitative appearance of the potential and Pearson correlation between experimental and simulated 4D STEM intensities (Fig. 7). The manuscript itself notes (Fig. 3 and accompanying text) that extending the apodisation range improves Pearson correlation even when the reconstruction is degrading due to overfitting to noise. Thus Pearson correlation alone is not a reliable accuracy metric. The authors should report how the final potential changes for neighboring grid points, particularly the parameter combinations that achieved similar Pearson correlations in Fig. 7b, and provide a quantitative accuracy metric against the known SrTiO3 structure.","section":"Experimental reconstruction"},{"comment":"The abstract and conclusion claim 'excellent quantitative agreement' with the expected structure, but the only quantitative evidence is the Fourier-coefficient visualisation in Fig. 8(a), which is not accompanied by numerical values or error bars. The reconstructed absorptive potential is acknowledged to disagree with simulation, which shows that model mismatch can be present without invalidating the visual fit. To support the quantitative claim, the authors should provide, for example, a table of reconstructed versus expected low-order structure factors with uncertainties.","section":"Experimental reconstruction"},{"comment":"The trial-and-error failure of Gaussian effective source models and success of Lorentzian models is surprising and unexplained; the physical effective source is more typically Gaussian- or Voigt-like. The Lorentzian may be absorbing other model errors, such as residual aberrations, scan noise, or sample mistilt. Please discuss or test whether the Lorentzian width is compensating for other parameter errors, for example by repeating the fine grid search with a Voigt profile or with a source model plus residual aberration parameters.","section":"Spatial incoherence"}],"minor_comments":[{"comment":"There is a typo in the Experimental reconstruction section: 'descibed' should be 'described'.","section":"Experimental reconstruction"},{"comment":"The caption says 'The starting guess for sample thickness is set via the antidiagonal symmetry constraint from the initial S reconstruction and the assumed defocus'; this should be made explicit, since the antidiagonal constraint only gives t+2Δf, so the thickness is obtained as t = (t+2Δf) − 2Δf.","section":"Fig. 7 caption"},{"comment":"The symbol Λ(R) for the effective source distribution is visually very close to λ for wavelength; please use a distinct notation or font to avoid confusion.","section":"Introduction and methods"},{"comment":"The Pearson correlation values (e.g., 0.94 in Fig. 7b) are cited without stating the number of data points or the reciprocal-space region used; please define the metric precisely.","section":"Experimental reconstruction"},{"comment":"The ptychographic reconstruction is initialised with the same defocus and thickness as the S-matrix reconstruction; because these parameters are not independently determined, the comparison should be described explicitly as a consistency check rather than a validation of the parameters.","section":"Fig. 8(b)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's algorithmic derivation is sound and the simulation study is extensive, but the experimental demonstration is the weak link. The paper's own simulations show that the reconstruction fails when the effective source is underestimated or too broad, yet the experimental source model is chosen by grid search on the same data without independent verification. If the authors can provide an independent measurement of the effective source or a stability analysis across source models and widths, the paper would be acceptable; without that, the central experimental claim remains conditional. I recommend major revision rather than rejection because the issue is fixable within the manuscript's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline: this paper delivers the first experimental scattering-matrix reconstruction from a single-defocus 4D STEM dataset, and it does so with an honest, well-tested pipeline. The new analytic result, the t+2Δf coupling in Eq. (14), is a genuine extension of Findlay et al. (2021) and is correctly derived. The simulations are thorough: 100 noise realisations, a dose ladder, systematic thickness/defocus sweeps, and clear reporting of when the method fails. The dark-field inclusion via iterative re-seeding is a practical, sensible fix. Credit is also due for testing against an independent mixed-state ptychographic reconstruction on the same data; the two agree and match the expected SrTiO3 potential.\n\nThe soft spot is the effective source model. The authors tried Gaussian sources, failed, switched to Lorentzian, and picked HWHM 0.3 Å by grid search on the same experimental data. Their own simulations show the method breaks when the source is underestimated by 50% and fails outright for a 0.6 Å HWHM source even when known exactly. So the experimental success really does hinge on the Lorentzian form and that width being close to correct. That is not a fatal flaw—the Pearson correlation and PACBED consistency give some support—but it is a real limitation that should be stated more prominently. The paper also reports the final reconstruction without uncertainty estimates, and calling it 'excellent quantitative agreement' outruns the visual evidence, especially since the reconstructed absorptive potential disagrees with simulation.\n\nNone of this undermines the central contribution. The t+2Δf constraint is parameter-free and stands on its own. The experimental demonstration is credible and, with the ptychography comparison, gives independent confirmation. The main fixes are easy: add error bars or at least a sensitivity analysis, release the data and reconstruction parameters, and soften the quantitative claim to match what is actually shown.\n\nThis paper is for people who work on quantitative 4D STEM structure retrieval of periodic samples. It deserves a serious referee: the topic is important, the new constraint is solid, and the experimental result, despite the source-model caveat, is the first of its kind. I would send it to review, with a request for those revisions.","headline":"First experimental S-matrix 4D-STEM structure retrieval, with a clean new t+2Δf constraint and honest simulations; the main caveat is an unverified Lorentzian effective source chosen by grid search.","tokens_in":24065,"tokens_out":2261,"would_cite":true,"duration_ms":23388,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A single 4D-STEM dataset can now reveal a crystal's structure, even with thick samples and multiple scattering.","keywords":["4D STEM","scattering matrix","structure determination","phase retrieval","partial spatial coherence","multiple scattering","projected potential","SrTiO3"],"falsifier":"Re-run the same single-defocus reconstruction pipeline on a well-characterized periodic specimen while independently measuring the effective source distribution (for example, by fitting the probe intensity in vacuum or the source size from a known structure), or by acquiring data with a deliberately measured wider source (HWHM ≥ 0.6 Å). If the reconstructed Fourier coefficients of the potential deviate substantially from the known structure, or if an independently measured non-Lorentzian source of comparable width fails to reproduce the reported reconstruction, the claim that the method is robust to partial spatial coherence on experimental data would be falsified.","tokens_in":22982,"feed_emoji":"🔬","tokens_out":3447,"duration_ms":29650,"temperature":0.7,"pith_summary":"This paper develops the scattering matrix method for quantitative structure determination from four-dimensional scanning transmission electron microscopy (4D STEM) data, targeting samples thick enough that multiple scattering cannot be neglected. Through simulation and experiment, it shows that three modifications — coherence-aware phase retrieval, a symmetry constraint linking thickness to defocus, and inclusion of dark-field data — make the reconstruction work on real experimental data. The paper's central experimental claim is that a single-defocus 4D STEM dataset from an imperfectly coherent microscope suffices to reconstruct the projected electrostatic potential of a periodic crystal with good quantitative agreement. If true, this would simplify a class of atomic-resolution structure determinations that previously relied on through-focal series or more complex experimental setups.","feed_headline":"One defocus, one dataset: crystal structure from 4D STEM","feed_subtitle":"A single 4D-STEM dataset now reconstructs a crystal's potential even with multiple scattering and partial coherence.","key_machinery":"The scattering matrix S = exp(iπtA/K), the quantum mechanical operator relating incident and exit plane-wave amplitudes through a crystalline sample, together with the structure matrix A whose Fourier coefficients U_g encode the projected electrostatic potential. The reconstruction is a two-step inversion: phase retrieval reconstructs S from 4D STEM intensities, then nonlinear conjugate gradient descent refines A, thickness, defocus, phase factors, and (in the experimental case) mistilt. Three named mechanisms carry the new developments: the Clark–Peele-style amplitude update (a square-root intensity-ratio update that incorporates the effective source convolution into the phase retrieval step); the antidiagonal symmetry constraint S_{h,g} = S_{-g,-h}, whose enforcement yields the thickness–defocus estimate t + 2Δf; and the iteration loop that re-seeds phase retrieval from a simulated S, which supplies phase relations that let dark-field rows enter the cost function.","core_discovery":"The paper demonstrates experimental reconstruction of the projected electrostatic potential of a monolithic SrTiO3 crystal from a single-defocus 4D STEM dataset, via the scattering matrix. The reconstruction pipeline first retrieves the scattering matrix S from measured intensities using phase retrieval, then solves for the structure matrix A (whose off-diagonal elements are the Fourier coefficients of the projected potential) by gradient descent. Three developments make this work on imperfect experimental data: (1) a modified phase retrieval amplitude update that accounts for partial spatial coherence as an effective source convolution; (2) use of the antidiagonal symmetry of S to estimate the linear combination t + 2Δf of thickness and defocus when defocus is unknown; and (3) inclusion of dark-field diffraction data after the first iteration re-seeds phase relations in the dark-field rows. The paper claims that combining these advances yields a reconstruction with the Sr, Ti, and weakly-scattering O columns visible at their expected coordinates and Fourier coefficients of the potential in quantitative agreement with the expected structure out to beyond the bright-field disk.","pith_inferences":["The same coherence-aware update and dark-field inclusion strategy could be transplanted into multislice ptychography pipelines, since the amplitude update of Eq. (7) is not specific to the scattering-matrix basis.","The demonstrated sensitivity to effective source shape suggests that independent, structure-free measurements of the effective source (e.g., from vacuum or amorphous regions of the same dataset) would remove the largest empirical uncertainty of the method and could be tested before applying the reconstruction to an unknown structure.","Because the method recovers only the projected potential (with periodicity built in), it is a natural building block for a phased workflow: scattering-matrix reconstruction for a periodic region of interest, followed by inverse multislice ptychography with the reconstructed potential as an initialization for non-periodic parts of the field of view.","The reported inability to reconstruct with a 0.6 Å HWHM source at single defocus even when known exactly suggests an approximate coherence cutoff — beyond which no amount of parameter tuning will recover high-order Fourier coefficients — that could be quantified as a practical specification for this technique."],"forward_implications":["A single-defocus 4D STEM dataset, averaged over a unit cell, can serve as the sole input for quantitative projected-potential reconstruction of a periodic sample, removing the need for through-focal series data with their associated dose and alignment burdens.","The t + 2Δf constraint from antidiagonal symmetry gives an experimental route to jointly bracket thickness and defocus before the full structure optimization, shrinking the parameter search space for unknown samples.","Including dark-field intensities extends the reliable range of reconstructed Fourier coefficients of the potential, which should help at small probe-forming apertures and for oxygen columns or other weak scatterers.","The scattering-matrix formulation provides a sparse, periodic, thickness-independent forward calculation, so for periodic samples it offers a computationally distinct alternative to inverse multislice ptychography, with which the paper demonstrates comparable experimental reconstruction quality.","When the effective source width is uncertain, overestimating it yields reliable low-order Fourier coefficients whereas underestimating it fails, so safe practice for this method is to err toward larger assumed source widths."],"supporting_citations":[{"why":"Established the scattering-matrix-based phase retrieval and antidiagonal symmetry thickness estimate at a single defocus, which this paper extends and experimentally validates.","marker":"Findlay et al. (2021)"},{"why":"Introduced the optimization-based A-matrix reconstruction that this paper adopts and extends with dark-field information and effective-source handling.","marker":"Sadri and Findlay (2023)"},{"why":"Earlier experimental scattering-matrix structure retrieval that required through-focal series and used only bright-field data; the baseline this paper's single-defocus, dark-field-inclusive method must match or beat.","marker":"Brown et al. (2018)"},{"why":"Supplied the coherence-aware amplitude update used in the modified phase retrieval step.","marker":"Clark and Peele (2011)"},{"why":"Derived the inversion and antidiagonal symmetry properties of the scattering matrix on which the potential reconstruction and thickness estimate rest.","marker":"Allen et al. (1999, 2000)"},{"why":"Demonstrated that thermal scattering model mismatch can bias ptychographic structure retrieval; this paper checks the analogous effect for the scattering-matrix approach.","marker":"Diederichs et al. (2024)"},{"why":"Showed defocus-series phase retrieval of the scattering matrix, providing the multi-defocus comparison against which single-defocus performance is assessed.","marker":"Pelz et al. (2021)"}],"fun_headline_variants":["Scattering matrix recovers crystal potential from single 4D STEM set","Single-defocus 4D STEM reveals full crystal potential","4D STEM with scattering matrix handles real-world aberrations","Quantitative crystal structure from one 4D STEM dataset","Scattering matrix turns imperfect 4D STEM into clear structure"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The experimental reconstruction presumes that the true partial spatial coherence is faithfully represented by a Lorentzian effective source with a half-width-at-half-maximum near 0.3 Å — a model chosen by trial-and-error grid search after Gaussian models failed.","fun_headline_variants_meta":{"raw":{"variants":["Scattering matrix recovers crystal potential from single 4D STEM set","Single-defocus 4D STEM reveals full crystal potential","4D STEM with scattering matrix handles real-world aberrations","Quantitative crystal structure from one 4D STEM dataset","Scattering matrix turns imperfect 4D STEM into clear structure"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000749,"raw_usage":{"total_tokens":3280,"prompt_tokens":835,"completion_tokens":2445,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":451,"completion_tokens_details":{"reasoning_tokens":2360}},"tokens_in":451,"tokens_out":2445,"duration_ms":18301,"temperature":1.0,"reasoning_tokens":2360,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:36:34.595803+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Re-run the same single-defocus reconstruction pipeline on a well-characterized periodic specimen while independently measuring the effective source distribution (for example, by fitting the probe intensity in vacuum or the source size from a known structure), or by acquiring data with a deliberately measured wider source (HWHM ≥ 0.6 Å). If the reconstructed Fourier coefficients of the potential deviate substantially from the known structure, or if an independently measured non-Lorentzian source of comparable width fails to reproduce the reported reconstruction, the claim that the method is robust to partial spatial coherence on experimental data would be falsified.","supporting_citations":[],"review_version":1}