Section 4.2.6.3.3. Comparison of theory with experiment

In this case, there is some possibility of testing the validity of the relativistic dipole and relativistic multipole theories since, in the high-energy limit, the value of must approach a value related to the total self energy of the atom . That there is an atomic number dependent systematic error in the relativistic dipole approach has been demonstrated by Creagh (1984). The question of whether the relativistic multipole approach yields a result in better accord with the experimental data is answered in Table 4.2.6.4, where a comparison of values of is made for three theoretical data sets (this work; Cromer & Liberman, 1981; Wagenfeld, 1975) with a number of experimental results. These include the `direct' measurements using X-ray interferometers (Cusatis & Hart, 1975; Creagh, 1984), the Kramers–Kronig integration of X-ray attenuation data (Gerward et al., 1979), and the angle-of-the-prism data of Deutsch & Hart (1984b). Also included in the table are `indirect' measurements: those of Price et al. (1978), based on Pendellösung measurements, and those of Grimvall & Persson (1969). These latter data estimate and not . Table 4.2.6.4 details values of the real part of the dispersion correction for LiF, Si, Al and Ge for the characteristic wavelengths Ag , Mo and Cu . Of the atomic species listed, the first three are approaching the high-energy limit at Ag , whilst for germanium the K-shell absorption edge lies between Mo and Ag .

Table 4.2.6.4| top | pdf | Comparison of measurements of the real part of the dispersion correction for LiF, Si, Al and Ge for characteristic wavelengths Ag Kα1, Mo Kα1 and Cu Kα1 with theoretical predictions; the experimental accuracy claimed for the experiments is shown thus: (10) = 10% error Sample Reference Cu Mo Ag LiF Theory        This work 0.075 0.017 0.010  Cromer & Liberman (1981) 0.068 0.014 0.006  Wagenfeld (1975) 0.080 0.023 0.015 Experiment        Creagh (1984) 0.085 (5) 0.020 (10) 0.014 (10)  Deutsch & Hart (1984b) – 0.0217 (1) 0.0133 (1) Si Theory        This work 0.254 0.817 0.052  Cromer & Liberman (1981) 0.242 0.071 0.042  Wagenfeld (1975) 0.282 0.101 0.071 Experiment        Cusatis & Hart (1975) – 0.0863 (2) 0.0568 (2)  Price et al. (1978) – 0.085 (7) 0.047 (7)  Gerward et al. (1979) 0.244 (7) 0.099 (7) 0.070 (7)  Creagh (1984) 0.236 (5) 0.091 (5) 0.060 (5)  Deutsch & Hart (1984b) – 0.0847 (1) 0.0537 (1) Al Theory        This work 0.213 0.0645 0.041  Cromer & Liberman (1981) 0.203 0.0486 0.020  Wagenfeld (1975) 0.235 0.076 0.553 Experiment        Creagh (1985) – 0.065 (20) 0.044 (20)  Takama et al. (1982) 0.20 (5) 0.07 (5) 0.035 (10) Ge Theory        This work −1.089 0.155 0.302  Cromer & Liberman (1981) −1.167 0.062 0.197  Wagenfeld (1975) −1.80 −0.08 0.14 Experiment        Gerward et al. (1979) −1.04 0.30 0.43  Grimvall & Persson (1969) −1.79 0.08 0.27

The high-energy-limit case is considered first: both the relativistic dipole and relativistic multipole theories underestimate for LiF whereas the non-relativistic theory overestimates when compared with the experimental data. For silicon, however, the relativistic multipole yields values in good agreement with experiment. Further, the values derived from the work of Takama et al. (1982), who used a Pendellösung technique to measure the atomic form factor of aluminium are in reasonable agreement with the relativistic multipole approach. Also, some relatively imprecise measurements by Creagh (1985) are in better accordance with the relativistic multipole values than with the relativistic dipole values.

Further from the high-energy limit (smaller values of , the relativistic multipole approach appears to give better agreement with theory. It must be reported here that measurements by Katoh et al. (1985a) for lithium fluoride at a wavelength of 0.77366 Å yielded a value of 0.018 in good agreement with the relativistic multipole value 0.017.

At still smaller values of , the non-relativistic theory yields values considerably at variance with the experimental data, except for the case of LiF using Cu radiation. The relativistic multipole approach seems, in general, to be a little better than the relativistic approach, although agreement between experiment and theory is not at all good for germanium. Neither of the experiments cited here, however, has claims to high accuracy.

In Table 4.2.6.5, a comparison is made of measurements of derived from the results of the IUCr X-ray Attenuation Project (Creagh & Hubbell, 1987, 1990) with a number of theoretical predictions. The measurements were made on carbon, silicon and copper specimens at the characteristic wavelengths Cu , Mo and Ag . The principal conclusion that can be drawn from perusal of Table 4.2.6.5 is that only minor, non-systematic differences exist between the predictions of the several relativistic approaches and the experimental results. In contrast, the non-relativistic theory fails for higher values of atomic number.

Table 4.2.6.5| top | pdf | Comparison of measurements of f′(ω, 0) for C, Si and Cu for characteristic wavelengths Ag Kα1, Mo Kα1 and Cu Kα1 with theoretical predictions; the measurements are from the IUCr X-ray Attenuation Project Report (Creagh & Hubbell, 1987, 1990), corrected for the effects of Compton, Laue–Bragg, and small-angle scattering Sample Reference Cu Mo Ag 6C Theory        This work 0.0091 0.0016 0.0009  Cromer & Liberman (1981) 0.0091 0.0016 0.0009  Wagenfeld (1975) – – –  Scofield (1973) 0.0093 0.0016 0.0009  Storm & Israel (1970) 0.0090 0.0016 0.0009 Experiment        IUCr Project 0.0093 0.0016 0.0009 14Si Theory        This work 0.330 0.070 0.043  Cromer & Liberman (1981) 0.330 0.0704 0.0431  Wagenfeld (1975) 0.330 0.071 0.044  Scofield (1973) 0.332 0.0702 0.0431  Storm & Israel (1970) 0.331 0.0698 0.0429 Experiment        IUCr Project 0.332 0.0696 0.0429 29Cu Theory        This work 0.588 1.265 0.826  Cromer & Liberman (1981) 0.589 1.265 0.826  Scofield (1973) 0.586 1.256 0.826 Experiment        IUCr Project 0.588 1.267 0.826

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.

Experimentalists must be aware of two potential sources of error in the values of listed in Table 4.2.6.5. One is computational, arising from the error in calculating the relativistic correction. Stibius-Jensen (1980) has suggested that this error may be as large as . This means, for example, that the real part of the dispersion correction for lead at the wavelength of 0.55936 Å is −(1.168 ± 0.146). The effect of this error is to shift the dispersion curve vertically without distorting its shape. Note, however, that the direction of the shift is either up or down for all atoms: the effect of multipole cancellation and retardation will be in the same direction for all atoms.

The second possible source of error occurs because the position of the absorption edge varies somewhat depending on the oxidation state of the scattering atom. This has the effect of displacing the dispersion curve laterally. Large discrepancies may occur for those regions in which the dispersion corrections are varying rapidly with photon energy, i.e. near absorption edges.

It must also be borne in mind that in the neighbourhood of an absorption edge polarization effects may occur. The tables are valid only for average polarization.

The question of how best to describe the interaction of X-rays with crystalline materials is quite difficult to answer. In the form factor formalism, the atoms are supposed to scatter as though they are isolated atoms situated at fixed positions in the unit cell. In the vast majority of cases, the polarization on scattering is not detected, and only the scattered intensities are measured. From the scattered intensities, the distribution of the electron density within the unit cell is calculated, and the difference between the form-factor model and that calculated from the intensities is taken as a measure of the nature and location of chemical bonds between atoms in the unit cell. This is the zeroth-order approximation to a solution, but it is in fact the only way crystal structures are solved ab initio.

The existence of chemical bonding imposes additional restrictions on the symmetry of lattices, and, if the associated influence this has on the complexity of energy levels is taken into account, significant changes in the scattering factors may occur in the neighbourhood of the absorption edges of the atoms comprising the crystal structure. The magnitudes of the dispersion corrections are sensitive to the chemical state, particularly oxidation state, and phenomena similar to those observed in the XAFS case (Section 4.2.4) are observed.

The XAFS interaction arising from the presence of neighbouring atoms is proportional to and therefore is related to through the Kramers–Kronig integral. It is not surprising that these modulations are observed in diffracted intensities in those X-ray diffraction experiments where the photon energy is scanned through the absorption edge of an atomic species in the crystal lattice. Studies of this type are referred to as diffraction absorption fine structure (DAFS) experiments. A recent review of work performed using counter techniques has been given by Sorenson (1994). Creagh & Cookson (1995) have described the use of imaging-plate techniques to study the structure and site symmetry using the DAFS technique. This technique has the ability to discriminate between different lattice sites in the unit cell occupied by an atomic species. XAFS cannot make this discrimination. The DAFS modulations are small perturbations to the diffracted intensities. They are, however, significantly larger than the tensor effects described in the following paragraphs.

In the case where the excited state lacks high symmetry and is oriented by crystal bonding, the scattering can no longer be described by a scalar scattering factor but must be described by a symmetric second-rank tensor. The consequences of this have been described by Templeton (1994). It follows therefore that material media can be optically active in the X-ray region. Hart (1994) has used his unique polarizing X-ray optical devices to study, for example, Faraday rotation in such materials as iron, in the region of the iron K-absorption edge, and cobalt(III) bromide monohydrate in the region of the cobalt K-absorption edge.

The theory of anisotropy in anomalous scattering has been treated extensively by Kirfel (1994), and Morgenroth, Kirfel & Fischer (1994) have extended this to the description of kinematic diffraction intensities in lattices containing anisotropic anomalous scatterers. Their treatment was developed for space groups up to orthorhombic symmetry.

All the preceding treatments apply to scattering in the neighbourhood of an absorption edge, and to a fairly restricted class of crystals for which the local site symmetry of the electron density of states in the excited state is very different from the apparent crystal symmetry.

These approaches seek to treat the scattering from the crystal as though the scattering from each atomic position can be described by a symmetric second-rank tensor whose properties are determined by the point-group symmetries of those sites. Clearly, this procedure cannot be followed unless the structure has been solved by the usual method. The tensor approach can then be used to explain apparent deficiencies in that model such as the existence of `forbidden' reflections, birefringence, and circular dichroism.

Scattering of X-rays from the electron spins in anti-ferromagnetically ordered materials can also be described by imposing a tensor description on the form factor (Blume, 1994). The tensor in this case is a fourth-rank tensor, and the strength of the interaction, even for the favourable case of resonance scattering, is several orders of magnitude lower in intensity than the polarization effects. Nevertheless, studies have been made on holmium and uranium arsenide, and significant magnetic Bragg scattering has been observed.

All the cases cited above represent exciting, state-of-the-art, scientific studies. However, none of the work will assist in the solution of crystal structures directly. Researchers should avoid the temptation, in the first instance, to ascribe anything but a scalar value to the form factor.

For the imaginary part of the dispersion correction , the following observations can be made.

For the real part of the dispersion correction , the following observations can be made.