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International
Tables for Crystallography Volume I X-ray absorption spectroscopy and related techniques Edited by C. T. Chantler, F. Boscherini and B. Bunker © International Union of Crystallography 2024 |
International Tables for Crystallography (2024). Vol. I. ch. 7.4, pp. 867-925
https://doi.org/10.1107/S1574870721011010 Chapter 7.4. Tables and supplementary material for X-ray absorption spectroscopy, pre-edge, XANES and XAFSaSchool of Physics, University of Melbourne, Australia This chapter attempts to address two challenges: to present a representative set of reduced XAFS data sets, including illustrative pre-edge and XANES; and to discuss scientific purposes to which each may – or may not – properly be put, as we understand at the current time. Comparisons of standards of quality and portability of data sets for different purposes are therefore core to this. The user can happily extract ideas, applications or data for their purposes from this selection, and one hopes this chapter can thereby inform data-collection and quality-control procedures that more readily allow a particular experiment to address a particular scientific purpose. The illustrations herein are arranged mainly in time order. The illustrations and discussion included in tables and the online supporting information can be used as guides towards templates for XAFS, XANES and pre-edge data standards in CIF, eFEFFit or iFEFFit formats. Keywords: XAFS data; XANES data; data quality. |
X-ray diffraction and crystallography in general have for a long time defined standards for reporting crystal and diffraction data, allowing these to be checked automatically according to important criteria to judge the suitability, accuracy and self-consistency of the data with the scientific result that was sought and is claimed in a publication. X-ray absorption spectroscopy (XAS) and X-ray fluorescence spectroscopy (XFS) or X-ray emission spectroscopy (XES) have been developing in this respect, and it is the intention of this chapter to discuss and illustrate some of the opportunities, needs and requirements to develop this further, especially for X-ray absorption fine structure (XAFS) studies, X-ray absorption near-edge structure (XANES) studies and pre-edge studies.
The illustrations herein are clearly only a selection, which one hopes will nonetheless be useful and lead to further development and discussion. Since there are several thousand XAFS and XANES publications a year, there is no possibility that this can or should be exhaustive. Many publications do not provide reduced data in a form that is portable for other researchers to use, and in addition the dominant modes of presentation are χ(k) versus k plots or transformed χ(r) versus r plots, thus representing the data after extensive reduction and transformation, and without presentation of the uncertainties. For most purposes, these studies have made many assumptions regarding the science which could and should be questioned, including uncertainties, grid spacing, background subtraction, normalization, interpolation, definition of edge energy and spline fitting, among others. Less common are publications with plots of scaled or normalized μ versus E values or [μ/ρ] versus E values. For many purposes there remains a critical need for tabulations of pre-processed data which can be ported to and from a database, preferably with defined uncertainties. The present chapter might hopefully encourage further work towards this goal.
2. Pre-synchrotron and early synchrotron studies. Purpose – XAFS: developing ideas of spherical wavefronts and phase offsets, presenting processed figures and illustrations
XAFS began with local sources (Fricke, 1920![[link]](/graphics/greenarr.gif)
; Kievit & Lindsay, 1930; Lindsay, 1931; Coster & Veldkamp, 1931; Lytle, 1965, 1966; Lytle et al., 1975), but it was soon understood that synchrotron sources would dominate for the acquisition of high-flux, high-precision data sets (Lapeyre et al., 1983; Doniach et al., 1997; Lynch, 1997). Indirectly, this focus led to the development of some standard practices at synchrotron beamlines for sample preparation and data collection (Newville, 2004; Bunker, 2010). In early publications there are few tables. Papers report plots, usually without supplementary material, e.g. Chantler et al. (2001c). There tended to be individual, personal collections of spectra of mixed statistical quality and characterization, including the Farrel Lytle database (http://ixs.iit.edu/database/data/Farrel_Lytle_data
) and spectral profiles of Wong (1999) used as standards by numerous beamlines, but without absolute calibration. [Note that the results of Wong (1999) have a very fine grid giving the local structure with greater detail than most published results. These data, after scaling to give absolute results, are excellent for testing reproducibility of structure in XAFS.] Following the pioneers and founders of the modern XAFS data collection and analysis techniques, different beamlines developed raw or fully processed reference data sets, usually for their internal processes and usually not suitable for cross-portability. This early work and history was discussed recently by Chantler et al. (2018).
The International Union of Crystallography (IUCr) and the extended XAFS community began to develop a series of recommendations of methodology, in separate works and meetings and as part of the IUCr Attenuation Project. This produced a number of excellent works by Barnea, Creagh and others (Mika et al., 1985; Creagh & Hubbell, 1987, 1990), particularly focused on attenuation on the one hand, and on processed data on the other hand, as synchrotron science developed.
Some work, translated from high-accuracy fields, focused on the determination of absolute coefficients of attenuation or photoelectric absorption for ideal systems, with the early development of the X-ray extended range technique (XERT), such as shown in Table 2 (Gerward et al., 1989, 1981; Chantler et al., 2001a).
A particular purpose of these studies was to compare these data with recent theoretical tabulations of mass attenuation coefficients, mass absorption coefficients and atomic form factors, including comparisons with data from XCOM (Scofield, 1973; Berger & Hubbell, 1987; Gerward et al., 2004), FFAST (Chantler, 1995, 2000; Chantler et al., 2000), International Tables for Crystallography Volume C (Creagh, 1999) and other tabulations (Hubbell & Øverbø, 1979; Schaupp et al., 1983). The data were also compared with experimental data and mixed experimental–theoretical tabulations and databases of elemental attenuation coefficients (Hubbell et al., 1980; Hubbell, 1994, 1996; Perkins et al., 1991; Henke et al., 1993; Cullen et al., 1997) and to individual measurements (Wang et al., 1992; Sandiago et al., 1997; Stanglmeier et al., 1992). This effort culminated in achieving accuracies of attenuation coefficients to 0.27% with reproducibility (precision) to 0.02%. For many materials, and even for reference materials, a typical best level of accuracy is no better than 1–15% even now (2024). This led very clearly to a call for a round-robin project to investigate data quality, reproducibility, cross-portability, optimized experimental methodologies and analytic approaches (Chantler et al., 2018).
Comparison was made of the X-ray absorption fine structure, at least in the XANES region, with data (Wong, 1999; Aberdam et al., 1980) and theory (Joly et al., 1999; Joly, 2001). A key issue during this period was the selection of an optimal thickness of material or ideal foil (Chantler et al., 2001a) and the optimization of this remains ongoing. A good statistical accuracy of the data can be obtained over a wide range of thicknesses (![[0.5\leq\left[{{\mu} / {\rho}}\right]\left[\rho t\right]\leq 6]](/teximages/bz5029/bz5029fi1.svg)
) with similar data-collection times, well beyond the Nordfors attenuation criterion (![[2\leq\left[{{\mu} / {\rho}}\right]\left[\rho t\right]\leq 4]](/teximages/bz5029/bz5029fi2.svg)
) (Nordfors, 1960; Creagh & Hubbell, 1987; Chantler et al., 2001a). However, it is well known that dominant systematic errors arise either for thin or for thick samples. The nature of different systematic effects and errors and the magnitudes of corrections are a key theme of this volume of International Tables for Crystallography. An early conclusion was that multiple samples across a range of thicknesses will best achieve an accurate result in the face of common systematic errors.
For more recent work, see the remainder of this chaper and other chapters in this volume. One should also note the current joint efforts of the round-robin collaboration of the IUCr Commission on XAFS and the International XAFS Society (Chantler et al., 2018), the compilation efforts of the Japanese XAFS Society (Asakura et al., 2018) and the IXAS Lytle compendium (http://ixs.iit.edu/database/
), restricted or open local synchrotron databases (http://cars.uchicago.edu/xaslib/search
, https://sp8dr.spring8.or.jp/portal/dspace
, https://www.esrf.eu/home/UsersAndScience/Experiments/XNP/ID21/php.html
), an ongoing effort to aggregate experimental spectra within the XAS part of the Materials Genome Initiative (Strange & Feiters, 2008; Chantler et al., 2019), and the recent development of immense populations of theoretical spectra that are being created, (usually) catalogued and made public, typically with the goal of use in machine-learning applications (Timoshenko et al., 2017, 2018; Mathew et al., 2018; Zheng et al., 2018; Guda et al., 2019, 2020; Martini et al., 2020).
All the data sets discussed herein are appropriate for use as reference standards and for calibration purposes, and all formats are suitable for deposition – indeed all have been deposited in these forms. However, discussions within the international community, the International XAFS Society, the IUCr Commission on XAFS and the Q2XAFS meetings, and the associated reports have led to the recommendation of text-based and computer-readable forms, in particular the .dat (eFEFFit or iFEFFit) or .cif formats described in later sections. All of the data sets are available as supplementary files at https://it.iucr.org/I
as specified in each section below. Some of these data sets have been measured following the principles of XERT; some following hybrid methodology [as explained in Chapter 3.14 (Best & Chantler, 2024)].
Some are primarily useful for measurement and determination of the mass attenuation coefficient, the mass absorption coefficient and the imaginary component of the (atomic) form factor. These XAS studies and data sets are also particularly useful for investigating theory and for comparison with form factors in crystallographic and attenuation databases. In general these data sets attempt to quantify an uncertainty in the X-ray energy, an uncertainty in the absolute value of the mass attenuation, a pointwise uncertainty in the mass attenuation coefficient, an uncertainty in the extracted mass absorption coefficient, and hence in the imaginary component of the (atomic) form factor. Near any absorption edge any extracted form factor is just that – solid-state effects typically exceed 1% and hence the accuracy of the form factor in an atomic sense is reduced. Far from an edge and in particular well above the K edge, the form factor has been shown to be accurate as an atomic form factor to within 1%.
A subset of these data sets are suitable or ideal for XAFS investigation, and are indicated as such in Table 1. This selection is illustrative and personal. It includes XAS data sets (i.e. absorption or transmission spectra) and fluorescence XAS data sets (labelled XFS to avoid confusion with the characteristic fluorescence spectra, but note that these both follow second-order Hamiltonian operators and selection rules instead of first order as for absorption/transmission XAS spectra). Most relate to K-edge XAFS, but some cover regions away from any edges, some cover L edges and some cover an edge but with sparse or limited energy steps in the XANES or XAFS region. The data sets cover examples of K-edge spectra, L-edge spectra, metal foils, elemental crystals, binary crystals, dilute solutions and cryostatic measurements.
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Checklist-style columns in Table 1 are given for key aspects of data collection, analysis and possible sources of systematic errors. More details on these are provided below.
Column (A). All detectors have a dark current or signal in the absence of the beam, which is usually energy-dependent and time-dependent during an experiment. Regular dark-current measurements are generally critical for assessing detector linearity and accuracy for all samples, but particularly for thicker foil or solution samples. This requires regular measurement during a long data collection method or if the sample is relatively thick. This is characterized and corrected for in all these data sets (Chantler et al., 2001a).
Column (B). Beamlines always have windows, air paths and an upstream detector, so it is important to regularly measure a `blank' or take measurements with the sample absent to correct for background attenuation, especially in transmission measurements. For solutions one should use a solvent blank or cell to calibrate the sample. All these data sets have been corrected for this.
Column (C). For absorption measurements, what is actually measured is always attenuation with feedback from fluorescence into upstream and downstream detectors, so this should be measured and corrected for to obtain the actual attenuation (and from thence to be able to obtain the absorption coefficient to compare with theory) (Chantler et al., 2001b). Conversely, for fluorescence measurements (XFS) the absorption and self-absorption corrections are often dominant and it is important to make these corrections before, for example, applying a spline to extract χ versus k values (Chantler et al., 2012b; Trevorah et al., 2019).
Column (D). If the data are not to have an arbitrary energy fitting offset they need to be calibrated using reference materials and not just a single edge (Rae et al., 2010c; Tantau et al., 2014). Where the energy or an uncertainty is determined, it has often been the case that a single reference edge defined by an inflection point defines a possible energy or energy offset with unknown error or uncertainty of the slope or the range of XAS. The energy and uncertainty can be defined directly by crystal or powder diffraction, or by multiple inflection points of reference materials; these each have their own accuracies and limitations.
Column (E). In the same manner, if the value of the attenuation or absorption is to be compared with theory or used to define the absorption profile with energy, it is important to characterize the samples both in the beam and off site, especially to measure and quantify distortions of amplitudes as a function of k (de Jonge et al., 2004a; Islam et al., 2010a; Rae et al., 2010b).
Column (F). Harmonics affect the measured attenuation and absorption coefficients, especially in the upstream and downstream ion chambers and for thicker samples. In XFS, harmonics affect the upstream detector, but with suitable choice of regions of interest harmonics should not directly affect the fluorescence detector (Tran et al., 2003a, 2004b; Glover & Chantler, 2009).
Column (G). The synchrotron bandwidth at the sample or upstream and downstream detectors particularly broadens the edge, pre-edge and white line at the beginning of the XANES, so acts somewhat like an augmented hole width, except that it is most significant in a critical region for chemical fingerprinting (de Jonge et al., 2004b; Sier et al., 2020).
Column (H). Sample roughness affects transmission measurements especially as a function of attenuation, so distorts high-k versus low-k oscillations and pre-edge versus above-edge structure (Glover et al., 2009; Ekanayake et al., 2021a).
The form of the material and factors like the temperature dictate the dominant systematic effects to be investigated or corrected for; there is certainly more than one approach to estimating an uncertainty or error or correcting for it. For an ideal solid metal foil, the blank normalization is particularly important for transmission (XAS) measurements; for a dilute liquid or frozen solution, the solvent measurement normalization is particularly critical for transmission (XAS) measurements; and for a fluorescence measurement (XFS), the absorption and self-absorption effects may be dominant. Some uncertainties may inevitably be unmeasured or unaccounted for or unknown. When a data set is deposited in a database or used as the basis of a study, these uncertainties should be noted where they may be significant. Whilst a summary checklist is provided in Table 1, there are potentially several other sources of systematic errors, including monochromator drift, Bragg glitches, thermal diffuse scattering etc., which may be important contributions in particular data sets.
In general, the data sets that are described in Sections 5 to 17 were presented as tables in pdf format in the associated publications, or deposited as text files and readme files (Section 8), or deposited as sets of pdfs of tabulated data of results (Sections 16 and 17). However, from Section 18 onwards (starting around 2018), following the work of Q2XAFS and the joint work of the IUCr Commission on XAFS and the International XAFS Society, it was seen to be important to develop some standard formats of data sets for direct input into XANES and XAFS fitting packages. Section 18 provides minimal examples of these for mu2chi, iFEFFit and eFEFFit. These examples were later developed much further, as described in the last sections of this chapter, to recommended formats for CIF and eFEFFit/iFEFFit.
The publication by Chantler et al. (2001a) is representative of some achievements of this early period (Table 2). The grid was designed for attenuation measurement and accuracy; with 84 points across an 11 keV energy range, the grid is far too coarse for XAFS or to determine any XAFS nanostructure. Nonetheless, it illustrates the need to provide an uncertainty in the energy E, the mass attenuation [μ/ρ] and the extracted form factors f′′. This work investigated ideal copper metal foils, and used data from foils of nominal thicknesses of 5 µm, 10 µm, 15 µm, 20 µm, 30 µm and 100 µm, using at least three foils at each energy whether above or below the edge. Typical accuracies were 0.27% to 0.467%, i.e. well below 1%, which had proven nearly impossible to achieve before. The precision (reproducibility) was 0.024% to 0.58%, the latter typically at the rising slope of the edge, as expected. This enabled a range of studies of systematic errors, statistics at a typical synchrotron beamline, harmonic determination and direct observation of fluorescence in XAS measurements. Despite the sparseness of this XAFS data set, it could be modelled and provided insight for more advanced theory on XAFS (Witte et al., 2006; Bourke & Chantler, 2010a) and photoelectron inelastic mean free paths (Bourke et al., 2007). The data are available in the supporting information to this chapter as file bz5029sup1.pdf.
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![[\left[{{\mu} / {\rho}}\right]]](/teximages/bz5029/bz5029fi59.svg)
(cm![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm rel}} = \sigma_{\mu,{\rm se}}+\mu_{i}]](/teximages/bz5029/bz5029fi60.svg)
. (![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi61.svg)
(%) [Note (![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi62.svg)
(%) [Note (![[\sigma_{f^{\prime\prime}}]](/teximages/bz5029/bz5029fi63.svg)
(Table 3 illustrates the discussion of specific sources of uncertainty and systematic errors and their dominant or typical contribution in different regions of the spectrum, leading towards the detailed tabulation. In principle, this provides a basis for cross-portability between beamlines, samples or data sets. Table 4 illustrates the use of powder diffraction standards to calibrate the monochromated energy at the sample, in the potential presence of variations due to monochromator heat load, hysteresis, drift or detuning.
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![[\left[{{\mu} / {\rho}}\right]_{\rm pe}]](/teximages/bz5029/bz5029fi64.svg)
and ![[\left[{{\mu} / {\rho}}\right]_{\rm pe}]](/teximages/bz5029/bz5029fi65.svg)
and
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||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
![[\sigma\times\sqrt{{\chi^{2}_{\rm r}}}]](/teximages/bz5029/bz5029fi66.svg)
) (Chantler ![[E_{w = \sigma\left({\chi^{2}_{\rm r}} \right)^{1/2}}]](/teximages/bz5029/bz5029fi67.svg)
(keV)![[\sigma\left({\chi^{2}_{r}} \right)^{1/2}]](/teximages/bz5029/bz5029fi68.svg)
(eV)![[E_{w = \sigma\left({\chi^{2}_{\rm r}} \right)^{1/2}}]](/teximages/bz5029/bz5029fi69.svg)
(keV)![[\sigma\left({\chi^{2}_{r}} \right)^{1/2}]](/teximages/bz5029/bz5029fi70.svg)
(eV)![[E_{w = \sigma\left({\chi^{2}_{\rm r}} \right)^{1/2}}]](/teximages/bz5029/bz5029fi71.svg)
(keV)![[\sigma\left({\chi^{2}_{\rm r}} \right)^{1/2}]](/teximages/bz5029/bz5029fi72.svg)
(eV)Another key example was presented in the work by Tran et al. (2003b,c) on silicon crystal samples (Table 5). In the X-ray energy range 5–20 keV there is no silicon edge and no XAFS, so this is both an XAS study and a study of attenuation coefficients and elemental form factors. The data are available in the supporting information to this chapter as file bz5029sup2.pdf. There were 123 measured data points.
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![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi73.svg)
– percentage precision of repeated measurements (one standard error); ![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi74.svg)
: total percentage accuracy in measured [![[\sigma_{f^{\prime\prime}}]](/teximages/bz5029/bz5029fi75.svg)
– absolute uncertainty in ![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi76.svg)
(%)![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi77.svg)
(%)![[\sigma_{f^{\prime\prime}}]](/teximages/bz5029/bz5029fi78.svg)
(These samples are near-perfect crystals, so Bragg diffraction from the sample was a key problem to be corrected for. Because of the low atomic number and the relatively high energies, these standard reference samples, which were also used for the IUCr Attenuation Project, covered the thickness range from to 50 µm to 4 mm, with three foils investigated at each energy.
Table 5 lists the energies and the corresponding uncertainties in columns 1 and 2, respectively. Columns 3, 4, 5 and 6 give the corresponding measured mass attenuation coefficient [μ/ρ], the experimental precision, the accuracy in the thickness determination and the total uncertainty of [μ/ρ], respectively. The last three columns list the imaginary part of the form factor after correction for scattering. Uncertainties in the theoretically calculated components of the scattering factor are indicated by the difference between the two model-dependent estimates of f′′. This latter uncertainty is clearly insignificant in the lower-energy region, and contributes at most 0.05% as one approaches 20 keV. Above 5.6 keV, the experimental values of [μ/ρ] are the weighted mean of the measurements obtained with three thicknesses, excluding those points affected by Bragg diffraction. The final uncertainty in the mass attenuation coefficient σ[μ/ρ] in this range is the root mean square of the contributions from the uncertainty in the thickness calibration σt, and from the consistency of the measurements of using different samples, σse, defined as![[\sigma_{\rm se} = \left({{{\sum_{\rm all}{{([\mu/\rho]_{t_{i}}-\overline{[\mu/ \rho]})^{2}} / {\sigma_{i}^{2}}}} \over {\sum_{\rm all}1/\sigma_{i}^{2}}}} \right)^{1/2},\eqno(1)]](/teximages/bz5029/bz5029fd1.svg)
where ![[[\mu/\rho]_{t_{i}}]](/teximages/bz5029/bz5029fi3.svg)
are the mass attenuation coefficients measured using wafers of different thicknesses ti, ![[\overline{[\mu/\rho]}]](/teximages/bz5029/bz5029fi4.svg)
is the weighted average of ![[[\mu/\rho]_{t_{i}}]](/teximages/bz5029/bz5029fi5.svg)
and σi are the corresponding statistical errors in the measurements of ![[[\mu/\rho]_{t_{i}}]](/teximages/bz5029/bz5029fi6.svg)
. Between 5.0 keV and 5.6 keV the values of [μ/ρ] are corrected for (significant) harmonic contamination. In this energy range, the final uncertainty σ[μ/ρ] is the root mean square of the contributions of σt and of the final error in the procedure of the harmonic correction σhar, calculated from ![[\sigma_{\rm har} = \left({\sigma_{\rm fit}^{2}+\sigma_{\rm stat}^{2}} \right)^{1/2},\eqno(2)]](/teximages/bz5029/bz5029fd2.svg)
where σfit is the fitting error and σstat is the minimum of the statistical errors σi. Mass attenuation coefficients and their uncertainties are not affected by the value of the density. It is for this reason that the mass attenuation coefficient [μ/ρ] rather than the linear attenuation coefficient μ should be used for comparisons of data from different sources.
Table 6 summarizes the major sources of uncertainty contributing to the final results. Major factors affecting the precision or the consistency of the measurements of [μ/ρ] using multiple foils are listed in the first part of Table 6. Apart from the intrinsic statistics – the intrinsic sources of statistical uncertainty of the system at the level of 0.02%, the other main factors affecting the consistency of the measurements in this experiment are Bragg diffraction and the harmonic contamination in the low-energy range of the measurements.
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![[\left[{{\mu} /{\rho}}\right]_{\rm pe}]](/teximages/bz5029/bz5029fi79.svg)
and ![[\left[{{\mu} / {\rho}}\right]_{\rm pe}]](/teximages/bz5029/bz5029fi80.svg)
and Uncertainties from the fitting of the harmonic contamination of the incident beam below 5.6 keV are at the level of 0.3%. In the high-energy range, measurements that were significantly affected by Bragg diffraction (where the measured values of [ln(I/I0)] were more than 0.5% higher than those from the other two specimens) were excluded from the calculations of the final results. The remaining points were consistent to better than 0.44% (the maximum discrepancy was at 18.226 keV, as in Tables 5 and 6).
Uncertainties in the correction of the (very significant) backlash hysteresis of the monochromator amounted to 1.3 eV or less between 5 keV and 6 keV. The effect of hysteresis is thus less than 0.07% in [μ/ρ] in this energy range. Errors in the energy determination of less than 1 eV elsewhere are equivalent to less than 0.01% in [μ/ρ]. These main components of the factors affecting the experimental precision resulted in the final experimental precision listed in the second part of Table 6. This contributed to the total experimental accuracy at levels of 0.3% below 5.6 keV, and up to 0.44% in the higher-energy region.
Uncertainty from the determination of the thicknesses of the specimens increased from 0.06% (at 20 keV) to 0.139% (at 5 keV) due to the additional contribution from the thickness transfer procedure of Tran et al. (2004a). Mika et al. (1985) and Gerward et al. (1981) reported similar accuracies [3 µm (0.075%) and 2 µm (0.05%), respectively] for specimens of similar (4 mm) thickness. However, their results were for the local thickness measured with a micrometer and not for the absolute accuracy of the determination. Baltazar-Rodrigues & Cusatis (2001) reported 0.3 µm accuracy in the thickness determination of their silicon specimens of between 100 µm and 800 µm, but they appear to have used the average thicknesses of the specimens. Both methods (micrometry and average thickness) differ from the local mass per unit area actually seen by the X-ray beam and determined by our technique. The error can be significant (Tran et al., 2004a).
This work led to a critical evaluation of how to account for Bragg/Laue reflections from the sample (Chantler et al., 2010), as well as the nature of the elastic scattering contribution and its impact upon XAS measurement and the determination of the form factor. In particular, the replacement of a Rayleigh scattering coefficient with a prediction of thermal diffuse scattering was investigated. Approaches to absolute coefficient measurement and the calibration of energy using powder diffraction were also investigated.
Tran et al. (2005) measured the X-ray mass attenuation coefficient of silver metal foils. In this case, the K edge was included. There were 146 data points in the data set, which was enough to see XAFS oscillations but only just enough to carry out standard XAFS analysis of the nanostructure. The energy range covered was 15.2–49.9 keV, so only about 60 data points covered a standard XAFS region, with a minimum energy step size of 5 eV. Whilst sufficient for a range of XAS applications, tests of theory and a simple analysis of the nanostructure, it is generally considered that analysis of sources of systematic errors and the many-body physics requires much closer point spacing near the edge.
Three foils were used at every energy, varying from 12 µm to 100 µm and 275 µm to cover an attenuation range of 0.1 < ln(I0/I) < 6.8, well beyond the Nordfors criterion, in order to interrogate sources of systematic errors. Table 7 shows the results. Columns E and σE show the energies (in keV) and the corresponding uncertainties (in eV) at which attenuation measurements were carried out. The columns ![[\left[{{\mu} / {\rho}}\right]_{\rm meas}]](/teximages/bz5029/bz5029fi7.svg)
, ![[\sigma_{[\mu/\rho]_{\rm rel}}]](/teximages/bz5029/bz5029fi8.svg)
and σ[μ/ρ] show the measured mass attenuation coefficient ![[\left[{{\mu} /{\rho}}\right]_{\rm meas}]](/teximages/bz5029/bz5029fi9.svg)
(in cm2 g−1), the weighted deviation and the uncertainties of ![[\left[{{\mu} / {\rho}}\right]_{\rm meas}]](/teximages/bz5029/bz5029fi10.svg)
. The uncertainty in the measured mass attenuation coefficient σ[μ/ρ] is the root mean square of the contributions from the uncertainty in the thickness calibration σt and from the consistency of the measurements obtained with the different foils ![[\sigma_{[\mu/\rho]_{\rm rel}}]](/teximages/bz5029/bz5029fi11.svg)
.
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![[\sigma_{\left[{{\mu}/ {\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi81.svg)
is the percentage precision of repeated measurements (one standard error); ![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi82.svg)
is the total percentage accuracy in the mass attenuation coefficient [![[\left[{{\mu} / {\rho}}\right]_{\rm pe} = \left[{{\mu} / {\rho}}\right]-\left(\left [{{\mu} / {\rho}}\right]_{\rm R}+\left[{{\mu} / {\rho}}\right]_{\rm C}\right)]](/teximages/bz5029/bz5029fi83.svg)
; and ![[\sigma_{f^{\prime\prime}}]](/teximages/bz5029/bz5029fi84.svg)
is the absolute uncertainty in ![[\left[{{\mu} / {\rho}}\right]_{\rm meas}]](/teximages/bz5029/bz5029fi85.svg)
(cm![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi86.svg)
(%)![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi87.svg)
(%)![[\sigma_{f^{\prime\prime}}]](/teximages/bz5029/bz5029fi88.svg)
(The column [μ/ρ] shows the total mass attenuation coefficients obtained by applying appropriate corrections to the ![[\left[{{\mu} / {\rho}}\right]_{\rm meas}]](/teximages/bz5029/bz5029fi12.svg)
values for the effects of fluorescence and scattering. Note that ![[\left[{{\mu} / {\rho}}\right]_{\rm meas}]](/teximages/bz5029/bz5029fi13.svg)
is ill-defined since it corrects for some of the sources of systematic errors but not, for example, scattering and fluorescence. However, it is an improvement compared with some earlier work. Columns f′′ and ![[\sigma_{f^{\prime\prime}}]](/teximages/bz5029/bz5029fi14.svg)
list the imaginary part of the complex atomic form factor, f, and the corresponding absolute uncertainties. f′′ was obtained from the optical theorem: ![[f^{\prime\prime} = {{E\sigma_{\rm pe}} \over {2hcr_{e}}} = {{E uA \left[{{\mu} /{\rho}}\right]_{\rm pe}} \over {2hcr_{e}}},\eqno(3)]](/teximages/bz5029/bz5029fd3.svg)
where E is the energy in eV, σpe is the (photoelectric) mass absorption cross section, h and c are Planck's constant and the speed of light, respectively, re is the classical electron radius, u is the atomic mass unit, A is the relative atomic mass, ma = uA is the (atomic) mass, and ![[\left[{{\mu} /{\rho}}\right]_{\rm pe}]](/teximages/bz5029/bz5029fi15.svg)
is the (photoelectric) mass absorption coefficient obtained by subtracting the total scattering coefficients ![[\left(\left[{{\mu} / {\rho}}\right]_{\rm R}\,+\,\left[{{\mu} / {\rho}}\right]_{\rm C}\right)]](/teximages/bz5029/bz5029fi16.svg)
(following Chantler, 1995, 2000; Chantler et al., 2000) from the mass attenuation coefficient [μ/ρ]. The use of equation (3) in the region of XAFS is clearly affected by non-atomic, i.e. solid-state, effects and hence includes processes other than just those due to the atomic form factor. This also represented one of the first clear observations of the `triangle effect' (Fig. 1): a discrepancy of absorption versus energy around the edge compared with all theory to date, which has the appearance of a triangle as a function of energy above the edge, or sometimes the appearance of a dispersion shape, like a double triangle (de Jonge et al., 2005, 2007; Sier et al., 2020). The data are available in the supporting information to this chapter as file bz5029sup3.pdf.
![[Figure 1]](/figures/bz5029fig1thm.gif)
The tabulated values of the measured mass attenuation coefficients ![[\left[{{\mu} / {\rho}}\right]_{\rm meas}]](/teximages/bz5029/bz5029fi17.svg)
were calculated from the weighted mean of all the measurements obtained with combinations of the three foils and the three apertures. The total mass attenuation coefficients [μ/ρ] were obtained by applying corrections to the measured attenuation coefficients for the effects of scattering and fluorescence. As this correction is small (less than 0.05%), the difference between applying this correction before or after taking the average of ![[\left[{{\mu} / {\rho}}\right]_{\rm meas}]](/teximages/bz5029/bz5029fi18.svg)
is insignificant.
Table 8 summarizes the major sources of uncertainty contributing to the tabulated values of ![[\left[{{\mu} / {\rho}}\right]_{\rm meas}]](/teximages/bz5029/bz5029fi19.svg)
. Major factors affecting the precision or the consistency of the measurements of ![[\left[{{\mu} / {\rho}}\right]_{\rm meas}]](/teximages/bz5029/bz5029fi20.svg)
using multiple foils are listed in the first part of Table 8. The main factors affecting the consistency of the measurements are the intrinsic sources of statistical errors of the system at the level of 0.02%, and the uncertainty in the energy.
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![[\left[{{\mu} / {\rho}}\right]_{\rm pe}]](/teximages/bz5029/bz5029fi89.svg)
and ![[\left[{{\mu} / {\rho}}\right]_{\rm pe}]](/teximages/bz5029/bz5029fi90.svg)
and The final uncertainty in [μ/ρ] (0.27–0.4% away from the K edge, 0.4–0.7% at the K edge) is dominated by the experimental precision (0.15% away from the K edge and 0.2% to 0.5% at the edge) and by the uncertainty in the local thickness (0.17–0.36%). Although limited in XAFS, this work was fully adequate for detailed comparison of advanced theoretical methods for computation of XAS and XAFS, including using FDMNES (Cosgriff et al., 2005) and variable-cluster-size computations. It also was able to calibrate the energy using powder sample standards (Rae et al., 2010c).
The article by de Jonge et al. (2005) was the first to provide an accurate XAS spectrum together with a detailed XAFS spectrum, in this case for molybdenum at and above the K edge at 20 keV. The foil thicknesses used were nominally 25 µm, 50 µm, 100 µm, 150 µm, 200 µm and 250 µm. Between three and five samples were used at each energy. This article defined significance as a measure of anomalies and unknown systematic errors, searched for a systematic error due to roughness, and found a systematic error due to bandwidth, allowing the bandwidth to be measured from the XAS data directly. This was the first example of its type to provide the pre-processed data as supplementary material – that is, the first to deposit a transferable data set. The data tabulated in the publication presented 94 points; the full set included 526 independent energies with 0.5 eV spacing above the edge. Hence this data set was the first high-accuracy data set amenable to detailed structural XAFS analysis and exploration of a range of independent systematic errors and new areas of physics.
Table 9 presents mass attenuation coefficients measured at 526 energies between 13.5 keV and 41.5 keV. The first column is the calibrated photon energy (in keV) with the uncertainty in the last significant figure presented in parentheses. The second column is the mass attenuation coefficient [μ/ρ] (in cm2 g−1; see Fig. 2) with uncertainty. The third column provides the percentage uncertainty in the mass attenuation coefficient. The values in the second and third columns were determined from the weighted mean of the measurements made with a variety of apertures and foil thicknesses, and using the values determined from the counts recorded in both of the downstream ion chambers. The weighted mean typically involved between 18 and 30 determinations. The uncertainty in the mass attenuation coefficient was evaluated from σse. The imaginary component of the atomic form factor f′′ was evaluated using equation (3). ![[\left[{{\mu} / {\rho}}\right]_{\rm pe}]](/teximages/bz5029/bz5029fi21.svg)
was evaluated by subtracting the average of the Rayleigh plus Compton contributions, as tabulated in XCOM (Scofield, 1973; Berger & Hubbell, 1987; Gerward et al., 2004) and FFAST (Chantler, 1995, 2000; Chantler et al., 2000).
|
![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi91.svg)
, follows. The uncertainty in ![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi92.svg)
(%)
![[Figure 2]](/figures/bz5029fig2thm.gif)
|
Molybdenum foil mass attenuation coefficients, 13.5–41.5 keV. Reprinted with permission from de Jonge et al. (2005 |
In parentheses following the reported values are the uncertainties in f′′, evaluated from ![[\sigma_{f^{\prime\prime}} = {{EuA} \over {2hcr_{e}}}\bigl{(}\sigma_{\left [{{\mu} / {\rho}}\right]}^{2}+\Delta_{\rm RC}^{2}\bigr{)}^{{{1} \over {2}}},\eqno(4)]](/teximages/bz5029/bz5029fd4.svg)
which includes an uncertainty contribution of half of the difference ΔRC between the tabulated values of the Rayleigh plus Compton contributions. The use of the photoelectric component of the attenuation determined in this manner is appropriate when Rayleigh and Compton scattering are the only significant other contributions to the total attenuation. This is the case apart from near the absorption edge and in the region of the XAFS. Near the edge, the influence of solid-state and bonding effects is difficult to evaluate or estimate. Values of f′′ in the energy range from 19.9–20.9 keV should be subject to a further uncertainty (hence correction) of the same order as the XAFS amplitude when alternative atomic environments are investigated. The mass attenuation coefficient can be written as a sum of the photoelectric absorption ![[\left[{{\mu} / {\rho}}\right]_{\rm pe}]](/teximages/bz5029/bz5029fi22.svg)
, Rayleigh scattering ![[\left[{{\mu} /{\rho}}\right]_{\rm R}]](/teximages/bz5029/bz5029fi23.svg)
and Compton scattering ![[\left[{{\mu} / {\rho}}\right]_{\rm C}]](/teximages/bz5029/bz5029fi24.svg)
according to ![[\left[{{\mu} \over {\rho}}\right]\simeq\left[{{\mu} \over {\rho}}\right]_{\rm pe}+\,\left [{{\mu} \over {\rho}}\right]_{\rm R}+\,\left[{{\mu} \over {\rho}}\right]_{\rm C}.\eqno(5)]](/teximages/bz5029/bz5029fd5.svg)
Further attenuating processes are negligible in the energy region of this experiment. The results of atomic form factor calculations can be assessed by comparing the calculated photoelectric absorption coefficients with the measured values. The authors estimated the Rayleigh plus Compton cross section to be equal to the average of the values reported by the FFAST and XCOM tabulations, and estimated the uncertainty in the Rayleigh plus Compton cross section to be half of the difference between these tabulations. They subtracted these scattering components from the measured values to determine the photoelectric absorption coefficients.
The information deposited with the original publication consisted of two text files: a README file representing header information, as required for portability to iFEFFit, eFEFFit or CIF formats (available in the supporting information for this present chapter as file bz5029sup4.txt); and the actual tabulation of data (file bz5029sup5.txt). In the interests of a compact notation for ease of use by other researchers, there were only four columns of data, but with uncertainties in the last significant figures of three of these given in parentheses (as is conventional). It is clearly important to report the evaluated mass attenuation coefficient (column 2 in Table 9), yet in much XAFS work the mass absorption coefficient is a more relevant quantity. This can be obtained from the fourth column, f′′, which represents the photoabsorption. Another potential deficiency of these data was the absence of the relative uncertainty versus the total absolute uncertainty of the mass attenuation coefficient. Often it is more useful to use the relative quantity in, for example, fitting of XAFS, as it separates independent point errors (uncertainties) from overall scaling or normalization uncertainties. This format is compact and in ASCII, but the use of parentheses for uncertainties does make the processing of the raw data for fitting by other researchers slightly more complicated. The separation of the header from the deposited spectrum also means that the two files could become separated or one could be lost, perhaps especially without direct connection to the details in the published manuscript.
Table 10 presents estimates of contributions of the individual errors to the reported values. One significant concern was the comparison of inflection points versus Bragg diffraction for the determination of energy. The accuracy of the energy determination can be assessed by comparing the absorption edge energy with the most accurate value in the literature. The first point of inflection of the mass attenuation coefficient on the absorption edge occurs at 19.9944 ± 0.0002 ± 0.0003 keV, where the first uncertainty reflects the ability to locate the position of the point of inflection and the second is the uncertainty in determining the energy. Comparison with the value reported by Kraft et al. (1996), 20.00036 ± 0.00002 keV, indicates a discrepancy of 6 eV, or 0.03%. The most likely causes of this discrepancy are a difference in the interpretation of the absorption-edge location, chemical or thermal effects on the edge location, or further errors in the energy determination. de Jonge et al. (2005) considered an upper limit on the accuracy of the determined energies to be half of the difference between these absorption-edge locations, at about 0.015%.
|
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As it stands, this implies agreement with FFAST to within a quoted uncertainty of 1% well above the edge, and 3–5% near the edge, and a confirmation of the triangle effect of magnitude 3–5% around the edge. Conversely, it suggests FFAST (Chantler, 1995, 2000) is significantly more accurate than XCOM (Berger & Hubbell, 1987; Berger et al., 1999) and Henke et al. (1993). This work also led to further investigations and publications on bandwidth, the integrated column density, and especially investigation of standard theoretical approaches and anomalies in theoretical broadening (Smale et al., 2006), advanced investigations of beamline-independent spectra and structure (Glover & Chantler, 2007), advanced investigations of theory using FDMX, and the development of experimental investigations of inelastic mean free paths of the photoelectron and plasmons (Chantler & Bourke, 2010, 2014b,c; Bourke & Chantler, 2015; Chantler & Bourke, 2019). It also led to detailed investigations of theory with the program FEFF (Kas et al., 2010).
The work of de Jonge et al. (2007) was unique in that for this single experiment the team implemented the energy calibration for the setup in situ and also installed a unique four-bounce monochromator particularly for the higher energies. This was a credit to these researchers and the beamline staff. The Sn foil thicknesses were nominally 25 µm, 50 µm, 100 µm, 150 µm, 200 µm, 250 µm and 500 µm. Sample thicknesses spanned the range of attenuation (0.1–0.9) ≤ [μ/ρ] ≤ (2–7.5) across the wide range of energies. Accuracies were 0.04–3%, and typically in the range 0.1–0.2%. This was used as a test case for new theory (Bourke et al., 2016a).
The X-ray energy was selected by tuning the upstream monochromator crystal so that the X-rays reflected from the (444) planes of silicon were of the desired energy. When this is done, X-rays of all allowed harmonic energies are also transmitted into the beam. Unwanted harmonic energies are then removed by reflecting this partially monochromated beam from the (333) planes of a second, downstream silicon crystal. The downstream channel-cut monochromator crystal was tuned to optimize the reflected X-ray intensity by scanning it through a small range of angles about the Bragg angle corresponding to the (333) planes. The peak intensity was identified from the scan, and the crystal was then set at the angle corresponding to the peak intensity.
Table 11 presents some of the data that were collected at 293 energies across and above the K edge of Sn (Fig. 3). The full data set is available as supporting information to this chapter as file bz5029sup6.pdf. The calibrated photon energy (in keV) is followed by the uncertainty in the last significant figures presented in parentheses. The mass attenuation coefficient [μ/ρ] (in cm2 g−1) is similarly given with its uncertainty. The third column is the percentage uncertainty in the mass attenuation coefficient. The second and third columns are determined from the weighted mean of the measurements made with a variety of apertures and foil thicknesses. The weighted mean typically involved about ten individual measurements, and hence if each had similar statistical quality and consistency, the precision of the pooled result could be reduced by just over a factor of three. At a number of energies in the XAFS region only one measurement is used for efficiency, and these naturally have larger uncertainties. The uncertainty in the mass attenuation coefficient was generally evaluated from σsd defined in equations A2 and A3 of the article by de Jonge et al. (2007).
|
![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi93.svg)
, is also given. Uncertainty in ![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi94.svg)
(%)![[Figure 3]](/figures/bz5029fig3thm.gif)
The imaginary component of the atomic form factor f′′ was evaluated from equation (3) and [μ/ρ]pe has been evaluated by subtracting the average of the Rayleigh plus Compton contribution as tabulated in XCOM (Scofield, 1973; Berger & Hubbell, 1987; Gerward et al., 2004) and FFAST (Chantler, 1995, 2000; Chantler et al., 2000). In parentheses following the reported values are uncertainties in f′′, evaluated from equation (4), which include an uncertainty contribution of half of the difference ΔRC between the two tabulated values of the Rayleigh plus Compton contribution.
The use of the photoelectric component of the attenuation determined in this manner is appropriate when Rayleigh and Compton scattering are the only significant other contributions to the total attenuation. This is certainly the case in the energy range covered by this experiment apart from near the absorption edge and in the region of the XAFS. In these regions the influence of solid-state and bonding effects is naturally substantial. Table 12 presents estimates of the individual error contributions to the reported values.
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Glover et al. (2008) attempted to test the beamline independence of earlier copper metal foil measurements of XAS and also attempted to investigate the XAFS structure directly. A total of 108 data points were collected across, above and below the K edge compared with the previous best synchrotron data set, which had 84 points across the energy range 8.9–20 keV. Samples of nominal thicknesses of 5 µm, 10 µm, 15 µm, 30 µm and 100 µm were used with three samples for every measured energy, as given in Table 13. The measurements are accurate to between 0.09% and 4.5%, with most measurements being accurate to better than 0.12%. A key systematic error due to monochromator drift during the measurements was characterized accurately. This enabled development of advanced theory of XAFS (Bourke & Chantler, 2010a), the development of the field of extracting photoelectron inelastic mean free paths from XAFS data sets (Bourke & Chantler, 2010b) and development of the theory of the inelastic mean free path of electrons (IMFP theory) (Bourke & Chantler, 2012; Chantler & Bourke, 2014a). It also allowed detailed exploration of developments of XAS theory (Kas et al., 2010).
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The imaginary component of the form factor quantifies the photoelectric absorption of a material. Photoelectric absorption is the dominant contributor to the X-ray mass attenuation coefficient for copper for the energies in Table 14, with scattering contributing less than 5%. The photoelectric mass absorption was calculated from the measured total mass attenuation coefficient by subtracting the contribution to the attenuation from Rayleigh and Compton scattering. The scattering contribution was calculated by taking the average of the FFAST (Chantler, 2000) and XCOM (Berger & Hubbell, 1987) tabulations of the Rayleigh plus Compton attenuation coefficient with the uncertainty assumed to be the difference between the two tabulations divided by ![[\sqrt 2]](/teximages/bz5029/bz5029fi25.svg)
. The scattering uncertainty contributed between 0.05% and 0.13% to the photoelectric absorption and is only significant in the region just below the edge. The imaginary component of the atomic form factor f′′ was calculated using equation (3).
|
This study used samples of metallic (solid-state) copper, but these gave an excellent approximation to the atomic values outside the edge regions. The equivalence of the solid-state and atomic mass attenuation coefficients outside the edge and XAFS regions has been suggested and illustrated for cadmium (Kodre et al., 2006). Therefore, this measurement of the form factor approximates the atomic form factor of copper, except at the edge and in the XAFS region between 8.9 keV and 9.5 keV where solid-state effects are dominant.
Measurements of the mass attenuation coefficient prove useful for XAFS and as a standard XAFS spectrum. XAFS analysis does not require absolute measurements of the mass attenuation coefficient; for current modelling it conventionally requires high-accuracy relative measurements. Therefore the uncertainty due to the absolute thickness determination (0.092%) can be subtracted from the total uncertainty when the data from Table 14 are used in XAFS analyses. The uncertainty in the mass attenuation coefficient was dominated by the contribution due to the absolute calibration, so subtracting this reduces the uncertainty greatly. The data are available as supporting information to this chapter as file bz5029sup7.pdf.
A particular success of this work was the proof of consistency within uncertainty to the earlier extensive XAS measurement, proving that stability and accuracy can be correctly measured and consistently determined. This was not a full proof of beamline portability and transferability but was a very welcome demonstration. Another useful development was the identification of effective harmonic contributions versus relative harmonic probability, its measurement and the determination of its impact (Glover & Chantler, 2009).
Table 14 gives the calibrated X-ray energy in keV with the uncertainty in the last significant digit(s) given in parentheses. The second column gives the value of the mass attenuation coefficient in cm2 g−1 with the uncertainty in parentheses. Column three gives the uncertainty in the mass attenuation coefficient as a percentage of its value. The fourth column lists the imaginary component of the form factor along with its uncertainty in parentheses. Table 15 gives a breakdown of the contributions to the uncertainty of the energy, mass attenuation coefficient and imaginary component of the form factor.
|
Rae et al. (2010a) reported a short XAS spectrum with a very large grid spacing and no XAFS. This had only 19 energy points across the energy range with an absolute accuracy of between 0.044% and 0.197%. This was the most accurate determination of any attenuation coefficient on a bending-magnet beamline at that time and reduced the absolute uncertainty by a factor of 3 compared with earlier work by using advances in integrated column density determination and the full-foil mapping technique.
Four zinc foils of nominal thicknesses of 10 µm, 25 µm, 50 µm and 100 µm provided a range of log attenuation values ![[0.5\leq\ln\left[{{I_{0}} / {I}}\right]\leq 6]](/teximages/bz5029/bz5029fi26.svg)
across the experimental energy range. This work defined and presented a `relative accuracy' of 0.006%, which is not the same as either the precision or the absolute accuracy. Relative accuracy is the appropriate parameter for standard implementation of analysis of near-edge spectra, so can be called the `XAFS accuracy'. These additional data are therefore particularly relevant for XAFS analysis. This work also provided estimates of the assumed (Rayleigh and Compton) scattering so that any errors in that assumption could be clarified or corrected for in later work, and clarified methods for absolute determination of the integrated column density and thickness, and hence for the mass attenuation coefficient (Rae et al., 2010b). The data are shown in Table 16 and are also available as supporting information to this chapter as file bz5029sup8.pdf.
|
![[\sigma_{\left[{{\mu} /{\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi95.svg)
and the percentage (absolute) accuracy ![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi96.svg)
are given. The (photoelectric) mass absorption coefficient ![[\left[{{\mu} / {\rho}}\right]_{\rm pe}]](/teximages/bz5029/bz5029fi97.svg)
is derived by subtracting the mass attenuation due to Raleigh and Compton scattering ![[\left[{{\mu} / {\rho}}\right]_{\rm R+C}]](/teximages/bz5029/bz5029fi98.svg)
from the total mass attenuation coefficient, to derive the form factor ![[\left[{{\mu} / {\rho}}\right]_{\rm R+C}]](/teximages/bz5029/bz5029fi99.svg)
estimated by half the discrepancy between tabulations in XCOM and FFAST is included in the uncertainty of ![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi100.svg)
(% relative)![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi101.svg)
(% absolute)![[\left[{{\mu} / {\rho}}\right]_{\rm pe}]](/teximages/bz5029/bz5029fi102.svg)
(cm![[\left[{{\mu} / {\rho}}\right]_{\rm R+C}]](/teximages/bz5029/bz5029fi103.svg)
(cmIslam et al. (2010b) reported an XAS spectrum with 9 points between the K and L edges, hence with no points covering XAFS. This data set is not useful for edge determination, reference calibration or for studying bonding and dynamic structure, and many other good sets of relative data are available along with a high-accuracy set of absolute data (Glover et al., 2010). However, the data set given by Islam et al. (2010b) represented a detailed study of systematic errors and XAS at higher energies, and gold metal foils give a good approximation to atomic form factors for use as a reference standard and for calibration. Four gold foils with nominal thicknesses of 9.3 µm, 100.6 µm, 116.5 µm and 275 µm were used for the measurements. The study clarified methods for the absolute determination of the integrated column density and thickness, especially for foil samples (Islam et al., 2010a). The method for energy analysis using powder diffraction standards was also detailed (Rae et al., 2010c). The data are shown in Table 17 and are also available as supporting information to this chapter as the file bz5029sup9.pdf.
|
![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi104.svg)
is the percentage uncertainty from the standard deviations of the measurements. ![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi105.svg)
is the total percentage uncertainty including the contribution from the uncertainty in the absolute value of [![[\%\sigma_{[\rho{t}]_{\rm c}} = 0.1\%]](/teximages/bz5029/bz5029fi106.svg)
. The (photoelectric) mass absorption coefficient ![[\left[{{\mu} / {\rho}}\right]_{\rm pe}]](/teximages/bz5029/bz5029fi107.svg)
and the imaginary part of the atomic scattering factor ![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi108.svg)
(% relative)![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi109.svg)
(% absolute)![[\left[{{\mu} / {\rho}}\right]_{\rm pe}]](/teximages/bz5029/bz5029fi110.svg)
(cmGlover et al. (2010) provided a detailed XAS and XAFS data set for gold metal foils across the LI edge, accurate to between 0.08% and 0.10%, dominated by the absolute calibration uncertainty. Four gold foil thicknesses were used, nominally 5 µm, 9 µm, 15 µm and 25 µm. This helped to develop the absolute method for integrated column density determination for foils: the multiple independent foil technique (Chantler et al., 2012b).
An analysis of the LI edge XAFS showed excellent agreement between the measured and simulated XAFS and yielded highly accurate values of the bond lengths of gold. This data set included 91 points across the LI edge, and the study included comparison with eFEFFit analysis following iFEFFit (i.e. providing and fitting input data uncertainties). This data set measured nanoroughness in 5 µm gold foils inside the spot size of the synchrotron beam (Glover et al., 2009), which is very important for nanostructure quality control, and modelled the fluorescence signature (Fig. 4).
![[Figure 4]](/figures/bz5029fig4thm.gif)
The photoelectric mass absorption coefficient was calculated by subtracting the contribution from Rayleigh and Compton scattering. The scattering attenuation coefficient was calculated from the average of the FFAST (Chantler, 2000) and XCOM (Berger & Hubbell, 1987) tabulations and the uncertainty was assumed to be the difference between the two tabulations divided by ![[\sqrt 2]](/teximages/bz5029/bz5029fi27.svg)
. The uncertainty in the scattering attenuation contributed less than 0.03% to the photoelectric absorption and was not a major source of error. These measurements should provide a good approximation to the imaginary part of the atomic form factor of gold, except at the edge and in the XAFS region between about 14.3 keV and 15 keV, where solid-state effects are significant. The data are available as supporting information to this chapter as file bz5029sup10.pdf. The experimental and fitted structure of the XAFS above the edge are shown in Fig. 5.
![[Figure 5]](/figures/bz5029fig5thm.gif)
![[\chi^{2}_{\rm r} = 1.94]](/teximages/bz5029/bz5029fi49.svg)
). Copyright IOP Publishing. Reproduced with permission. All rights reserved.Table 18 gives the values and uncertainties of the calibrated X-ray energy, mass attenuation coefficient, photoelectric mass absorption coefficient and imaginary component of the form factor. Column four lists the accuracy of the mass attenuation coefficient measurement excluding the contribution from the absolute calibration. This quantity is useful for XAFS, since most researchers use and analyse attenuation data on a relative scale. This quantity is referred to as σrel. A breakdown of the various contributions to the uncertainty in the energy, mass attenuation coefficient and imaginary part of the form factor is given in Table 19.
|
![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi111.svg)
is the precision of the mass attenuation coefficient measurements (the uncertainty excluding the contribution from full-foil mapping), which is useful for XAFS research. ![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi112.svg)
(%)![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi113.svg)
(cm![[\left[{{\mu} / {\rho}}\right]_{\rm pe}]](/teximages/bz5029/bz5029fi114.svg)
(cm
|
Islam et al. (2014) measured the XAS spectra from silver foils for 84 discrete energies between the LI and the K edges, hence their data showed no XAFS. Silver foils were used with nominal thicknesses of 5 µm (2 foils), 10 µm (2 foils), 12 µm, 50 µm (2 foils), 100 µm (2 foils) and 275 µm. The data had an accuracy of 0.01–0.2% on a relative scale down to 5.3 keV, and of 0.09–1.22% on an absolute scale down to 5.0 keV. This was the first high-accuracy measurement of X-ray mass attenuation coefficients of silver in the low-energy range, indicating the possibility of obtaining high-accuracy X-ray absorption fine structure down to the LI edge (3.8 keV) of silver. Comparison of these results with an earlier data set optimized for higher energies (Tran et al., 2005) confirmed the accuracy to within one standard error of each data set collected and analysed using the principles of the X-ray extended-range technique (XERT). Comparison with theory showed a slow divergence towards lower energies in this region away from absorption edges.
This analysis indicated that high accuracy is obtainable at lower energies by using comparatively thin (e.g. 5 µm) foils and by using dilute solutions of silver compounds if accurate transfer of thickness or concentration is obtained. This work thus indicated that measurements of L-edge XAFS of silver (theoretically at and above 3.8 keV) are possible using XERT by making use of thinner silver foils at lower energies (3–6 keV). Perhaps just as significant is the independent verification of the accuracy of the earlier work (Tran et al., 2005) to within one standard error, which confirms the potential accuracy of this technique and the portability and reproducibility across different diffracting monochromator crystals, energy ranges and foils. The data are shown in Table 20 and are available as supporting information to this chapter in the file bz5029sup11.pdf.
|
![[[{{\mu} / {\rho}}]]](/teximages/bz5029/bz5029fi115.svg)
, the photoelectric mass absorption coefficients ![[[{{\mu} / {\rho}}]_{\rm pe}]](/teximages/bz5029/bz5029fi116.svg)
, the imaginary components of the form factor ![[[{{\mu} / {\rho}}]_{\rm R+C}]](/teximages/bz5029/bz5029fi117.svg)
, with uncertainties![[[{{\mu} / {\rho}}]_{\rm R+C}]](/teximages/bz5029/bz5029fi118.svg)
was determined from half of the variation between the tabulated FFAST and XCOM values. The uncertainty of ![[[{{\mu} / {\rho}}]_{\rm pe}]](/teximages/bz5029/bz5029fi119.svg)
was determined from the uncertainty contributions of ![[[{{\mu} / {\rho}}]_{\rm R+C}]](/teximages/bz5029/bz5029fi120.svg)
and ![[[{{\mu} / {\rho}}]_{t}]](/teximages/bz5029/bz5029fi121.svg)
.![[[{{\mu} / {\rho}}]]](/teximages/bz5029/bz5029fi122.svg)
(cm![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi123.svg)
(% relative)![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi124.svg)
(% absolute)![[[{{\mu} / {\rho}}]_{\rm pe}]](/teximages/bz5029/bz5029fi125.svg)
(cm![[[{{\mu} / {\rho}}]_{\rm R+C}]](/teximages/bz5029/bz5029fi126.svg)
(cmTantau et al. (2015) investigated the XAFS region over 80 discrete energies including the K edge (Figs. 6 and 7), using six high-purity silver foils of nominal thicknesses 1 µm, 10 µm, 12.5 µm, 50 µm (2 foils) and 100 µm, chosen to ensure that for each energy at least one absorber would satisfy Nordfors' criterion (i.e. ![[2\,\lt\,\ln\left({{I} / {I_{0}}}\right)\,\lt\, 4]](/teximages/bz5029/bz5029fi28.svg)
) for counting statistics. All thicknesses were used at most energies. This study showed extremely good consistency between different data sets with different systematic errors collected in different years and with different experimental geometries, so presented a very strong argument for the possibility of beamline-independent, portable and reproducible measurements – that is, the potential to ask questions on an absolute and on a relative basis of theory.
![[Figure 6]](/figures/bz5029fig6thm.gif)
![[Figure 7]](/figures/bz5029fig7thm.gif)
![[\chi^{2}_{\rm r}]](/teximages/bz5029/bz5029fi50.svg)
for the reduced region of interest. Copyright IOP Publishing. Reproduced with permission from Tantau The results are accurate to better than 0.1%, permitting critical tests of atomic and solid-state theory. This is one of the most accurate demonstrations of cross-platform accuracy in synchrotron studies up to now. The data set can be fully analysed by conventional XAFS analysis techniques, but the analysis can also be extended to include error propagation and uncertainty, yielding bond lengths accurate to approximately 0.24% and Debye–Waller parameters accurate to 30%. It also enabled the investigation of advanced theory (using FDMX) for accurate analysis of such data across the full XAFS spectrum, built on full-potential theory, yielding a bond-length accuracy of the order of 0.1% and demonstrating that a single Debye–Waller parameter is inadequate and inconsistent across the XAFS range. The first ten oscillations of XAFS are very clear. Two effective Debye–Waller parameters are determined: a high-energy value based on the highly correlated motion of bonded atoms [σDW = 0.1413 (21) Å] and an uncorrelated bulk value [σDW = 0.1766 (9) Å], in good agreement with that derived from room-temperature crystallography. The data are shown in Table 21 and are also available in the supporting information to this chapter as file bz5029sup12.pdf.
|
![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi127.svg)
and ![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi128.svg)
are given. Uncertainties in ![[\left[{{\mu} / {\rho}}\right]_{\rm pe}]](/teximages/bz5029/bz5029fi129.svg)
and ![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi130.svg)
(%)![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi131.svg)
(%)![[\left[{{\mu} / {\rho}}\right]_{\rm {R+C}}]](/teximages/bz5029/bz5029fi132.svg)
(cm![[\left[{{\mu} /{\rho}}\right]_{\rm pe}]](/teximages/bz5029/bz5029fi133.svg)
(cm16. Dilute solutions of nickel(II) complexes with no long-range order. Hybrid technique, XAS, XAFS and nanostructure
The article by Chantler et al. (2015) presents a completely different type of study: the samples are dilute solutions of bis(N-isopropylsalicylaldiminato)nickel(II) (i-pr Ni for short) and bis(N-n-propylsalicylaldiminato)nickel(II) (n-pr Ni for short). Conventional wisdom would suggest measuring the XAS in fluorescence mode, but these data sets were measured in transmission mode, so in principle can yield an absolute measurement of XAS and XAFS without direct normalization to an additional reference standard.
However, the techniques of XERT are not appropriate for a solution or a dilute system. We cannot have several carefully calibrated thicknesses to check for systematic errors or the linearity of the detection chain. In principle, this can be addressed by using an ideal reference standard measured under identical conditions. Hence these were the first sets of data following the hybrid technique. This used different concentrations of a solution on the assumption that the local molecular structure would be unchanged (or at least similar), and, in principle, it can use a measurement of the solvent instead of (or as well as) an air path or blank measurement. The most important characteristic behind the accuracy, utility and success of these data sets is the concept of the signal-to-noise ratio versus the signal-to-background ratio. The signal-to-background ratio is very small in this case and in transmission mode in general, yet with good experimental design the signal-to-noise ratio can be very strong and such a study can provide useful insights. The hybrid technique can be used to collect data in either transmission or fluorescence mode. The data for this paper related to measurements collected in transmission mode. The measurements were made using a cryostat at ca 80 K. The solvent measurement was able to determine the solvent attenuation and that for the airpath, window adhesive (silicone), window (Kapton) and detector gas.
In this experiment, three samples provided three independent data sets: data for 226 discrete energies for 15.26 mM i-pr Ni, 199 independent energies for 1.515 mM i-pr Ni and 194 energies for 15.33 mM n-pr Ni were collected. A mixed solvent of 60% butyronitrile (BCN) + 40% acetonitrile (ACN) was used to prepare the solutions to avoid microcrystallization at low temperatures in the cryostat. Both complexes have the same composition, NiN2O2C20H24, so the ability to distinguish between the two isomers is ideally a matter for spectroscopy, isomer isolation and even machine learning. These complexes are already used as standards and reference materials by XAS researchers, because it is well believed that the i-pr Ni complex is tetrahedral and the n-pr Ni complex is square planar with identical coordination number at the Ni atom and identical atoms at very similar distances. However, it is perceived as potentially very hard or even impossible to establish the Ni conformation from XAFS.
The data that were deposited as supporting information to the original publication are also available here in Tables 22 to 24 and also as supporting information to this chapter as file bz5029sup13.pdf. These tables of data have detailed column headings and header information, so can be used for diverse applications.
|
![[\left[{{\mu} / {\rho}}\right]_{\rm S}]](/teximages/bz5029/bz5029fi134.svg)
from intensity measurements (![[\left[{{\mu} / {\rho}}\right]_{\rm S}]](/teximages/bz5029/bz5029fi135.svg)
of the solute S are given with the associated relative and percentage uncertainties and the absolute uncertainties including the uncertainty contributions from the thickness ratio, ![[\left[{{\mu} /{\rho}}\right]_{\rm pe}]](/teximages/bz5029/bz5029fi136.svg)
is determined by subtracting the tabulated X-ray mass attenuation values of the complex for Rayleigh and Compton scattering from the total experimental X-ray mass attenuation coefficients ![[\left[{{\mu} / {\rho}}\right]_{\rm S}]](/teximages/bz5029/bz5029fi137.svg)
at the measured energies, with uncertainties. The X-ray mass attenuation values for Rayleigh and Compton scattering are estimated, with the uncertainty of ![[\left[{{\mu} / {\rho}}\right]_{\rm R+C}]](/teximages/bz5029/bz5029fi138.svg)
determined from half of the variation between the FFAST and XCOM tabulated values. (Some rows of values have been omitted for brevity. The full version is available in the supporting information.)![[\left[{{\mu} / {\rho}}\right]_{\rm S}]](/teximages/bz5029/bz5029fi139.svg)
(cm![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi140.svg)
(cm![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi141.svg)
(%)![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi142.svg)
(cm![[\left[{{\mu} / {\rho}}\right]_{\rm pe}]](/teximages/bz5029/bz5029fi143.svg)
(cm![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm pe}}]](/teximages/bz5029/bz5029fi144.svg)
(cm![[\left[{{\mu} / {\rho}}\right]_{\rm R+C}]](/teximages/bz5029/bz5029fi145.svg)
(cm
|
![[\left[{{\mu} / {\rho}}\right]_{\rm S}]](/teximages/bz5029/bz5029fi146.svg)
from intensity measurements (![[\left[{{\mu} / {\rho}}\right]_{\rm S}]](/teximages/bz5029/bz5029fi147.svg)
of the solute S are given with the associated relative and percentage uncertainties and the absolute uncertainties including the uncertainty contributions from the thickness ratio, ![[\left[{{\mu} /{\rho}}\right]_{\rm pe}]](/teximages/bz5029/bz5029fi148.svg)
is determined by subtracting the tabulated X-ray mass attenuation values of the complex for Rayleigh and Compton scattering from the total experimental X-ray mass attenuation coefficients ![[\left[{{\mu} / {\rho}}\right]_{\rm S}]](/teximages/bz5029/bz5029fi149.svg)
at the measured energies, with uncertainties. The X-ray mass attenuation values for Rayleigh and Compton scattering are estimated, with the uncertainty of ![[\left[{{\mu} / {\rho}}\right]_{\rm R+C}]](/teximages/bz5029/bz5029fi150.svg)
determined from half of the variation between the FFAST and XCOM tabulated values. (Some rows of values have been omitted for brevity. The full version is available in the supporting information.)![[\left[{{\mu} / {\rho}}\right]_{\rm S}]](/teximages/bz5029/bz5029fi151.svg)
(cm![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi152.svg)
(cm![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi153.svg)
(%)![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi154.svg)
(cm![[\left[{{\mu} / {\rho}}\right]_{\rm pe}]](/teximages/bz5029/bz5029fi155.svg)
(cm![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm pe}}]](/teximages/bz5029/bz5029fi156.svg)
(cm![[\left[{{\mu} / {\rho}}\right]_{\rm R+C}]](/teximages/bz5029/bz5029fi157.svg)
(cm
|
![[\left[{{\mu} / {\rho}}\right]_{\rm S}]](/teximages/bz5029/bz5029fi158.svg)
determined from the intensity measurements (![[\left[{{\mu} / {\rho}}\right]_{\rm S}]](/teximages/bz5029/bz5029fi159.svg)
of the solute S are given with the associated relative and percentage uncertainties and the absolute uncertainties including the uncertainty contributions from the thickness ratio, ![[\left[{{\mu} /{\rho}}\right]_{\rm pe}]](/teximages/bz5029/bz5029fi160.svg)
is determined by subtracting the tabulated X-ray mass attenuation values of the complex for Rayleigh and Compton scattering from the total experimental X-ray mass attenuation coefficients ![[\left[{{\mu} / {\rho}}\right]_{\rm S}]](/teximages/bz5029/bz5029fi161.svg)
at the measured energies, with uncertainties. The X-ray mass attenuation values for Rayleigh and Compton scattering are estimated, with the uncertainty of ![[\left[{{\mu} / {\rho}}\right]_{\rm R+C}]](/teximages/bz5029/bz5029fi162.svg)
determined from half of the variation between the FFAST and XCOM tabulated values. (Some rows of values have been omitted for brevity. The full version is available in the supporting information.)![[\left[{{\mu} / {\rho}}\right]_{\rm S}]](/teximages/bz5029/bz5029fi163.svg)
(cm![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi164.svg)
(cm![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi165.svg)
(%)![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi166.svg)
(cm![[\left[{{\mu} / {\rho}}\right]_{\rm pe}]](/teximages/bz5029/bz5029fi167.svg)
(cm![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm pe}}]](/teximages/bz5029/bz5029fi168.svg)
(cm![[\left[{{\mu} / {\rho}}\right]_{\rm R+C}]](/teximages/bz5029/bz5029fi169.svg)
(cmFig. 8 shows a schematic of the experimental setup at the ANBF, Tsukuba, Japan, using the hybrid technique that was used to collect transmission and fluorescence XAS from these multiple dilute solutions of nickel(II) complexes and absorption spectra from a 5 µm Ni foil. Fig. 9 illustrates the multi-chambered solution cell suitable for cryostat use, and the filling of the chambers. This was the first implementation of the hybrid technique and led to further developments of the technique in later publications.
![[Figure 8]](/figures/bz5029fig8thm.gif)
|
A schematic diagram of the experimental setup using the hybrid technique at the ANBF, Tsukuba, Japan. A multi-chambered solution cell was used in the cryostat containing two dilute solutions and the pure solvent in three chambers. The cryostat was translated vertically to record intensities attenuated by each of the solutions at each energy. Two daisy wheels containing 14 aluminium filters (of different thicknesses) were employed upstream and downstream to monitor harmonic contamination from higher-order reflections. Different aperture sizes on the daisy wheel allowed aperture-dependent measurements to be collected in order to characterize scattering effects for correction. Three ion chambers were employed to record unattenuated intensities, I0, intensities attenuated by the dilute solutions, I1, and intensities further attenuated by the corresponding metallic sample mounted with a cantilever as shown, I2. |
![[Figure 9]](/figures/bz5029fig9thm.gif)
|
A three-chambered cryostat cell is filled with three solutions (15 mM and 1.5 mM nickel(II) complex, and pure solvent). A precise flow of solution into the chambers was achieved using three 10 ml syringe pumps, with fittings, and 1/16 inch Teflon tubes. The cell, mounted on a cryostat stick, was clamped in the horizontal plane. Each syringe pump was controlled by a stepper-motor controller programmed by software to pulse the syringe pumps. |
Fig. 10 presents full plots of measured ![[\left[{{\mu} / {\rho}}\right]\left[\rho t\right]]](/teximages/bz5029/bz5029fi29.svg)
values for the two i-pr Ni solutions and the pure solvent (top pane) and the statistical accuracy of these three data sets in the bottom panes. Despite the dilution and the very low signal-to-background ratio, the accuracy of each data set is at or below 0.01%, so the solvent can be subtracted to find the attenuation by the active species and the absorption edge. Similarly, Fig. 11 presents full plots of measured ![[\left[{{\mu} / {\rho}}\right]\left[\rho t\right]]](/teximages/bz5029/bz5029fi30.svg)
values for the two n-pr Ni solutions and the pure solvent (top pane) with the statistical accuracy of these three in the bottom panes.
![[Figure 10]](/figures/bz5029fig10thm.gif)
![[Figure 11]](/figures/bz5029fig11thm.gif)
Because of the high signal-to-noise ratio and the characterization of the solvent and windows, highly accurate values for the attenuation due to the active species can be obtained (Figs. 12, 13). In these figures, the attenuation by the active species, i.e. the 15 mM solute, ![[\left\{\left[{{\mu} /{\rho}}\right]\left[\rho t\right]\right\rbrace_{\rm S}]](/teximages/bz5029/bz5029fi31.svg)
, is plotted in the top panel, with four subpanels showing contributions to the total absolute uncertainty (second panel, δtotal) from the fitting/filling fraction (third panel, δfrac), the statistical uncertainty from the sample and solvent measurements (fourth panel, ![[{\delta_{\left[{{\mu} / {\rho}}\right]\left[\rho t\right]}}_{\rm S,stat} = {\sigma_{ \left\lbrace\left[{{\mu} / {\rho}}\right]\left[\rho t\right]\right\rbrace}}_{\rm S,stat}]](/teximages/bz5029/bz5029fi32.svg)
) and the contribution to the uncertainty from the background and dark current (bottom panel, δbkg+dc). Although the signals only correspond to a peak magnitude of 0.15 attenuation, or 0.12 above the below-edge value, the uncertainty is less than 0.0003 from statistical sources of error and about 0.0003 from statistical and system-related sources of error, so in fact the attenuation has an uncertainty of only about 0.2% across the edge, XANES and XAFS region.
![[Figure 12]](/figures/bz5029fig12thm.gif)
|
Corrected and normalized XAS (attenuation) of the i-pr Ni isomer (solute) |
![[\left\{\left[{{\mu} / {\rho}}\right]\left[\rho t\right]\right\rbrace_{\rm S}]](/teximages/bz5029/bz5029fi51.svg)
determined from the attenuations of 15 m![[{\delta_{\left[{{\mu} / {\rho}}\right]\left[\rho t\right]}}_{\rm S,stat} = {\sigma_{ \left\lbrace\left[{{\mu} / {\rho}}\right]\left[\rho t\right]\right\rbrace}}_{\rm S,stat}]](/teximages/bz5029/bz5029fi52.svg)
), and the background and dark-current corrections (bottom panel,
![[Figure 13]](/figures/bz5029fig13thm.gif)
|
Corrected and normalized XAS (attenuation) of 15 mM n-pr Ni (solute) |
![[\left\{\left[{{\mu} / {\rho}}\right]\left[\rho t\right]\right\rbrace_{\rm S}]](/teximages/bz5029/bz5029fi53.svg)
. The corresponding total uncertainty (second panel, ![[{\delta_{\left[{{\mu} /{\rho}}\right]\left[\rho t\right]}}_{\rm S,stat} = {\sigma_{ \left\lbrace\left[{{\mu} / {\rho}}\right]\left[\rho t\right]\right\rbrace}}_{\rm S,stat}]](/teximages/bz5029/bz5029fi54.svg)
), and the background and dark-current corrections (bottom panel, The eFEFFit package was used with weighting of the data derived from the experimental uncertainty, and in each case converged to a well defined XAFS model (Islam et al., 2015). Structural refinement of the i-pr Ni complex gave an excellent fit to a tetrahedral geometry. n-pr Ni showed a distorted square planar geometry. This demonstrates the insight that can be obtained from the propagation of uncertainties in XAFS analysis and the consequent confidence in hypothesis testing and in the ab initio analysis of alternative structures (Fig. 14).
![[Figure 14]](/figures/bz5029fig14thm.gif)
17. Dilute solutions with no long-range order: mM solutions of ferrocene and decamethylferrocene. Hybrid technique, XAS, XAFS and nanostructure
Islam et al. (2016) reported another example of the hybrid technique applied to the relatively small molecules ferrocene [Fc, Fe(C5H5)2] and decamethylferrocene [DmFc or Fc*, Fe(C5(CH3)5)2], also as dilute solutions, at two concentrations and with a solvent measurement. Ferrocene has long been the archetypal metallocene and was the first known example of organometallic bonding. As well as having many applications and derivatives, it is also used as a key reference material for chemistry and XAS but, interestingly, one for which the structure is ill-determined. The experimental setup used by Islam et al. was equivalent to that described in Section 16 for hybrid measurement in transmission (and fluorescence) modes.
An earlier experiment looked at 10 mM Fc in fluorescence mode at 10–20 K (Chantler et al., 2012b). This primarily led to development of the understanding of modelling absorption and self-absorption in multipixel fluorescence detectors.
Four data sets are presented in Tables 25–28, and also in the supporting information to this chapter as the file bz5029sup14.pdf. These are for nominal concentrations of 15 mM Fc, 3 mM Fc, 15 mM DmFc and 3 mM DmFc. The actual concentations were 15.26 mM and 3.07 mM (Fc) and 15.29 mM and 3.06 mM (DmFc), respectively. The columns of data and the header information for these tables were detailed, allowing use of the data for diverse applications. There are 297, 249, 234 and 236 independent energies across the Fe K edge, respectively, for these transmission XAS measurements of frozen solutions at 10–20 K, with accuracies (for the solute) from 0.2% to 2%, observing statistically significant fine structure to k > 12 Å−1.
|
![[\left[{{\mu} / {\rho}}\right]_{\rm S}]](/teximages/bz5029/bz5029fi170.svg)
from the absolute intensities (![[\left[{{\mu} / {\rho}}\right]_{\rm S}]](/teximages/bz5029/bz5029fi171.svg)
of the solute S are given with associated relative and percentage uncertainties, and the absolute uncertainties including the uncertainty contributions from the thickness ratio ![[\left[{{\mu} / {\rho}}\right]_{\rm pe}]](/teximages/bz5029/bz5029fi172.svg)
is determined by subtracting the tabulated X-ray mass attenuations of the complex for the Rayleigh and Compton scattering from the total experimental X-ray mass attenuation coefficients ![[\left[{{\mu} / {\rho}}\right]_{\rm S}]](/teximages/bz5029/bz5029fi173.svg)
at the measured energies, with uncertainties. The X-ray mass attenuation due to Rayleigh and Compton scattering is estimated, with the uncertainty of ![[\left[{{\mu} / {\rho}}\right]_{\rm R+C}]](/teximages/bz5029/bz5029fi174.svg)
determined from half of the variation between the FFAST and XCOM values. (Some rows of values have been omitted for brevity. The full version is available in the supporting information.)![[\left[{{\mu} / {\rho}}\right]_{\rm S}]](/teximages/bz5029/bz5029fi175.svg)
(cm![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi176.svg)
(cm![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi177.svg)
(%)![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi178.svg)
(cm![[\left[{{\mu} /{\rho}}\right]_{\rm pe}]](/teximages/bz5029/bz5029fi179.svg)
(cm![[\sigma_{\left[{{\mu} /{\rho}}\right]_{\rm pe}}]](/teximages/bz5029/bz5029fi180.svg)
(cm![[\left[{{\mu} / {\rho}}\right]_{\rm R+C}]](/teximages/bz5029/bz5029fi181.svg)
(cm
|
![[\left[{{\mu} / {\rho}}\right]_{\rm S}]](/teximages/bz5029/bz5029fi182.svg)
obtained from the absolute intensities for a 3 m![[\left[{{\mu} / {\rho}}\right]_{\rm S}]](/teximages/bz5029/bz5029fi183.svg)
(cm![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi184.svg)
(cm![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi185.svg)
(%)![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi186.svg)
(cm![[\left[{{\mu} / {\rho}}\right]_{\rm pe}]](/teximages/bz5029/bz5029fi187.svg)
(cm![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm pe}}]](/teximages/bz5029/bz5029fi188.svg)
(cm![[\left[{{\mu} / {\rho}}\right]_{\rm R+C}]](/teximages/bz5029/bz5029fi189.svg)
(cm
|
![[\left[{{\mu} / {\rho}}\right]_{\rm S}]](/teximages/bz5029/bz5029fi190.svg)
using absolute intensities (![[\left[{{\mu} / {\rho}}\right]_{\rm S}]](/teximages/bz5029/bz5029fi191.svg)
(cm![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi192.svg)
(cm![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi193.svg)
(%)![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi194.svg)
(cm![[\left[{{\mu} / {\rho}}\right]_{\rm pe}]](/teximages/bz5029/bz5029fi195.svg)
(cm![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm pe}}]](/teximages/bz5029/bz5029fi196.svg)
(cm![[\left[{{\mu} / {\rho}}\right]_{\rm R+C}]](/teximages/bz5029/bz5029fi197.svg)
(cm
|
![[[{{\mu} / {\rho}}]_{\rm S}]](/teximages/bz5029/bz5029fi198.svg)
from the intensity measurements (![[\left[{{\mu} / {\rho}}\right]_{\rm S}]](/teximages/bz5029/bz5029fi199.svg)
(cm![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi200.svg)
(cm![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi201.svg)
(%)![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi202.svg)
(cm![[\left[{{\mu} / {\rho}}\right]_{\rm pe}]](/teximages/bz5029/bz5029fi203.svg)
(cm![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm pe}}]](/teximages/bz5029/bz5029fi204.svg)
(cm![[\left[{{\mu}/ {\rho}}\right]_{\rm R+C}]](/teximages/bz5029/bz5029fi205.svg)
(cmThe solvent was 50% butyronitrile (CH3CH2CH2CN) and 50% acetonitrile (CH3CN), the mixture being less susceptible to crystallization during freezing. Uncertainties from the repeated measurements including the solvent were from 0.005% to 0.03%, indicating a very high signal-to-noise ratio (see Figs. 15 and 16), enabling extraction of the isolated solute spectra (see Figs. 17 and 18). Key challenges were the absolute calibration and characterization of the solvent and air path, and the volume change of freezing, together with the treatment of uncertainty within the hybrid technique.
![[Figure 15]](/figures/bz5029fig15thm.gif)
![[Figure 16]](/figures/bz5029fig16thm.gif)
![[Figure 17]](/figures/bz5029fig17thm.gif)
![[\delta_{{t_{\rm frac}}}]](/teximages/bz5029/bz5029fi55.svg)
), statistical uncertainties (![[\delta_{\left[{{\mu} / {\rho}}\right]\left[\rho{t}\right]}]](/teximages/bz5029/bz5029fi56.svg)
), and uncertainties due to background attenuation and dark current (![[Figure 18]](/figures/bz5029fig18thm.gif)
![[\delta_{{t_{\rm frac}}}]](/teximages/bz5029/bz5029fi57.svg)
), statistical uncertainties (![[\delta_{\left[{{\mu} /{\rho}}\right]\left[\rho{t}\right]}]](/teximages/bz5029/bz5029fi58.svg)
), and uncertainties due to background attenuation and dark current (There is moderately strong evidence for three crystal phases of Fc, but more challenging is the determination of their structures (Fig. 19). Note that in this study, it was not required that these frozen solutions (nor the higher-temperature solutions) should show the same structures as either an isolated molecule or a molecule in a crystal.
![[Figure 19]](/figures/bz5029fig19thm.gif)
The accuracy of these data sets was a major achievement of the hybrid technique and led to advanced theoretical investigations using the finite-difference method and the new theory FDMX (Bourke et al., 2016b). However, there is still much more to understand about Fc and DmFc in crystalline and solution forms.
18. n-pr Ni and i-pr Ni complexes, 7.9–9.5 keV. XAS, XAFS, structure, Ni conformation, error analysis and grid spacing
The study by Schalken & Chantler (2018) revealed one key problem of many data sets – the grid spacing in energy and the spacing translated to k-space are almost never uniform. Even if the spacing is more-or-less uniform in k-space, there are likely to be some points missing due to data processing and removal due to Bragg glitches (three-beam interactions of the monochromator, unnormalized by an unmatched secondary detector). It is also assumed that the E0 offset energy for the edge and continuum is defined exactly in the process of the transformation to k-space and hence that there is no need for an additional offset parameter for E0. More commonly, the near-edge region has a finer spacing, e.g. 0.5 eV, increasing at some distance above the XANES region, so the grid is not uniform in k-space. Therefore to obtain data points with independent uncertainties and the (correct) propagation of errors, one must process each data point individually (Fig. 20). Then one can model the detailed structure with theory (Fig. 21). Conversely, this study also assessed how to interpolate the data, possibly onto a uniform grid in k-space, post facto with `minimal' distortion of the data and well defined statistical significance. Quite often, and indeed for any current processing in R-space, a fast Fourier transform (FFT) routine is used which assumes and requires a uniform grid in k-space. This study also discussed data sets where the beamtime was spent pursuing high-accuracy data points at fewer energies, versus a strategy where a much higher point density was pursued but with lower accuracy for each point.
![[Figure 20]](/figures/bz5029fig20thm.gif)
|
Quality of the data and background spline for the absorption spectrum with the solvent contribution removed for 15 mM i-pr Ni (top) and n-pr Ni (bottom) for the high-point-accuracy data sets. |
![[Figure 21]](/figures/bz5029fig21thm.gif)
|
k2-weighted mu2chi (non-interpolated) output for 15 mM i-pr Ni and n-pr Ni from the high-point-accuracy hybrid experiment. |
Around this time, discussions arose through the joint Q2XAFS meetings coordinated by the IUCr Commission on XAFS and the International XAFS Society about appropriate data formats for deposition and tabulations for cross-platform portability (Ravel et al., 2012; Chantler et al., 2012a; Hester, 2016; Abe et al., 2018; Chantler et al., 2018, 2019). A key recommendation was that the format should be text-based and readable by humans, but also suitable for input to fitting, analysis and theory software. Secondly, it was recommended that information about the different columns and key issues should be noted in a header. Potential formats could be a .dat format similar to that used for iFEFFit or a format similar to the .cif format, already in use for the description of crystal structures and diffraction experiments. Other formats discussed included the binary HDF5 format, the European XDI format and later a XIF format (Sarangi, 2018).
The data sets discussed in the previous sections of this chapter were presented as typeset tables of data within published articles (e.g. in pdf format), or deposited as text files and readme files (Section 8), or deposited as pdfs of tabulated data (Sections 16 and 17). However, by this point and in the following sections, discussions at the Q2XAFS meetings and the joint work of the IUCr Commission on XAFS and the International XAFS Society had shown that it was important to develop standard formats of data sets for direct input into XANES and XAFS fitting packages, for cross-platform portability and for deposition. For the study described in this section a minimalist template for mu2chi, iFEFFit and eFEFFit formats was used, as shown in Table 29, which is also available the supporting information to this chapter as the file bz5029sup15.txt. `E' is the energy in eV; `MU' is [μ/ρ] in cm2 g−1 and `MU_ERR' is ![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi33.svg)
in cm2 g−1. The table contains no discussion of relative errors, but presents data for 194 independent energies for 15 mM n-pr Ni under the same conditions as discussed in Section 16. This template focused only on standard processing and packages, and not on the presentation of additional derived results. As a .txt file it was directly machine-readable. Comments, header information and metadata as recommended by the discussions at Q2XAFS are not included here, so the reader should refer to the article by Schalken & Chantler (2018) for these details. This template was later developed into the recommended formats of .cif and .dat (for eFEFFit/iFEFFit), as described in the sections below.
|
19. n-pr Ni and i-pr Ni complexes, 8.14–9.32 keV. XFS, XAFS, nanostructure, fluorescence reference data sets and Ni conformation
Trevorah et al. (2019) presented the first example of data sets collected using fluorescence detection. These data sets were of the same quality as the earlier transmission (XAS) data sets discussed in Section 16, and like them include detailed error analysis and uncertainty propagation. These data sets were presented in both eFEFFit/iFEFFit data format (.dat) and also in CIF (.cif) format. These potential `standard formats' have not yet been agreed to by the international community, but are illustrative and can be used and ported as presented. In this section we explicitly present portable data with metadata and headers as discussed by the international community. We recommend this practice for future supplementary information, standards, deposition and storing of data as fulfilling the requirements decided upon during the Q2XAFS meetings.
X-ray absorption is a first-order process (in lowest order) in the Hamiltonian, whereas fluorescence is a second-order process (in lowest order) [see Chapters 2.8 by Chantler (2024) and 3.49 by Glatzel & Ghiringhelli (2024)]. Hence X-ray emission spectroscopy follows the same Hamiltonian whether it is detecting fluorescence in high-resolution fluorescence detection `XAS' (HERFD-XAS) or observing emission fluorescence spectra, often called XES. Technologies such as resonant inelastic X-ray scattering also observe emission and are also types of X-ray emission spectroscopy, although conventionally one axis of the spectrum is labelled as the incident energy (`XAS' or perhaps `XFS') and one is labelled as the emission energy (`XES').
The fluoresence data for 572 independent energies for 15 mM n-pr Ni are presented in Table 30 in .dat format. Table 31 shows the i-pr Ni fluorescence data in .cif format, with the same columns in the same order. Four files (one in .dat format and one in .cif format for each complex) are available the supporting information to this chapter as the files bz5029sup16.txt to bz5029sup19.txt.
|
![[A\left[{{\mu} / {\rho}}\right]^{*}_{\rm pe}]](/teximages/bz5029/bz5029fi206.svg)
(Ni) in cm![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi207.svg)
, ![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi208.svg)
, ![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm cal}}]](/teximages/bz5029/bz5029fi209.svg)
, and ![[Scheme scheme1]](/schemes/bz5029scheme1.gif)
|
![[A\left[{{\mu} / {\rho}}\right]^{*}_{\rm pe}]](/teximages/bz5029/bz5029fi210.svg)
(Ni) in cm![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi211.svg)
, ![[\sigma_{\left[{{\mu} /{\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi212.svg)
, ![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm cal}}]](/teximages/bz5029/bz5029fi213.svg)
, and ![[Scheme scheme2]](/schemes/bz5029scheme2.gif)
One can convert a data file in .cif format to .dat format by commenting out some of the header items and perhaps clarifying some of the descriptions in other parts of the header. Similarly one can convert the .dat header fields into a CIF-like data format. In Tables 30 and 31, which correspond to Fig. 22, the energy E is in eV and is followed by ![[A\left[{{\mu} / {\rho}}\right]^{*}_{\rm pe}]](/teximages/bz5029/bz5029fi34.svg)
(for Ni) in cm2 g−1. ![[A\left[{{\mu} / {\rho}}\right]^{*}_{\rm pe}]](/teximages/bz5029/bz5029fi35.svg)
is not the same as [μ/ρ] for transmission or XAS experiments because there is intrinsically a scaling and efficiency correction, even if the results of this analysis are in excellent agreement with the results of the corresponding transmission experiments. The raw data for a transmission experiment are attenuation values, from which the photoelectric absorption coefficient [μ/ρ]pe may be extracted, whereas only a core–hole ionization can lead to fluorescence, so fluorescence measurements intrinsically measure the photoelectric coefficient ![[A\left[{{\mu} / {\rho}}\right]^{*}_{\rm pe}]](/teximages/bz5029/bz5029fi36.svg)
. Similarly a transmission experiment will measure the attenuation or absorption from all shells and subshells, whereas fluorescence experiments will only measure the observed active edge or subshell, indicated by the label *. Because fluorescence measurements have significant nonlinearity from detector and self-absorption effects and directly measure I/I0, the extracted values typically need to be scaled by a factor A to be consistent with attenuation or absorption measurements. Finally, the selection rules for a first-order process (attenuation) and a second-order process (fluorescence) are in general different. Additional columns in the tables give σtotal (cm2 g−1) or ![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi37.svg)
; σExperimental (cm2 g−1), i.e. ![[\sigma_{\left[{{\mu} /{\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi38.svg)
; σpixel or the scaling/absolute transfer uncertainty or ![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm cal}}]](/teximages/bz5029/bz5029fi39.svg)
; and σE in eV.
![[Figure 22]](/figures/bz5029fig22thm.gif)
These data were of sufficient quality to be used for nanostructural analysis, studies of the propagation of uncertainties and for reference standards (Trevorah et al., 2020), see Fig. 23. For data deposition and processing, we recommend a column order with E first, then either [μ/ρ] for transmission measurements (i.e. the attenuation in XAS measurements) or ![[A\left[{{\mu} / {\rho}}\right]^{*}_{\rm pe}]](/teximages/bz5029/bz5029fi40.svg)
in cm2 g−1 for fluorescence measurements. These are the typical data processed without uncertainties or uncertainty estimation. Then we recommend that the third column is ![[\sigma_{\left[{{\mu} /{\rho}}\right]}]](/teximages/bz5029/bz5029fi41.svg)
, which is the uncertainty of the second column, also in cm2 g−1 for simplicity of processing further with programs such as mu2chi, eFEFFit, iFEFFit, Larch or other programs that use this key uncertainty. The fourth column should then represent the estimate of the relative uncertainty σExperimental (cm2 g−1) or ![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi42.svg)
, also in cm2 g−1; subsequent columns could contain data related to particular contributions to the uncertainty from different sources of systematic error, e.g. σpixel. For transmission XAS data there should be estimated values of ![[\left[{{\mu} /{\rho}}\right]_{\rm pe}]](/teximages/bz5029/bz5029fi43.svg)
and their uncertainties, possibly with an explicit scattering estimate; and perhaps the last column should be σE, preferably with an indication of how the energy was estimated. If values for the imaginary component of the atomic form factor are extracted from the data, these values should also be presented in a column with their uncertainties. Common usage of such data sets might not distinguish important terms. However, distinguishing the use of the term `measured' from a mass attenuation measurement and from a measurement or extraction of a (photoelectric) mass absorption coefficient helps to clarify the importance of the terms, and is certainly important for comparison with and development of theory. The first two columns should be adequate for simple data processing, the first three for processing with uncertainties, and the later columns for investigating and interrogating this and other science more closely. Of course in CIF format every column is labelled, so the columns of data of interest can be easily extracted for any purpose. This is also the case for data in .dat format as long as the columns are labelled.
![[Figure 23]](/figures/bz5029fig23thm.gif)
|
Transmission versus fluorescence spectra for i-pr Ni. The first peak within the Hanning window is in good agreement between the spectra; note the absorption spectrum has a larger amplitude than the fluorescence spectra elsewhere. |
Sier et al. (2020) present XERT measurements and analysis of the binary crystal ZnSe, which has been the focus of a long-standing debate about the origin of anomalous Bijvoet ratios in its diffraction data. For a binary crystal, the concept of Rayleigh scattering is meaningless and instead Bragg/Laue diffraction or thermal diffuse scattering (TDS) dominate the elastic scattering with very different probabilities. Inelastic scattering might be thought of as conventional, but oriented crystals can also possess tensorial properties. The tensorial properties of ZnSe predict that there will be a major anomaly in the inelastic mean free path, especially with respect to the comparative additivity of the photoelectric mass absorption coefficient (Bourke & Chantler, 2014). This work derived and proved the dominance of thermal diffuse scattering (TDS) (Fig. 24) when Bragg/Laue scattering was avoided. There is, of course, another debate as to whether TDS is elastic or inelastic. Finding the answers to some of these questions will require further accurate experiments; other questions might turn out to be semantic.
![[Figure 24]](/figures/bz5029fig24thm.gif)
|
ZnSe binary crystal foils: attenuation of thermal diffuse scattering (blue) and Compton scattering (red) for zinc selenide across the measured energy range. |
Three high-purity zinc selenide foils were chosen for this experiment with nominal thicknesses of 25 µm, 50 µm and 100 µm. 561 attenuation coefficient data points were recorded in the energy range 6.818–15.073 keV, with measurements concentrated at the zinc and selenium pre-edge, near-edge and absorption fine-structure regions (Fig. 25). Steps of 0.5 eV were used near the edges.
![[Figure 25]](/figures/bz5029fig25thm.gif)
|
ZnSe binary crystal foils: absolute mass attenuation coefficients [μ/ρ] in cm2 g−1 and structure in the XAFS regions of the zinc (top) and selenium (bottom) K edges. |
There were two possible calibrations of the energy: that of 9.6638 (1) keV and 12.6578 (1) keV for the zinc and the selenium K-absorption edges, respectively, from external energy calibration, versus the values from Kraft et al. (1996) for zinc metal [9.66047 (8) keV] and pure selenium (Bearden, 1967; Bearden & Burr, 1967) [12.6578 (7) keV]; and taking the operational experimental edge energy as the lowest energy inflection point, yielding 9.6667 (12) keV and 12.6631 (13) keV for zinc and selenium, respectively. The accuracy was <0.13%, and led to detailed nanostructural analysis of room-temperature ZnSe with full propagation of the uncertainties. Systematic errors due to fluorescence, bandwidth, monochromator hysteresis and drift were all significant (Table 32).
|
||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
![[\left[{{\mu} /{\rho}}\right]_{\rm rel}]](/teximages/bz5029/bz5029fi214.svg)
![[\left[{{\mu} / {\rho}}\right]_{\rm abs}]](/teximages/bz5029/bz5029fi215.svg)
This data set yielded detailed information on non-atomic and atomic behaviour across wide energy ranges compared with the theory current at that time (Fig. 26). The bond lengths, which were accurate to 0.003 Å to 0.009 Å, or 0.1% to 0.3%, are plausible and physically meaningful (Fig. 27). Importantly, the structures determined independently from the Zn and the Se K edges were in excellent agreement. Small variations from the structure determined by single-crystal diffraction suggest local dynamic motion beyond that usual for a crystal lattice (note that XAFS is sensitive to dynamic correlated motion). The results obtained in this work are the most accurate to date, and comparisons with theoretically determined values of the attenuation show discrepancies from theory of up to 4%, motivating further investigations into the origin of such discrepancies.
![[Figure 26]](/figures/bz5029fig26thm.gif)
![[Figure 27]](/figures/bz5029fig27thm.gif)
|
ZnSe binary crystal foils: fitted model output (red) with experimental data (blue) and uncertainties for the (top) zinc and (bottom) selenium K edges. The black box indicates the k Hanning window. |
Five files, bz5029sup20.txt to bz5029sup24.txt, are available as supporting information to this chapter. bz5029sup20.txt is in .cif format and includes all of the data set across both edges, whereas .dat files conventionally relate to a single edge for processing by mu2chi, eFEFFit, iFEFFit or Athena, for example, so one file for each edge is provided in .dat format (bz5029sup21.txt, bz5029sup22.txt). Similarly the χ versus k spectra are specific to an edge and the relevant data depend on assumptions about E0, spline fitting and background removal. The extracted χ versus k spectra for both the Zn and Se K edges are provided for comparison and fitting with different theoretical approaches (bz5029sup23.txt, bz5029sup24.txt).
Compared with Table 33, in the .cif and .dat format files the columns of data are in a different order, so that the values most usually used for data processing are in the earlier columns. For example, in the .cif format file the ordering is E, [μ/ρ], ![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi44.svg)
, ![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi45.svg)
, ![[\left[{{\mu} / {\rho}}\right]_{\rm pe}]](/teximages/bz5029/bz5029fi46.svg)
, ![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm pe}}]](/teximages/bz5029/bz5029fi47.svg)
and σE, where all coefficient values are given in cm2 g−1, i.e. in the same units as the coefficient for ease of processing and error propagation. Note that having multiple edges in the data raises a question about how best to deposit the data, as authors may wish to provide mutiple sets of data, for example one data set for each edge, data sets for k2χ versus k or spectra transformed to R-space. Within the .cif format these could in principle all be in the one file, but it might be easiest to collect the data sets together in a dedicated folder or use a hierarchical data format (HDF) to include all processing information and outputs, even including fitting, for example.
|
![[\left[{{\mu} / {\rho}}\right]_{\rm {\rm pe}}]](/teximages/bz5029/bz5029fi216.svg)
with one standard deviation![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm {rel}}}]](/teximages/bz5029/bz5029fi217.svg)
and ![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi218.svg)
are presented with the latter also given in absolute units. The percentage uncertainty in ![[\left[{{\mu} / {\rho}}\right]_{\rm {pe}}]](/teximages/bz5029/bz5029fi219.svg)
includes uncertainty in the measurements and in the calculations of thermal diffuse and Compton scattering attenuation. (Some rows of values have been omitted for brevity. The full version is available in the supporting information.)![[\left[{{\mu} / {\rho}}\right]_{\rm pe}]](/teximages/bz5029/bz5029fi220.svg)
(cm![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi221.svg)
(%)![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi222.svg)
(%)![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi223.svg)
(cm![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm pe}}]](/teximages/bz5029/bz5029fi224.svg)
(%)Ekanayake et al. (2021a) continued the detailed investigation of zinc metal, both as XAS and XAFS, with a detailed investigation across the K edge. The analysis of the nanostructure was described in Ekanayake et al. (2021b). This required a new model for and understanding of fluorescence and fluorescence scattering (Sier et al., 2022). This was the first X-ray extended range technique (XERT)-like experiment carried out at the Australian Synchrotron, and high-accuracy measurements were recorded at 496 energies from 8.51 keV to 11.59 keV. The `relative' accuracy (neglecting the absolute calibration) is better than 0.01–0.027%; the `absolute' accuracy (including all pointwise and scaling uncertainties that were determined) is 0.023–0.036%. The XERT protocol requires that measurements related to dark-current nonlinearities, corrections for blank measurements, full-foil mapping to characterize the absolute value of the attenuation, scattering, harmonics and roughness are collected over an extended range of experimental parameter space.
This resulted in better data for analysis, culminating in measurement of mass attenuation coefficients across the zinc K edge to 0.023–0.036% accuracy (Table 34; Figs. 28, 29 and 30). Dark-current corrections are energy- and structure-dependent, and the magnitude of the corrections reached 57% for thicker samples, but was still large and significant for thin samples. Blank measurements scaled the thin-foil attenuation coefficients by 60% to 500%, and by up to even 90% for thicker foils. Full-foil mapping and characterization corrected discrepancies between foils of up to 20%, rendering the possibility of absolute measurements of attenuation. Fluorescence scattering was also significant. Harmonics, roughness and bandwidth were explored. These corrections are of course thickness-, sample-, composition-, energy-, temperature- and form-dependent, but are likely to be typical for most beamlines.
|
![[[{{\mu} / {\rho}}]_{\rm rel}]](/teximages/bz5029/bz5029fi225.svg)
if they contribute in particular to the relative structure of adjacent points, ![[[{{\mu}/ {\rho}}]_{\rm abs}]](/teximages/bz5029/bz5029fi226.svg)
if they primarily scale all values with a slowly varying function. Hence there are two final uncertainties, relating to the absolute value of the mass attenuation coefficient ![[[{{\mu} / {\rho}}]]](/teximages/bz5029/bz5029fi227.svg)
and relating to the pointwise and local structure, ![[[{{\mu} /{\rho}}]_{\rm rel}]](/teximages/bz5029/bz5029fi228.svg)
.
![[Figure 28]](/figures/bz5029fig28thm.gif)
|
Zn metal foils 8.51–11.59 keV. Mass attenuation coefficients: (a) over the energy range 8.51 keV to 11.59 keV; (b) covering the edge and XAFS region; (c) in the central XAFS region; and (d) absolute and relative percentage uncertainties. The zinc K absorption edge is observed at 9.66 keV and the associated XAFS lies between 9.66 keV and 10.10 keV. |
![[Figure 29]](/figures/bz5029fig29thm.gif)
![[Figure 30]](/figures/bz5029fig30thm.gif)
|
Zinc metal foils, 8.51–11.59 keV: data (black) with absolute uncertainties for the fine structure function above the zinc K edge produced by the mu2chi non-interpolation background subtraction software in eFEFFit and the fitted model (red) over a Hanning window k = 4.5 Å−1 to 17 Å−1. |
Four light-tight zinc foils from Goodfellow 25 mm × 25 mm in size with nominal thicknesses of 10 µm, 25 µm, 50 µm and 100 µm were chosen such that the log attenuation of the material fell between 0.5 and 6 over the energy range of the measurements at room temperature (Chantler et al., 2001a). One could in principle present data sets for each of these thicknesses, either as `raw' or `corrected' data; however, the figures presented by Ekanayake et al. (2021a) prove that this would in no way be structurally consistent with one another, and of course therefore that none of the data sets would be accurate to the level needed for detailed analysis. Hence pre-processing is important. The mass attenuation coefficient of zinc metal and the mass absorption coefficient were determined to high accuracy using an advanced wiggler beamline, and are in good agreement with values from earlier data sets collected on a bending-magnet beamline. The imaginary component of the atomic form factor and the zinc K-edge jump ratio and jump factor were determined and compared with widely varying results in the literature, representing two attempts at linking XAS and XAFS theory (for the linear or mass absorption coefficient or photoelectric effect) with experimental data. The XAFS analysis shows excellent agreement between the measured and tabulated values, and yields bond lengths and the nanostructure of zinc with uncertainties from 0.1% to 0.3%, or 0.003 Å to 0.008 Å. We observed significant variation from the reported crystal structure, suggesting local dynamic motion of the Zn atoms. XAFS is sensitive to dynamic correlated motion and in principle is capable of observing local dynamic motion beyond the reach of conventional crystallography.
Four files, bz5029sup25.txt to bz5029sup28.txt, are available as supporting information. The data shown in Table 35 are available in .dat format in bz5029sup25.txt. The file bz5029sup26.txt, also in .dat format, provides ![[[{{\mu} / {\rho}}]_{\rm tot}]](/teximages/bz5029/bz5029fi48.svg)
(cm2 g−1) versus E values with uncertainties suitable for input to eFEFFit (Smale et al., 2006; Schalken & Chantler, 2018), iFEFFit (Newville, 2001), Athena (Ravel & Newville, 2005) and mu2chi (Schalken & Chantler, 2018). bz5029sup27.txt contains the data in .cif format, and bz5029sup28.txt contains χ versus k values with uncertainties for input to eFEFFit, iFEFFit and Athena.
|
![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm {rel}}}]](/teximages/bz5029/bz5029fi229.svg)
and ![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi230.svg)
are presented with the latter also given in absolute units. The percentage uncertainty in ![[\left[{{\mu} / {\rho}}\right]_{\rm pe}]](/teximages/bz5029/bz5029fi231.svg)
includes uncertainty in the measurements and in the calculations of thermal diffuse and Compton scattering attenuation. (Some rows of values have been omitted for brevity. The full version is available in the supporting information.)![[\sigma_{\left[{{\mu} / {\rho}}\right]}]](/teximages/bz5029/bz5029fi232.svg)
(%)![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm rel}}]](/teximages/bz5029/bz5029fi233.svg)
(%)![[\left[{{\mu}/ {\rho}}\right]_{\rm pe}]](/teximages/bz5029/bz5029fi234.svg)
(cm![[\sigma_{\left[{{\mu} / {\rho}}\right]_{\rm pe}}]](/teximages/bz5029/bz5029fi235.svg)
(%)All of the data sets which report detailed XAFS also report detailed XANES and pre-edge spectra. It might seem obvious that transmission (i.e. XAS) data sets with a fine grid spacing will provide the best XANES and pre-edge detail and structure. However, many XANES and pre-edge spectra are currently collected using fluorescence spectroscopy; care must be taken to define the region of interest explicitly or implicitly by the detector chain, and also the lower cut-off for onset of characteristic fluorescence. Many experiments measure and report XANES/pre-edge spectra only, especially in reaction or evolution environments, so while the individual data sets might be small, data collection results in a significant number of data files. These can be extremely valuable, if difficult to calibrate or normalize (Streltsov et al., 2018). A key question about any data set, on a par with the challenge of spacing and grid uniformity, is the beamline or extracted resolution (the resolution after data-uncertainty determination and propagation, ready for fitting in k- or R-space). Separately, there are exciting developments for modalities such as RIXS or HERFD that are ongoing.
Acknowledgements
The author acknowledges all experimental and analytical coauthors and conceptual contributions to this work, and acknowledges the Australian National Beamline Facility, the Australian Synchrotron and the Advanced Photon Source in particular.
Supporting information
Data for Cu foils, 8.9-20 keV (Section 5). DOI: 10.1107/S1574870721011010/bz5029sup1.pdf
Data for Si crystals, 5-20 keV (Section 6). DOI: 10.1107/S1574870721011010/bz5029sup2.pdf
Data for Ag foils, 15-50 keV (Section 7). DOI: 10.1107/S1574870721011010/bz5029sup3.pdf
README file for data for Mo foils, 13.5-41.5 keV (Section 8). DOI: 10.1107/S1574870721011010/bz5029sup4.txt
Data for Mo foils, 13.5-41.5 keV (Section 8). DOI: 10.1107/S1574870721011010/bz5029sup5.txt
Data for Sn foils, 29-60 keV (Section 9). DOI: 10.1107/S1574870721011010/bz5029sup6.pdf
Data for Cu foils, 5-20 keV (Section 10). DOI: 10.1107/S1574870721011010/bz5029sup7.pdf
Data for Zn foils, 7.2-15.2 keV (Section 11). DOI: 10.1107/S1574870721011010/bz5029sup8.pdf
Data for Au foils, 38-50 keV (Section 12). DOI: 10.1107/S1574870721011010/bz5029sup9.pdf
Data for Au foils, 14.2-21.1 keV (Section 13). DOI: 10.1107/S1574870721011010/bz5029sup10.pdf
Data for Ag foils, 5-20.1 keV (Section 14). DOI: 10.1107/S1574870721011010/bz5029sup11.pdf
Data for Ag foils, 11.0-28.1 keV (Section 15). DOI: 10.1107/S1574870721011010/bz5029sup12.pdf
Dilute solutions of Ni(II) complexes (Section 16). DOI: 10.1107/S1574870721011010/bz5029sup13.pdf
Dilute solutions of ferrocene and decamethylferrocene (Section 17). DOI: 10.1107/S1574870721011010/bz5029sup14.pdf
Dilute solution of bis(N-n-propylsalicylaldiminato)nickel(II), .dat format (Section 18). DOI: 10.1107/S1574870721011010/bz5029sup15.txt
Dilute solution of bis(N-isopropylsalicylaldiminato)nickel(II), .cif format (Section 19). DOI: 10.1107/S1574870721011010/bz5029sup16.txt
Dilute solution of bis(N-isopropylsalicylaldiminato)nickel(II), .dat format (Section 19). DOI: 10.1107/S1574870721011010/bz5029sup17.txt
Dilute solution of bis(N-n-propylsalicylaldiminato)nickel(II), .cif format (Section 19). DOI: 10.1107/S1574870721011010/bz5029sup18.txt
Dilute solution of bis(N-n-propylsalicylaldiminato)nickel(II), .dat format (Section 19). DOI: 10.1107/S1574870721011010/bz5029sup19.txt
ZnSe crystal foils, .cif format (Section 20). DOI: 10.1107/S1574870721011010/bz5029sup20.txt
ZnSe crystal foils, Zn edge, .dat format (Section 20). DOI: 10.1107/S1574870721011010/bz5029sup21.txt
ZnSe crystal foils, Se edge, .dat format (Section 20). DOI: 10.1107/S1574870721011010/bz5029sup22.txt
ZnSe crystal foils, chi vs k for Zn edge, .dat format (Section 20). DOI: 10.1107/S1574870721011010/bz5029sup23.txt
ZnSe crystal foils, chi vs k for Se edge, .dat format (Section 20). DOI: 10.1107/S1574870721011010/bz5029sup24.txt
Data for Zn foils 8.51-11.59 keV, .cif format (Section 21). DOI: 10.1107/S1574870721011010/bz5029sup25.txt
Data for Zn foils 8.51-11.59 keV, .dat format (Section 21). DOI: 10.1107/S1574870721011010/bz5029sup26.txt
Zn foils 8.51-11.59 keV, chi vs k, .cif format (Section 21). DOI: 10.1107/S1574870721011010/bz5029sup27.txt
Zn foils 8.51-11.59 keV, chi vs k, .dat format (Section 21). DOI: 10.1107/S1574870721011010/bz5029sup28.txt
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