Nicholas A. Rae

Stereochemical analysis of ferrocene and the uncertainty of fluorescence XAFS data

Methods for the quantification of statistically valid measures of the uncertainties associated with X-ray absorption fine structure (XAFS) data obtained from dilute solutions using fluorescence measurements are developed. Experimental data obtained from 10 mM solutions of the organometallic compound ferrocene, Fe(C(5)H(5))(2), are analysed within this framework and, following correction for various electronic and geometrical factors, give robust estimates of the standard errors of the individual measurements. The reliability of the refinement statistics of standard current XAFS structure approaches that do not include propagation of experimental uncertainties to assess subtle structural distortions is assessed in terms of refinements obtained for the staggered and eclipsed conformations of the C(5)H(5) rings of ferrocene. Standard approaches (XFIT, IFEFFIT) give refinement statistics that appear to show strong, but opposite, preferences for the different conformations. Incorporation of experimental uncertainties into an IFEFFIT-like analysis yield refinement statistics for the staggered and eclipsed forms of ferrocene which show a far more realistic preference for the eclipsed form which accurately reflects the reliability of the analysis. Moreover, the more strongly founded estimates of the refined parameter uncertainties allow more direct comparison with those obtained by other techniques. These XAFS-based estimates of the bond distances have accuracies comparable with those obtained using single-crystal diffraction techniques and are superior in terms of their use in comparisons of experimental and computed structures.

Measurement of the x-ray mass-attenuation coefficients of gold, derived quantities between 14 keV and 21 keV and determination of the bond lengths of gold

The x-ray mass-attenuation coefficients of gold are measured at 91 energies between 14 keV and 21 keV using synchrotron radiation. The measurements are accurate to between 0.08% and 0.1%. The photoelectric mass-absorption coefficients and the imaginary component of the form factors of gold are also determined. The results include the LI edge and are the most accurate and extensive gold dataset available in this energy range. An analysis of the LI edge XAFS showed excellent agreement between the measured and simulated XAFS and yielded highly accurate values of the bond lengths of gold. When our results are compared with earlier measurements and with predictions of major theoretical tabulations, significant discrepancies are noted. The comparison raises questions about the nature of discrepancies between experimental and theoretical values of mass-attenuation coefficients.

Accurate determination and correction of the lattice parameter of LaB6 (standard reference material 660) relative to that of Si (640b)

X-ray powder diffraction and synchrotron radiation have been used to determine the lattice parameter of the NIST standard reference material (SRM 660) LaB6 as 4.156468 A with an accuracy of 12 parts per million (p.p.m.), calibrated relative to the lattice parameter of the Si powder standard [a0 = 5.430940 (11) A, Si 640b]. A discrepancy of 0.00048 (5) A, or nine standard deviations from the NIST reference, is observed between the currently accepted lattice spacing of LaB6 and the measured value. Twelve different measurements of the lattice parameter were made at beam energies between 10 and 20 keV. The observed discrepancy in the lattice parameter is consistent for the different energies used. The absolute values of the mean difference between the measured and calculated 2θ centroids, \overline{\left| \delta 2 \theta \right|}, are highly consistent, between 0.0002 and 0.0004° for energies from 5 to 14 keV, and between 0.0005 and 0.0008° for energies from 15 to 20 keV. In order to determine the peak positions with high precision, account must be taken of the observed peak asymmetry. It is shown that significant asymmetry is due to peak broadening and must be taken into account in order to determine accurate peak locations and lattice spacings. The approach shows significant advantages over conventional analysis. The analysis of peak broadening is compared with models used in Rietveld analysis.

New consistency tests for high-accuracy measurements of X-ray mass attenuation coefficients by the X-ray extended-range technique.

An extension of the X-ray extended-range technique is described for measuring X-ray mass attenuation coefficients by introducing absolute measurement of a number of foils - the multiple independent foil technique. Illustrating the technique with the results of measurements for gold in the 38-50 keV energy range, it is shown that its use enables selection of the most uniform and well defined of available foils, leading to more accurate measurements; it allows one to test the consistency of independently measured absolute values of the mass attenuation coefficient with those obtained by the thickness transfer method; and it tests the linearity of the response of the counter and counting chain throughout the range of X-ray intensities encountered in a given experiment. In light of the results for gold, the strategy to be ideally employed in measuring absolute X-ray mass attenuation coefficients, X-ray absorption fine structure and related quantities is discussed.

Structure determination from XAFS using high-accuracy measurements of x-ray mass attenuation coefficients of silver, 11 keV–28 keV, and development of an all-energies approach to local dynamical analysis of bond length, revealing variation of effective thermal contributions across the XAFS spectrum

We use the x-ray extended range technique (XERT) to experimentally determine the mass attenuation coefficient of silver in the x-ray energy range 11 kev-28 kev including the silver K absorption edge. The results are accurate to better than 0.1%, permitting critical tests of atomic and solid state theory. This is one of the most accurate demonstrations of cross-platform accuracy in synchrotron studies thus far. We derive the mass absorption coefficients and the imaginary component of the form factor over this range. We apply conventional XAFS analytic techniques, extended to include error propagation and uncertainty, yielding bond lengths accurate to approximately 0.24% and thermal Debye-Waller parameters accurate to 30%. We then introduce the FDMX technique for accurate analysis of such data across the full XAFS spectrum, built on full-potential theory, yielding a bond length accuracy of order 0.1% and the demonstration that a single Debye parameter is inadequate and inconsistent across the XAFS range. Two effective Debye-Waller parameters are determined: a high-energy value based on the highly-correlated motion of bonded atoms (σ(DW) = 0.1413(21) Å), and an uncorrelated bulk value (σ(DW) = 0.1766(9) Å) in good agreement with that derived from (room-temperature) crystallography.

A step toward standardization: development of accurate measurements of X-ray absorption and fluorescence.

This paper explains how to take the counting precision available for XAFS (X-ray absorption fine structure) and attenuation measurements, of perhaps one part in 10(6) in special cases, to produce a local variance below 0.01% and an accuracy of attenuation of the order 0.01%, with an XAFS accuracy at a similar level leading to the determination of dynamical bond lengths to an accuracy similar to that obtained by standard and experienced crystallographic measurements. This includes the necessary corrections for the detector response to be linear, including a correction for dark current and air-path energy dependencies; a proper interpretation of the range of sample thicknesses for absorption experiments; developments of methods to measure and correct for harmonic contamination, especially at lower energies without mirrors; the significance of correcting for the actual bandwidth of the beam on target after monochromation, especially for the portability of results and edge structure from one beamline to another; definitions of precision, accuracy and XAFS accuracy suitable for theoretical model analysis; the role of additional and alternative high-accuracy procedures; and discusses some principles regarding data formats for XAFS and for the deposition of data sets with manuscripts or to a database. Increasingly, the insight of X-ray absorption and the standard of accuracy needed requires data with high intrinsic precision and therefore with allowance for a range of small but significant systematic effects. This is always crucial for absolute measurements of absorption, and is of equal importance but traditionally difficult for (usually relative) measurements of fluorescence XAFS or even absorption XAFS. Robust error analysis is crucial so that the significance of conclusions can be tested within the uncertainties of the measurements. Errors should not just include precision uncertainty but should attempt to include estimation of the most significant systematic error contributions to the results. This is essential if the results are to be subject to deposition in a central accessible reference database; it is also crucial for specifying a standard data format for portability and ease of use by depositors and users. In particular this will allow development of theoretical formulations to better serve the world-wide XAFS community, and a higher and more easily comparable standard of manuscripts.

Measuring the linearity of X-ray detectors: consequences for absolute attenuation, scattering and absolute Bragg intensities

The linearity of response of X-ray detectors is tested. Examples of linearity tests demonstrate the remarkable range of linear response of flowing-gas ion chambers in the synchrotron environment. The diagnostic is also highly sensitive to the presence in the X-ray beam of harmonic X-rays diffracted by a higher-order reflection of the monochromator. The remarkable range of linearity of ion chambers has enabled the accurate measurement of the absolute X-ray attenuation of a number of elements. It should now be possible to measure the absolute intensity of Bragg reflections, provided such measurements are carried out with extended-face single crystals. The advantages of the extended-face crystal technique for Bragg intensity measurements are summarized and a number of approaches to absolute Bragg intensity measurement are discussed.

Measurement of the X-ray mass attenuation coefficients of silver in the 5-20 keV range

The X-ray mass attenuation coefficients of silver were measured in the energy range 5-20 keV with an accuracy of 0.01-0.2% on a relative scale down to 5.3 keV, and of 0.09-1.22% on an absolute scale to 5.0 keV. This analysis confirms that with careful choice of foil thickness and careful correction for systematics, especially including harmonic contents at lower energies, the X-ray attenuation of high-Z elements can be measured with high accuracy even at low X-ray energies (<6 keV). This is the first high-accuracy measurement of X-ray mass attenuation coefficients of silver in the low energy range, indicating the possibility of obtaining high-accuracy X-ray absorption fine structure down to the L1 edge (3.8 keV) of silver. Comparison of results reported here with an earlier data set optimized for higher energies confirms accuracy to within one standard error of each data set collected and analysed using the principles of the X-ray extended-range technique (XERT). Comparison with theory shows a slow divergence towards lower energies in this region away from absorption edges. The methodology developed can be used for the XAFS analysis of compounds and solutions to investigate structural features, bonding and coordination chemistry.