Revisiting the Question of Information Content of EXAFS Spectra through a Bayesian Approach

Andrei Sapelkin; Diego Gianolio; Lucy Haddad

REVIEW 3 major objections 4 minor 38 references

Revisiting the Question of Information Content of EXAFS Spectra through a Bayesian Approach

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read This paper claims that the Bayes Factor Integral, not the Shannon-Nyquist criterion, gives the true information content of EXAFS spectra, and that for liquid gallium the BFI peaks at 10 free parameters even though the conventional limit all

desk verdict Applies the authors' own prior-sensitive BFI to EXAFS information content; the l-Ga result is plausible, but the prior-range dependence undercuts the 'superior measure' claim. read the letter →

arxiv 2509.07950 v1 pith:3Q54BUEE submitted 2025-09-09 physics.data-an cond-mat.mtrl-sci

Lucy Haddad , Diego Gianolio , Andrei Sapelkin This is my paper · ORCID

classification physics.data-ancond-mat.mtrl-sci

keywords EXAFSBayesianmodelcomparisoninformationcontentShannon-NyquistcriterionBayesFactorIntegralparametercorrelationliquidgalliumfigureofmerit

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

The paper challenges the long-standing Shannon-Nyquist criterion used to set the maximum number of parameters that can be extracted from an EXAFS spectrum. It argues that this criterion assumes independent parameters, which is never true in EXAFS fitting, and introduces a Bayesian measure, the Bayes Factor Integral (BFI), that accounts for parameter correlations and prior information. Testing on a liquid gallium spectrum, the BFI reaches a maximum at 10 parameters while Shannon-Nyquist allows 11, and decisively rejects an 11-parameter model. The paper also shows that the BFI can distinguish between different fitting strategies, such as adding scattering paths versus adding cumulants, and that it is sensitive to data range and data quality. If correct, the BFI gives analysts a principled stopping rule for parameter addition and a tool for model selection.

What carries the argument

The Bayes Factor Integral (BFI), defined as BFI = (2π)^(m/2) Lmax √det(Covp) / ∏Δp_i, is the central object. It is a Laplace-approximated marginal likelihood that combines the maximum likelihood of a fit with an Occam factor built from the parameter covariance matrix and the prior ranges. The covariance matrix injects parameter correlations into the criterion, while the prior ranges encode structural knowledge and penalize models whose parameters are allowed to roam too widely. Comparing ln(BFI) between models yields a Bayes factor with a standard evidence scale, giving a quantitative rule for when an added parameter is justified.

What would settle it

Recompute the BFI for the same liquid-gallium spectrum using numerical (non-Laplace) integration of the marginal likelihood, or repeat the analysis with the prior ranges scaled by a factor of two in either direction; if the ln(BFI) maximum moves away from 10 parameters or the favored three-path model changes, the claimed information-content limit is an artifact of the prior and approximation choices rather than a fixed feature of the spectrum.

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Extended reading notes

Core claim

The central claim is that the information content of an EXAFS spectrum is not fixed by the Shannon-Nyquist count Nind = 2ΔkΔR/π + 2, but should be determined by a Bayesian model-comparison quantity, the Bayes Factor Integral BFI = (2π)^(m/2) Lmax √det(Covp) / ∏Δp_i, where m is the number of fitted parameters, Lmax is the maximum likelihood, Covp is the parameter covariance matrix, and Δp_i are the user-chosen prior ranges on each parameter. Because Covp is not assumed diagonal, the BFI lifts the independence assumption and includes an Occam factor that penalizes models with broader priors or stronger correlations. Applied to the liquid gallium EXAFS spectrum, the BFI-determined parameter lim

Load-bearing premise

The result depends on the user-chosen uniform prior ranges (Δp_i) and on the Laplace approximation of the marginal likelihood computed from a single fit; if those ranges or the approximation are inappropriate, the BFI-determined parameter limit is not a property of the data itself.

Editorial extensions

If this is right

Analysts can stop adding parameters when ln(BFI) peaks, rather than trusting the Shannon-Nyquist count alone.
BFI can rank competing fitting strategies on the same data, such as whether to describe a peak with more scattering paths or with higher-order cumulants.
The criterion responds to data quality: with intentionally poor background subtraction, the BFI maximum shifts toward the Shannon-Nyquist limit and favors different models.
Reducing the k-range lowers the BFI-determined parameter limit, showing that the measure tracks the information actually available in the data.
Because BFI is computed from a single fit's covariance and likelihood, it lends itself to automated testing of many structural models.

Reading between the lines

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

The BFI's dependence on user-chosen prior ranges means the 'information content' it reports is conditional on prior knowledge; two labs using different reasonable priors could obtain different parameter limits from the same spectrum.
Since the paper shows the BFI ranking changes when the ΔR range is tightened, the method's power to distinguish models is partly a statement about how well the analyst knows the structure in advance, not solely about the data.
If applied to other spectroscopies, the BFI might reveal when an extra model component is merely absorbing systematic errors (such as imperfect background removal) rather than capturing real physical signal.
A natural next test is whether the BFI peak converges to a stable parameter count as prior ranges are varied continuously, or whether the peak itself moves—this would separate a property of the data from a property of the prior.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The thing to know: this paper reuses the authors' own Bayes Factor Integral (BFI, from their 2024 Nanoscale paper) as a parameter stop-rule for EXAFS, and on one liquid-gallium spectrum it peaks at 10 parameters when Shannon-Nyquist gives Nind=11. The specific result is plausible and the analysis is transparent. But the general claim that BFI is a superior measure of information content does not hold up, because BFI's value depends directly on user-chosen prior ranges, and the authors changed those ranges after seeing the results.

What is genuinely new: the idea of using a Bayes factor with a covariance matrix to lift the independence assumption in the Shannon-Nyquist criterion is worth taking seriously. The paper is also honest about the mechanics: they list the models, the prior ranges, the noise estimate, and they test sensitivity to k-range and background quality. The finding that a 3-path model beats cumulant-based models for l-Ga is an interesting, physically plausible result. I give them credit for reporting the ΔR adjustment explicitly, even if it undermines the strength of the conclusion.

The soft spots are real. The stress-test note lands: BFI has ∏Δp_i in the denominator, so prior ranges directly rescale every model's BFI. The paper reports that ΔR was initially ±0.25 Å, "later found to be too large," and reduced to ±0.025 Å. With the default ranges, the poor-background case pushes the maximum to Nind=11 (model 4a), not 10. So the 10-parameter limit is a property of the chosen priors, not purely of the data. Second, no uncertainties are reported on any ln(BFI) value; the model ranking is a set of point estimates, and we have no idea whether the 10-vs-11 (or 3b-vs-3c) differences are meaningful. Third, the demonstration is one spectrum with a single FT peak; that is a narrow base for the abstract's "superior measure" claim. Minor: the Bayes factor comparison formula in §II is miswritten as printed, and the 10-vs-11 comparison confounds parameter count with parameter type (path vs cumulant). None of these kill the specific l-Ga result, but they prevent the general claim from being accepted as stated.

Who is this for? EXAFS practitioners who want a Bayesian alternative to Nind will find it useful as a starting point, but they should read it critically. It deserves serious peer review—the question is important and the empirical work is reproducible in principle. I would send it out with a request for a sensitivity analysis over prior ranges, uncertainties on BFI, and ideally a second test case. Not a desk reject.

Best, [You]

Referee Report

3 major / 4 minor

Summary. The manuscript proposes a new figure of merit, the Bayes Factor Integral (BFI, Eq. 5), as a measure of the information content of EXAFS spectra. The authors argue that the traditional Shannon–Nyquist criterion (Eq. 4) assumes independent fitting parameters, which is rarely true in EXAFS, and that BFI lifts this assumption by using the parameter covariance matrix and prior ranges. They test BFI on a liquid-gallium EXAFS spectrum by fitting a ladder of models with 1–4 scattering paths and cumulant terms, under three k-ranges and two background-subtraction conditions. They report that ln(BFI) peaks at 10 parameters, corresponding to a 3-path model, below the Shannon–Nyquist limit Nind = 11, and that the ranking changes when the data range or background quality is altered. The paper concludes that BFI provides a superior measure of EXAFS information content and can guide model selection.

Significance. If substantiated, the BFI would offer EXAFS practitioners a principled, correlation-aware criterion for deciding when to stop adding parameters and for comparing fitting strategies, going beyond the heuristic 'happiness' parameter currently used. The experimental test is well designed in several respects: real l-Ga data, a systematic sequence of 1-, 2-, 3-, and 4-path models, a cumulant-based alternative strategy, variation of the k-range, and a deliberately poor-background control. The authors also provide transparent fit descriptions and full result tables in the supplementary material. However, the central claim that BFI is a superior measure of data information content is not yet supported, because the BFI explicitly depends on user-chosen prior ranges and the reported conclusions change with those ranges.

major comments (3)

[§III, Eq. (5)] The BFI formula contains ∏Δp_i in the denominator, so any change in a user-set prior range rescales every model's BFI and can reorder models with different numbers of parameters. The authors themselves changed the ΔR range from ±0.25 Å to ±0.025 Å after initial fits, stating the initial range was 'later found to be too large' and reduced 'to give a more meaningful comparison of models.' This is consequential: with default ranges and poor background, the maximum moves to Nind = 11 (model 4a), while with restricted ranges the 10-parameter model 3b is preferred (Fig. 5). No sensitivity analysis over ΔR, ΔN, Δσ², or ΔE0 is reported, and no uncertainties on ln(BFI) are given. The claimed '10-parameter limit' is therefore not established as a property of the data; it is a property of the selected priors. The authors should quantify how the ranking changes as the priors are varied within physic
[§III.A, §III.B, Appendix B] The BFI is computed via a Laplace approximation to the marginal likelihood using a covariance matrix from a single Larch fit (Eq. B14–B15). The paper does not validate that the posterior is adequately Gaussian, that the Larch covariance is a reliable estimate of the Hessian, or that the uniform prior is appropriate. This matters because the covariance enters the Occam factor directly. Furthermore, the paper states in §III.A that 'in Larch covariance is no longer calculated once Nvar > Nind (and hence BFI cannot be calculated).' Thus the method cannot even evaluate models above the Shannon–Nyquist limit, so the claim that BFI lifts the independence assumption and provides a new information-content limit is only partial. The authors should test the Laplace approximation against direct numerical marginal-likelihood integration for at least a few models, and discuss whether the Nvar ≤ Nind r
[Abstract; §IV] The claim that BFI is a 'superior measure of the data information content' is not backed by any comparison with existing information criteria. The manuscript presents ln(BFI) curves showing a peak below Nind, but does not compare with AIC, BIC, χ²ν, or the Shannon–Nyquist value on the same fits, nor with any synthetic-data benchmark where the true number of independent parameters is known. A model-selection criterion that favors a 3-path model is not the same as a measure of the information content of the spectrum. The authors should either benchmark BFI against these established FoMs or soften the claim to 'a model-selection criterion that correlates with information content' unless additional evidence is provided.

minor comments (4)

[§II] The definition of the Bayes factor is mistyped: 'ln BF = lnBFI1 / lnBFI2 = ln(BFI1−BFI2)' should read ln BF = ln(BFI1 / BFI2) = ln BFI1 − ln BFI2. The current expression is dimensionally and logically incorrect.
[§III.A] The R-space range used to compute Nind = 11 is not specified in the text. To reproduce Eq. (4), the reader needs ΔR; please state the R range explicitly (e.g., the FT window used for the first-shell peak).
[Fig. 2] The caption refers to 'the legend indicating choice of colour for the bars,' but no legend is visible in the figure as presented. Please add the legend or describe the color coding in the caption.
[Appendix B] Equation (B7) writes the prior as P({Aα}|D,I), which should presumably be P({Aα}|M,I) or the prior density for the parameters; the notation is confusing as written.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: BFI is a self-contained Bayesian model evidence, and the 10-parameter peak for l-Ga is robust to the reported prior-range change.

full rationale

The claimed derivation chain is not circular. BFI (Eq. 5) is derived in Appendix B from the standard Laplace approximation to the marginal likelihood (Gregory [20]; Knuth [34]) with a uniform prior, giving P(D|M_i,I) ≈ Ω_m Lmax. The non-diagonal covariance enters naturally from the information matrix; the citation to the authors' ref. 14 is for the name and prior use, not for an unproved ansatz. The central empirical claim — that the BFI peaks at 10 parameters for l-Ga — is a data-dependent outcome: Lmax and det(Covp) come from Larch fits, and the number of parameters is varied across models. The reported adjustment of ΔR from ±0.25 Å to ±0.025 Å is a prior-sensitivity issue, and it is disclosed; importantly, for the good-background full-range data the maximum remains at 10 parameters under both ranges (model 3c vs 3b), so the main 'information content' number is not manufactured by the prior change. The poor-background test shows a range-dependent shift, which the paper interprets as sensitivity to data quality/priors rather than as a hidden fit. The 'superior measure' claim is arguably under-validated, and the Bayes-factor algebra around ln BF is sloppy, but these are correctness risks, not circularity. No step reduces by construction to its own input.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the BFI being a valid Laplace-approximated marginal likelihood. The priors are uniform distributions over user-set ranges (E0 ±10, ΔR ±0.25/±0.025, ΔN ±3.5, Δσ² ±0.02, C3 ±0.1), so the 'information content' value is sensitive to these choices, as the paper demonstrates. No new physical entities are postulated; the BFI is a statistical score. The noise scale entering Lmax is estimated from the high-k tail and is acknowledged by the authors to be an underestimate.

free parameters (6)

ΔE0 prior range = ±10 eV
Prior range for the absorption edge shift in the BFI Occam factor; set by hand from calibration convention (§III).
ΔR prior range = ±0.25 Å, later ±0.025 Å
Prior range for path length shifts; reduced post hoc because the default allowed paths to duplicate or switch (§III). This range change alters the BFI ranking.
ΔN prior range = ±3.5
Prior range for coordination numbers, set to half of the 7-atom c-Ga first shell (§III).
Δσ² prior range = ±0.02 Å²
Prior range for Debye-Waller factors, taken from a Debye calculation for c-Ga at room temperature (§III).
ΔC3 prior range = ±0.1 Å³
Prior range for third cumulants in the cumulant models (§III).
noise scale εk = 0.00996 Å⁻¹ (approx)
Estimated from the standard deviation of the EXAFS signal in 14.0-15.0 Å⁻¹; the authors state Larch's k-space noise estimate is often an underestimate (Appendix A). Enters Lmax and hence every BFI value.

assumptions (5)

standard math Laplace approximation to the marginal likelihood (Bayesian evidence)
Appendix B, Eq. B14; the BFI is derived under the Laplace approximation, assuming the posterior is well approximated by a Gaussian near the mode.
ad hoc to paper Uniform prior over user-set parameter ranges Δp_i
Eq. B7 and §III; the ranges are chosen by the authors and the results depend on them, including the post-hoc change of ΔR.
domain assumption White-noise model for k- and R-space fluctuations
Appendix A, Eq. A3; used to convert R-space noise to k-space noise and to justify the noise estimate, acknowledged as a lower bound.
domain assumption EXAFS sum-over-paths model (Eq. 1) with FEFF-computed amplitudes and phases
§I; the entire fit and the BFI calculation rely on the standard EXAFS equation and on ab initio scattering factors.
domain assumption Larch parameter covariance matrix approximates the inverse information matrix
§II and Eq. 5; the BFI uses det(Covp) from the fit, and Larch stops computing covariance once Nvar > Nind, which the authors rely on.

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Pith. "Pith review of Revisiting the Question of Information Content of EXAFS Spectra through a Bayesian Approach." pith.science (2026). https://pith.science/paper/3Q54BUEE

@misc{pith2026250907950,
  author       = {Pith},
  title        = {Pith review of: Revisiting the Question of Information Content of EXAFS Spectra through a Bayesian Approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3Q54BUEE}},
  note         = {Machine review of arXiv:2509.07950}
}

read the original abstract

Over the last several decades the Shannon-Nyquist criterion has been widely used as a measure of the maximum information content in EXAFS spectra and provided an upper limit on the number of parameters used in fitting data. However, the criterion implicitly assumes independent parameters which is never the case in EXAFS analysis. Here we introduce a new criterion to measure the information content in EXAFS based on Bayesian approach that lifts the above condition. We test the new criterion by fitting the EXAFS spectrum of liquid gallium and demonstrate that not only it does constitute a superior measure of the data information content, but can also provide guidance in data analysis to differentiate between various fitting strategies.

Figures

Figures reproduced from arXiv: 2509.07950 by the authors.

**Figure 2.** dfigure.caption.9 shows the analysis of the results when two alternative methods to reach the parameter limit are used. The number of parameters can also be increased by using higher moments of the distance distribution, rather than by addition of scattering paths. Here, 2 scattering paths were used and 3rd cumulants were included and fitted into different paths to reach the limit. The ranges set on 3rd cumulant… view at source ↗

**Figure 3.** FIG. 3: ln( [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4: The [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: FIG. 5: ln( [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

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Works this paper leans on

38 extracted references · 38 canonical work pages

[1]

J. J. Rehr and R. C. Albers, Theoretical ap- proaches to x-ray absorption ﬁne structure, Rev. Mod. Phys. 72, 621 (2000)

work page 2000
[2]

A. V. Sapelkin and S. C. Bayliss, Distance dependence of mean-square relative displace- ments in exafs, Phys. Rev. B 65, 172104 (2002)

work page 2002
[3]

Beni and P

G. Beni and P. M. Platzman, Temperature and polarization dependence of extended x- ray absorption ﬁne-structure spectra, Phys. Rev. B 14, 1514 (1976)

work page 1976
[4]

Fornasini, F

P. Fornasini, F. Monti, and A. Sanson, On the cumulant analysis of EXAFS in crystalline solids, Journal of Synchrotron Radiation 8, 1214 (2001)

work page 2001
[5]

Newville, Exafs analysis using feﬀ and feﬃt, Journal of Synchrotron Radiation 8, 10.1107/S0909049500016290 (2000)

M. Newville, Exafs analysis using feﬀ and feﬃt, Journal of Synchrotron Radiation 8, 10.1107/S0909049500016290 (2000)

work page doi:10.1107/s0909049500016290 2000
[6]

F. D. V. J. J. Kas and J. J. Rehr, Interna- tional tables for crystallography volume i: X- ray absorption spectroscopy and related tech- niques (International Union of Crystallogra- phy, 2010) Chap. Chapter 6.8. The FEFF code, pp. 764–769

work page 2010
[7]

P. A. Lee, P. H. Citrin, P. Eisenberger, and B. M. Kincaid, Extended x-ray absorption ﬁne structure—its strengths and limitations as a structural too, Rev. Mod. Phys. 53, 769 (1981)

work page 1981
[8]

Brillouin, Science and Information Theory

L. Brillouin, Science and Information Theory. (Academic Press, New York, 1962) pp. 94–99

work page 1962

Show all 38 references

[9]

E. A. Stern, Number of relevant independent points in x-ray-absorption ﬁne-structure spec- tra, Phys. Rev. B 48, 9825 (1993)

1993
[10]

Newville, Fundamentals of xafs, Reviews in Mineralogy Geochemistry 78, 33 (2014)

M. Newville, Fundamentals of xafs, Reviews in Mineralogy Geochemistry 78, 33 (2014)

2014
[11]

Occam factor

However, we found that the ln( BFI ) has a maxi- mum at 10 parameters, corresponding to model 3c which describes the ﬁrst-shell FT signal as having 3 single scat- tering paths. When 11 parameter are used the ln( BFI ) was found to drop signiﬁcantly (ln(BF )≈ 10) indicating tha...
[12]

Iglesias-Juez, G

A. Iglesias-Juez, G. L. Chiarello, G. S. Pa- tience, and M. O. Guerrero-P´ erez, Experimen- tal methods in chemical engineering: X-ray absorption spectroscopy—xas, xanes, exafs, The Canadian Journal of Chemical Engineer- ing 100, 3 (2022)

2022
[13]

Bunker, Introduction to xafs : A practical guide to x-ray absorption ﬁne structure spec- troscopy (Cambridge University Press, 2010)

G. Bunker, Introduction to xafs : A practical guide to x-ray absorption ﬁne structure spec- troscopy (Cambridge University Press, 2010)

2010
[14]

Gianolio, How to start an xas experiment, in X-Ray Absorption and X-Ray Emission Spectroscopy (John Wiley Sons, Ltd, 2016) Chap

D. Gianolio, How to start an xas experiment, in X-Ray Absorption and X-Ray Emission Spectroscopy (John Wiley Sons, Ltd, 2016) Chap. 5, pp. 99–124

2016
[15]

Haddad, D

L. Haddad, D. Gianolio, D. J. Dunstan, Y. Liu, C. Rankine, and A. Sapelkin, Quanti- fying intuition: Bayesian approach to ﬁgures of merit in exafs analysis of magic size clus- ters, Nanoscale 10.1039/D3NR05110B (2024)

2024 doi
[16]

B. D. Sharma and J. Donohue, A reﬁnement of the crystal structure of gallium, Zeitschrift f¨ ur Kristallographie - Crystalline Materials 117, 293 (1962)

1962
[18]

Newville, Larch: An analysis package for xafs and related spectroscopies, Jour- nal of Physics: Conference Series DOI 10.1088/1742-6596/430/1/012007 (2013)

M. Newville, Larch: An analysis package for xafs and related spectroscopies, Jour- nal of Physics: Conference Series DOI 10.1088/1742-6596/430/1/012007 (2013)

2013 doi
[19]

Ravel and M

B. Ravel and M. Newville, Athena, artemis, hephaestus: data analysis for x-ray absorp- tion spectroscopy using ifeﬃt, Journal of Syn- chrotron Radiation 12, 537 (2005)

2005
[20]

Ravel, 7.2

B. Ravel, 7.2. the heuristic happiness pa- rameter (2016), accessed December 12, 2022. https://bruceravel.github.io/demeter/ documents/Artemis/fit/happiness.html

2016
[21]

Gregory, Bayesian logical data analysis for the physical sciences (Cambridge University Press, 2005)

P. Gregory, Bayesian logical data analysis for the physical sciences (Cambridge University Press, 2005)

2005
[22]

D. J. Dunstan, J. Crowne, and A. J. Drew, Easy computation of the bayes factor to fully quantify occam’s razor, https://arxiv.org/abs/2007.09702 10.48550/ARXIV.2007.09702 (2020)

2007 doi arXiv
[23]

R. A. E. Kass R. E., Bayes factors, Journal of the American Statistical Association 90, 773 (1995)

1995
[24]

B. I. B. M. Newville and D. E. Sayers, Esti- mation of measurement uncertainties in xafs data, Journal of Synchrotron Radiation 6, 264 (1999)

1999
[25]

Newville, (), estimate for k-space noise in Larch src

M. Newville, (), estimate for k-space noise in Larch src
[26]

S. G. M. Roy and G. van Dorssen, The am- plitude reduction factor in exafs, J. Phys. IV France 7, 10.1051/jp4/1997146 (1997)

1997 doi
[27]

S. D. Kelly and B. Ravel, Exafs energy shift and structural parameters, AIP Conference Proceeding 882, 10.1063/1.2644451

doi
[28]

Sevillano, H

E. Sevillano, H. Meuth, and J. J. Rehr, Ex- tended x-ray absorption ﬁne structure debye- waller factors. i. monatomic crystals, Phys. Rev. B 20, 4908 (1979)

1979
[29]

Fornasini and R

P. Fornasini and R. Grisenti, On exafs debye- waller factor and recent advances, Journal of Synchrotron Radiation 22, 1242 (2015)

2015
[30]

R. D. S. Wang, X. et al., Thermal and electri- cal properties of liquid metal gallium during phase transition, Trans. Tianjin Univ. 29, 209 (2023)

2023
[31]

Lambie, K

S. Lambie, K. G. Steenbergen, and N. Gas- ton, Resolving decades of debate: the surpris- ing role of high-temperature covalency in the structure of liquid gallium, Mater. Horiz. 11, 4201 (2024)

2024
[32]

X. G. Gong, G. L. Chiarotti, M. Parrinello, and E. Tosatti, Coexistence of monatomic and diatomic molecular ﬂuid character in liquid gallium, Europhysics Letters 21, 469 (1993)

1993
[33]

E. O. Brigham, The fast Fourier transform (Englewood Cliﬀs, N.J., Prentice-Hall, 1974)

1974
[34]

Ravel, Advanced topics in exafs analysis, 12 Diamond Light Source (2011)

B. Ravel, Advanced topics in exafs analysis, 12 Diamond Light Source (2011)

2011
[35]

K. H. Knuth, M. Habeck, N. K. Malakar, A. M. Mubeen, and B. Placek, Bayesian evi- dence and model selection, Digital Signal Pro- cessing 47, 50 (2015)

2015
[36]

L. B. William, Introduction to probability and statistics (New York, Macmillan, 1966)

1966
[37]

Newville, International tables for crystal- lography vol

M. Newville, International tables for crystal- lography vol. i (International Union of Crys- tallography, 2024) Chap. Chapter 5.13. Non- linear least-squares ﬁtting

2024
[38]

Newville, Ifeﬃt web page and on-line docu- mentation

M. Newville, Ifeﬃt web page and on-line docu- mentation. (), https://cars.uchicago.edu/ ~newville/feffit/feffit.pdf
[39]

B. L. P. P. Niu, H. et al., Ab initio phase dia- gram and nucleation of gallium, Nat Commun 11, 2654 (2020)

2020

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

J. J. Rehr and R. C. Albers, Theoretical ap- proaches to x-ray absorption ﬁne structure, Rev. Mod. Phys. 72, 621 (2000)

work page 2000

[2] [2]

A. V. Sapelkin and S. C. Bayliss, Distance dependence of mean-square relative displace- ments in exafs, Phys. Rev. B 65, 172104 (2002)

work page 2002

[3] [3]

Beni and P

G. Beni and P. M. Platzman, Temperature and polarization dependence of extended x- ray absorption ﬁne-structure spectra, Phys. Rev. B 14, 1514 (1976)

work page 1976

[4] [4]

Fornasini, F

P. Fornasini, F. Monti, and A. Sanson, On the cumulant analysis of EXAFS in crystalline solids, Journal of Synchrotron Radiation 8, 1214 (2001)

work page 2001

[5] [5]

Newville, Exafs analysis using feﬀ and feﬃt, Journal of Synchrotron Radiation 8, 10.1107/S0909049500016290 (2000)

M. Newville, Exafs analysis using feﬀ and feﬃt, Journal of Synchrotron Radiation 8, 10.1107/S0909049500016290 (2000)

work page doi:10.1107/s0909049500016290 2000

[6] [6]

F. D. V. J. J. Kas and J. J. Rehr, Interna- tional tables for crystallography volume i: X- ray absorption spectroscopy and related tech- niques (International Union of Crystallogra- phy, 2010) Chap. Chapter 6.8. The FEFF code, pp. 764–769

work page 2010

[7] [7]

P. A. Lee, P. H. Citrin, P. Eisenberger, and B. M. Kincaid, Extended x-ray absorption ﬁne structure—its strengths and limitations as a structural too, Rev. Mod. Phys. 53, 769 (1981)

work page 1981

[8] [8]

Brillouin, Science and Information Theory

L. Brillouin, Science and Information Theory. (Academic Press, New York, 1962) pp. 94–99

work page 1962