Section 4.2.6.3.3.2. Measurements in the vicinity of an absorption edge

The advent of the synchrotron-radiation source as a routine experimental tool and the deep interest that many crystallographers have in both XAFS and the anomalous-scattering determinations of crystal structures have stimulated considerable interest in the determination of the dispersion corrections in the neighbourhood of absorption edges. In this region, the inter­action of the ejected photoelectron with electrons belonging to neighbouring atoms causes the modulations that are referred to as XAFS. Both (which is directly proportional to the X-ray scattering cross section) and [which is linked to through the Kramers–Kronig integral] exhibit these modulations. It is at this point that one must realize that the theoretical tabulations are for the interactions of photons with isolated atoms. At best, a comparison of theory and experiment can show that they follow the same trend.

Measurements have been made in the neighbourhood of the absorption edges of a variety of atoms using the `direct' techniques interferometry, Kramers–Kronig, refraction of a prism and critical-angle techniques, and by the `indirect' refinement techniques. In Table 4.2.6.6, a comparison is made of experimental values taken at or near the absorption edges of copper, nickel and niobium with theoretical predictions. These have not been adjusted for any energy window that might be thought to exist in any particular experimental configuration. The theoretical values for niobium have been calculated at the energy at which the experimentalists claimed the experiment was conducted.

Table 4.2.6.6| top | pdf | Comparison of for copper, nickel, zirconium, and niobium for theoretical and experimental data sets; in this table: BR Bragg reflection; IN interferometer; KK Kramers–Kronig; CA = critical angle; and REF = reflectivity; measurements have been made for the K-absorption edges of copper and nickel and near the K-absorption edges of zirconium and niobium; claimed experimental errors are not worse than 5% Reference Method Cu Ni Nb Zr Experiment             Freund (1975) BR −8.2         Begum, Hart, Lea & Siddons (1986) IN −7.84 −7.66       Bonse & Materlik (1972) IN   −8.1       Bonse, Hartmann-Lotsch & Lotsch (1983a) IN −8.3         Hart & Siddons (1981) IN −9.3 −9.2 −4.396 −6.670   Kawamura & Fukimachi (1978; cited in Bonse & Hartmann-Lotsch, 1984) KK   −7.9       Dreier et al. (1984) KK −8.2 −7.8   −7.83   IN −8.3 −8.1       Bonse & Hartmann-Lotsch (1984) KK −8.3 −7.7       Fukamachi et al. (1978; cited in Bonse & Hartmann-Lotsch, 1984) KK −8.8         CA −10.0                     Bonse & Henning (1986) IN     −7.37; −7.73     KK     −7.21; −7.62     Stanglmeier, Lengeler, Weber, Gobel & Schuster (1992) REF −8.5 −8.1       Creagh (1990, 1993) REF −8.2 −7.7   −6.8 Theory             Cromer & Liberman (1981)   −13.50 −9.45 −4.20; −7.39 −6.207   This work   −9.5 −9.40 −4.04; −7.23 −6.056   Averaged values (5 eV window)   −9.0 −7.53 −8.18 −6.04

Despite the considerable experimental difficulties and the wide variety of experimental apparatus, there appears to be close agreement between the experimental data for each type of atom. There appears to be, however, for both copper and nickel, a large discrepancy between the theoretical values and the experimental values. It must be remembered that the experimental values are averages of the value of , the average being taken over the range of photon energies that pass through the device when it is set to a particular energy value. Furthermore, the exact position of the wavelength chosen may be in doubt in absolute terms, especially when synchrotron-radiation sources are used. Therefore, to be able to make a more realistic comparison between theory and experiment, the theoretical data gained using the relativistic multipole approach (this work) were averaged over a rectangular energy window of 5 eV width in the region containing the absorption edge. The rectangular shape arises because of the shape of the reflectivity curve and 5 eV was chosen as a result of (i) analysis of the characteristics of the interferometers used by Bonse et al. and Hart et al., and (ii) a statement concerning the experimental bandpass of the interferometer used by Bonse & Henning (1986). It must also be borne in mind that mechanical vibrations and thermal fluctuations can broaden the energy window and that 5 eV is not an overestimate of the width of this window. Note that for elements with atomic numbers less than 40 the experimental width is greater than the line width.

For the Bonse & Henning (1986) data, two values are listed for each experiment. Their experiment demonstrates the effect the state of polarization of the incoming photon has on the value of . Similar X-ray dichroism has been shown for sodium bromate by Templeton & Templeton (1985b) and Chapuis et al. (1985). The theoretical values are for averaged polarization in the incident photon beam. Another important feature is the difference of 0.16 electrons between the Kramers–Kronig and the interferometer values. Bonse & Henning (1986) did not add the relativistic correction term to their Kramers–Kronig values. Inclusion of this term would have reduced the quoted values by 0.20, bringing the two data sets into close agreement with one another.

Katoh et al. (1985b) have made measurements spanning the K-absorption edge of germanium using the deviation by a prism method, and these data have been shown to be in excellent agreement with the theory on which these tables are based (Creagh, 1993). In contrast, the theoretical approach of Pratt, Kissel & Bergstrom (1994) does not agree so well, especially near to, and at higher photon energies, than the K-edge energy. Also, Chapuis et al. (1985) have measured the dispersion corrections for holmium in [HoNa(edta)]·8H₂O for the characteristic emission lines Cu , Cu , Cu Kβ, and Mo using a refinement technique. Their results are in reasonable agreement with the relativistic multipole theory, e.g. for at the wavelength of Cu experiment gives −(16.0 ± 0.2) whereas the relative multipole approach yields −15.0. For Cu , experiment yields −(13.9 ± 0.3) and theory gives −13.67. The discrepancy between theory and experiment may well be explained by the oxidation state of the holmium ion, which is in the form Ho³⁺. The oxidation state of an atom affects both the position of the absorption edge and the magnitude of the relativistic correction. Both of these will have a large influence on the value of in the neighbourhood of the absorption edge, Another problem that may be of some significance is the natural width of the absorption edge, about 60 eV. What is remarkable is the extent of the agreement between theory and experiment given the nature of the experiment. In these experiments, the intensities of many reflections (usually nearly 1000) are analysed and compared. Such a procedure can be followed only if there is no dependence of on .

It had often been thought that the dispersion corrections should exhibit some functional dependence on scattering angle. Indeed, some texts ascribe to these corrections the same functional dependence on angle of scattering as the form factor. A fundamental dependence was also predicted theoretically on the basis of non-relativistic quantum mechanics (Wagenfeld, 1975). This prediction is not supported by modern approaches using relativistic quantum mechanics [see, for example, Kissel et al. (1980)]. Reference to Tables 4.2.6.4 and 4.2.6.6 shows that the agreement between experimental values derived from diffraction experiments and those derived from `direct' experiments is excellent. They are also in excellent agreement with the recent calculations, using relativistic quantum mechanics, so that it may be inferred that there is indeed no functional dependence of the dispersion corrections on scattering angle. Moreover, Suortti, Hastings & Cox (1985) have recently demonstrated that was independent of in a powder-diffraction experiment using a nickel specimen.