X-ray extended-range technique for precision measurement of the X-ray mass attenuation coefficient and Im(f) for copper using synchrotron radiation
Introduction
X-ray diffraction and medical transmission imaging have been major scientific developments following the first X-ray attenuation investigation by Röntgen. Quantum mechanical determination of atomic form factors has combined with the dynamical theory of X-ray diffraction to allow quantitative prediction of X-ray interactions. By the 1970's, this work had yielded tabulations of X-ray scattering from all elements, complemented by a large body of experimental literature [1], [2], [3], [4]. Recent major developments have concentrated on applications for structural determination near absorption edges, including the use of Bijvoet ratios [5], multiple-wavelength anomalous dispersion (MAD) techniques [6], X-ray absorption fine structure (XAFS) investigations [7] and diffraction anomalous fine structure (DAFS) [8].
Methods for experimentally determining the fundamental atomic form factor have included X-ray interferometry [9], [10], reflection and refraction [11], [12], diffraction intensities [13], [14] and pendellösung fringes [15], [16], with developing efforts at synchrotron facilities over the last two decades. These innovative approaches have been complemented by more traditional investigations of linear attenuation to provide experimental values for the complex atomic form factor over an extensive range of X-ray energies.
Every attenuation experiment is, in principle, of simple design. An absorbing sample is inserted into the beam and the difference in the transmitted signal is measured as a function of energy. The Beer–Lambert law allows extraction of a mass attenuation coefficient, and an assumption regarding scattering is made to derive from it the imaginary component of the form factor Im(f). Traditional attenuation measurements in studies over three decades have claimed accuracies of 1%. It has also been claimed that, even in ideal experiments, absolute accuracies below 1% were unlikely to be achieved.
Many theoretical issues have been raised in the past decades. Differentiation between alternative relativistic correction factors for Re(f), the separability of the scattering coefficients, the use of Dirac–Hartree–Fock or Hartree–Slater wavefunctions with or without a normalisation correction, and the independent particle assumption are all key issues for isolated atom computations in atomic physics. Computations investigating these issues theoretically find differences varying from 1% to 10% or more. Hence an experimental accuracy of 0.5% should be able to distinguish between wavefunction and orbital precision using alternate computational techniques for neutral systems, and to probe solid state and resonant atomic physics near edges.
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Difficulties with theory and experiment
Previous high-precision experimental investigations [17], [18], [19] have raised concerns about particular implementations of theory. Recent tabulations of theoretical results are discrepant from one another by up to 10σ across energy ranges for a range of elements [20], [21].
Critical experimental compilations and syntheses show inconsistencies of 10–30% in regions of interest. Most applications using atomic wavefunctions or X-ray form factors have relied upon one selected theoretical
Experimental technique
A sample is interposed between a downstream detector and an upstream beam monitor in an X-ray beam monochromatized by a double-reflection silicon 111 crystal, where second-order harmonics are forbidden and only third-order harmonics may be significant. Two apertures define the beam size of 1 mm×1 mm with a vertical divergence of 0.12±0.03 mrad. The monochromator is detuned to minimise higher-order harmonic contamination. Between the foil mounting stage and the detector is a wheel on whose rim a
Absolute determination of column thickness using the X-ray extended range technique
In addition to statistical optimisation, direct analysis of linearity and harmonics, and calibration of energy, the X-ray extended range technique calibrates the thickness of thin samples against those of thick samples of the same element and purity. Thin samples are required to satisfy the attenuation criterion at lower energies. Thick samples are needed for high energies. The mechanical accuracy with which thin sample thicknesses can be measured is usually poor, and thin samples are usually
Application to copper
High-Z impurities in copper, typically due to silver and lead, yield a maximum correction less than 0.013% for the samples discussed. Oxidation on the surface of order 3.5 nm [11] leads to a correction less than 0.02% for the thinnest (5 μm) samples.
The Beer–Lambert law relates strictly to photoelectric absorption excluding coherent and incoherent scattering. In fact, in our experiments, when a sample is inserted, the upstream monitor shows an increase in the flux of 0.1–0.2% as a function of
Results
We determine Im(f) to be 3.8±0.013 e/atom (electrons/atom) at 9 keV and 1±0.003 e/atom at 20 keV, compared to corresponding theoretical values of 3.8±0.38 e/atom and 1.00±0.01 e/atom [21]. This sensitivity in electrons per atom enables critical investigation of large contributions to Im(f) from atomic or bound near-edge resonances (XANES), local X-ray absorption fine structure (XAFS) and small relativistic 0.1 e/atom contributions to the real component of the atomic form factor. Recent
Solid state structure
The EXAFS structure illustrated can only be explained by a combination of accurate relativistic atomic and solid state computations. Modelling of these systems has often used an atomic multiplet approach [34], a local density approximation using infinite crystals (a band structure approach) [35], [36], [37], or a cluster approach using multiple scattering theory [7], [39]. These codes are contemporaneous with the latest general atomic calculations just discussed, but they are qualitative
Further discussion and conclusions
The large number and appropriate distribution of our experimental results over the energy range of investigation (8.84–20 keV) is a major advantage in comparison with other results. Our results are among the first of sufficient accuracy to probe and distinguish between alternative theoretical calculations and to quantify solid-state contributions near the Cu Kα edge. In particular, the data provide high-precision profiles of structure while simultaneously giving high-accuracy results, and
Acknowledgements
We acknowledge encouragement from D.C. Creagh and R.F. Garrett. The work was performed at the Australian National Beamline Facility with support from the Australian Synchrotron Research Program, funded by the Commonwealth of Australia under the Major National Research Facilities program. We acknowledge the helpful comments of referees and the kind assistance of Y. Joly in supplying original theoretical data and the unpublished work of Aberdam et al. for reference in this manuscript.
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