An ideal starting point for any measurement is the presentation of a well defined, well characterized homogeneous sample of appropriate thickness, density, composition and concentration. Overviews of the requirements and strategies for sample preparation are available in monographs (Calvin, 2013), which are commonly supplemented by guidance from the different beamline teams. Clearly, samples which are inhomogeneous on the micrometre scale (roughness, wedges or curved surfaces) or discontinuous (pinholes or cracking) will give more challenging spectra subject to ill-defined distortions arising from time-dependent translation of the sampling volume during the collection of the spectrum. Solution or frozen solution samples can give homogeneous samples, but photolysis/photoreduction of the solute and crystallization of solvent during the generation of frozen solutions are potential complications. In general, a flat metal foil or a well defined crystal is still inhomogeneous on the micrometre scale, so methodologies of XERT are required to characterize them and the ensuing systematics to yield high accuracy. This is no less the case for solution or frozen solution samples.

A critical requirement in hybrid techniques is assessment of the error of the individual points of the spectrum. In effect, this typically requires point-by-point measurement in which blank, background, scattering and multiple sample measurements are conducted at each energy step using samples with multiple concentrations or thicknesses. Further, measurements related to beam quality (scattering, harmonic component and energy) are needed to correct for key systematic errors. Repeated measurement of data points is a common requirement for all XERT and hybrid applications, with preferably towards ten repeat measurements of each sample,blank or reference. Published protocols for data collection and error propagation for transmission measurements are summarized in Section 3.

The measurement of dilute samples is a requirement that is common to the majority of XAS investigations extending over all discipline areas. In this context, `dilute' is interpreted to span the range from 0.5 to 10 mM for first-row transition metals (i.e. ∼25–500 p.p.m.). In conventional wisdom, this corresponds to samples for which transmission measurements are either not practicable or give very poor spectra compared with those obtained using fluorescence detection. Notwithstanding this prejudice, transmission measurements can remain competitive at lower concentrations and, importantly, provide the experimental information needed for normalization of the fluorescence signal. Multi-element germanium detectors are most commonly employed at XAS beamlines, largely due to their sensitivity, energy discrimination and large solid area of detection. In conventional experiments, detectors of this sort are used to provide a single averaged output, although the processing of this output signal is not entirely straightforward. In hybrid experiments the individual detector pixel elements are independent observations which are used to extract both the signal and its associated uncertainty. As discussed in later sections, the energy-dependent normalization of individual detector elements complicates data processing but has the advantage of providing a statistical basis for assessing the uncertainty of the individual spectral data points.

The strategies outlined here apply to transmission and to fluorescence measurements, to high-accuracy (XERT) and to intermediate advances and developments (hybrid), and to second-generation (bending-magnet) beamlines and to advanced third-generation beamlines, for example wigglers and undulators. The implementation of the principles described herein are, of course, highly beamline-specific and need to be commensurate with the objectives of the application.

Photodamage, principally photoreduction, is an ever-present hazard in XAS measurements, especially for dilute, fragile or biological samples, and cryogenic methods are commonly used to mitigate these effects sufficient to allow the collection of spectra over a useful k-range. A critical requirement of hybrid measurements is the illumination of multiple sample positions at each energy point. This can generally be achieved by physical translation of the cryostat to allow the interrogation of sample positions within the clear aperture provided by the entrance and exit windows (typically ∼30 mm in diameter). The pressure differential implies that exterior cryostat windows are not flat, and cryostat designs often expect a curved profile for the radiation-shield windows. These effects lead to small differences in the background absorption which are dependent on the cryostat position. The differences are small and are typically 1–2% of the absorption due to the Kapton windows and the air path which, in turn, are small corrections to the absorption cross-section.

The physical restrictions on the area available for samples/references imposed by the cryostat geometry impact significantly on the experiment design. Individual samples will ideally have dimensions sufficient to allow the measurement of spectra from at least three non-overlapping regions, limiting the number of samples able to be studied in a single experiment. For solution samples, it is challenging to achieve a compact placement of solution wells, which is compounded when it might be necessary to choose solvents which prevent the use of self-adhesive Kapton windows. Strategies for dealing with this problem are available in the literature (Chantler et al., 2015). Alternate approaches include the assembly of pre-frozen solution samples or the implementation of a strategy in which the sample is translated while the cryostat remains static.

Unlike conventional measurements, which involve sample movement at the completion of spectral acquisition, hybrid measurements are typically conducted in a single energy scan with translation of the cryostat typically to 4–8 different positions at each energy with repeats (three sample, three dilute sample, one background and one blank). While the travel for each movement is comparatively small, the reliability of the stepper motors driving the cryostat stage is an important and beamline-dependent consideration. Further, each measurement should consist of sufficient repeat observations, preferably ∼10, so as to allow statistical estimation of the measurement variance.

An advantage of hybrid measurements is the potential to optimize the data-collection strategy according to the requirements of the experiment and as a function of k or E. At each energy point measurements of beam or sample diagnostics may be conducted, and this can be implemented at an energy spacing that matches the strengths or limitations of the beamline and the information content of the sample XAS spectrum being collected (i.e. the XAFS oscillations, resolution, structure etc.). This approach can, for example, include measurement of multiple calibration foils (Chantler, Islam et al., 2012), recognize measurements where high I₀ (for example monochromator glitches) or I_f (for example Bragg diffraction) intensity would otherwise lead to unusable data, and either apply correction protocols or alter the energy of the measurement (Tran, Chantler et al., 2003; Sutter et al., 2016) or simply provide reliable stop and restart protocols following beam loss (Chantler, Tran, Paterson, Barnea et al., 2000).

A beamline or facility manager may question the cost of building and designing an endstation to achieve accuracy or precision, pre-processing software and investing time in the pre-processing of data to automate the determination of precision and pointwise variance. The benefit is often well defined data quality and potentially accuracies some 10–100 times previous convention. This seems to be a simple argument and a valid priority. Once implemented, as a specialized experiment or as a routine facility option, operational considerations come into play. There is a real cost in experimental beamtime allocation and investigation for defining data quality using hybrid or high-accuracy techniques, for the proposer, for the expected time for a de novo experiment and for the increased time for a routine well set up beamline mode for XAFS; for example, in comparison to a standard beamtime XAFS, XANES or a pre-edge fingerprint comparison. In many cases a simple XANES fingerprinting is effective and valuable, although conclusions without EXAFS analysis are more liable to give incorrect answers. Hence, the quest for greater insight and greater validity of conclusions should lead us to new technology and implementation.

Theoretical errors can still often dominate in particular regions, especially at low k, which can be cited as a reason for not bothering to collect data of defined quality; yet many experiments seeking to develop our understanding of particular systems or to develop theory or find new physics need this information and insight. Equally, extracting the XAFS using E₀, splines and other techniques can introduce large and systematic errors in parameters, especially at low k. It remains very important to try to better define uncertainties or optimize data-collection techniques in order to define data quality more carefully. The discussions of this chapter do not address all possible systematics (if indeed they were known) so one could be concerned that for a particular application there may be a specific failure of current or developing methods. The solution to this meta-question is to fix known systematic issues and then to fix more. This is always a work in progress.

The cost–benefit analysis of the data-collection strategies hinge on the time side of the equation (Table 1). Whilst any such comparison is highly dependent on the sample, a reasonable estimate of the increase in time required for optimized hybrid measurements of XAS spectra relative to conventional methods is a factor of approximately 3 or 6, depending on whether the diluted form of the sample is counted separately. Approximately half of this difference is accounted for by the time required to collect beam and detector diagnostics and these carry the greatest time overhead in the first measured spectrum. With the improved beamline diagnostics provided by experiments of this type, it is possible to more efficiently utilize observations of these diagnostics between runs and lower the time penalty. The major gains in the accuracy and reliability of the data can be achieved without a significant increase in beam-on-sample time. Case studies which illustrate both the protocols for establishing the uncertainties of the individual data points and potential improvements in the quality and the reliability of the analysis are given in Section 6.

Beam harmonic energy composition (contamination) leads to nonlinear changes in the apparent absorption coefficient with thickness. This affects the monitor signal in both transmission and fluorescence experiments. Often, the monochromator design dictates that the third harmonic is the dominant contamination, but this can vary. The second harmonic, while forbidden for many monochromator reflections, can be present. Clearly, this would affect the monitor signal in both transmission and fluorescence experiments. The fluorescence detector usually sets a region of interest (ROI) to exclude this effect in the detector, but also causes a systematic error in the coefficient due to the upstream signal. Hence, measurements with different thicknesses of a flat homogeneous sample will yield different attenuation coefficients. In transmission this can lead to a damping of the oscillations (with increasing harmonic or thickness; Lu et al., 1989; Ovsyannikov & Shuvaev, 1995), while in fluorescence it will tend to lead to a loss of efficiency and hence measured co­efficient, although this thickness effect will saturate towards higher thicknesses. At lower energies a harmonic rejection mirror can be very effective, but a level of harmonics after the rejection mirror is often present but not calibrated. The harmonics usually have a lower attenuation coefficient, usually with no XAFS structure or edge, so under these circumstances the harmonic contamination adds an energy-dependent background function to the spectrum, and in general the oscillations are therefore effectively damped. For a single harmonic in transmission, the relation is given bywhere α is the effective harmonic fraction and [μ/ρ]_f and [μ/ρ]_h are the attenuation coefficients at the fundamental energy (usually that being investigated) and at the energy of the harmonic contamination, respectively. Under some circumstances an extremely thin sample limit can be used to minimize the impact of harmonics, although this depends upon a small α and high attenuation for the fundamental. The harmonic coefficient can often be represented to high accuracy by theoretical tables (Chantler, 1995, 2000) to enable clear quantification of α and extraction of the true attenuation coefficient from the data. This is most easily computed and corrected using multiple samples of a reference foil (Glover & Chantler, 2009) or a material such as aluminium (Tran, Barnea et al., 2003) that likewise probes the functional of the harmonic spectrum. Uncertainties in α can be below 0.01%. In fluorescence, with multipixel detectors or otherwise, the harmonic frequencies will not fall within the ROI and an apparent attenuated fluorescence yield will result.

Bandwidth is defined as the spread of energies across the upstream X-ray beam entering the monitor detector. Transmission and fluorescence spectra of a sample recorded using sources with different bandwidths will exhibit differences in the edge location and the resolution, with thickness-dependent differences in attenuation coefficients and the apparent edge structure (Parratt et al., 1957; Pease, 1976). Bandwidth effects can have an arbitrary functional form, especially, for example, off-axis. A general expression for the broadening of the spectrum is given as (Hayakawa et al., 1991)where the relation between each component I is given above and m is the measured value of the central energy of the distribution function given by E_mono. Compared with the value obtained for a single energy component (the `true' value ), the correction of the measured attenuation co­efficient, say across an edge, in a linear regime (de Jonge et al., 2004b; Tantau et al., 2015; Glover & Chantler, 2009) can be given or approximated by

The functional form can be deduced from multiple thickness measurements, although the effect does not disappear in the limit of zero thickness. If the distribution is approximately Gaussian or truncated Gaussian with a given FWHM, then the second-order approximation of equation (14) is compact and valid across a broader range, allowing the efficient extraction of the ideal edge profile with higher resolution (Tantau et al., 2015),

Monochromator glitches (secondary Bragg diffraction peaks) primarily yield a change in incident intensity and secondarily in energy. The intensity variation should be fully corrected for in transmission if the detectors are perfectly linear, in turn requiring valid dead-time correction and avoidance of saturation (Diaz-Moreno & Strange, 2018; Pease, 1976; Koningsberger & Prins, 1988; Krappe & Rossner, 1999; Tran, Barnea et al., 2003; Pettifer et al., 2005; Bunker, 2010). Such glitches have been documented by beamline scientists for a particular monocrystal at a particular beam position, for example on the Diamond Light Source beamlines (Diaz-Moreno, 2012; Sutter et al., 2016; Diaz-Moreno & Strange, 2018). However, such glitches in general are large and dispersive. The upstream (monitor) detector and the downstream transmission detector are usually matched in attenuation and gas type, so these changes in efficiencies normalize using XERT (de Jonge et al., 2004a, 2005; Tran, Chantler et al., 2004; Chantler, 2009; Glover et al., 2010; Rae et al., 2010; Chantler, Barnea et al., 2012; Tantau et al., 2014; Chantler et al., 2015). However, in fluorescence the changes in efficiency and linearity (with energy or flux, for example, towards dead-time corrections and saturation) do not in general match those of the monitor, so we usually observe that glitches in transmission normalize out (for example using XERT), while glitches in fluorescence remain in the data (Trevorah et al., 2019) and most users delete these points (see Sutter, 2021; Ciatto et al., 2023).

Bragg diffraction peaks will also arise from the sample, whether due to low-temperature frozen-sample crystallization of solvent or oriented crystalline samples such as perfect silicon. In this case these `Bragg glitches' can destroy the value of the data for XAFS investigation, but they can be uniquely identified from the multiple samples with small changes of orientation and dealt with accordingly (de Jonge et al., 2004a, 2005; Tran, Chantler et al., 2004; Chantler, 2009; Glover et al., 2010; Rae et al., 2010; Chantler, Barnea et al., 2012; Tantau et al., 2014; Chantler et al., 2015).

The spatial beam structure may exhibit non-uniformity and divergence or convergence in both horizontal and vertical axes, leading to distortion of the spectra of heterogeneous or structured (variable-thickness) samples. In both transmission and fluorescence measurements this yields an averaging or broadening of the spectrum. In a flat uniform sample, the spatial (x, y) beam structure cancels in transmission, while the fluorescence signature can lead to a broadening due to the consequential angular dependence. The angular structure of the beam (divergence or convergence in θ_x and θ_y and non-uniformity) can lead to broadening and distortion of the spectrum even for a uniform homogeneous sample in both transmission and fluorescence.

Any deviation of the sample from a perfect flat plate, and any variation of thickness or impurity or composition across the beam profile, will yield a different value for the mass absorption coefficient from the sample. These `thickness' effects depend upon the variation of sample thickness or orientation in the beam as opposed to the `thickness effects' discussed in Sections 5.1–5.4, which are due to the beam distribution and profile and would exist for any sample thickness. Most extreme are non-light-tight samples with pinholes or micro-cracks, but this effect can arise from samples in the form of a perfect wedge, a curved surface or thickness profile, or any variation of the integrated column density thickness or composition (Stern & Kim, 1981; Babanov et al., 2001). The most general form of this is a definition of the roughness of the sample. Both transmission and fluorescence spectra are affected and distorted and may be rendered unusable unless there is minimal relative movement of the X-ray beam and sample during the measurement and there is no energy-dependent change in the spatial beam structure.

Any homogeneous sample will have nonuniformities of the sample thickness on the nanoscale which can be characterized by mapping the absorption coefficient over an area of the sample. In XERT, thickness mapping is used to define an absolute attenuation compared with the point of the sample through which the X-ray beam passes, compared with the average thickness determined separately (Rae et al., 2010). Roughness in transmission then represents the moments of the integrated column density functional within the beam footprint through the sample. Measurement of the roughness of the sample under the beam can explain the nature of the sample origin, for example, from crystal growth, from tooling, from a local wedge and other patterns. We define roughness as the variability in (ρt), the integrated column density, seen by the X-ray beam passing through the sample (Chantler, Tran, Paterson, Cookson et al., 2000), where P is the path of the X-ray beam and ρ(x) is the density of the sample at point x. We define the magnitude of the roughness of a sample (in the region of the X-ray beam) using the variance of the integrated column density given bywhere S corresponds to the region of the sample being illuminated with area A, assuming that the foil sample is oriented normal to the beam. Then, assuming uniform incident intensity across the beam profile (see above), the roughness changes the measured mass attenuation coefficient,from which the change of the measured mass attenuation coefficient from that of a perfectly flat surface is given byHence, the measurement of attenuation coefficients relative to a smooth reference foil will yield a measurement of, and a correction for, nano-roughness which has a clear dependence on the mass attenuation coefficient.

Because two angles are involved, the impact of roughness on fluorescence measurements is more complicated than that for the transmission case, and indeed is much more significant for any of the grazing- or normal-incidence geometries in which the incident or outgoing angles are small. However, in conventional (45°) fluorescence measurements roughness per se may be a minor systematic, particularly for solution cells, which will often have smooth (flat or curved) windows. While curved windows give a sample position-dependent non-uniform background and can be different for the blank sample, a blank measurement is generally sufficient to address the large first-order effect of this correction. Of course, this consideration is sample-dependent.

The primary beam is absorbed and attenuated as a function of thickness or penetration depth. In transmission, the attenuated beam gives the mass attenuation coefficient from the measurement and can be corrected to give the mass absorption coefficient by removing the (nonlinear) contribution to attenuation from scattering if known or measured; if not, this contributes a general background loss function due to scattering, i.e. to attenuation versus absorption, and to deviations from the Beer–Lambert relation. Corrections for these effects are straightforward based on measurements from samples of different thickness. In transmission there is often a significant fluorescence signal of characteristic radiation relating to the absorption edge involved; this can be corrected for using different calibrated apertures. However, in fluorescent detection this lower energy photon is the primary detected event in the fluorescence detector, and there will be self-absorption (self-attenuation) as the fluorescent photons are created in the sample and propagate towards the detector. Fluorescence measurements present significant complications owing to the geometry dependence of the path length for the incident and fluorescent photons (Tan et al., 1989; Hayakawa et al., 1991; Tröger et al., 1992; Iida & Noma, 1993; Meitzner & Fischer, 2002). The impact of this systematic is especially obvious in multi-pixel detection in fluorescence measurements, where self-absorption is responsible for the dominant angular dependence of the spectra from each pixel.

Fluorescence measurement, which was developed as a particularly valuable approach for dilute systems in the 1970s and 1980s (Lee et al., 1981; Jaklevic et al., 1977, 1993; Pease et al., 1989), has since become the technique of choice for most XAS studies, but suffers from detector nonlinearities, dead-time correction, ROI issues and sample-thickness effects (Goulon et al., 1982). It has also spawned such techniques as resonant inelastic X-ray scattering (RIXS), as described in Glatzel et al. (2022). Individual pixels will have individual dead-time corrections and individual amplification and this should generally have been calibrated at the beginning of the experiment or during pre-processing (see De Panfilis & Bardelli, 2023). Fluorescence measurements present significant complications owing to the geometry dependence of the path length of the incident and fluorescent photons (Tan et al., 1989; Hayakawa et al., 1991; Tröger et al., 1992; Iida & Noma, 1993; Meitzner & Fischer, 2002). The impact of this systematic is especially obvious in multi-pixel detection in fluorescence measurements, where self-absorption is responsible for the dominant angular dependence of the spectra from each pixel. After early work on this using a known stoichiometry (Goulon et al., 1982; Tröger et al., 1992), multiple measurements at different detection angles (Eisebitt et al., 1993) or known energy dependence of absorption coefficients (Pfalzer et al., 1999), general approaches have been investigated (Booth & Bridges, 2005; Chantler, Rae et al., 2012; Trevorah et al., 2019).

The core equation for a fluorescence-detected measurement is given as (Jaklevic et al., 1977; Pease et al., 1989; Trevorah et al., 2019; Booth & Bridges, 2005; Chantler, Rae et al., 2012)where I_f is the number of fluorescent photons emitted from the surface of the material, the asterisk indicates that only the component absorbed in the absorber producing a fluorescent photon is relevant, `pe' signifies that only the photoelectric component can yield a fluorescent photon, Ω represents the solid angle subtended by each detector pixel, θ<sub>inc</sub> is the incident angle of the X-rays onto the sample, θ<sub>out</sub> is the emission angle of the fluorescent X-rays from the sample and [μ<sub>f</sub>/ρ] is the mass attenuation coefficient of the material at the energy of the fluorescent photon. Because this functional depends upon the path length and angle, it also depends upon any thickness inhomogeneity and the angle of alignment of both the sample and the detector (pixel) location. It is assumed that this formula is corrected for dark current at least in the upstream detector, and it is assumed that any ROI (region of interest) set includes all of these photons and excludes any photons from different processes which might overlap in energy and space. To consider the number of detected measured photons, we in general consider the functional for the energy-dependent factors (upstream, A, for absorption corrections) as given in equation (20). Note that the ROI may overlap with the elastic peak tail, which can complicate the analysis further.where ɛ<sub>det</sub> and ɛ<sub>mon</sub> represent the energy-dependent detector quantum efficiency, t<sub>air</sub>/cosθ<sub>air</sub> is the path length of the photons from the monitor to the front surface of the sample, and similarly for any window materials w, the monitor gas path m and the cryostat or other gas path c. Curvature of the Kapton windows at the air–vacuum interface of the cryostat has been treated as a minor perturbation. Additives such as silicone adhesive are included in the calculation of [μ/ρ]<sub>w</sub>. The [μ/ρ] X-ray mass absorption coefficient of each component material is energy-dependent. For the fluorescent photons emitted from the surface and measured using a multi-pixel detector, a second functional is required,where all symbols have their previous meaning except that F, downstream, self-absorption corrections, describes the path lengths from the sample to the detected photon at a particular pixel and the fluorescent photons must use attenuation co­efficients for the fluorescent photon [μ/ρ]<sub>f</sub>. Since the energy of each fluorescent photon is not dependent upon the incident energy, these factors are not energy-dependent but they are geometry-dependent, i.e. angle-dependent and pixel-dependent. Numerous sources give idealized simplifications of a very thin, concentrated solution limit and a thick, dilute solution limit which, however, hide the significant nonlinearities of most real spectra. It is also important to recognize the formal lack of an inversion formula, which however has been addressed, for example, in the SeAFFlux package (Trevorah et al., 2019). In particular, as soon as one sees the fluorescent spectra diverging for different pixels away from the edge, or increasing with energy or k above the edge, one knows that the nonlinearities and `thickness effects' are dominant. Nonlinearities include X-ray scattering and X-ray Raman scattering (Glatzel et al., 2022), including matrix effects (Bunker, 2023a; Heald, 2023; d'Acapito, 2023), and the positive slope should be extracted in order to achieve a linear response, which is then interpretable as an absorption or attenuation coefficient (Trevorah et al., 2019). Analytic and computational estimates of the correction for this in the mass absorption coefficient and hence in the XAFS functional are treated in detail elsewhere (Trevorah et al., 2019).

Particular experimental systematic errors may invite double transforms with filtering, which is discussed and commended by Bunker (2023c) and Newville (2023).

Fits in either k-space or r-space must look at all paths within the range. It is certainly necessary to consider outer shells and multiple paths, but this can be performed intelligently in k-space. In r-space the FT limitations, offsets and images of lower paths increase the systematics, make the analysis more complex and problematic, and certainly decrease the information content, especially when there are no propagated error bars. Further, in r-space it is common to `clean' the transform with low and high filters, which both destroys data and introduces artefacts. In k-space the real data are used directly (see Bunker, 2023c,d; Liu & Frenkel, 2022; Kavčič & Žitnik, 2023; Newville, 2023).

For later first-row transition metals the relative energies of square-planar (D_4h) and tetrahedral geometries of four-coordinate complexes can be finely balanced, and this can impact on the chemistry and catalysis of these metal ions in chemical and biological environments. Distortions from the limiting high-symmetry geometry will depend both on intramolecular and intermolecular interactions, and EXAFS can play an important role in establishing the geometry in cases where crystalline diffraction cannot be used to solve the structure, for example for complexes in disordered environments.

The differing geometries of four-coordinate metal complexes reflect an electronic preference for the D_4h form for particular d-electron counts (for example d⁸) and the minimization of ligand–ligand repulsion by the adoption of tetrahedral geometry. A good example of this effect is provided by the N-propylsalicylaldiminato complexes of nickel(II), which give neutral compounds with N₂O₂ coordination of the metal (Sacconi, 1966; Fox et al., 1963; Holm & Swaminathan, 1963). The geometry of the compound can be tuned by steric effects associated with the N-alkyl group. In the solid state, the n-propyl compound (Ni-ⁿPr-Sal) has an approximately planar geometry, while the more sterically demanding i-propyl form (Ni-ⁱPr-Sal) has distorted tetrahedral coordination (Fox et al., 1964; Britton & Pignolet, 1989; Fig. 3). The difference in geometry is reflected by their spectroscopic and magnetic properties (planar and diamagnetic or tetrahedral and paramagnetic) and impacts on chemical properties such as their tendency to form dimers (Benelli et al., 1972; Hill et al., 1973; Knorr et al., 2007). Molecular-weight determination for the solution species suggests that the monomeric species dominates (Sacconi et al., 1964). The temperature dependence of the magnetization has been interpreted in terms of square planar–tetrahedral isomerism, in which the complexes with a secondary C atom bound to the N atom (Ni-ⁱPr-Sal) yield predominantly tetrahedral coordination, particularly for solutions at higher temperatures (Sacconi et al., 1963, 1964; Sacconi, 1966).

Figure 3 X-ray structures of bis(N-propylsalicylaldiminato)nickel(II) (Ni-nPr-Sal; Britton & Pignolet, 1989) and bis(N-isopropylsalicylaldiminato)nickel(II) (Ni-iPr-Sal; Fox et al., 1964). For clarity, H atoms are omitted.

6.1.1. EXAFS data extraction for transmission and fluorescence measurements

XAS spectra of frozen solutions of Ni-ⁿPr-Sal and Ni-ⁱPr-Sal using an acetonitrile (40%)/butyro­nitrile (60%) mixed solvent contained in a multiposition solution cell allow the measurement of samples with differing concentrations together with the requisite background blank channels. Corrections for the transmittance spectra followed the approach shown in Fig. 2 and the resulting spectra for 15 and 1.5 mM solutions of pure (Ni-ⁿPr-Sal) at T = 80 K together with the point-by-point estimate of errors of order 0.01% are given in Fig. 4.

Figure 4 X-ray absorption spectra using two solutions (15 and 1.5 mM) of Ni-nPr-Sal and the acetonitrile/butyronitrile mixed solvent together with determination of their corresponding uncertainty or the independent quality of the data. Past conventions would reject these absorption data on the basis of poor signal-to-background ratios and neglect the estimation of data quality of about 0.01% or better. Reproduced with permission from Chantler et al. (2015). Copyright 2015 International Union of Crystallography.

Measurements of the fluorescence have been normalized and statistical analysis conducted in the manner outlined in Section 5.6 and this provides [μ/ρ] values and their associated uncertainties, which are consistent with the transmission measurements. In addition to the hybrid measurements, which give data with high point accuracy (HPA), a more conventional XAS scan was conducted to give a data set with high point density (HPD). In this case the measurements with HPA can be used to provide corrections for the HPD data and, for the fluorescence measurements, statistical analysis of the spectra obtained from the multi-element detector (Section 5.6) allows a statistically based assessment of the point-by-point uncertainty (Fig. 5).

Figure 5 Transmission versus fluorescence spectra from Schalken & Chantler (2018). An excellent agreement is found between the two spectra within the Hanning window, indicative of the success of the self-absorption correction methodology. Some discrepancies remain between the two spectra, particularly in the application of background subtraction, edge removal and spline removal, an observation made possible by propagation of statistically robust experimental uncertainties.

6.1.2. XANES analysis

The logical starting point for the analysis is to focus on the pre-edge structure, where the intensity of the 1s–3d transition is expected to be significantly more intense for the noncentrosymmetric tetrahedral form. While this expectation is reflected in the XANES of tetrahedral and octahedral cobalt(II) complexes (Best, Kolev et al., 2016) and of high-valent chromium complexes (Harris et al., 2005; Parsons et al., 2007), the spectra of Ni-ⁿPr-Sal and Ni-ⁱPr-Sal have pre-edge features of similar intensity (Fig. 6). Differences in the pre-edge and edge spectra are indicative of differences in the geometry of the two isomers in the solution state, but it is clear that a simple interpretation of the XANES is not sufficient to allow assignment of the geometry.

Figure 6 Ni-nPr-Sal and Ni-iPr-Sal in the pre-edge and edge regions. The relative intensities of the 1s–3d pre-edge transitions do not give a clear assignment of the geometry.

6.1.3. Modelling the EXAFS data

Owing to the limited extent of the EXAFS data, it is sensible first to consider the complexity of the model needed to give a good reproduction of the experimental observations. The simplest model is limited to the atoms that are directly bonded to the nickel absorber (region A in Fig. 7), where the inclusion of three-leg paths leads to differences in the computed EXAFS for the square-planar and tetrahedral forms. Inclusion of more atoms into the model by extending to regions B and C will increase the number of paths where, importantly, only the multiple-scattering paths involving both ligands in the calculated EXAFS will be sensitive to the rotation angle φ. The most simple structure refinement would use the crystallographically determined intra-ligand and metal–ligand bond lengths and optimize E₀, , Debye–Waller factors and, possibly, a global scale factor for models with φ equal to 0° and 90°.

Figure 7 Models of increasing complexity can be constructed using the atoms bounded within regions A, B and C. The H atoms are assumed to make an insignificant contribution to the spectra and are omitted from the models. Distortion from the square-planar geometry (shown) to tetrahedral involves a rotation of the plane of one of the bidentate ligands (φ) by 90°. The two bidentate ligands are assumed to have identical geometries.

A summary of the refinement of the transmission EXAFS data from Ni-ⁿPr-Sal and Ni-ⁱPr-Sal is given in Table 2, where the reference point is a model based on the X-ray structures of the two compounds with optimization of the Ni–O and Ni–N distances. It is important to note that the refinement was conducted using a development of iFEFFIT, which takes account of the uncertainties associated with the individual data points (eFEFFIT), and the values are statistically based and are not directly comparable with the conventional EXAFS analysis normalization (Islam et al., 2015). There are several points that can be made from an analysis of differences or improvements of refinement parameters (Table 2). Firstly, the statistical significance of changes to the complexity of the model are much more clearly articulated through a properly determined value of . Secondly, the small differences in bond lengths associated with a change in state (from crystal XRD to solution XAFS) show up strongly in the XAFS, so care and insight needs to be exercised when transferring or evolving bond-length data from crystal to solution. Thirdly, fragmentary models can provide a useful way of identifying the parts of the structure that are most critical for analysis. Clearly, it is important that the model fragment has sufficient complexity to capture the main contributions to the XAFS. Here, the contributions of the first, second and third shells are significant, and the symmetrized structure gives at least as good a fit of the solution-state EXAFS as the solid-state X-ray structure. It is well known that > 1 is a consequence of (i) poor estimates or underestimates of point uncertainty, (ii) a loss of information in interpolation of data or transformation to the fitting axis of χ(k) or χ(r) and (iii) a poor or incomplete theoretical model encompassing the parameterization of the nuclear and thermal coordinates (see Abe, 2023; Booth, 2022a,b; Newville, 2023; Fernandez-Garcia & Kubacka, 2023). It is also common that a `significant' reduction in is indicative of a significantly better structure, model, theory etc. and this is in part formalized by the F-test, amongst others, which characterizes the χ² difference relative to the change of the number of degrees of freedom (see Booth, 2022a; Newville, 2023; Timoshenko & Kuzmin, 2023; Krappe et al., 2023).

Table 2 Summary of the main refinement parameters obtained using IFEFFIT-like (eFEFFIT) analysis of the transmission EXAFS data from Ni-iPr-Sal and Ni-nPr-Sal   Ni-iPr-Sal data; XRD or φ = 90° models Ni-nPr-Sal data; XRD or φ = 0° models Model E0 (eV) α σ2 (µm2) E0 (eV) α σ2 (µm2) XRDopt† −2.6 (1.0) 1.27 (0.31) 1.0010 (0.0003) 0.005 (0.003) 5.47 −2.9 (1.3) 1.31 (0.34) 1.048 (0.015) 0.010 (0.004) 6.17 Fig. 7, A‡ −8.6 (2.3) 0.98 (0.11) 0.990 (0.030) 0.005 (0.004) 6.82 −13.9 (6.5) 1.11 (0.31) 0.980 (0.020) 0.007 (0.005) 8.73 Fig. 7, B −2.6 (1.3) 1.00 (0.11) 0.996 (0.006) 0.004 (0.003) 4.41 0.1 (1.4) 0.95 (0.13) 1.008 (0.007) 0.005 (0.003) 4.30 Fig. 7, C§ 0.87 (1.16) 1.01 (0.03) 0.998 (0.003) 0.005 (0.002) 3.25 2.6 (1.5) 0.89 (0.04) 1.004 (0.006) 0.007 (0.003) 3.62 Fig. 7, C*¶ 0.62 (0.28) 1.02 (0.02) 1.001 (0.003) 0.003 (0.002) 2.94 2.3 (1.1) 0.91 (0.02) 1.007 (0.003) 0.006 (0.003) 3.27 †Structural parameters were obtained from the XRD crystal structures (Fox et al., 1964; Britton & Pignolet, 1989) but with optimized Ni–O and Ni–N distances (using eFEFFIT). ‡Optimized Ni–N and Ni–O distances; C–C and C–N distances and the N–Ni–O angle (93°) were based on the crystallographic database distribution function of similar groups. §Debye–Waller factors were independently refined for the Ni–O [0.0010 (5)] and Ni–N [0.0015 (1)] interactions. ¶Extension of the model by adjustment of the N—Ni—O angle of the tetrahedral (89.2°) and square-planar (89.5°) species; the parameter α is the scale of bond lengths from the initial model structure.

It is well understood that the complexity of model able to be supported by XAFS analysis is dependent on the quality and the extent of the experimental data, and for dilute samples fluorescence measurements can offer significant advantages. EXAFS analysis using fluorescence measurements collected from the same sample with HPA and HPD have been shown to support a more nuanced analysis, with a 2D parameter search giving optimized Ni—O and Ni—N bond distances and O—N—N angle. The optimized model provides a basis for evaluating the solution structures of the Ni-ⁿPr-Sal and Ni-ⁱPr-Sal complexes, where differences in the fits of the experimental data to the φ = 0° and 90° models allow a statistically based assignment of the structure (Fig. 8; Schalken & Chantler, 2018). While it is apparent that there is a distortion of the EXAFS data for both compounds in the range 8.5–9 Å⁻¹, there is a clear distinction between the fits of the EXAFS of the two compounds to either the φ = 0° or 90° models, allowing the assignment of Ni-ⁿPr-Sal to a distorted square-planar geometry and Ni-ⁱPr-Sal to a distorted tetrahedral geometry. Further, there are significant distortions of the solid-state structures for the complexes in solution. The most important outcome from this analysis is that the differences in geometry can be assessed from a solid statistical basis and that improvements in the beamline and the time available for the measurement of spectra will lead to quantifiable improvements in the quality of the data, which then translates into more precise and reliable structure determination.

Figure 8 High-point-density fluorescence EXAFS of frozen solutions of Ni-nPr-Sal and Ni-iPr-Sal fitted to the optimized `square-planar' (φ = 0°) or optimized `tetrahedral' (φ = 90°) geometry. The data, model and are indicated for each of the eFEFFIT fits. It is clear that the actual uncertainties are not uniform in χ space or in k2χ space but are in general much smaller at low k, so that fits with defined uncertainties put a premium on accurate theory and experiment for lower k regions. Reproduced with permission from Schalken & Chantler et al. (2018). Copyright 2018 International Union of Crystallography.

Ferrocene (Fc) is an example of a high-symmetry molecule which has long been a focus of fascination for reasons related to its electronic and molecular structure (Dunitz, 1993; Coriani et al., 2006). Fc was the first example of side-on bonding of the delocalized orbitals of an aromatic molecule to a transition metal, a motif that is common among organometallic compounds and catalytic systems. The delocalized bonding within the molecule underlies its well defined redox chemistry, a property exploited in sensors such as glucose monitors (Sakakida et al., 1993). The structure of Fc can be described in terms of the high-symmetry rotameric forms differing according to the relative orientation of the two cyclopentadienyl (Cp) rings. When viewed along the fivefold axis, the Cp rings may adopt a staggered conformation to give a molecule with D_5d symmetry or an eclipsed conformation with D_5h molecular symmetry. Phase transitions, static and dynamic disorder complicate the interpretation of the crystallography (Fischer & Pfab, 1952; Wilkinson et al., 1952), but all phases can be interpreted in terms of a disordered ensemble of molecules individually of D_5h symmetry (Seiler & Dunitz, 1979). The low barrier to interconversion between the D_5h and D_5d forms, first quantified as 1–2 kJ mol⁻¹ by gas-phase electron diffraction measurements (Haaland & Nilsson, 1968), is a feature of the molecules in all phases (Freyberg et al., 1979; Best, Wang et al., 2016; Struchkov et al., 1978). A staggered (D_5d) conformation of the C₅ rings can be achieved by replacement of the Cp rings by pentamethylcyclopentadienyl (Cp*) groups (Mochida & Yoza, 2010; Sanjuan-Szklarz et al., 2016). Decamethylferrocene (DMFc) is available to provide a companion system with its staggered isomeric structure dictated by steric forces. Thus, Fc and DMFc present the opportunity to study both isomerism and molecular dynamics.

Fc represents an extremely challenging problem for EXAFS. In common with the Ni-Pr-Sal complexes, the differences in EXAFS between the isomeric forms are only in terms of multiple-scattering interactions. For Fc the differences in scattering paths are subtle, as conversion between the D_5h and D_5d forms is associated with a relative rotation of the Cp rings by 36°, and this is reflected by small differences in the calculated EXAFS of the isomeric forms (Bourke, Islam et al., 2016). In this example discrimination between the alternate isomeric forms is a very serious test of both theory and data quality. Indeed, the two-leg paths are identical and the signature differences arise from three-leg and higher n-leg paths and remain a small percentage of the total interference signal.

Interestingly, there have been two good prior studies of Fc using XAFS. The first (Cramer et al., 1976) compared the structural effects of background removal rather than a detailed determination of the structure or conformer. The second (Ruiz-Lopez et al., 1988) used muffin-tin theory similar to current popular approaches, with apparently much higher data quality. However, they found no theoretical difference between the conformer XAFS paths or theoretical predictions and therefore concluded that (based on the available data) it was impossible to distinguish the conformers even if one were significantly more stable.

XAS measurements of frozen solutions of Fc and DMFc [15 and 3 mM in acetonitrile (40%):butyronitrile (60%)] were conducted using the protocols described for Ni-Pr-Sal (Section 6.1; Bourke, Islam et al., 2016; Islam et al., 2016). Model building for Fc can take the form of the reported crystal structure with a relative rotation of one of the Cp rings to generate approximate D_5h and D_5d forms, or the nuclear coordinates can be used to generate the coordinates of the single, independent C and H atoms needed to construct a high-symmetry molecule in either isomeric form. Whilst models based on the crystal structure coordinates for the D_5h geometry gave refinements which were satisfactory ( values of 3.97–3.99), significantly better fits were obtained using high-symmetry regularized structures ( values of 3.44 and 3.28), which were further improved by eFEFFIT refinement of the ratio of the Fe—C and C—C bond lengths ( = 3.17), and the addition of a separate Debye–Waller factor for the H atom ( = 3.06) results in further small improvements to the fit (Islam et al., 2016). A similar approach applied to DMFc returned values of 4.15 using models based on the X-ray structures which were regularized (C₅ symmetry of the Cp* rings) and refined (Islam et al., 2016). The poorer DMFc fit may relate to a loss of C₅ symmetry from distortions due to steric repulsions between methyl groups.

The question concerning the conformation of Fc and DMFc is subtle and in view of the similarity of the calculated EXAFS for the D_5h and D_5d forms (Fig. 9) it is unsurprising that the differences in the refinement statistics are small. Whilst the lower values are obtained for the D_5h form for Fc and the D_5d form for DMFc are consistent with expectations, these differences are only suggestive and it is difficult to discern model-dependent systematic differences from inspection of the calculated and observed EXAFS (Fig. 10). A frustrating element of conventional FEFF analysis is that the data with the greatest information content are at low k, where this region has both the largest contribution from multiple-scattering contributions and the smallest relative error of the measurement. Data within the k-range below 3 or 3.5 Å⁻¹ are, however, those most poorly defined by standard theory.

Figure 9 Top: the near-linear three-leg multiple-scattering paths for the D5h (left) and D5d (right) forms of Fc (for clarity, H atoms are not shown). Similar three-leg paths can be formed which involve each of the five C atoms of the top Cp ring. Bottom: calculated EXAFS for the D5h and D5d forms of Fc. Structures were based either on X-ray structure determinations or ab initio density-functional theory calculations with different functional optimizations. Reproduced with permission from Bourke et al. (2016). Copyright 2016 American Chemical Society.

Figure 10 k2-weighted EXAFS spectra of Fc (upper row) and DMFc (lower row) using eclipsed (left) and staggered (right) models of the geometry. The experimental conditions and values of are explained in Islam et al. (2016). The fit of the DMFc data above k = 9 Å may be limited by the constraint that the Cp* ring have C5 symmetry. Modified with permission from Islam et al. (2016). Copyright 2016 American Chemical Society.

A limitation of EXAFS analysis based on calculations conducted within a `muffin-tin' approximation of the inter­atomic potential is a failure to accurately model the EXAFS in the low-k region (k < 3–3.5 Å⁻¹), this being critical to the stereochemical analysis of Fc. Full potential modelling, conducted within a quantum-mechanical framework, is needed to address the problem. Finite-difference methods (FDMs) have been developed to allow the modelling of XANES (FDMNES; FDM near-edge structure; Bunău & Joly, 2009; Bourke & Chantler, 2010; Guda et al., 2015). More recently, this has been extended into the FDMX package to fit the EXAFS range (Bourke, Chantler et al., 2016). Although very subtle, small differences in the shape of the first oscillations of the EXAFS are evident for FDMX calculations of the D_5h and D_5d conformers of Fc (Fig. 11; Bourke, Islam et al., 2016). The excellent agreement between the FDMX-calculated EXAFS and experiment supports extension of the analysis to k = 1 Å⁻¹; however, statistical evaluation of the difference between the structural models relies heavily on a reliable assessment of the uncertainty of the individual measurements as is provided by the hybrid approach.

Figure 11 χ(k) XAFS spectra of ferrocene using eclipsed and staggered conformations based on the aggregated crystal structure. Differences lie mainly at lower k from multiple-leg paths. Reproduced with permission from Bourke et al. (2016). Copyright 2016 American Chemical Society.

The FDMX approach is much more computationally demanding than FEFF-based approaches and analogous least-squares parameter optimization is challenging. The approach taken to assess the performance of D_5h-type and D_5d-type models has been to calculate the goodness of fit of computationally or X-ray structure-derived models of Fc and DMFc and assess their relative merits over different regions of k-space (Table 3). The structure of Fc derived from DFT with the B3LYP functional returns the lowest value of if the k-range is between 3 and 7 Å⁻¹ and is highly sensitive to the conformation in the range 1–3 Å⁻¹. This difference relates to the relative importance of single- and multiple-scattering contributions to the EXAFS in the different k-ranges. In terms of the structural question relating to Fc, it is most significant that for the low-k data the values are significantly lower for the D_5h conformer of Fc for all of the models considered. This conclusion is consistent with a broad range of independent experimental and theoretical conclusions and, most importantly, the comparative performance of the models based on the D_5h and D_5d geometries appears not to be dependent on small differences in the geometric details of the model; for example, the internuclear contacts differ for the CCSD(T), B3LYP and ACS models. Further, refinement of the EXAFS of DMFc leads to a small, but now significant, preference for the expected staggered D_5d conformation. Central to the development of the approach is a strong statistical basis for the evaluation of the performance of the different models, where this depends on the measurement of data with well defined precision and relative accuracy.

Table 3 Goodness of fit for eclipsed and staggered conformers based on prototype geometries over different k-ranges ACS refers to the aggregated crystal structure reported in Islam et al. (2016) and derived as a refinement of the structure given by Seiler & Dunitz (1979). Note that within each k-range the core parameters were optimized. The best-fit geometry is identified in cases where |(D5h) − (D5d)| > 1 and is indicated by *. These are suggestive or conclusive determinations, whilst other subsets are inconclusive.   Fc  k-range (Å−1) 1.0–3.0 1.0–7.0 3.0–7.0  CCSD(T), D5h 14.4* 14.3 7.43  CCSD(T), D5d 19.6 14.0 7.01  B3LYP, D5h 20.1* 24.4* 5.12  B3LYP, D5d 35.2 35.2 5.14  ACS, D5h 12.8* 14.0* 8.02  ACS, D5d 17.0 15.6 8.01 DMFc  k-range (Å−1) 1.5–3.5 1.5–7.0 3.5–7.0  Islam et al. (2016), D5h 9.02 11.4 6.37  Islam et al. (2016), D5d 8.16 10.4* 5.76  Makal et al. (2010), D5h 7.69 6.71 3.49  Makal et al. (2010), D5d 6.91 5.32* 2.97