research papers
Highaccuracy measurement of mass attenuation coefficients and the imaginary component of the atomic form factor of zinc from 8.51 keV to 11.59 keV, and Xray absorption fine structure with investigation of zinc theory and nanostructure
^{a}School of Physics, University of Melbourne, Australia, ^{b}ANSTO, Australian Synchrotron, Melbourne, Australia, and ^{c}La Trobe University, Australia
^{*}Correspondence email: chantler@unimelb.edu.au
Highaccuracy Xray mass attenuation coefficients were measured from the first Xray Extended Range Technique (XERT)like experiment at the Australian Synchrotron. Experimentally measured mass attenuation coefficients deviate by ∼50% from the theoretical values near the zinc Kedge jump ratio and jump factor are determined and results raise significant questions regarding the definitions of quantities used and best practice for background subtraction prior to Xray absorption finestructure (XAFS) analysis. The analysis shows excellent agreement between the measured and tabulated values and yields bond lengths and nanostructure of zinc with uncertainties of from 0.1% to 0.3% or 0.003 Å to 0.008 Å. Significant variation from the reported was observed, suggesting local dynamic motion of the standard is sensitive to dynamic correlated motion and in principle is capable of observing local dynamic motion beyond the reach of conventional crystallography. These results for the zinc and structure are the most accurate structural refinements of zinc at room temperature.
suggesting that improvements in theoretical tabulations of mass attenuation coefficients are required to bring them into better agreement with experiment. Using these values the imaginary component of the atomic form factor of zinc was determined for all the measured photon energies. The zincKeywords: materials science; nanoscience; XAFS; computational modelling; density functional theory; materials modelling; nanostructure; Xray mass attenuation coefficients; atomic form factor; XERT; zinc.
1. Introduction
Xray attenuation coefficients and atomic form factors explain how Xrays interact with matter and atoms (Rontgen, 1896). Xray absorption finestructure (XAFS) spectra display the energydependent fine structure in the mass attenuation coefficients, sensitive to many structural details of materials (Rehr & Albers, 2000). Thus, Xray mass attenuation coefficients, particularly across regions, are widely used in many techniques and fields including biological (Coulthard & Sham, 1996; Streltsov et al., 2018), molecular, condensed matter and solid state fields (Palmberg & Rhodin, 1968; Bohic et al., 2001) to study structural parameters including bond lengths, thermal parameters, and electron inelastic mean free paths of materials and molecules (Ignatov et al., 2001; Han et al., 2002; Islam et al., 2015, 2016). Attenuation coefficients and atomic form factors are important in the study of the density of electron states (Joly et al., 1999; Hossain et al., 2005), roughness of sample surfaces (Glover et al., 2009), dynamics of atomic processes, and other scattering of Xray interactions with matter (Hopersky et al., 2004) and selfabsorption of materials (Trevorah et al., 2019). Mass attenuation across the can be used to confirm or determine nearneighbour coordination lengths of materials (Sayers et al., 1972; Glover et al., 2010).
Many improvements and developments in analytic techniques and experiment have been introduced in ). The spectrum consists of oscillations at energies just after the due to the interference of the photoelectron wave with the neighbouring electrons (Joly, 2001). The oscillations can be investigated using FEFF (Rehr et al., 1991; Rehr & Albers, 2000) and eFEFFIT (Smale et al., 2006; Schalken & Chantler, 2018), from modified IFEFFIT (Newville, 2001) analysis propagating robust uncertainties of data over the entire energy spectrum. The implementation of the XERTlike experiment, the first in Australia, is presented by Ekanayake et al. (2021), hereafter called Part 1.
analysis including the XERT protocol for highaccuracy data with uncertainties below 0.1% and removal of systematic errors (Chantler, 2009This paper presents Xray mass attenuation coefficients and derivations of the imaginary component of the atomic form factor of zinc to a high accuracy. Experimental details and systematic corrections are provided in Part 1. The photoelectric mass XCOM (Berger & Hubbell, 1987) and FFAST (Chantler, 1995, 2000) are the two theoretical tabulations recommended by the National Institute of Standards and Technology (NIST). XCOM tabulated unrenormalized values with Hartree–Slater nonrelativistic wavefunctions (Berger & Hubbell, 1987) and renormalized values with relativistic Hartree–Fock potential (Scofield, 1973) are compared with experiment. The use of renormalization has been an ongoing discussion in the field.
is also calculated from the after subtracting the scattering contribution. These measurements can be used as standards in XAFS and are also useful in crystallography and medical fields. These highaccuracy data permit direct access to critical measurements of electron inelastic mean free paths (IMFPs) and other key parameters. Discrepancies between the tabulated and the highaccuracy experimental values are investigated. Xray mass attenuation coefficients are calculated using relativistic quantum mechanics under several assumptions.The edge jump ratio r is the ratio of the mass photoabsorption coefficient just above the to the mass photoabsorption coefficient just below the (McMaster et al., 1969; Veigele, 1973; Broll, 1986; Bennal & Badiger, 2007; Kaya et al., 2011). Investigation of the experimental edge jump J, or J_{K} for the Kedge, and the edge jump ratio r, or r_{K} here, is important for better understanding of absorption spectra since it checks the predictions of theory which are often used from tabulations for many applications. The edge jump is generally important for radiology, shielding, Xray filters, reference materials, (Budak et al., 2003) and in particular applications for individual elements or materials including, for example, characterization of dopants in superconductors (Mallikarjuna et al., 2002). In many cases oxides or other compounds have been used which complicated any analysis. The zinc Kedge jump ratio and jump factor raise significant questions about how to define the edge and the background subtraction prior to analysis.
XAFS analysis was performed on the above calculated measurements and the bond lengths of zinc were determined. These highaccuracy values can be used as a baseline to discriminate different IMFP theories for zinc data.
2. The mass attenuation coefficients and imaginary component of the atomic form factor
The mass attenuation coefficients were obtained by considering the mean of the results from measurements using different foils and apertures. Evaluated values of mass attenuation coefficients with the uncertainties along the interested energy region are plotted in Fig. 1.
The imaginary component of the form factor can be derived from the photoelectric mass absorption coefficients
where E is the energy, u is the A is the R_{e} is the classic electron radius, h is the plank constant and c is the speed of light. The photoelectric absorption is the dominant contribution for the total mass attenuation and it can be derived by subtracting the attenuation contributions due to Coherent (Rayleigh) and incoherent (Compton) scatterings. These scattering contributions were obtained from the FFAST calculations.
Additional uncertainties were estimated to be the half of the discrepancy between the XCOM and FFAST tabulated values. The scattering uncertainties are always less than 0.036% of the photoelectric The uncertainty of f_{2} is
This work demonstrates the attenuation measurements of zinc over an extended energy range including Kedge spectrum from XERT and solidstate properties of the sample on an absolute scale. The energydependent mass attenuation coefficients from 34.77 to 327.76 (cm^{2} g^{−1}) were calculated over the energy range from 8.51 keV to 11.5914 keV. The absolute experimental uncertainties range from 0.023% to 0.036%, limited by various experimental systematic errors such as energy drifts during the monochromator tuning, monochromator glitches, and ion chamber current fluctuations.
Quantifying uncertainties is crucial in identifying any data affected by any systematics. Often badly affected points are deleted by researchers; yet a quantitative uncertainty including variance can validate them with an appropriate weighting from the inconsistency. The imaginary component of the atomic form factor was determined to be from 0.492 (e atom^{−1}) to 4.898 (e atom^{−1}). Uncertainty contributions and the dominant systematic corrections in the current work are summarized in Table 1. Table S1 of the supporting information provides: the measured mass attenuation coefficients or as a function of energy with energy E and estimated energy uncertainty σ_{E}; the total (or absolute) estimated percentage uncertainties and estimated relative percentage uncertainties ; the mass photoabsorption coefficient with percentage uncertainties , particularly after subtraction of the estimated scattering coefficients (Rayleigh and Compton), are provided in later columns, together with the estimated imaginary component of the atomic form factor f_{2} (e atom^{−1}).
Supplementary documents are provided including the data of Table S1 in text format (see ZnTable2.dat in the supporting zip file). The format of versus E with uncertainties (Zn_K_muoverrhovsE.dat) provides input for efeffit (Smale et al., 2006; Schalken & Chantler, 2018), IFEFFIT (Newville, 2001), Athena (Ravel & Newville, 2005) and particularly mu2chi (Schalken & Chantler, 2018). Corresponding files are represented as a template for both a standard (Zn_K.cif) and χ versus k with uncertainties (Zn_K_eFEFFIT.dat) data for an efeffit, IFEFFIT and Athena input standard.
3. Comparison with theory
Figure 2 compares our measured mass attenuation coefficients with FFAST tabulated values (Chantler, 1995, 2000), XCOM tabulated values with Hartree–Slater nonrelativistic wavefunctions including Scofield relativistic perturbations (Berger & Hubbell, 1987) (unrenomalized) and also renormalized values in a relativistic Hartree–Fock potential as defined by Scofield (1973). There is an agreement in experimental results with FFAST values in the range of 8.5 keV to the edge energy and above 12 keV. The deviation of calculated values from the FFAST values begins at around the edge energy (9.66 keV) and decreases and converges to a 1% discrepancy above 12 keV (see Fig. 1 of Part 1).
At lower energies the variation of our experimental measurements is smooth (Fig. 1). The deviation from the XCOM values also begins at the edge with 5–10% uncertainty or error. Above the the experimental values deviate up and down significantly due to the oscillatory as expected. There is a dramatic discrepancy of XCOM theory with the experimental measurement at the white line, likely due to the inaccuracy of the edge energy predicted by XCOM. Whilst the experimental claimed accuracy is much better than this, the theoretical claimed uncertainty far above the (K) edge is about 1%, as observed. Perhaps surprisingly, given other highaccuracy results, XCOM has a 2–5% discrepancy from experimental results below the edge, favouring FFAST; though the two theoretical predictions are in very good agreement with one another above about 10.3 keV. Both disagree with both highaccuracy experiments.
Scofield (1973) discussed and provided two theoretical predictions for atomic absorption coefficients – the unrenormalized values in XCOM and another renormalized set, obtained by renormalizing the cross sections by the amplitude of the potential at the nucleus, in a relativistic Hartree–Fock potential compared with XCOM's Hartree–Slater potential. The decision as to whether or not to perform this renormalization has been an ongoing discussion in the field. Some previous investigations have favoured the unrenormalized results (Saloman & Hubbell, 1987), whilst some have claimed that the difference in the estimates should be used as an uncertainty, and that the correct value should lie between the two approaches (Gerward, 1992).
Those comparisons were made with significantly less accurate experimental data than that presented in this work, making a definitive resolution to this discussion impossible at that time. At all energies, XCOM [using Hartree–Slater (HS) nonrelativistic wavefunctions] is more consistent with experimental data than the renormalized data referencing Hartree–Fock (HF) calculations. At lower energies, the discrepancy from the current experimental results is about 3% for XCOM data with HS calculations whereas data with HF calculations have a (much) higher (12%) discrepancy below 9.6 keV. This current evidence justifies the lessfully relativistic unrenormalized HS results [XCOM] over the HF renormalized data. Note that FFAST is a better relativistic and selfconsistent model and in turn appears in better agreement with the experimental data.
The experimentally calculated data deviate from the theory by about 10–20% at the edge, 2.5% after the edge and 1–2% far above and below the edge. About 70% of the experimentally calculated data are in agreement (within 1–2σ) with FFAST. The oscillations near the edge are absent in theory because the theory was developed as a freeatom model and so deviations are therefore large near the edge energy. More importantly, the nearedge region shows a generally triangular discrepancy quite separate from the oscillatory solid state effects. This is a general observation for many materials such as silver (Tran et al., 2005; Islam et al., 2014; Tantau et al., 2015), molybdenum (de Jonge et al., 2005; Chantler & Bourke, 2010), tin (de Jonge et al., 2007), copper (Chantler et al., 2001; Glover et al., 2008) and gold (Glover et al., 2010; Islam et al., 2010). Improvements in theoretical understanding are definitely required in the future especially for the baseline.
Data from Rae et al. (2010) are compared with these results over the common energy range, as it has the highest previous claimed accuracy of 0.044% to 0.197%. Rae et al. (2010) performed an experiment for zinc material on a different beamline using a similar technique. However, these detailed approaches in analysis were quite different. At lower energies our current results and Rae et al. (2010) are in excellent agreement (1σ). Rae et al. (2010) provide no and the spacing is unsuitable for any detailed nanostructure determination. However, it ought to be useful for an absolute measurement of the attenuation. These experimental results are in fair agreement with the earlier highaccuracy results to about 0.1% or 1–3σ of the earlier highaccuracy data set. The signature for the deviation can be investigated from the original paper, and may be due to harmonic, roughness or bandwidth in either analysis. Blank and darkcurrent normalization were performed in both experiments and the pattern of discrepancy does not match these possible sources of systematic error. The energy calibration could provide this signature noting that the absence of and nearedge structure prevents the confirmation of this systematic or the confirmation of its absence. Perhaps the most likely cause is a systematic error of the previous analysis of scattering which decreases from 3% to 6% just above the edge to 1–2% or 2–4% around 11.5 keV. The current work corrected for significant direct scattering but also for significant background scattering whereas the earlier work only investigated direct scattering (Ekanayake et al., 2021).
Other existing experimental data of Unonius & Suortti (1989) and Hopkins (1959) were compared in Fig. 1 of Part 1. Due to the lack of correction for numerous systematics, it is not surprising that discrepancies from current experiment and theory are 10–13%. These measurements are also sparse and provide no to confirm energy analysis. Highaccuracy measurements with absolute uncertainties can therefore be used in understanding theoretical tabulations and limiting atomic physics modelling and nanostructural models. The derived mass attenuation coefficients are useful in solidstate and atomic fields in investigating manybody effects such as corehole lifetime, interatomic interactions, and thermal effects.
4. Xray jumps and edge jump ratios
The σ_{tot} (barns atom^{−1}) by
is related to the total attenuationwhere N_{A} is the Avogadro number and A is the The difference between the mass photoabsorption coefficient above and below the edge gives the Kshell mass photoabsorption coefficient and then the jump ratio is
where = (cm^{2} g^{−1}) is the photoelectric for the shell i. The Kshell jump factor is then defined as (Latyshev, 1947; Schmickley & Pratt, 1967)
Four main methods for calculating the edge jump ratio and jump factor are reported:
(1) A gammaray attenuation method, where gammarays are attenuated through a sample and the et al., 1981; Ouseph et al., 1982; Mallikarjuna et al., 2002). The early key idea was to have a with lines on either side of the and use the relative attenuation to determine the jump ratio and jump factor for the edge. Uncertainties of order 5% are reported; however, in this approach, the energies of the two lines are often significantly (very) far apart, e.g. 7 keV or 12 keV or 20 keV away from the edge; and hence interpretation of the edge jump is seriously limited by the extrapolation required.
is determined (Puttaswamy(2) A Compton peak attenuation method, where the broad Comptonscattered spectral photons are allowed to attenuate through an elemental solid to measure attenuation coefficients above and below the edge for the continuum spectrum (Ayala & Mainardi, 1996; Budak & Polat, 2004; Polat et al., 2004). Measurement step sizes are typically 50 eV or 58 eV. Uncertainties are estimated as 1.5%, although discrepancies of 5% are observed.
In methods (1) and (2), no influence of scattering or fluorescence on the measurement is normally corrected for. This introduces a somewhat illdefined uncertainty and systematic error. In general, this is similar to replacing with , so that typically the numerator and denominator of equation (4) are increased by a scattering offset; and the denominator of equation (5) is increased by the same amount. In both methods (1) and (2), normally only one experimental sample thickness is measured. Although in an ideal situation this might yield a correct result, this manuscript and Part 1 prove that the thickness dependence of systematics is quite significant.
(3) An energydispersive et al., 2002; Budak et al., 2003; Polat et al., 2005; Bennal & Badiger, 2007; Kaya et al., 2011; Singh Sidhu et al., 2011). This method derives the jump factor using the fairly complex expression
(EDXRF) method, where Xray photons are allowed to pass through a sample and then the attenuation is measured (ErtugrulWhilst this equation quotes the cross sections in barns atom^{−1}, this is exactly equivalent to using the mass attenuation coefficients in cm^{2} g^{−1}. The method measures the intensity in fluorescence of the two characteristic radiations I_{Kα}, I_{Kβ}; and separately measures the mass attenuation coefficients. The method requires: that the selfabsorption of the two characteristic radiations (Kα, Kβ) are computed in fluorescence using theoretical mass attenuation coefficients (Trevorah et al., 2019); that the Kshell ω_{K} is taken from an available tabulation (e.g. Broll, 1986; Bambynek et al., 1972); that the efficiencies of the detector at the two characteristic energies are known; and that separately the scattering coefficients σ_{ts} are taken from a suitable tabulation (e.g. Berger & Hubbell, 1987; Chantler, 1995). Typical uncertainties are estimated as 3–8%, yet the uncertainties in the contributing theoretical corrections are significant. Scattering may or may not be corrected for in the extracted numerator and denominator.
A variation of this method produces secondary characteristic Kα, Kβ spectra from a incident on several targets. Spectra are then measured using an EDXRF spectrometer to measure the of the sample across the energy resolution of the detector with good statistics in the region of the characteristic radiation (Kaya et al., 2007). Without correction for scattering, this will measure ratios based on attenuation coefficients rather than photoelectric coefficients. Kaya et al. (2011) comment that the absorption curves are often distorted above the edge leading to major challenges in defining a clear attenuation or (XAFS); and similarly that scattering is quite significant at these energies and will generally lead to ratios of being measured instead of photoabsorption.
(4) A Bremsstrahlung transmission method, where a continuous bremsstrahlung radiation is allowed to pass through a sample to determine a sudden drop in intensity (Nayak & Badiger, 2006) across the edge, and then used for measuring the absorption jump ratio and factors. In principle, this is a direct attenuation measurement with potentially fine spacing (50–600 eV) around the edge, and by fitting the aboveedge and belowedge region (neglecting XAFS) one can derive the relevant attenuation coefficients, from which the photoelectric absorption coefficients can be extracted.
We, however, measure the zinc Kedge jump, jump ratio and jump factor from the current data herein and hence use an alternative XERTlike method. Synchrotron Xrays are allowed to pass through the sample to measure attenuation coefficients, from which photoabsorption coefficients and edgejump ratios can be derived (Chantler et al., 2001; Rae et al., 2010; Sier et al., 2020). Following Fig. 3, we initially removed the preedge background. Then we investigated the structure of the oscillation, defined spline and removed the spline. Far above the edge, the theory and experiment presented here are in good agreement. A key issue is the challenge to extrapolate the curves from below and from above the edge to define or extract the edge jump and the relevant cross sections.
The `triangle effect' implies that near the edge both XCOM and FFAST underestimate the average experimental FFAST performs better than XCOM. However, both theoretical predictions continue to increase towards the edge from above the edge, roughly as (E − E_{K})^{−3}; however, the imputed spline from experimental data passing through the and XANES structure curves downwards towards the edge. The correct theoretical prediction, or equivalently the correct form for the spline near the edge, is currently not yet known.
The theoretical predictions are well defined and have low uncertainty per se. However, there is likely a discrepancy separate from the complexity of the structure in matching the edge jump ratios and edge jump factors.
On the experimental side above the edge for many materials including ideal metals and elemental solids, the selection of one particular spline and form can easily change the aboveedge estimate of the ^{2} g^{−1} in this case or by about 1% in general.
by about 3 cmBelow the edge the issue of extrapolating to the edge whilst avoiding any preedge structure or indeed the beginning of the edge is a challenge even if very fine point spacing is used, so we estimate an uncertainty from this of about 0.3 cm^{2} g^{−1} in this case or perhaps 1% in general. Therefore both the experimental and the theoretical uncertainty of (E < E_{K}), `A', and (E > E_{K}), `B', in Table 2 can be estimated by the difference between, for example, two estimates from different splines or different extrapolation methods, or different theoretical predictions. In Table 2 we have propagated the uncertainties assuming that the cause of a systematic in `A' or `B' are uncorrelated. In expressions involving `A'/`A', for example, the correlated uncertainty cancels. The statistical propagation of uncertainty neglecting this potential systematic (i.e. uncorrelated uncertainty) is a full factor of ten smaller (±0.012 for r_{K} and ±0.0015 for J_{K}).

We then compare the data directly from XCOM and FFAST with that from this work. Discrepancies are real and yet there is a good consensus. Comparing with the literature for Zn edge jump ratios and edge jump factors in the literature reveals some good consistency. However, even for the stable theory with no structure, reference a should be identical to reference f (see footnote of Table 2) and is at least an underestimate of the associated error in extrapolation for the theory. Above the edge, XCOM is anomalously low compared with experiment or FFAST; below the edge it is anomalously high; hence the edge jump ratio is anomalously low.
There is a particularly strong discrepancy for McMaster et al. (1969) (reference c in Table 2) between the adjacent elements which is not replicated in other theory or experiment, which could be due to a typographic or convergence error; or could be due to the issue of scattering. Data from McMaster et al. (1969) are taken from a variety of sources, and therefore inherently inconsistent. Given the definitions and equations above, we recompute these ratios in Table 3 after subtracting off elastic and (Rayleigh and Compton scattering contributions to attenuation) so that we can report ratios of photoabsorption as defined. In this paper, we have used Chantler (2000) for scattering components, ideal for amorphous systems or metals. In oriented crystals the coherent follows Bragg–Laue scattering or thermal diffuse scattering; these coefficients are very different from Chantler (2000) or Scofield, 1973) and Berger & Hubbell (1987). In any case, the edge jump (`A' − `B') is independent of this scattering contribution as seen in the tables. Correction for scattering increases the values of J_{K} from the previous table by about 0.006; and similarly increases the values of r_{K} by about 0.39, much larger than claimed respective uncertainties. With the exception of the anomalous value for McMaster et al. (1969), none of the experimental or other theoretical values appear to follow the formulae stated using . Instead all of them appear to take ratios of attenuation coefficients . In light of this, we define (or redefine) the attenuation edge jump ratio and factor to be r_{K,tot} and J_{K,tot}.

With these difficulties, one might question the utility of the edge jump ratios and edge jump factors. However, they remain extremely important if nontrivial to extract experimentally. The widespread use of XRF and EDXRF argues strongly for increased understanding of these ratios and how to use and interpret them; the propagation errors and uncertainties of XANES and k, depend upon the stability and uniqueness of the definition of the edge and spline or other background subtraction; and more theoretical and experimental work is required in this area in attenuation, absorption and fluorescence measurements. Immediately, we can use this analysis to justify the data presented herein; and to emphasize or demonstrate the error of Singh Sidhu et al. (2011) observed in Table 2 [references Singh Sidhu et al. (2011) and Rae et al. (2010)].
analysis, especially at lowPrevious work to obtain the edge jumps of the zinc Kedge using this method (Sier et al., 2020) on zinc selenide found the jump ratio to be r_{K,tot} = 3.380. However, as this work was conducted on a compound, this value must be scaled by the relative atomic composition of zinc within zinc selenide (0.453) which converts the value to r_{K,tot} = 7.461 as the attenuation ratio; and to r_{K} = 3.406 as the photoabsorption ratio in the compound, and r_{K} = 7.519 as the photoabsorption ratio for the Zn. Note that ZnSe does not have but rather thermal diffuse scattering, so the coefficients are necessarily different from the tabulations. Similarly, in that work r_{K,tot} = 2.153 was reported for the Se Kedge jump ratio with r_{K} = 2.161 for the compound, and r_{K,tot} = 3.936 and r_{K} = 3.951 for the selenium edge.
Rae et al. (2010) measured the of pure zinc metal. The data are far too sparse to enable effective spline fitting of the above the edge; or to extract a good belowedge background. Taking a ratio of the photoabsorption data points closest to the edge results in an r_{K} of 7.426. The sparse nature of this data means that the postedge point is significantly lower in attenuation than the true preedge, thus leading to a lower r_{K}.
Overall for Zn metal we are able to report absorption ratios r_{K} = 8.482 (12) (124) where the first uncertainty is statistical and the second is systematic; and J_{K} = 0.882 (15) (149); and we are able to report r_{K,tot} = 8.091 (12) (116); and J_{K,tot} = 0.8764 (15) (148) following past evaluations using the attenuation coefficient ratios. These measurements are and must be higher than the theoretical predictions of either FFAST or XCOM, though FFAST is almost within uncertainty of the experiment, which is promising for future work. This confirmation allows us to proceed with some confidence to investigating the nanostructure from the signal.
5. Xray absorption fine structure
The Kedge (Fig. 1) probes the oscillations of the photoabsorption coefficients above the edge due to the photoelectron interference. The oscillatory part of the spectrum was isolated using the finestructure function
of zinc above the zincwhere μ_{0}(E) is a smooth atomlike background, E_{0} is the edge energy – ideally the onset of the open channel to the continuum but often (as here) estimated by the maximum of the first derivative across the edge. χ is then expressed as a function of k, the effective photoelectron wavenumber,
where m_{e} is the mass of an electron, E is the photon energy and E_{0} is the edge energy.
The background function μ_{0}(E) is obtained from an approximated spline using the eFEFFIT package (Smale et al., 2006; Schalken & Chantler, 2018). The spline approach has been widely used to isolate the oscillatory part in the community for several decades (Newville et al., 1993; Smale et al., 2006; Islam et al., 2016). The uncertainty of the oscillatory measurements was quantified using
where σ_{χ(k)} is the uncertainty of χ(k) and σ_{μ(E)} is the experimental uncertainties of the Xray absorption measurements. Figure 4 illustrates the oscillatory part of the spectrum with fully propagated uncertainties.
The χ(k) as a sum over the multiple scattering paths,
equation describes the oscillatory partwhere j denotes the jth path, N_{j} is the degeneracy, S^{ 2}_{0} is the manybody reduction factor, F_{j}(k) is the backscattering amplitude function, r_{j} is half the path length, ϕ_{j}(k) is the total phase shift function, σ_{j} is the Debye–Waller factor and λ(k) is the photoelectron function. A dominant standard approach for analysis uses equation (10), perhaps particularly via the IFEFFIT computational package (Newville, 2001) together with FEFF (Rehr & Albers, 2000; Rehr et al., 1991), to investigate the local nanostructure of materials. The lattice constants of zinc have been determined by singlecrystal Xray diffraction, to be a = 266.36 (1) pm and c = 494.57 (2) pm with c/a = 1.857 at a temperature of 300 K (Hull & Davey, 1921; Nuss et al., 2010). Zinc is located at the special 2c ( 1/3,2/3,1/4) in P6_{3}/mmc(194). The theoretical prediction from FEFF was used in eFEFFIT (Smale et al., 2006; Schalken & Chantler, 2018), modified from the IFEFFIT package to propagate uncertainties. Data were fitted across several kranges to investigate robustness. Most effective was the range of k from 4.5 Å ^{−1} to 17 Å ^{−1} using absolute uncertainties of input data. Below this range, the theory underpinning the fitting function begins to break down. Above this range the uncertainties prove that the data are now dominated by noise.
After extensive analysis, the two nearestneighbour bond lengths, also with the highest ranks from FEFF, were given two separate fitting parameters, whilst all other bond lengths were scaled by the usual α (Ravel, 2005). Independent were used for the two nearestneighbour bond lengths (Ravel & Kelly, 2007; Kelly et al., 2002; Ravel, 2009). Linear paths were quantified in terms of the values of the singlescattering paths that contain the end atoms (Hudson et al., 1995). Particular paths for double forwardscattering followed recommendations from Ravel (2016) about increased thermal parameters expected ( and depending upon path combination). FEFF6 and FEFF8 gave similar results, parameters, uncertainties and . Path information and ordering does change between the versions, so parameter files need to be checked carefully in going from one to the other. Results presented here are for FEFF6.
This analysis expanded the scattering to 45 paths, i.e. out to a maximum R ≃ 6.7 Å. A model with a single scale α and a single thermal parameter σ^{2} fails with = 290, and a model with an α and σ^{2} assigned to the nearestneighbour bond length and another α and σ^{2} to all other paths also fails with = 16.8. Increasing the number of σ^{2} parameters to three is necessary and significant and a model with a single scale α yielded = 18.1, and a model with an α assigned to the nearestneighbour bond length and another to all other paths yielded a relatively poor = 14.4. A developed model with three α and three σ^{2} values for the relevant paths led to = 9.5. Finally, incorporating a value for two singlescattering paths (`8' and `9') led to the final = 6.66. Other investigations of further independent parameters including a separated α showed insignificant improvement of the fits. However, note that the space is highly correlated in this area.
The incorporation of the path lengths and σ^{2} parameters led to uncertainties in these parameters reducing from 0.5% to between 0.10% to 0.29% for the bond lengths or to ±0.003 Å to ±0.008 Å; and led to σ^{2} thermal parameters with uncertainties around 3–5%. Perhaps more significantly, S_{0}^{ 2} converged from the simpler fits with fewer parameters of 1.7 ± 0.3 to 0.904 ± 0.037 with four independent parameters.
This demonstrates the need for four independent σ^{2} thermal parameters, and the need for three independent path length scalings. Whilst the parameters are moderately robust in these simpler analyses, the fits are poor until the two shortest bond lengths are given independent path lengths and thermal broadening.
Zinc could have a single scaling (expansion) parameter relative to a crystallographic determination, if the metal is ideal at zero temperature; or of course if the crystallographic structure, the difference between mean lattice sites, were fully consistent with the dynamical bond lengths and path lengths. In this experiment, however, dynamical motion at room temperature might be expected to lead to significant variation of expansion of the path lengths relative to a corresponding crystallographic determination. Fitted parameters are summarized in Table 4 and results are illustrated in Fig. 5. Fits give good physical agreement and parameters are consistent within 1σ.

An interesting result of this analysis is the different expansions obtained for the nearest and secondnearest scattering paths. The reported nearestneighbour and the secondnearestneighbour Zn—Zn bond lengths are 2.6636 Å and 2.9120 Å (Nuss et al., 2010) (Fig. 6). This work determines bond lengths 0.11% (±0.10%) and 2.7% (±0.3%) smaller, respectively. The scaling parameter obtained for the other bond lengths are 0.62% ± 0.13% larger than the scaling of the nearest neighbour bond length.
These 5–7 standard error variations can be due to dynamic motion, or to the pathlength including motion perpendicular to the bond length.
is sensitive to atomic vibrations perpendicular to the bond direction, and to (positively or negatively) correlated motion of the atoms of the corresponding bond to increase or decrease the bond length compared with the crystallographic lattice site separation. Hence a bond length 0.004 Å to 0.08 Å smaller than Xray diffraction (XRD) results at room temperature would be no surprise especially if the motion of both atoms was positively correlated.The shift of nextneighbour bond lengths compared with XRD may alternatively be due to the sensitivity with temperature to the c/a axis. According to crystallographic determinations, a 5 K error in temperature would have only a shift of c/a of order 0.0003 Å so is inadequate to explain the lattice variation observations. Similarly, rolled foils have some static disorder from or stacking faults, compared for example with an ideal Changing ABABAB to ABABCAB compromises the assigned Wykoff positions in the earlier calculations. This static disorder would then confuse the separation of static disorder from thermal disorder. This question can explicitly be clarified with a temperature series. To first order this will increase the disorder, i.e. the σ on the bond lengths, rather than the bond lengths themselves. In particular, this will increase the σ or mean isotropic thermal parameters for farther atomic pairs. To second order this would increase the bond lengths or lattice separations especially with a high density of phase boundaries, stacking faults or microstructure such as However, the largest observed effect with respect to the nearestneighbour bond length is a decrease rather than an increase, so is the incorrect sign. The bond lengths fitted for the farther Zn moieties is an increase so may be augmented by such details. In fact, we know that thermal parameters increase with path length and distance, and this is naturally due to the increasing disorder, static and dynamic, as seen in most studies. The uncertainty of the larger bond lengths is about 0.13% suggesting significant variation of bond lengths, and supported by the increased thermal parameters.
In this case the evidence is inconclusive, but quite suggestive of local dynamical motion and local order combined with distant increasing static disorder. Hence this suggests dynamic motion in the
inaccessible to other techniques.6. Conclusion
Xray mass attenuation coefficients and the imaginary component of the atomic form factor of zinc were measured over the energy range from 8.51 keV to 11.5914 keV including the zinc Kedge and with 496 data points at different energies. The accuracy of measured mass attenuation coefficients is about 0.023% to 0.036% and some 50% better than previous claims. These measurements are the most accurate zinc Kedge data set.
Comparison of the experimental data with the tabulated FFAST and XCOM theory shows very good agreement. The edge jump and jump ratio analysis shows some discrepancies due to the challenges of defining both the belowedge and aboveedge so that most of the preceding literature uses ratios and jump factors for the thereby including scattering – we separate these ideas by defining (or redefining) the attenuation ratio and factor to be r_{K,tot} and J_{K,tot}. These measures are also challenging because of the experimental preedge and edgewidth below the edge; and of the typical use of splines to attempt to deal with the aboveedge Nonetheless, this is a valuable confirmation of the scaling and absolute value of the coefficients and can be used for any experiment as a test of the quality of the data and data processing.
The highaccuracy measurements were improved for structural investigations. Robust investigation of zinc nanostructure was carried out using eFEFFIT with propagated uncertainties from XERT. Xray absorption finestructure analysis shows an excellent agreement between the measured and tabulated values and yields bond lengths and nanostructure of zinc with uncertainties of 0.11% to 0.27% or 0.003 Å to 0.009 Å, respectively. Thermal σ^{2} parameters were fitted with an accuracy of from approximately 5%, and S_{0}^{ 2} is determined to be 0.904 ± 0.037. Observed significant variation from reported crystal structures suggests local dynamic motion compared with a standard and increasing static disorder as expected in a nonideally crystalline environment such as a metal foil. is sensitive to dynamic correlated motion so is in principle able to observe local dynamic motion beyond the reach of conventional crystallography.
These highaccuracy studies also allow one to investigate solid state effects including inelastic mean free paths, and inelastic and
crosssections.Supporting information
Table S1: Mass attenuation coefficients and imaginary componen. DOI: https://doi.org//10.1107/S1600577521005981/rv5147sup1.pdf
Data files. DOI: https://doi.org//10.1107/S1600577521005981/rv5147sup2.zip
Acknowledgements
We sincerely acknowledge and are very grateful for the collaboration with Zwi Barnea who has driven the development of XERT with us and also has driven the investigation of Zn. Part of this research was undertaken on the
beamline at the Australian Synchrotron, part of ANSTO. We thank beamline scientists in the Australian Synchrotron for their efforts and dedication to build up part of this methodology. We thank all of the synchrotron team including Jeremy Wykes, Chris Glover and Susan Cumberland. We thank our attending team members Stephen P. Best and Victor A. Streltsov for their contributions here and in earlier experiments. We thank Hamish A. Melia for his support for this work and we thank Ryan J. Trevorah and Geoffrey P. Cousland for helpful reading of the manuscript.References
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